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3.1.59 \(\int \frac {(c+d x)^3}{(a+b \tan (e+f x))^2} \, dx\) [59]

Optimal. Leaf size=848 \[ -\frac {2 i b^2 (c+d x)^3}{\left (a^2+b^2\right )^2 f}+\frac {2 b^2 (c+d x)^3}{(a+i b) (i a+b)^2 \left (i a-b+(i a+b) e^{2 i e+2 i f x}\right ) f}+\frac {(c+d x)^4}{4 (a-i b)^2 d}+\frac {b (c+d x)^4}{(i a-b) (a-i b)^2 d}-\frac {b^2 (c+d x)^4}{\left (a^2+b^2\right )^2 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1+\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{\left (a^2+b^2\right )^2 f^2}+\frac {2 b (c+d x)^3 \log \left (1+\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{(a-i b)^2 (a+i b) f}-\frac {2 i b^2 (c+d x)^3 \log \left (1+\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{\left (a^2+b^2\right )^2 f}-\frac {3 i b^2 d^2 (c+d x) \text {PolyLog}\left (2,-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{\left (a^2+b^2\right )^2 f^3}+\frac {3 b d (c+d x)^2 \text {PolyLog}\left (2,-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{(i a-b) (a-i b)^2 f^2}-\frac {3 b^2 d (c+d x)^2 \text {PolyLog}\left (2,-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{\left (a^2+b^2\right )^2 f^2}+\frac {3 b^2 d^3 \text {PolyLog}\left (3,-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{2 \left (a^2+b^2\right )^2 f^4}+\frac {3 b d^2 (c+d x) \text {PolyLog}\left (3,-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{(a-i b)^2 (a+i b) f^3}-\frac {3 i b^2 d^2 (c+d x) \text {PolyLog}\left (3,-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{\left (a^2+b^2\right )^2 f^3}-\frac {3 b d^3 \text {PolyLog}\left (4,-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{2 (i a-b) (a-i b)^2 f^4}+\frac {3 b^2 d^3 \text {PolyLog}\left (4,-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{2 \left (a^2+b^2\right )^2 f^4} \]

[Out]

-2*I*b^2*(d*x+c)^3/(a^2+b^2)^2/f+2*b^2*(d*x+c)^3/(a+I*b)/(I*a+b)^2/(I*a-b+(I*a+b)*exp(2*I*e+2*I*f*x))/f+1/4*(d
*x+c)^4/(a-I*b)^2/d+b*(d*x+c)^4/(I*a-b)/(a-I*b)^2/d-b^2*(d*x+c)^4/(a^2+b^2)^2/d+3*b^2*d*(d*x+c)^2*ln(1+(a-I*b)
*exp(2*I*e+2*I*f*x)/(a+I*b))/(a^2+b^2)^2/f^2+2*b*(d*x+c)^3*ln(1+(a-I*b)*exp(2*I*e+2*I*f*x)/(a+I*b))/(a-I*b)^2/
(a+I*b)/f-3*I*b^2*d^2*(d*x+c)*polylog(2,-(a-I*b)*exp(2*I*e+2*I*f*x)/(a+I*b))/(a^2+b^2)^2/f^3-3*I*b^2*d^2*(d*x+
c)*polylog(3,-(a-I*b)*exp(2*I*e+2*I*f*x)/(a+I*b))/(a^2+b^2)^2/f^3+3*b*d*(d*x+c)^2*polylog(2,-(a-I*b)*exp(2*I*e
+2*I*f*x)/(a+I*b))/(I*a-b)/(a-I*b)^2/f^2-3*b^2*d*(d*x+c)^2*polylog(2,-(a-I*b)*exp(2*I*e+2*I*f*x)/(a+I*b))/(a^2
+b^2)^2/f^2+3/2*b^2*d^3*polylog(3,-(a-I*b)*exp(2*I*e+2*I*f*x)/(a+I*b))/(a^2+b^2)^2/f^4+3*b*d^2*(d*x+c)*polylog
(3,-(a-I*b)*exp(2*I*e+2*I*f*x)/(a+I*b))/(a-I*b)^2/(a+I*b)/f^3-2*I*b^2*(d*x+c)^3*ln(1+(a-I*b)*exp(2*I*e+2*I*f*x
)/(a+I*b))/(a^2+b^2)^2/f-3/2*b*d^3*polylog(4,-(a-I*b)*exp(2*I*e+2*I*f*x)/(a+I*b))/(I*a-b)/(a-I*b)^2/f^4+3/2*b^
2*d^3*polylog(4,-(a-I*b)*exp(2*I*e+2*I*f*x)/(a+I*b))/(a^2+b^2)^2/f^4

________________________________________________________________________________________

Rubi [A]
time = 1.38, antiderivative size = 848, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {3815, 2216, 2215, 2221, 2611, 6744, 2320, 6724, 2222} \begin {gather*} \frac {b (c+d x)^4}{(i a-b) (a-i b)^2 d}+\frac {(c+d x)^4}{4 (a-i b)^2 d}-\frac {b^2 (c+d x)^4}{\left (a^2+b^2\right )^2 d}+\frac {2 b \log \left (\frac {e^{2 i e+2 i f x} (a-i b)}{a+i b}+1\right ) (c+d x)^3}{(a-i b)^2 (a+i b) f}-\frac {2 i b^2 \log \left (\frac {e^{2 i e+2 i f x} (a-i b)}{a+i b}+1\right ) (c+d x)^3}{\left (a^2+b^2\right )^2 f}+\frac {2 b^2 (c+d x)^3}{(a+i b) (i a+b)^2 \left (i a+(i a+b) e^{2 i e+2 i f x}-b\right ) f}-\frac {2 i b^2 (c+d x)^3}{\left (a^2+b^2\right )^2 f}+\frac {3 b^2 d \log \left (\frac {e^{2 i e+2 i f x} (a-i b)}{a+i b}+1\right ) (c+d x)^2}{\left (a^2+b^2\right )^2 f^2}+\frac {3 b d \text {Li}_2\left (-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right ) (c+d x)^2}{(i a-b) (a-i b)^2 f^2}-\frac {3 b^2 d \text {Li}_2\left (-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right ) (c+d x)^2}{\left (a^2+b^2\right )^2 f^2}-\frac {3 i b^2 d^2 \text {Li}_2\left (-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right ) (c+d x)}{\left (a^2+b^2\right )^2 f^3}+\frac {3 b d^2 \text {Li}_3\left (-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right ) (c+d x)}{(a-i b)^2 (a+i b) f^3}-\frac {3 i b^2 d^2 \text {Li}_3\left (-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right ) (c+d x)}{\left (a^2+b^2\right )^2 f^3}+\frac {3 b^2 d^3 \text {Li}_3\left (-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{2 \left (a^2+b^2\right )^2 f^4}-\frac {3 b d^3 \text {Li}_4\left (-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{2 (i a-b) (a-i b)^2 f^4}+\frac {3 b^2 d^3 \text {Li}_4\left (-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{2 \left (a^2+b^2\right )^2 f^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3/(a + b*Tan[e + f*x])^2,x]

[Out]

((-2*I)*b^2*(c + d*x)^3)/((a^2 + b^2)^2*f) + (2*b^2*(c + d*x)^3)/((a + I*b)*(I*a + b)^2*(I*a - b + (I*a + b)*E
^((2*I)*e + (2*I)*f*x))*f) + (c + d*x)^4/(4*(a - I*b)^2*d) + (b*(c + d*x)^4)/((I*a - b)*(a - I*b)^2*d) - (b^2*
(c + d*x)^4)/((a^2 + b^2)^2*d) + (3*b^2*d*(c + d*x)^2*Log[1 + ((a - I*b)*E^((2*I)*e + (2*I)*f*x))/(a + I*b)])/
((a^2 + b^2)^2*f^2) + (2*b*(c + d*x)^3*Log[1 + ((a - I*b)*E^((2*I)*e + (2*I)*f*x))/(a + I*b)])/((a - I*b)^2*(a
 + I*b)*f) - ((2*I)*b^2*(c + d*x)^3*Log[1 + ((a - I*b)*E^((2*I)*e + (2*I)*f*x))/(a + I*b)])/((a^2 + b^2)^2*f)
- ((3*I)*b^2*d^2*(c + d*x)*PolyLog[2, -(((a - I*b)*E^((2*I)*e + (2*I)*f*x))/(a + I*b))])/((a^2 + b^2)^2*f^3) +
 (3*b*d*(c + d*x)^2*PolyLog[2, -(((a - I*b)*E^((2*I)*e + (2*I)*f*x))/(a + I*b))])/((I*a - b)*(a - I*b)^2*f^2)
- (3*b^2*d*(c + d*x)^2*PolyLog[2, -(((a - I*b)*E^((2*I)*e + (2*I)*f*x))/(a + I*b))])/((a^2 + b^2)^2*f^2) + (3*
b^2*d^3*PolyLog[3, -(((a - I*b)*E^((2*I)*e + (2*I)*f*x))/(a + I*b))])/(2*(a^2 + b^2)^2*f^4) + (3*b*d^2*(c + d*
x)*PolyLog[3, -(((a - I*b)*E^((2*I)*e + (2*I)*f*x))/(a + I*b))])/((a - I*b)^2*(a + I*b)*f^3) - ((3*I)*b^2*d^2*
(c + d*x)*PolyLog[3, -(((a - I*b)*E^((2*I)*e + (2*I)*f*x))/(a + I*b))])/((a^2 + b^2)^2*f^3) - (3*b*d^3*PolyLog
[4, -(((a - I*b)*E^((2*I)*e + (2*I)*f*x))/(a + I*b))])/(2*(I*a - b)*(a - I*b)^2*f^4) + (3*b^2*d^3*PolyLog[4, -
(((a - I*b)*E^((2*I)*e + (2*I)*f*x))/(a + I*b))])/(2*(a^2 + b^2)^2*f^4)

Rule 2215

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[(c
+ d*x)^(m + 1)/(a*d*(m + 1)), x] - Dist[b/a, Int[(c + d*x)^m*((F^(g*(e + f*x)))^n/(a + b*(F^(g*(e + f*x)))^n))
, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2216

Int[((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.))^(p_)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Dis
t[1/a, Int[(c + d*x)^m*(a + b*(F^(g*(e + f*x)))^n)^(p + 1), x], x] - Dist[b/a, Int[(c + d*x)^m*(F^(g*(e + f*x)
))^n*(a + b*(F^(g*(e + f*x)))^n)^p, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && ILtQ[p, 0] && IGtQ[m, 0
]

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
 (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
 - Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2222

Int[((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((a_.) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.))^(p_.)*
((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^m*((a + b*(F^(g*(e + f*x)))^n)^(p + 1)/(b*f*g*n*(p + 1
)*Log[F])), x] - Dist[d*(m/(b*f*g*n*(p + 1)*Log[F])), Int[(c + d*x)^(m - 1)*(a + b*(F^(g*(e + f*x)))^n)^(p + 1
), x], x] /; FreeQ[{F, a, b, c, d, e, f, g, m, n, p}, x] && NeQ[p, -1]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 3815

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Int[ExpandIntegrand[(
c + d*x)^m, (1/(a - I*b) - 2*I*(b/(a^2 + b^2 + (a - I*b)^2*E^(2*I*(e + f*x)))))^(-n), x], x] /; FreeQ[{a, b, c
, d, e, f}, x] && NeQ[a^2 + b^2, 0] && ILtQ[n, 0] && IGtQ[m, 0]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 6744

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(p_.)], x_Symbol] :> Simp
[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Dist[f*(m/(b*c*p*Log[F])), Int[(e +
f*x)^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c, d, e, f, n, p}, x] && GtQ[m,
0]

Rubi steps

\begin {align*} \int \frac {(c+d x)^3}{(a+b \tan (e+f x))^2} \, dx &=\int \left (\frac {(c+d x)^3}{(a-i b)^2}-\frac {4 b^2 (c+d x)^3}{(i a+b)^2 \left (i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i e+2 i f x}\right )^2}+\frac {4 b (c+d x)^3}{(a-i b)^2 \left (i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i e+2 i f x}\right )}\right ) \, dx\\ &=\frac {(c+d x)^4}{4 (a-i b)^2 d}+\frac {(4 b) \int \frac {(c+d x)^3}{i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i e+2 i f x}} \, dx}{(a-i b)^2}-\frac {\left (4 b^2\right ) \int \frac {(c+d x)^3}{\left (i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i e+2 i f x}\right )^2} \, dx}{(i a+b)^2}\\ &=\frac {(c+d x)^4}{4 (a-i b)^2 d}+\frac {b (c+d x)^4}{(i a-b) (a-i b)^2 d}+\frac {\left (4 b^2\right ) \int \frac {(c+d x)^3}{i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i e+2 i f x}} \, dx}{(i a-b) (a-i b)^2}-\frac {(4 b) \int \frac {e^{2 i e+2 i f x} (c+d x)^3}{i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i e+2 i f x}} \, dx}{a^2+b^2}-\frac {\left (4 b^2\right ) \int \frac {e^{2 i e+2 i f x} (c+d x)^3}{\left (i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i e+2 i f x}\right )^2} \, dx}{a^2+b^2}\\ &=-\frac {2 b^2 (c+d x)^3}{(a-i b)^2 (a+i b) \left (i a-b+(i a+b) e^{2 i e+2 i f x}\right ) f}+\frac {(c+d x)^4}{4 (a-i b)^2 d}+\frac {b (c+d x)^4}{(i a-b) (a-i b)^2 d}-\frac {b^2 (c+d x)^4}{\left (a^2+b^2\right )^2 d}+\frac {2 b (c+d x)^3 \log \left (1+\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{(a-i b)^2 (a+i b) f}-\frac {\left (4 b^2\right ) \int \frac {e^{2 i e+2 i f x} (c+d x)^3}{i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i e+2 i f x}} \, dx}{(a+i b)^2 (i a+b)}-\frac {(6 b d) \int (c+d x)^2 \log \left (1+\frac {\left (1-\frac {i b}{a}\right ) e^{2 i e+2 i f x}}{1+\frac {i b}{a}}\right ) \, dx}{(a-i b)^2 (a+i b) f}+\frac {\left (6 b^2 d\right ) \int \frac {(c+d x)^2}{i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i e+2 i f x}} \, dx}{(a-i b)^2 (a+i b) f}\\ &=-\frac {2 i b^2 (c+d x)^3}{\left (a^2+b^2\right )^2 f}-\frac {2 b^2 (c+d x)^3}{(a-i b)^2 (a+i b) \left (i a-b+(i a+b) e^{2 i e+2 i f x}\right ) f}+\frac {(c+d x)^4}{4 (a-i b)^2 d}+\frac {b (c+d x)^4}{(i a-b) (a-i b)^2 d}-\frac {b^2 (c+d x)^4}{\left (a^2+b^2\right )^2 d}+\frac {2 b (c+d x)^3 \log \left (1+\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{(a-i b)^2 (a+i b) f}-\frac {2 i b^2 (c+d x)^3 \log \left (1+\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{\left (a^2+b^2\right )^2 f}+\frac {3 b d (c+d x)^2 \text {Li}_2\left (-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{(i a-b) (a-i b)^2 f^2}-\frac {\left (6 b d^2\right ) \int (c+d x) \text {Li}_2\left (-\frac {\left (1-\frac {i b}{a}\right ) e^{2 i e+2 i f x}}{1+\frac {i b}{a}}\right ) \, dx}{(i a-b) (a-i b)^2 f^2}-\frac {\left (6 b^2 d\right ) \int \frac {e^{2 i e+2 i f x} (c+d x)^2}{i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i e+2 i f x}} \, dx}{(a-i b) (a+i b)^2 f}+\frac {\left (6 i b^2 d\right ) \int (c+d x)^2 \log \left (1+\frac {\left (1-\frac {i b}{a}\right ) e^{2 i e+2 i f x}}{1+\frac {i b}{a}}\right ) \, dx}{\left (a^2+b^2\right )^2 f}\\ &=-\frac {2 i b^2 (c+d x)^3}{\left (a^2+b^2\right )^2 f}-\frac {2 b^2 (c+d x)^3}{(a-i b)^2 (a+i b) \left (i a-b+(i a+b) e^{2 i e+2 i f x}\right ) f}+\frac {(c+d x)^4}{4 (a-i b)^2 d}+\frac {b (c+d x)^4}{(i a-b) (a-i b)^2 d}-\frac {b^2 (c+d x)^4}{\left (a^2+b^2\right )^2 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1+\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{\left (a^2+b^2\right )^2 f^2}+\frac {2 b (c+d x)^3 \log \left (1+\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{(a-i b)^2 (a+i b) f}-\frac {2 i b^2 (c+d x)^3 \log \left (1+\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{\left (a^2+b^2\right )^2 f}+\frac {3 b d (c+d x)^2 \text {Li}_2\left (-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{(i a-b) (a-i b)^2 f^2}-\frac {3 b^2 d (c+d x)^2 \text {Li}_2\left (-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{\left (a^2+b^2\right )^2 f^2}+\frac {3 b d^2 (c+d x) \text {Li}_3\left (-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{(a-i b)^2 (a+i b) f^3}-\frac {\left (3 b d^3\right ) \int \text {Li}_3\left (-\frac {\left (1-\frac {i b}{a}\right ) e^{2 i e+2 i f x}}{1+\frac {i b}{a}}\right ) \, dx}{(a-i b)^2 (a+i b) f^3}-\frac {\left (6 b^2 d^2\right ) \int (c+d x) \log \left (1+\frac {\left (1-\frac {i b}{a}\right ) e^{2 i e+2 i f x}}{1+\frac {i b}{a}}\right ) \, dx}{\left (a^2+b^2\right )^2 f^2}+\frac {\left (6 b^2 d^2\right ) \int (c+d x) \text {Li}_2\left (-\frac {\left (1-\frac {i b}{a}\right ) e^{2 i e+2 i f x}}{1+\frac {i b}{a}}\right ) \, dx}{\left (a^2+b^2\right )^2 f^2}\\ &=-\frac {2 i b^2 (c+d x)^3}{\left (a^2+b^2\right )^2 f}-\frac {2 b^2 (c+d x)^3}{(a-i b)^2 (a+i b) \left (i a-b+(i a+b) e^{2 i e+2 i f x}\right ) f}+\frac {(c+d x)^4}{4 (a-i b)^2 d}+\frac {b (c+d x)^4}{(i a-b) (a-i b)^2 d}-\frac {b^2 (c+d x)^4}{\left (a^2+b^2\right )^2 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1+\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{\left (a^2+b^2\right )^2 f^2}+\frac {2 b (c+d x)^3 \log \left (1+\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{(a-i b)^2 (a+i b) f}-\frac {2 i b^2 (c+d x)^3 \log \left (1+\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{\left (a^2+b^2\right )^2 f}-\frac {3 i b^2 d^2 (c+d x) \text {Li}_2\left (-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{\left (a^2+b^2\right )^2 f^3}+\frac {3 b d (c+d x)^2 \text {Li}_2\left (-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{(i a-b) (a-i b)^2 f^2}-\frac {3 b^2 d (c+d x)^2 \text {Li}_2\left (-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{\left (a^2+b^2\right )^2 f^2}+\frac {3 b d^2 (c+d x) \text {Li}_3\left (-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{(a-i b)^2 (a+i b) f^3}-\frac {3 i b^2 d^2 (c+d x) \text {Li}_3\left (-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{\left (a^2+b^2\right )^2 f^3}-\frac {\left (3 b d^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {(a-i b) x}{a+i b}\right )}{x} \, dx,x,e^{2 i e+2 i f x}\right )}{2 (i a-b) (a-i b)^2 f^4}+\frac {\left (3 i b^2 d^3\right ) \int \text {Li}_2\left (-\frac {\left (1-\frac {i b}{a}\right ) e^{2 i e+2 i f x}}{1+\frac {i b}{a}}\right ) \, dx}{\left (a^2+b^2\right )^2 f^3}+\frac {\left (3 i b^2 d^3\right ) \int \text {Li}_3\left (-\frac {\left (1-\frac {i b}{a}\right ) e^{2 i e+2 i f x}}{1+\frac {i b}{a}}\right ) \, dx}{\left (a^2+b^2\right )^2 f^3}\\ &=-\frac {2 i b^2 (c+d x)^3}{\left (a^2+b^2\right )^2 f}-\frac {2 b^2 (c+d x)^3}{(a-i b)^2 (a+i b) \left (i a-b+(i a+b) e^{2 i e+2 i f x}\right ) f}+\frac {(c+d x)^4}{4 (a-i b)^2 d}+\frac {b (c+d x)^4}{(i a-b) (a-i b)^2 d}-\frac {b^2 (c+d x)^4}{\left (a^2+b^2\right )^2 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1+\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{\left (a^2+b^2\right )^2 f^2}+\frac {2 b (c+d x)^3 \log \left (1+\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{(a-i b)^2 (a+i b) f}-\frac {2 i b^2 (c+d x)^3 \log \left (1+\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{\left (a^2+b^2\right )^2 f}-\frac {3 i b^2 d^2 (c+d x) \text {Li}_2\left (-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{\left (a^2+b^2\right )^2 f^3}+\frac {3 b d (c+d x)^2 \text {Li}_2\left (-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{(i a-b) (a-i b)^2 f^2}-\frac {3 b^2 d (c+d x)^2 \text {Li}_2\left (-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{\left (a^2+b^2\right )^2 f^2}+\frac {3 b d^2 (c+d x) \text {Li}_3\left (-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{(a-i b)^2 (a+i b) f^3}-\frac {3 i b^2 d^2 (c+d x) \text {Li}_3\left (-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{\left (a^2+b^2\right )^2 f^3}-\frac {3 b d^3 \text {Li}_4\left (-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{2 (i a-b) (a-i b)^2 f^4}+\frac {\left (3 b^2 d^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {(a-i b) x}{a+i b}\right )}{x} \, dx,x,e^{2 i e+2 i f x}\right )}{2 \left (a^2+b^2\right )^2 f^4}+\frac {\left (3 b^2 d^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {(a-i b) x}{a+i b}\right )}{x} \, dx,x,e^{2 i e+2 i f x}\right )}{2 \left (a^2+b^2\right )^2 f^4}\\ &=-\frac {2 i b^2 (c+d x)^3}{\left (a^2+b^2\right )^2 f}-\frac {2 b^2 (c+d x)^3}{(a-i b)^2 (a+i b) \left (i a-b+(i a+b) e^{2 i e+2 i f x}\right ) f}+\frac {(c+d x)^4}{4 (a-i b)^2 d}+\frac {b (c+d x)^4}{(i a-b) (a-i b)^2 d}-\frac {b^2 (c+d x)^4}{\left (a^2+b^2\right )^2 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1+\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{\left (a^2+b^2\right )^2 f^2}+\frac {2 b (c+d x)^3 \log \left (1+\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{(a-i b)^2 (a+i b) f}-\frac {2 i b^2 (c+d x)^3 \log \left (1+\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{\left (a^2+b^2\right )^2 f}-\frac {3 i b^2 d^2 (c+d x) \text {Li}_2\left (-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{\left (a^2+b^2\right )^2 f^3}+\frac {3 b d (c+d x)^2 \text {Li}_2\left (-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{(i a-b) (a-i b)^2 f^2}-\frac {3 b^2 d (c+d x)^2 \text {Li}_2\left (-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{\left (a^2+b^2\right )^2 f^2}+\frac {3 b^2 d^3 \text {Li}_3\left (-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{2 \left (a^2+b^2\right )^2 f^4}+\frac {3 b d^2 (c+d x) \text {Li}_3\left (-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{(a-i b)^2 (a+i b) f^3}-\frac {3 i b^2 d^2 (c+d x) \text {Li}_3\left (-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{\left (a^2+b^2\right )^2 f^3}-\frac {3 b d^3 \text {Li}_4\left (-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{2 (i a-b) (a-i b)^2 f^4}+\frac {3 b^2 d^3 \text {Li}_4\left (-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{2 \left (a^2+b^2\right )^2 f^4}\\ \end {align*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2857\) vs. \(2(848)=1696\).
time = 11.03, size = 2857, normalized size = 3.37 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(c + d*x)^3/(a + b*Tan[e + f*x])^2,x]

[Out]

(b*(12*a*b*c^2*d*E^((2*I)*e)*f^3*x - (12*I)*b^2*c^2*d*E^((2*I)*e)*f^3*x + 8*a^2*c^3*E^((2*I)*e)*f^4*x - (8*I)*
a*b*c^3*E^((2*I)*e)*f^4*x + 12*a*b*c*d^2*E^((2*I)*e)*f^3*x^2 - (12*I)*b^2*c*d^2*E^((2*I)*e)*f^3*x^2 + 12*a^2*c
^2*d*E^((2*I)*e)*f^4*x^2 - (12*I)*a*b*c^2*d*E^((2*I)*e)*f^4*x^2 + 4*a*b*d^3*E^((2*I)*e)*f^3*x^3 - (4*I)*b^2*d^
3*E^((2*I)*e)*f^3*x^3 + 8*a^2*c*d^2*E^((2*I)*e)*f^4*x^3 - (8*I)*a*b*c*d^2*E^((2*I)*e)*f^4*x^3 + 2*a^2*d^3*E^((
2*I)*e)*f^4*x^4 - (2*I)*a*b*d^3*E^((2*I)*e)*f^4*x^4 + (12*I)*a*b*c*d^2*f^2*x*Log[1 + ((a - I*b)*E^((2*I)*(e +
f*x)))/(a + I*b)] - 12*b^2*c*d^2*f^2*x*Log[1 + ((a - I*b)*E^((2*I)*(e + f*x)))/(a + I*b)] + (12*I)*a*b*c*d^2*E
^((2*I)*e)*f^2*x*Log[1 + ((a - I*b)*E^((2*I)*(e + f*x)))/(a + I*b)] + 12*b^2*c*d^2*E^((2*I)*e)*f^2*x*Log[1 + (
(a - I*b)*E^((2*I)*(e + f*x)))/(a + I*b)] + (12*I)*a^2*c^2*d*f^3*x*Log[1 + ((a - I*b)*E^((2*I)*(e + f*x)))/(a
+ I*b)] - 12*a*b*c^2*d*f^3*x*Log[1 + ((a - I*b)*E^((2*I)*(e + f*x)))/(a + I*b)] + (12*I)*a^2*c^2*d*E^((2*I)*e)
*f^3*x*Log[1 + ((a - I*b)*E^((2*I)*(e + f*x)))/(a + I*b)] + 12*a*b*c^2*d*E^((2*I)*e)*f^3*x*Log[1 + ((a - I*b)*
E^((2*I)*(e + f*x)))/(a + I*b)] + (6*I)*a*b*d^3*f^2*x^2*Log[1 + ((a - I*b)*E^((2*I)*(e + f*x)))/(a + I*b)] - 6
*b^2*d^3*f^2*x^2*Log[1 + ((a - I*b)*E^((2*I)*(e + f*x)))/(a + I*b)] + (6*I)*a*b*d^3*E^((2*I)*e)*f^2*x^2*Log[1
+ ((a - I*b)*E^((2*I)*(e + f*x)))/(a + I*b)] + 6*b^2*d^3*E^((2*I)*e)*f^2*x^2*Log[1 + ((a - I*b)*E^((2*I)*(e +
f*x)))/(a + I*b)] + (12*I)*a^2*c*d^2*f^3*x^2*Log[1 + ((a - I*b)*E^((2*I)*(e + f*x)))/(a + I*b)] - 12*a*b*c*d^2
*f^3*x^2*Log[1 + ((a - I*b)*E^((2*I)*(e + f*x)))/(a + I*b)] + (12*I)*a^2*c*d^2*E^((2*I)*e)*f^3*x^2*Log[1 + ((a
 - I*b)*E^((2*I)*(e + f*x)))/(a + I*b)] + 12*a*b*c*d^2*E^((2*I)*e)*f^3*x^2*Log[1 + ((a - I*b)*E^((2*I)*(e + f*
x)))/(a + I*b)] + (4*I)*a^2*d^3*f^3*x^3*Log[1 + ((a - I*b)*E^((2*I)*(e + f*x)))/(a + I*b)] - 4*a*b*d^3*f^3*x^3
*Log[1 + ((a - I*b)*E^((2*I)*(e + f*x)))/(a + I*b)] + (4*I)*a^2*d^3*E^((2*I)*e)*f^3*x^3*Log[1 + ((a - I*b)*E^(
(2*I)*(e + f*x)))/(a + I*b)] + 4*a*b*d^3*E^((2*I)*e)*f^3*x^3*Log[1 + ((a - I*b)*E^((2*I)*(e + f*x)))/(a + I*b)
] + (6*I)*a*b*c^2*d*f^2*Log[I*a - b + (I*a + b)*E^((2*I)*(e + f*x))] - 6*b^2*c^2*d*f^2*Log[I*a - b + (I*a + b)
*E^((2*I)*(e + f*x))] + (6*I)*a*b*c^2*d*E^((2*I)*e)*f^2*Log[I*a - b + (I*a + b)*E^((2*I)*(e + f*x))] + 6*b^2*c
^2*d*E^((2*I)*e)*f^2*Log[I*a - b + (I*a + b)*E^((2*I)*(e + f*x))] + (4*I)*a^2*c^3*f^3*Log[I*a - b + (I*a + b)*
E^((2*I)*(e + f*x))] - 4*a*b*c^3*f^3*Log[I*a - b + (I*a + b)*E^((2*I)*(e + f*x))] + (4*I)*a^2*c^3*E^((2*I)*e)*
f^3*Log[I*a - b + (I*a + b)*E^((2*I)*(e + f*x))] + 4*a*b*c^3*E^((2*I)*e)*f^3*Log[I*a - b + (I*a + b)*E^((2*I)*
(e + f*x))] + 6*d*((-I)*b*(-1 + E^((2*I)*e)) + a*(1 + E^((2*I)*e)))*f*(c + d*x)*(b*d + a*f*(c + d*x))*PolyLog[
2, -(((a - I*b)*E^((2*I)*(e + f*x)))/(a + I*b))] + 3*d^2*(b*(-1 + E^((2*I)*e)) + I*a*(1 + E^((2*I)*e)))*(b*d +
 2*a*f*(c + d*x))*PolyLog[3, -(((a - I*b)*E^((2*I)*(e + f*x)))/(a + I*b))] - 3*a^2*d^3*PolyLog[4, -(((a - I*b)
*E^((2*I)*(e + f*x)))/(a + I*b))] - (3*I)*a*b*d^3*PolyLog[4, -(((a - I*b)*E^((2*I)*(e + f*x)))/(a + I*b))] - 3
*a^2*d^3*E^((2*I)*e)*PolyLog[4, -(((a - I*b)*E^((2*I)*(e + f*x)))/(a + I*b))] + (3*I)*a*b*d^3*E^((2*I)*e)*Poly
Log[4, -(((a - I*b)*E^((2*I)*(e + f*x)))/(a + I*b))]))/(2*(a^2 + b^2)^2*(b*(-1 + E^((2*I)*e)) + I*a*(1 + E^((2
*I)*e)))*f^4) + (3*x^2*(a*c^2*d - I*b*c^2*d + a*c^2*d*Cos[2*e] + I*b*c^2*d*Cos[2*e] + I*a*c^2*d*Sin[2*e] - b*c
^2*d*Sin[2*e]))/(2*(a - I*b)*(a + I*b)*(a + I*b + a*Cos[2*e] - I*b*Cos[2*e] + I*a*Sin[2*e] + b*Sin[2*e])) + (x
^3*(a*c*d^2 - I*b*c*d^2 + a*c*d^2*Cos[2*e] + I*b*c*d^2*Cos[2*e] + I*a*c*d^2*Sin[2*e] - b*c*d^2*Sin[2*e]))/((a
- I*b)*(a + I*b)*(a + I*b + a*Cos[2*e] - I*b*Cos[2*e] + I*a*Sin[2*e] + b*Sin[2*e])) + (x^4*(a*d^3 - I*b*d^3 +
a*d^3*Cos[2*e] + I*b*d^3*Cos[2*e] + I*a*d^3*Sin[2*e] - b*d^3*Sin[2*e]))/(4*(a - I*b)*(a + I*b)*(a + I*b + a*Co
s[2*e] - I*b*Cos[2*e] + I*a*Sin[2*e] + b*Sin[2*e])) + x*(c^3/(a^2 + (2*I)*a*b - b^2 + a^2*Cos[4*e] - (2*I)*a*b
*Cos[4*e] - b^2*Cos[4*e] + I*a^2*Sin[4*e] + 2*a*b*Sin[4*e] - I*b^2*Sin[4*e]) + ((-a - I*b + a*Cos[2*e] - I*b*C
os[2*e] + I*a*Sin[2*e] + b*Sin[2*e])*((-4*I)*a*b*c^3*Cos[2*e] + 4*a*b*c^3*Sin[2*e]))/((a - I*b)*(a + I*b)*(a +
 I*b + a*Cos[2*e] - I*b*Cos[2*e] + I*a*Sin[2*e] + b*Sin[2*e])*(a^2 + (2*I)*a*b - b^2 + a^2*Cos[4*e] - (2*I)*a*
b*Cos[4*e] - b^2*Cos[4*e] + I*a^2*Sin[4*e] + 2*a*b*Sin[4*e] - I*b^2*Sin[4*e])) + (c^3*Cos[4*e] + I*c^3*Sin[4*e
])/(a^2 + (2*I)*a*b - b^2 + a^2*Cos[4*e] - (2*I)*a*b*Cos[4*e] - b^2*Cos[4*e] + I*a^2*Sin[4*e] + 2*a*b*Sin[4*e]
 - I*b^2*Sin[4*e])) + (b^2*c^3*Sin[f*x] + 3*b^2*c^2*d*x*Sin[f*x] + 3*b^2*c*d^2*x^2*Sin[f*x] + b^2*d^3*x^3*Sin[
f*x])/((a - I*b)*(a + I*b)*f*(a*Cos[e] + b*Sin[e])*(a*Cos[e + f*x] + b*Sin[e + f*x]))

________________________________________________________________________________________

Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3511 vs. \(2 (763 ) = 1526\).
time = 0.66, size = 3512, normalized size = 4.14

method result size
risch \(\text {Expression too large to display}\) \(3512\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^3/(a+b*tan(f*x+e))^2,x,method=_RETURNVERBOSE)

[Out]

-2*I*b^2*(d^3*x^3+3*c*d^2*x^2+3*c^2*d*x+c^3)/(b-I*a)/f/(I*a+b)^2/(b*exp(2*I*(f*x+e))+I*a*exp(2*I*(f*x+e))-b+I*
a)+6/(I*a+b)^2/f^2/(b-I*a)*b*a*c^2*d/(a+I*b)*e^2+3/(I*a+b)^2/f^2/(b-I*a)*b*a*c^2*d/(a+I*b)*polylog(2,(I*b-a)*e
xp(2*I*(f*x+e))/(a+I*b))-3/(I*a+b)^2/f^2/(b-I*a)*b^3*c^2*d/(a+I*b)/(I*b-a)*ln(I*exp(2*I*(f*x+e))*b-a*exp(2*I*(
f*x+e))-I*b-a)-6*I/(I*a+b)^2/f^4/(b-I*a)*b^2*d^3*e^2/(a+I*b)*ln(exp(I*(f*x+e)))+3*I/(I*a+b)^2/f^2/(b-I*a)*b^2*
d^3/(a+I*b)*ln(1-(I*b-a)*exp(2*I*(f*x+e))/(a+I*b))*x^2-6*I/(I*a+b)^2/f^2/(b-I*a)*b^2*c^2*d/(a+I*b)*ln(exp(I*(f
*x+e)))-4*I/(I*a+b)^2/f/(b-I*a)*b*a*c^3/(a+I*b)*ln(exp(I*(f*x+e)))-1/4*d^3/(2*I*a*b-a^2+b^2)*x^4-1/(2*I*a*b-a^
2+b^2)*c^3*x-1/4/d/(2*I*a*b-a^2+b^2)*c^4-3*I/(I*a+b)^2/f^4/(b-I*a)*b^2*d^3*e^2/(a+I*b)*ln(1-(I*b-a)*exp(2*I*(f
*x+e))/(a+I*b))+4/(I*a+b)^2/(b-I*a)*b*a*c*d^2/(a+I*b)*x^3+6/(I*a+b)^2/(b-I*a)*b*a*c^2*d/(a+I*b)*x^2+3/(I*a+b)^
2/f^3/(b-I*a)*b^2*d^3/(a+I*b)*polylog(2,(I*b-a)*exp(2*I*(f*x+e))/(a+I*b))*x+3/2*I/(I*a+b)^2/f^4/(b-I*a)*b^2*d^
3/(a+I*b)*polylog(3,(I*b-a)*exp(2*I*(f*x+e))/(a+I*b))-6/(I*a+b)^2/f^3/(b-I*a)*b^2*d^3/(a+I*b)*e^2*x+6/(I*a+b)^
2/f/(b-I*a)*b^2*c*d^2/(a+I*b)*x^2+6/(I*a+b)^2/f^3/(b-I*a)*b^2*c*d^2/(a+I*b)*e^2+3/(I*a+b)^2/f^4/(b-I*a)*b*a*d^
3*e^4/(a+I*b)-3/2/(I*a+b)^2/f^4/(b-I*a)*b*a*d^3/(a+I*b)*polylog(4,(I*b-a)*exp(2*I*(f*x+e))/(a+I*b))+3/(I*a+b)^
2/f^3/(b-I*a)*b^2*c*d^2/(a+I*b)*polylog(2,(I*b-a)*exp(2*I*(f*x+e))/(a+I*b))-12/(I*a+b)^2/f^2/(b-I*a)*b*a*c*d^2
*e^2/(a+I*b)*x+12/(I*a+b)^2/f/(b-I*a)*b*a*c^2*d/(a+I*b)*e*x+2/(I*a+b)^2/f^4/(b-I*a)*b^2*a*d^3*e^3/(a+I*b)/(I*b
-a)*ln(I*exp(2*I*(f*x+e))*b-a*exp(2*I*(f*x+e))-I*b-a)+6/(I*a+b)^2/f^3/(b-I*a)*b^3*c*d^2*e/(a+I*b)/(I*b-a)*ln(I
*exp(2*I*(f*x+e))*b-a*exp(2*I*(f*x+e))-I*b-a)-6/(I*a+b)^2/f^3/(b-I*a)*b^2*a*c*d^2*e^2/(a+I*b)/(I*b-a)*ln(I*exp
(2*I*(f*x+e))*b-a*exp(2*I*(f*x+e))-I*b-a)+6/(I*a+b)^2/f^2/(b-I*a)*b^2*a*c^2*d*e/(a+I*b)/(I*b-a)*ln(I*exp(2*I*(
f*x+e))*b-a*exp(2*I*(f*x+e))-I*b-a)-6*I/(I*a+b)^2/f^3/(b-I*a)*b*a^2*c*d^2*e^2/(a+I*b)/(I*b-a)*ln(I*exp(2*I*(f*
x+e))*b-a*exp(2*I*(f*x+e))-I*b-a)+6*I/(I*a+b)^2/f^2/(b-I*a)*b*a^2*c^2*d*e/(a+I*b)/(I*b-a)*ln(I*exp(2*I*(f*x+e)
)*b-a*exp(2*I*(f*x+e))-I*b-a)+6*I/(I*a+b)^2/f^3/(b-I*a)*b^2*c*d^2*e/(a+I*b)/(I*b-a)*ln(I*exp(2*I*(f*x+e))*b-a*
exp(2*I*(f*x+e))-I*b-a)*a-3/(I*a+b)^2/f^4/(b-I*a)*b^3*d^3*e^2/(a+I*b)/(I*b-a)*ln(I*exp(2*I*(f*x+e))*b-a*exp(2*
I*(f*x+e))-I*b-a)-2/(I*a+b)^2/f/(b-I*a)*b^2*a*c^3/(a+I*b)/(I*b-a)*ln(I*exp(2*I*(f*x+e))*b-a*exp(2*I*(f*x+e))-I
*b-a)+4/(I*a+b)^2/f^3/(b-I*a)*b*a*d^3*e^3/(a+I*b)*x+12/(I*a+b)^2/f^2/(b-I*a)*b^2*c*d^2/(a+I*b)*e*x-8/(I*a+b)^2
/f^3/(b-I*a)*b*a*c*d^2*e^3/(a+I*b)+3/(I*a+b)^2/f^2/(b-I*a)*b*a*d^3/(a+I*b)*polylog(2,(I*b-a)*exp(2*I*(f*x+e))/
(a+I*b))*x^2+1/(I*a+b)^2/(b-I*a)*b*a*d^3/(a+I*b)*x^4+2/(I*a+b)^2/f/(b-I*a)*b^2*d^3/(a+I*b)*x^3-4/(I*a+b)^2/f^4
/(b-I*a)*b^2*d^3/(a+I*b)*e^3-6*I/(I*a+b)^2/f^3/(b-I*a)*b*a*c*d^2*e^2/(a+I*b)*ln(1-(I*b-a)*exp(2*I*(f*x+e))/(a+
I*b))+12*I/(I*a+b)^2/f^2/(b-I*a)*b*a*c^2*d*e/(a+I*b)*ln(exp(I*(f*x+e)))+6*I/(I*a+b)^2/f/(b-I*a)*b*a*c^2*d/(a+I
*b)*ln(1-(I*b-a)*exp(2*I*(f*x+e))/(a+I*b))*x+6*I/(I*a+b)^2/f/(b-I*a)*b*a*c*d^2/(a+I*b)*ln(1-(I*b-a)*exp(2*I*(f
*x+e))/(a+I*b))*x^2+6*I/(I*a+b)^2/f^2/(b-I*a)*b*a*c^2*d/(a+I*b)*ln(1-(I*b-a)*exp(2*I*(f*x+e))/(a+I*b))*e-3*I/(
I*a+b)^2/f^4/(b-I*a)*b^2*d^3*e^2/(a+I*b)/(I*b-a)*ln(I*exp(2*I*(f*x+e))*b-a*exp(2*I*(f*x+e))-I*b-a)*a-3*I/(I*a+
b)^2/f^2/(b-I*a)*b^2*c^2*d/(a+I*b)/(I*b-a)*ln(I*exp(2*I*(f*x+e))*b-a*exp(2*I*(f*x+e))-I*b-a)*a+2*I/(I*a+b)^2/f
^4/(b-I*a)*b*a^2*d^3*e^3/(a+I*b)/(I*b-a)*ln(I*exp(2*I*(f*x+e))*b-a*exp(2*I*(f*x+e))-I*b-a)-12*I/(I*a+b)^2/f^3/
(b-I*a)*b*a*c*d^2*e^2/(a+I*b)*ln(exp(I*(f*x+e)))+6/(I*a+b)^2/f^2/(b-I*a)*b*a*c*d^2/(a+I*b)*polylog(2,(I*b-a)*e
xp(2*I*(f*x+e))/(a+I*b))*x+6*I/(I*a+b)^2/f^2/(b-I*a)*b^2*c*d^2/(a+I*b)*ln(1-(I*b-a)*exp(2*I*(f*x+e))/(a+I*b))*
x+6*I/(I*a+b)^2/f^3/(b-I*a)*b^2*c*d^2/(a+I*b)*ln(1-(I*b-a)*exp(2*I*(f*x+e))/(a+I*b))*e+4*I/(I*a+b)^2/f^4/(b-I*
a)*b*a*d^3*e^3/(a+I*b)*ln(exp(I*(f*x+e)))+3*I/(I*a+b)^2/f^3/(b-I*a)*b*a*c*d^2/(a+I*b)*polylog(3,(I*b-a)*exp(2*
I*(f*x+e))/(a+I*b))-2*I/(I*a+b)^2/f/(b-I*a)*b*a^2*c^3/(a+I*b)/(I*b-a)*ln(I*exp(2*I*(f*x+e))*b-a*exp(2*I*(f*x+e
))-I*b-a)+12*I/(I*a+b)^2/f^3/(b-I*a)*b^2*c*d^2*e/(a+I*b)*ln(exp(I*(f*x+e)))+3*I/(I*a+b)^2/f^3/(b-I*a)*b*a*d^3/
(a+I*b)*polylog(3,(I*b-a)*exp(2*I*(f*x+e))/(a+I*b))*x+2*I/(I*a+b)^2/f/(b-I*a)*b*a*d^3/(a+I*b)*ln(1-(I*b-a)*exp
(2*I*(f*x+e))/(a+I*b))*x^3+2*I/(I*a+b)^2/f^4/(b-I*a)*b*a*d^3*e^3/(a+I*b)*ln(1-(I*b-a)*exp(2*I*(f*x+e))/(a+I*b)
)-d^2/(2*I*a*b-a^2+b^2)*c*x^3-3/2*d/(2*I*a*b-a^2+b^2)*c^2*x^2

________________________________________________________________________________________

Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 4521 vs. \(2 (707) = 1414\).
time = 2.81, size = 4521, normalized size = 5.33 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/(a+b*tan(f*x+e))^2,x, algorithm="maxima")

[Out]

-1/12*(36*c^2*d*(2*a*b*log(b*tan(f*x + e) + a)/((a^4 + 2*a^2*b^2 + b^4)*f) - a*b*log(tan(f*x + e)^2 + 1)/((a^4
 + 2*a^2*b^2 + b^4)*f) - b/((a^2*b + b^3)*f*tan(f*x + e) + (a^3 + a*b^2)*f) + (a^2 - b^2)*(f*x + e)/((a^4 + 2*
a^2*b^2 + b^4)*f))*e - 12*(2*a*b*log(b*tan(f*x + e) + a)/(a^4 + 2*a^2*b^2 + b^4) - a*b*log(tan(f*x + e)^2 + 1)
/(a^4 + 2*a^2*b^2 + b^4) + (a^2 - b^2)*(f*x + e)/(a^4 + 2*a^2*b^2 + b^4) - b/(a^3 + a*b^2 + (a^2*b + b^3)*tan(
f*x + e)))*c^3 - (3*(a^3 - I*a^2*b + a*b^2 - I*b^3)*(f*x + e)^4*d^3 - 72*(-I*a*b^2*e^2 + b^3*e^2)*c*d^2*f + 12
*((a^3 - I*a^2*b + a*b^2 - I*b^3)*c*d^2*f - (a^3*e - I*a^2*b*e + a*b^2*e - I*b^3*e)*d^3)*(f*x + e)^3 - 24*(I*a
*b^2*e^3 - b^3*e^3)*d^3 + 18*((a^3 - I*a^2*b + a*b^2 - I*b^3)*c^2*d*f^2 - 2*(a^3*e - I*a^2*b*e + a*b^2*e - I*b
^3*e)*c*d^2*f + (a^3*e^2 - I*a^2*b*e^2 + a*b^2*e^2 - I*b^3*e^2)*d^3)*(f*x + e)^2 + 12*(3*(a^3*e^2 - I*a^2*b*e^
2 + a*b^2*e^2 - I*b^3*e^2)*c*d^2*f - (a^3*e^3 - I*a^2*b*e^3 + a*b^2*e^3 - I*b^3*e^3)*d^3)*(f*x + e) - 12*(3*(-
I*a*b^2 + b^3)*c^2*d*f^2 + 6*(a*b^2*(e^2 + I*e) - I*a^2*b*e^2 - b^3*e)*c*d^2*f - (a*b^2*(2*e^3 + 3*I*e^2) - 2*
I*a^2*b*e^3 - 3*b^3*e^2)*d^3 + (3*(-I*a*b^2 - b^3)*c^2*d*f^2 - 6*(a*b^2*(e^2 - I*e) + I*a^2*b*e^2 - b^3*e)*c*d
^2*f + (a*b^2*(2*e^3 - 3*I*e^2) + 2*I*a^2*b*e^3 - 3*b^3*e^2)*d^3)*cos(2*f*x + 2*e) + (3*(a*b^2 - I*b^3)*c^2*d*
f^2 + 6*(a*b^2*(-I*e^2 - e) + a^2*b*e^2 + I*b^3*e)*c*d^2*f + (a*b^2*(2*I*e^3 + 3*e^2) - 2*a^2*b*e^3 - 3*I*b^3*
e^2)*d^3)*sin(2*f*x + 2*e))*arctan2(-b*cos(2*f*x + 2*e) + a*sin(2*f*x + 2*e) + b, a*cos(2*f*x + 2*e) + b*sin(2
*f*x + 2*e) + a) - 4*(8*(I*a^2*b - a*b^2)*(f*x + e)^3*d^3 + 9*(2*(I*a^2*b - a*b^2)*c*d^2*f + (a*b^2*(2*e + I)
- 2*I*a^2*b*e - b^3)*d^3)*(f*x + e)^2 + 18*((I*a^2*b - a*b^2)*c^2*d*f^2 + (a*b^2*(2*e + I) - 2*I*a^2*b*e - b^3
)*c*d^2*f - (a*b^2*(e^2 + I*e) - I*a^2*b*e^2 - b^3*e)*d^3)*(f*x + e) + (8*(I*a^2*b + a*b^2)*(f*x + e)^3*d^3 +
9*(2*(I*a^2*b + a*b^2)*c*d^2*f - (a*b^2*(2*e - I) + 2*I*a^2*b*e - b^3)*d^3)*(f*x + e)^2 + 18*((I*a^2*b + a*b^2
)*c^2*d*f^2 - (a*b^2*(2*e - I) + 2*I*a^2*b*e - b^3)*c*d^2*f + (a*b^2*(e^2 - I*e) + I*a^2*b*e^2 - b^3*e)*d^3)*(
f*x + e))*cos(2*f*x + 2*e) - (8*(a^2*b - I*a*b^2)*(f*x + e)^3*d^3 + 9*(2*(a^2*b - I*a*b^2)*c*d^2*f - (a*b^2*(-
2*I*e - 1) + 2*a^2*b*e + I*b^3)*d^3)*(f*x + e)^2 + 18*((a^2*b - I*a*b^2)*c^2*d*f^2 - (a*b^2*(-2*I*e - 1) + 2*a
^2*b*e + I*b^3)*c*d^2*f - (a*b^2*(I*e^2 + e) - a^2*b*e^2 - I*b^3*e)*d^3)*(f*x + e))*sin(2*f*x + 2*e))*arctan2(
(2*a*b*cos(2*f*x + 2*e) - (a^2 - b^2)*sin(2*f*x + 2*e))/(a^2 + b^2), (2*a*b*sin(2*f*x + 2*e) + a^2 + b^2 + (a^
2 - b^2)*cos(2*f*x + 2*e))/(a^2 + b^2)) + 3*((a^3 - 3*I*a^2*b - 3*a*b^2 + I*b^3)*(f*x + e)^4*d^3 + 4*((a^3 - 3
*I*a^2*b - 3*a*b^2 + I*b^3)*c*d^2*f + (a*b^2*(3*e - 2*I) - b^3*(I*e + 2) - a^3*e + 3*I*a^2*b*e)*d^3)*(f*x + e)
^3 + 6*((a^3 - 3*I*a^2*b - 3*a*b^2 + I*b^3)*c^2*d*f^2 + 2*(a*b^2*(3*e - 2*I) - b^3*(I*e + 2) - a^3*e + 3*I*a^2
*b*e)*c*d^2*f - (a*b^2*(3*e^2 - 4*I*e) + b^3*(-I*e^2 - 4*e) - a^3*e^2 + 3*I*a^2*b*e^2)*d^3)*(f*x + e)^2 - 4*(6
*(I*a*b^2 + b^3)*c^2*d*f^2 + 3*(a*b^2*(3*e^2 - 4*I*e) + b^3*(-I*e^2 - 4*e) - a^3*e^2 + 3*I*a^2*b*e^2)*c*d^2*f
- (3*a*b^2*(e^3 - 2*I*e^2) - b^3*(I*e^3 + 6*e^2) - a^3*e^3 + 3*I*a^2*b*e^3)*d^3)*(f*x + e))*cos(2*f*x + 2*e) -
 12*(4*(I*a^2*b - a*b^2)*(f*x + e)^2*d^3 + 3*(I*a^2*b - a*b^2)*c^2*d*f^2 + 3*(a*b^2*(2*e + I) - 2*I*a^2*b*e -
b^3)*c*d^2*f - 3*(a*b^2*(e^2 + I*e) - I*a^2*b*e^2 - b^3*e)*d^3 + 3*(2*(I*a^2*b - a*b^2)*c*d^2*f + (a*b^2*(2*e
+ I) - 2*I*a^2*b*e - b^3)*d^3)*(f*x + e) + (4*(I*a^2*b + a*b^2)*(f*x + e)^2*d^3 + 3*(I*a^2*b + a*b^2)*c^2*d*f^
2 - 3*(a*b^2*(2*e - I) + 2*I*a^2*b*e - b^3)*c*d^2*f + 3*(a*b^2*(e^2 - I*e) + I*a^2*b*e^2 - b^3*e)*d^3 + 3*(2*(
I*a^2*b + a*b^2)*c*d^2*f - (a*b^2*(2*e - I) + 2*I*a^2*b*e - b^3)*d^3)*(f*x + e))*cos(2*f*x + 2*e) - (4*(a^2*b
- I*a*b^2)*(f*x + e)^2*d^3 + 3*(a^2*b - I*a*b^2)*c^2*d*f^2 - 3*(a*b^2*(-2*I*e - 1) + 2*a^2*b*e + I*b^3)*c*d^2*
f - 3*(a*b^2*(I*e^2 + e) - a^2*b*e^2 - I*b^3*e)*d^3 + 3*(2*(a^2*b - I*a*b^2)*c*d^2*f - (a*b^2*(-2*I*e - 1) + 2
*a^2*b*e + I*b^3)*d^3)*(f*x + e))*sin(2*f*x + 2*e))*dilog((I*a + b)*e^(2*I*f*x + 2*I*e)/(-I*a + b)) + 6*(3*(a*
b^2 + I*b^3)*c^2*d*f^2 - 6*(a*b^2*(-I*e^2 + e) - a^2*b*e^2 + I*b^3*e)*c*d^2*f - (a*b^2*(2*I*e^3 - 3*e^2) + 2*a
^2*b*e^3 - 3*I*b^3*e^2)*d^3 + (3*(a*b^2 - I*b^3)*c^2*d*f^2 - 6*(a*b^2*(I*e^2 + e) - a^2*b*e^2 - I*b^3*e)*c*d^2
*f - (a*b^2*(-2*I*e^3 - 3*e^2) + 2*a^2*b*e^3 + 3*I*b^3*e^2)*d^3)*cos(2*f*x + 2*e) - (3*(-I*a*b^2 - b^3)*c^2*d*
f^2 - 6*(a*b^2*(e^2 - I*e) + I*a^2*b*e^2 - b^3*e)*c*d^2*f + (a*b^2*(2*e^3 - 3*I*e^2) + 2*I*a^2*b*e^3 - 3*b^3*e
^2)*d^3)*sin(2*f*x + 2*e))*log((a^2 + b^2)*cos(2*f*x + 2*e)^2 + 4*a*b*sin(2*f*x + 2*e) + (a^2 + b^2)*sin(2*f*x
 + 2*e)^2 + a^2 + b^2 + 2*(a^2 - b^2)*cos(2*f*x + 2*e)) + 2*(8*(a^2*b + I*a*b^2)*(f*x + e)^3*d^3 + 9*(2*(a^2*b
 + I*a*b^2)*c*d^2*f - (a*b^2*(2*I*e - 1) + 2*a^2*b*e - I*b^3)*d^3)*(f*x + e)^2 + 18*((a^2*b + I*a*b^2)*c^2*d*f
^2 - (a*b^2*(2*I*e - 1) + 2*a^2*b*e - I*b^3)*c*d^2*f - (a*b^2*(-I*e^2 + e) - a^2*b*e^2 + I*b^3*e)*d^3)*(f*x +
e) + (8*(a^2*b - I*a*b^2)*(f*x + e)^3*d^3 + 9*(...

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2555 vs. \(2 (707) = 1414\).
time = 0.51, size = 2555, normalized size = 3.01 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/(a+b*tan(f*x+e))^2,x, algorithm="fricas")

[Out]

1/4*((a^3 - a*b^2)*d^3*f^4*x^4 - 4*b^3*c^3*f^3 - 4*(b^3*d^3*f^3 - (a^3 - a*b^2)*c*d^2*f^4)*x^3 - 6*(2*b^3*c*d^
2*f^3 - (a^3 - a*b^2)*c^2*d*f^4)*x^2 - 4*(3*b^3*c^2*d*f^3 - (a^3 - a*b^2)*c^3*f^4)*x - 6*(-I*a^2*b*d^3*f^2*x^2
 - I*a^2*b*c^2*d*f^2 - I*a*b^2*c*d^2*f - I*(2*a^2*b*c*d^2*f^2 + a*b^2*d^3*f)*x + (-I*a*b^2*d^3*f^2*x^2 - I*a*b
^2*c^2*d*f^2 - I*b^3*c*d^2*f - I*(2*a*b^2*c*d^2*f^2 + b^3*d^3*f)*x)*tan(f*x + e))*dilog(2*((I*a*b - b^2)*tan(f
*x + e)^2 - a^2 - I*a*b + (I*a^2 - 2*a*b - I*b^2)*tan(f*x + e))/((a^2 + b^2)*tan(f*x + e)^2 + a^2 + b^2) + 1)
- 6*(I*a^2*b*d^3*f^2*x^2 + I*a^2*b*c^2*d*f^2 + I*a*b^2*c*d^2*f + I*(2*a^2*b*c*d^2*f^2 + a*b^2*d^3*f)*x + (I*a*
b^2*d^3*f^2*x^2 + I*a*b^2*c^2*d*f^2 + I*b^3*c*d^2*f + I*(2*a*b^2*c*d^2*f^2 + b^3*d^3*f)*x)*tan(f*x + e))*dilog
(2*((-I*a*b - b^2)*tan(f*x + e)^2 - a^2 + I*a*b + (-I*a^2 - 2*a*b + I*b^2)*tan(f*x + e))/((a^2 + b^2)*tan(f*x
+ e)^2 + a^2 + b^2) + 1) + 2*(2*a^2*b*d^3*f^3*x^3 + 2*a^2*b*d^3*e^3 + 3*(2*a^2*b*c*d^2*f^3 + a*b^2*d^3*f^2)*x^
2 + 6*(a^2*b*c^2*d*f^3 + a*b^2*c*d^2*f^2)*x - 3*(2*a^2*b*c*d^2*f + a*b^2*d^3)*e^2 + 6*(a^2*b*c^2*d*f^2 + a*b^2
*c*d^2*f)*e + (2*a*b^2*d^3*f^3*x^3 + 2*a*b^2*d^3*e^3 + 3*(2*a*b^2*c*d^2*f^3 + b^3*d^3*f^2)*x^2 + 6*(a*b^2*c^2*
d*f^3 + b^3*c*d^2*f^2)*x - 3*(2*a*b^2*c*d^2*f + b^3*d^3)*e^2 + 6*(a*b^2*c^2*d*f^2 + b^3*c*d^2*f)*e)*tan(f*x +
e))*log(-2*((I*a*b - b^2)*tan(f*x + e)^2 - a^2 - I*a*b + (I*a^2 - 2*a*b - I*b^2)*tan(f*x + e))/((a^2 + b^2)*ta
n(f*x + e)^2 + a^2 + b^2)) + 2*(2*a^2*b*d^3*f^3*x^3 + 2*a^2*b*d^3*e^3 + 3*(2*a^2*b*c*d^2*f^3 + a*b^2*d^3*f^2)*
x^2 + 6*(a^2*b*c^2*d*f^3 + a*b^2*c*d^2*f^2)*x - 3*(2*a^2*b*c*d^2*f + a*b^2*d^3)*e^2 + 6*(a^2*b*c^2*d*f^2 + a*b
^2*c*d^2*f)*e + (2*a*b^2*d^3*f^3*x^3 + 2*a*b^2*d^3*e^3 + 3*(2*a*b^2*c*d^2*f^3 + b^3*d^3*f^2)*x^2 + 6*(a*b^2*c^
2*d*f^3 + b^3*c*d^2*f^2)*x - 3*(2*a*b^2*c*d^2*f + b^3*d^3)*e^2 + 6*(a*b^2*c^2*d*f^2 + b^3*c*d^2*f)*e)*tan(f*x
+ e))*log(-2*((-I*a*b - b^2)*tan(f*x + e)^2 - a^2 + I*a*b + (-I*a^2 - 2*a*b + I*b^2)*tan(f*x + e))/((a^2 + b^2
)*tan(f*x + e)^2 + a^2 + b^2)) + 2*(2*a^2*b*c^3*f^3 + 3*a*b^2*c^2*d*f^2 - 2*a^2*b*d^3*e^3 + 3*(2*a^2*b*c*d^2*f
 + a*b^2*d^3)*e^2 - 6*(a^2*b*c^2*d*f^2 + a*b^2*c*d^2*f)*e + (2*a*b^2*c^3*f^3 + 3*b^3*c^2*d*f^2 - 2*a*b^2*d^3*e
^3 + 3*(2*a*b^2*c*d^2*f + b^3*d^3)*e^2 - 6*(a*b^2*c^2*d*f^2 + b^3*c*d^2*f)*e)*tan(f*x + e))*log(((I*a*b + b^2)
*tan(f*x + e)^2 - a^2 + I*a*b + (I*a^2 + I*b^2)*tan(f*x + e))/(tan(f*x + e)^2 + 1)) + 2*(2*a^2*b*c^3*f^3 + 3*a
*b^2*c^2*d*f^2 - 2*a^2*b*d^3*e^3 + 3*(2*a^2*b*c*d^2*f + a*b^2*d^3)*e^2 - 6*(a^2*b*c^2*d*f^2 + a*b^2*c*d^2*f)*e
 + (2*a*b^2*c^3*f^3 + 3*b^3*c^2*d*f^2 - 2*a*b^2*d^3*e^3 + 3*(2*a*b^2*c*d^2*f + b^3*d^3)*e^2 - 6*(a*b^2*c^2*d*f
^2 + b^3*c*d^2*f)*e)*tan(f*x + e))*log(((I*a*b - b^2)*tan(f*x + e)^2 + a^2 + I*a*b + (I*a^2 + I*b^2)*tan(f*x +
 e))/(tan(f*x + e)^2 + 1)) - 3*(I*a*b^2*d^3*tan(f*x + e) + I*a^2*b*d^3)*polylog(4, ((a^2 + 2*I*a*b - b^2)*tan(
f*x + e)^2 - a^2 - 2*I*a*b + b^2 - 2*(-I*a^2 + 2*a*b + I*b^2)*tan(f*x + e))/((a^2 + b^2)*tan(f*x + e)^2 + a^2
+ b^2)) - 3*(-I*a*b^2*d^3*tan(f*x + e) - I*a^2*b*d^3)*polylog(4, ((a^2 - 2*I*a*b - b^2)*tan(f*x + e)^2 - a^2 +
 2*I*a*b + b^2 - 2*(I*a^2 + 2*a*b - I*b^2)*tan(f*x + e))/((a^2 + b^2)*tan(f*x + e)^2 + a^2 + b^2)) + 3*(2*a^2*
b*d^3*f*x + 2*a^2*b*c*d^2*f + a*b^2*d^3 + (2*a*b^2*d^3*f*x + 2*a*b^2*c*d^2*f + b^3*d^3)*tan(f*x + e))*polylog(
3, ((a^2 + 2*I*a*b - b^2)*tan(f*x + e)^2 - a^2 - 2*I*a*b + b^2 - 2*(-I*a^2 + 2*a*b + I*b^2)*tan(f*x + e))/((a^
2 + b^2)*tan(f*x + e)^2 + a^2 + b^2)) + 3*(2*a^2*b*d^3*f*x + 2*a^2*b*c*d^2*f + a*b^2*d^3 + (2*a*b^2*d^3*f*x +
2*a*b^2*c*d^2*f + b^3*d^3)*tan(f*x + e))*polylog(3, ((a^2 - 2*I*a*b - b^2)*tan(f*x + e)^2 - a^2 + 2*I*a*b + b^
2 - 2*(I*a^2 + 2*a*b - I*b^2)*tan(f*x + e))/((a^2 + b^2)*tan(f*x + e)^2 + a^2 + b^2)) + ((a^2*b - b^3)*d^3*f^4
*x^4 + 4*a*b^2*c^3*f^3 + 4*(a*b^2*d^3*f^3 + (a^2*b - b^3)*c*d^2*f^4)*x^3 + 6*(2*a*b^2*c*d^2*f^3 + (a^2*b - b^3
)*c^2*d*f^4)*x^2 + 4*(3*a*b^2*c^2*d*f^3 + (a^2*b - b^3)*c^3*f^4)*x)*tan(f*x + e))/((a^4*b + 2*a^2*b^3 + b^5)*f
^4*tan(f*x + e) + (a^5 + 2*a^3*b^2 + a*b^4)*f^4)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c + d x\right )^{3}}{\left (a + b \tan {\left (e + f x \right )}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**3/(a+b*tan(f*x+e))**2,x)

[Out]

Integral((c + d*x)**3/(a + b*tan(e + f*x))**2, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/(a+b*tan(f*x+e))^2,x, algorithm="giac")

[Out]

integrate((d*x + c)^3/(b*tan(f*x + e) + a)^2, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c+d\,x\right )}^3}{{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^3/(a + b*tan(e + f*x))^2,x)

[Out]

int((c + d*x)^3/(a + b*tan(e + f*x))^2, x)

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