3.1.60 \(\int \frac {(c+d x)^2}{(a+b \tan (e+f x))^2} \, dx\) [60]

Optimal. Leaf size=654 \[ -\frac {2 i b^2 (c+d x)^2}{\left (a^2+b^2\right )^2 f}+\frac {2 b^2 (c+d x)^2}{(a+i b) (i a+b)^2 \left (i a-b+(i a+b) e^{2 i e+2 i f x}\right ) f}+\frac {(c+d x)^3}{3 (a-i b)^2 d}+\frac {4 b (c+d x)^3}{3 (i a-b) (a-i b)^2 d}-\frac {4 b^2 (c+d x)^3}{3 \left (a^2+b^2\right )^2 d}+\frac {2 b^2 d (c+d x) \log \left (1+\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{\left (a^2+b^2\right )^2 f^2}+\frac {2 b (c+d x)^2 \log \left (1+\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{(a-i b)^2 (a+i b) f}-\frac {2 i b^2 (c+d x)^2 \log \left (1+\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{\left (a^2+b^2\right )^2 f}-\frac {i b^2 d^2 \text {PolyLog}\left (2,-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{\left (a^2+b^2\right )^2 f^3}+\frac {2 b d (c+d x) \text {PolyLog}\left (2,-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{(i a-b) (a-i b)^2 f^2}-\frac {2 b^2 d (c+d x) \text {PolyLog}\left (2,-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{\left (a^2+b^2\right )^2 f^2}+\frac {b d^2 \text {PolyLog}\left (3,-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{(a-i b)^2 (a+i b) f^3}-\frac {i b^2 d^2 \text {PolyLog}\left (3,-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{\left (a^2+b^2\right )^2 f^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 1.02, antiderivative size = 654, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3815, 2216, 2215, 2221, 2611, 2320, 6724, 2222, 2317, 2438} \begin {gather*} -\frac {2 b^2 d (c+d x) \text {Li}_2\left (-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{f^2 \left (a^2+b^2\right )^2}+\frac {2 b^2 d (c+d x) \log \left (1+\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{f^2 \left (a^2+b^2\right )^2}-\frac {2 i b^2 (c+d x)^2 \log \left (1+\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{f \left (a^2+b^2\right )^2}-\frac {2 i b^2 (c+d x)^2}{f \left (a^2+b^2\right )^2}-\frac {4 b^2 (c+d x)^3}{3 d \left (a^2+b^2\right )^2}-\frac {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 )}{f^3 \left (a^2+b^2\right )^2}-\frac {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 )}{f^3 \left (a^2+b^2\right )^2}+\frac {2 b^2 (c+d x)^2}{f (a+i b) (b+i a)^2 \left ((b+i a) e^{2 i e+2 i f x}+i a-b\right )}+\frac {2 b d (c+d x) \text {Li}_2\left (-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{f^2 (-b+i a) (a-i b)^2}+\frac {2 b (c+d x)^2 \log \left (1+\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{f (a-i b)^2 (a+i b)}+\frac {4 b (c+d x)^3}{3 d (-b+i a) (a-i b)^2}+\frac {(c+d x)^3}{3 d (a-i b)^2}+\frac {b d^2 \text {Li}_3\left (-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{f^3 (a-i b)^2 (a+i b)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 2317

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

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 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

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]

Rubi steps

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

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Mathematica [A]
time = 7.76, size = 535, normalized size = 0.82 \begin {gather*} \frac {\frac {2 b \left (-2 f \left ((a-i b) e^{2 i e} f x \left (3 b d (2 c+d x)+2 a f \left (3 c^2+3 c d x+d^2 x^2\right )\right )+3 d \left (b \left (-1+e^{2 i e}\right )+i a \left (1+e^{2 i e}\right )\right ) x (b d+a f (2 c+d x)) \log \left (1+\frac {(a-i b) e^{2 i (e+f x)}}{a+i b}\right )+3 c \left (b \left (-1+e^{2 i e}\right )+i a \left (1+e^{2 i e}\right )\right ) (b d+a c f) \log \left (i a-b+(i a+b) e^{2 i (e+f x)}\right )\right )-3 d \left (-i b \left (-1+e^{2 i e}\right )+a \left (1+e^{2 i e}\right )\right ) (b d+2 a f (c+d x)) \text {PolyLog}\left (2,-\frac {(a-i b) e^{2 i (e+f x)}}{a+i b}\right )+3 a d^2 \left (b-b e^{2 i e}-i a \left (1+e^{2 i e}\right )\right ) \text {PolyLog}\left (3,-\frac {(a-i b) e^{2 i (e+f x)}}{a+i b}\right )\right )}{\left (a^2+b^2\right ) \left (b-b e^{2 i e}-i a \left (1+e^{2 i e}\right )\right )}+\frac {f^2 \left (\left (a^2-b^2\right ) f x \left (3 c^2+3 c d x+d^2 x^2\right ) \cos (f x)+\left (a^2+b^2\right ) f x \left (3 c^2+3 c d x+d^2 x^2\right ) \cos (2 e+f x)+2 b \left (3 b (c+d x)^2+a f x \left (3 c^2+3 c d x+d^2 x^2\right )\right ) \sin (f x)\right )}{(a \cos (e)+b \sin (e)) (a \cos (e+f x)+b \sin (e+f x))}}{6 \left (a^2+b^2\right ) f^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((2*b*(-2*f*((a - I*b)*E^((2*I)*e)*f*x*(3*b*d*(2*c + d*x) + 2*a*f*(3*c^2 + 3*c*d*x + d^2*x^2)) + 3*d*(b*(-1 +
E^((2*I)*e)) + I*a*(1 + E^((2*I)*e)))*x*(b*d + a*f*(2*c + d*x))*Log[1 + ((a - I*b)*E^((2*I)*(e + f*x)))/(a + I
*b)] + 3*c*(b*(-1 + E^((2*I)*e)) + I*a*(1 + E^((2*I)*e)))*(b*d + a*c*f)*Log[I*a - b + (I*a + b)*E^((2*I)*(e +
f*x))]) - 3*d*((-I)*b*(-1 + E^((2*I)*e)) + a*(1 + E^((2*I)*e)))*(b*d + 2*a*f*(c + d*x))*PolyLog[2, -(((a - I*b
)*E^((2*I)*(e + f*x)))/(a + I*b))] + 3*a*d^2*(b - b*E^((2*I)*e) - I*a*(1 + E^((2*I)*e)))*PolyLog[3, -(((a - I*
b)*E^((2*I)*(e + f*x)))/(a + I*b))]))/((a^2 + b^2)*(b - b*E^((2*I)*e) - I*a*(1 + E^((2*I)*e)))) + (f^2*((a^2 -
b^2)*f*x*(3*c^2 + 3*c*d*x + d^2*x^2)*Cos[f*x] + (a^2 + b^2)*f*x*(3*c^2 + 3*c*d*x + d^2*x^2)*Cos[2*e + f*x] +
2*b*(3*b*(c + d*x)^2 + a*f*x*(3*c^2 + 3*c*d*x + d^2*x^2))*Sin[f*x]))/((a*Cos[e] + b*Sin[e])*(a*Cos[e + f*x] +
b*Sin[e + f*x])))/(6*(a^2 + b^2)*f^3)

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2183 vs. \(2 (588 ) = 1176\).
time = 0.56, size = 2184, normalized size = 3.34

method	result	size
risch	\(\text {Expression too large to display}\)	\(2184\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2556 vs. \(2 (543) = 1086\).
time = 1.24, size = 2556, normalized size = 3.91 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(6*c*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 - 3*(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^2 - ((a^3 - I*a^2*b + a*b^2 - I*b^3)*(f*x + e)^3*d^2 + 3*(a^3*e^2 - I*a^2*b*e^2 + a*b^2*e^2 - I*b^3*e^
2)*(f*x + e)*d^2 + 3*((a^3 - I*a^2*b + a*b^2 - I*b^3)*c*d*f - (a^3*e - I*a^2*b*e + a*b^2*e - I*b^3*e)*d^2)*(f*
x + e)^2 - 6*(-I*a*b^2*e^2 + b^3*e^2)*d^2 - 6*((-I*a*b^2 + b^3)*c*d*f + (a*b^2*(e^2 + I*e) - I*a^2*b*e^2 - b^3
*e)*d^2 + ((-I*a*b^2 - b^3)*c*d*f - (a*b^2*(e^2 - I*e) + I*a^2*b*e^2 - b^3*e)*d^2)*cos(2*f*x + 2*e) + ((a*b^2
- I*b^3)*c*d*f + (a*b^2*(-I*e^2 - e) + a^2*b*e^2 + I*b^3*e)*d^2)*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) - 6*((I*a^2*b - a*b^2)*(f*x + e)^2*d^2
+ (2*(I*a^2*b - a*b^2)*c*d*f + (a*b^2*(2*e + I) - 2*I*a^2*b*e - b^3)*d^2)*(f*x + e) + ((I*a^2*b + a*b^2)*(f*x
+ e)^2*d^2 + (2*(I*a^2*b + a*b^2)*c*d*f - (a*b^2*(2*e - I) + 2*I*a^2*b*e - b^3)*d^2)*(f*x + e))*cos(2*f*x + 2
*e) - ((a^2*b - I*a*b^2)*(f*x + e)^2*d^2 + (2*(a^2*b - I*a*b^2)*c*d*f - (a*b^2*(-2*I*e - 1) + 2*a^2*b*e + I*b^
3)*d^2)*(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)) + ((a^3 - 3*I*a^2*b - 3*a
*b^2 + I*b^3)*(f*x + e)^3*d^2 + 3*((a^3 - 3*I*a^2*b - 3*a*b^2 + I*b^3)*c*d*f + (a*b^2*(3*e - 2*I) - b^3*(I*e +
2) - a^3*e + 3*I*a^2*b*e)*d^2)*(f*x + e)^2 - 3*(4*(I*a*b^2 + b^3)*c*d*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^2)*(f*x + e))*cos(2*f*x + 2*e) - 3*(2*(I*a^2*b - a*b^2)*(f*x + e)*d^2 +
2*(I*a^2*b - a*b^2)*c*d*f + (a*b^2*(2*e + I) - 2*I*a^2*b*e - b^3)*d^2 + (2*(I*a^2*b + a*b^2)*(f*x + e)*d^2 + 2
*(I*a^2*b + a*b^2)*c*d*f - (a*b^2*(2*e - I) + 2*I*a^2*b*e - b^3)*d^2)*cos(2*f*x + 2*e) - (2*(a^2*b - I*a*b^2)*
(f*x + e)*d^2 + 2*(a^2*b - I*a*b^2)*c*d*f - (a*b^2*(-2*I*e - 1) + 2*a^2*b*e + I*b^3)*d^2)*sin(2*f*x + 2*e))*di
log((I*a + b)*e^(2*I*f*x + 2*I*e)/(-I*a + b)) + 3*((a*b^2 + I*b^3)*c*d*f - (a*b^2*(-I*e^2 + e) - a^2*b*e^2 + I
*b^3*e)*d^2 + ((a*b^2 - I*b^3)*c*d*f - (a*b^2*(I*e^2 + e) - a^2*b*e^2 - I*b^3*e)*d^2)*cos(2*f*x + 2*e) - ((-I*
a*b^2 - b^3)*c*d*f - (a*b^2*(e^2 - I*e) + I*a^2*b*e^2 - b^3*e)*d^2)*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)) + 3*((a^2*b + I*a*b^2)*(f*x + e)^2*d^2 + (2*(a^2*b + I*a*b^2)*c*d*f - (a*b^2*(2*I*e - 1) + 2*a^2*b*e - I*
b^3)*d^2)*(f*x + e) + ((a^2*b - I*a*b^2)*(f*x + e)^2*d^2 + (2*(a^2*b - I*a*b^2)*c*d*f - (a*b^2*(-2*I*e - 1) +
2*a^2*b*e + I*b^3)*d^2)*(f*x + e))*cos(2*f*x + 2*e) - ((-I*a^2*b - a*b^2)*(f*x + e)^2*d^2 + (2*(-I*a^2*b - a*b
^2)*c*d*f + (a*b^2*(2*e - I) + 2*I*a^2*b*e - b^3)*d^2)*(f*x + e))*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))/(a^2 + b^2)) + 3*((a^2*b - I*a*b^2)*d^2*cos(2*f*x + 2*e) - (-I*a^2*b - a*b^2)*d^2*sin(2*f*x + 2*e) + (a^2*
b + I*a*b^2)*d^2)*polylog(3, (I*a + b)*e^(2*I*f*x + 2*I*e)/(-I*a + b)) + ((I*a^3 + 3*a^2*b - 3*I*a*b^2 - b^3)*
(f*x + e)^3*d^2 - 3*((-I*a^3 - 3*a^2*b + 3*I*a*b^2 + b^3)*c*d*f - (b^3*(e - 2*I) - a*b^2*(-3*I*e - 2) - I*a^3*
e - 3*a^2*b*e)*d^2)*(f*x + e)^2 + 3*(4*(a*b^2 - I*b^3)*c*d*f - (b^3*(e^2 - 4*I*e) + a*b^2*(3*I*e^2 + 4*e) - I*
a^3*e^2 - 3*a^2*b*e^2)*d^2)*(f*x + e))*sin(2*f*x + 2*e))/((a^5 - I*a^4*b + 2*a^3*b^2 - 2*I*a^2*b^3 + a*b^4 - I
*b^5)*f^2*cos(2*f*x + 2*e) - (-I*a^5 - a^4*b - 2*I*a^3*b^2 - 2*a^2*b^3 - I*a*b^4 - b^5)*f^2*sin(2*f*x + 2*e) +
(a^5 + I*a^4*b + 2*a^3*b^2 + 2*I*a^2*b^3 + a*b^4 + I*b^5)*f^2))/f

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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. 1606 vs. \(2 (543) = 1086\).
time = 0.43, size = 1606, normalized size = 2.46 \begin {gather*} \frac {2 \, {\left (a^{3} - a b^{2}\right )} d^{2} f^{3} x^{3} - 6 \, b^{3} c^{2} f^{2} - 6 \, {\left (b^{3} d^{2} f^{2} - {\left (a^{3} - a b^{2}\right )} c d f^{3}\right )} x^{2} - 6 \, {\left (2 \, b^{3} c d f^{2} - {\left (a^{3} - a b^{2}\right )} c^{2} f^{3}\right )} x - 3 \, {\left (-2 i \, a^{2} b d^{2} f x - 2 i \, a^{2} b c d f - i \, a b^{2} d^{2} + {\left (-2 i \, a b^{2} d^{2} f x - 2 i \, a b^{2} c d f - i \, b^{3} d^{2}\right )} \tan \left (f x + e\right )\right )} {\rm Li}_2\left (\frac {2 \, {\left ({\left (i \, a b - b^{2}\right )} \tan \left (f x + e\right )^{2} - a^{2} - i \, a b + {\left (i \, a^{2} - 2 \, a b - i \, b^{2}\right )} \tan \left (f x + e\right )\right )}}{{\left (a^{2} + b^{2}\right )} \tan \left (f x + e\right )^{2} + a^{2} + b^{2}} + 1\right ) - 3 \, {\left (2 i \, a^{2} b d^{2} f x + 2 i \, a^{2} b c d f + i \, a b^{2} d^{2} + {\left (2 i \, a b^{2} d^{2} f x + 2 i \, a b^{2} c d f + i \, b^{3} d^{2}\right )} \tan \left (f x + e\right )\right )} {\rm Li}_2\left (\frac {2 \, {\left ({\left (-i \, a b - b^{2}\right )} \tan \left (f x + e\right )^{2} - a^{2} + i \, a b + {\left (-i \, a^{2} - 2 \, a b + i \, b^{2}\right )} \tan \left (f x + e\right )\right )}}{{\left (a^{2} + b^{2}\right )} \tan \left (f x + e\right )^{2} + a^{2} + b^{2}} + 1\right ) + 6 \, {\left (a^{2} b d^{2} f^{2} x^{2} - a^{2} b d^{2} e^{2} + {\left (2 \, a^{2} b c d f^{2} + a b^{2} d^{2} f\right )} x + {\left (2 \, a^{2} b c d f + a b^{2} d^{2}\right )} e + {\left (a b^{2} d^{2} f^{2} x^{2} - a b^{2} d^{2} e^{2} + {\left (2 \, a b^{2} c d f^{2} + b^{3} d^{2} f\right )} x + {\left (2 \, a b^{2} c d f + b^{3} d^{2}\right )} e\right )} \tan \left (f x + e\right )\right )} \log \left (-\frac {2 \, {\left ({\left (i \, a b - b^{2}\right )} \tan \left (f x + e\right )^{2} - a^{2} - i \, a b + {\left (i \, a^{2} - 2 \, a b - i \, b^{2}\right )} \tan \left (f x + e\right )\right )}}{{\left (a^{2} + b^{2}\right )} \tan \left (f x + e\right )^{2} + a^{2} + b^{2}}\right ) + 6 \, {\left (a^{2} b d^{2} f^{2} x^{2} - a^{2} b d^{2} e^{2} + {\left (2 \, a^{2} b c d f^{2} + a b^{2} d^{2} f\right )} x + {\left (2 \, a^{2} b c d f + a b^{2} d^{2}\right )} e + {\left (a b^{2} d^{2} f^{2} x^{2} - a b^{2} d^{2} e^{2} + {\left (2 \, a b^{2} c d f^{2} + b^{3} d^{2} f\right )} x + {\left (2 \, a b^{2} c d f + b^{3} d^{2}\right )} e\right )} \tan \left (f x + e\right )\right )} \log \left (-\frac {2 \, {\left ({\left (-i \, a b - b^{2}\right )} \tan \left (f x + e\right )^{2} - a^{2} + i \, a b + {\left (-i \, a^{2} - 2 \, a b + i \, b^{2}\right )} \tan \left (f x + e\right )\right )}}{{\left (a^{2} + b^{2}\right )} \tan \left (f x + e\right )^{2} + a^{2} + b^{2}}\right ) + 6 \, {\left (a^{2} b c^{2} f^{2} + a b^{2} c d f + a^{2} b d^{2} e^{2} - {\left (2 \, a^{2} b c d f + a b^{2} d^{2}\right )} e + {\left (a b^{2} c^{2} f^{2} + b^{3} c d f + a b^{2} d^{2} e^{2} - {\left (2 \, a b^{2} c d f + b^{3} d^{2}\right )} e\right )} \tan \left (f x + e\right )\right )} \log \left (\frac {{\left (i \, a b + b^{2}\right )} \tan \left (f x + e\right )^{2} - a^{2} + i \, a b + {\left (i \, a^{2} + i \, b^{2}\right )} \tan \left (f x + e\right )}{\tan \left (f x + e\right )^{2} + 1}\right ) + 6 \, {\left (a^{2} b c^{2} f^{2} + a b^{2} c d f + a^{2} b d^{2} e^{2} - {\left (2 \, a^{2} b c d f + a b^{2} d^{2}\right )} e + {\left (a b^{2} c^{2} f^{2} + b^{3} c d f + a b^{2} d^{2} e^{2} - {\left (2 \, a b^{2} c d f + b^{3} d^{2}\right )} e\right )} \tan \left (f x + e\right )\right )} \log \left (\frac {{\left (i \, a b - b^{2}\right )} \tan \left (f x + e\right )^{2} + a^{2} + i \, a b + {\left (i \, a^{2} + i \, b^{2}\right )} \tan \left (f x + e\right )}{\tan \left (f x + e\right )^{2} + 1}\right ) + 3 \, {\left (a b^{2} d^{2} \tan \left (f x + e\right ) + a^{2} b d^{2}\right )} {\rm polylog}\left (3, \frac {{\left (a^{2} + 2 i \, a b - b^{2}\right )} \tan \left (f x + e\right )^{2} - a^{2} - 2 i \, a b + b^{2} - 2 \, {\left (-i \, a^{2} + 2 \, a b + i \, b^{2}\right )} \tan \left (f x + e\right )}{{\left (a^{2} + b^{2}\right )} \tan \left (f x + e\right )^{2} + a^{2} + b^{2}}\right ) + 3 \, {\left (a b^{2} d^{2} \tan \left (f x + e\right ) + a^{2} b d^{2}\right )} {\rm polylog}\left (3, \frac {{\left (a^{2} - 2 i \, a b - b^{2}\right )} \tan \left (f x + e\right )^{2} - a^{2} + 2 i \, a b + b^{2} - 2 \, {\left (i \, a^{2} + 2 \, a b - i \, b^{2}\right )} \tan \left (f x + e\right )}{{\left (a^{2} + b^{2}\right )} \tan \left (f x + e\right )^{2} + a^{2} + b^{2}}\right ) + 2 \, {\left ({\left (a^{2} b - b^{3}\right )} d^{2} f^{3} x^{3} + 3 \, a b^{2} c^{2} f^{2} + 3 \, {\left (a b^{2} d^{2} f^{2} + {\left (a^{2} b - b^{3}\right )} c d f^{3}\right )} x^{2} + 3 \, {\left (2 \, a b^{2} c d f^{2} + {\left (a^{2} b - b^{3}\right )} c^{2} f^{3}\right )} x\right )} \tan \left (f x + e\right )}{6 \, {\left ({\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} f^{3} \tan \left (f x + e\right ) + {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} f^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*(a^3 - a*b^2)*d^2*f^3*x^3 - 6*b^3*c^2*f^2 - 6*(b^3*d^2*f^2 - (a^3 - a*b^2)*c*d*f^3)*x^2 - 6*(2*b^3*c*d*
f^2 - (a^3 - a*b^2)*c^2*f^3)*x - 3*(-2*I*a^2*b*d^2*f*x - 2*I*a^2*b*c*d*f - I*a*b^2*d^2 + (-2*I*a*b^2*d^2*f*x -
2*I*a*b^2*c*d*f - I*b^3*d^2)*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) - 3*(2*I*a^2*b*d^2*f*x + 2*I*a^2*b*c*d
*f + I*a*b^2*d^2 + (2*I*a*b^2*d^2*f*x + 2*I*a*b^2*c*d*f + I*b^3*d^2)*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*(a^2*b*d^2*f^2*x^2 - a^2*b*d^2*e^2 + (2*a^2*b*c*d*f^2 + a*b^2*d^2*f)*x + (2*a^2*b*c*d*f + a*b^2*d^2)*e
+ (a*b^2*d^2*f^2*x^2 - a*b^2*d^2*e^2 + (2*a*b^2*c*d*f^2 + b^3*d^2*f)*x + (2*a*b^2*c*d*f + b^3*d^2)*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)) + 6*(a^2*b*d^2*f^2*x^2 - a^2*b*d^2*e^2 + (2*a^2*b*c*d*f^2 + a*b^2*d^2*f)*x + (2*a
^2*b*c*d*f + a*b^2*d^2)*e + (a*b^2*d^2*f^2*x^2 - a*b^2*d^2*e^2 + (2*a*b^2*c*d*f^2 + b^3*d^2*f)*x + (2*a*b^2*c*
d*f + b^3*d^2)*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)) + 6*(a^2*b*c^2*f^2 + a*b^2*c*d*f + a^2*b*d^2*e^2 - (2
*a^2*b*c*d*f + a*b^2*d^2)*e + (a*b^2*c^2*f^2 + b^3*c*d*f + a*b^2*d^2*e^2 - (2*a*b^2*c*d*f + b^3*d^2)*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))
+ 6*(a^2*b*c^2*f^2 + a*b^2*c*d*f + a^2*b*d^2*e^2 - (2*a^2*b*c*d*f + a*b^2*d^2)*e + (a*b^2*c^2*f^2 + b^3*c*d*f
+ a*b^2*d^2*e^2 - (2*a*b^2*c*d*f + b^3*d^2)*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*(a*b^2*d^2*tan(f*x + e) + a^2*b*d^2)*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*(a*b^2*d^2*tan(f*x + e) + a^2*b*d^2)*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)) + 2*((a^2*b - b^3)*d^2*f^3*x^3 + 3*a*b^2*c^2*f^2 + 3*(a*b^2*d^2*f^2 + (a^2*b - b^3)*c*d*f^3)*x^2 + 3*(2*
a*b^2*c*d*f^2 + (a^2*b - b^3)*c^2*f^3)*x)*tan(f*x + e))/((a^4*b + 2*a^2*b^3 + b^5)*f^3*tan(f*x + e) + (a^5 + 2
*a^3*b^2 + a*b^4)*f^3)

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

[Out]

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

[Out]

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

[Out]

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

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