\(\int \frac {(c+d x)^3}{a+b \tan (e+f x)} \, dx\) [54]

Optimal result

Integrand size = 20, antiderivative size = 243 \[ \int \frac {(c+d x)^3}{a+b \tan (e+f x)} \, dx=\frac {(c+d x)^4}{4 (a+i b) d}+\frac {b (c+d x)^3 \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) f}-\frac {3 i b d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{2 \left (a^2+b^2\right ) f^2}+\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{2 \left (a^2+b^2\right ) f^3}+\frac {3 i b d^3 \operatorname {PolyLog}\left (4,-\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{4 \left (a^2+b^2\right ) f^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3813, 2221, 2611, 6744, 2320, 6724} \[ \int \frac {(c+d x)^3}{a+b \tan (e+f x)} \, dx=\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{2 f^3 \left (a^2+b^2\right )}-\frac {3 i b d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{2 f^2 \left (a^2+b^2\right )}+\frac {b (c+d x)^3 \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{f \left (a^2+b^2\right )}+\frac {3 i b d^3 \operatorname {PolyLog}\left (4,-\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{4 f^4 \left (a^2+b^2\right )}+\frac {(c+d x)^4}{4 d (a+i b)} \]

[In]

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

[Out]

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

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 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 3813

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c + d*x)^(m + 1)/(d*
(m + 1)*(a + I*b)), x] + Dist[2*I*b, Int[(c + d*x)^m*(E^Simp[2*I*(e + f*x), x]/((a + I*b)^2 + (a^2 + b^2)*E^Si
mp[2*I*(e + f*x), x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 + b^2, 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*} \text {integral}& = \frac {(c+d x)^4}{4 (a+i b) d}+(2 i b) \int \frac {e^{2 i (e+f x)} (c+d x)^3}{(a+i b)^2+\left (a^2+b^2\right ) e^{2 i (e+f x)}} \, dx \\ & = \frac {(c+d x)^4}{4 (a+i b) d}+\frac {b (c+d x)^3 \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) f}-\frac {(3 b d) \int (c+d x)^2 \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right ) \, dx}{\left (a^2+b^2\right ) f} \\ & = \frac {(c+d x)^4}{4 (a+i b) d}+\frac {b (c+d x)^3 \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) f}-\frac {3 i b d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{2 \left (a^2+b^2\right ) f^2}+\frac {\left (3 i b d^2\right ) \int (c+d x) \operatorname {PolyLog}\left (2,-\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right ) \, dx}{\left (a^2+b^2\right ) f^2} \\ & = \frac {(c+d x)^4}{4 (a+i b) d}+\frac {b (c+d x)^3 \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) f}-\frac {3 i b d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{2 \left (a^2+b^2\right ) f^2}+\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{2 \left (a^2+b^2\right ) f^3}-\frac {\left (3 b d^3\right ) \int \operatorname {PolyLog}\left (3,-\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right ) \, dx}{2 \left (a^2+b^2\right ) f^3} \\ & = \frac {(c+d x)^4}{4 (a+i b) d}+\frac {b (c+d x)^3 \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) f}-\frac {3 i b d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{2 \left (a^2+b^2\right ) f^2}+\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{2 \left (a^2+b^2\right ) f^3}+\frac {\left (3 i b d^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,-\frac {\left (a^2+b^2\right ) x}{(a+i b)^2}\right )}{x} \, dx,x,e^{2 i (e+f x)}\right )}{4 \left (a^2+b^2\right ) f^4} \\ & = \frac {(c+d x)^4}{4 (a+i b) d}+\frac {b (c+d x)^3 \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) f}-\frac {3 i b d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{2 \left (a^2+b^2\right ) f^2}+\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{2 \left (a^2+b^2\right ) f^3}+\frac {3 i b d^3 \operatorname {PolyLog}\left (4,-\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{4 \left (a^2+b^2\right ) f^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.68 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.22 \[ \int \frac {(c+d x)^3}{a+b \tan (e+f x)} \, dx=\frac {1}{4} b \left (-\frac {2 (c+d x)^4}{(i a+b) d \left (-i b \left (-1+e^{2 i e}\right )+a \left (1+e^{2 i e}\right )\right )}+\frac {4 (c+d x)^3 \log \left (1+\frac {(a+i b) e^{-2 i (e+f x)}}{a-i b}\right )}{\left (a^2+b^2\right ) f}+\frac {3 d \left (2 i f^2 (c+d x)^2 \operatorname {PolyLog}\left (2,\frac {(-a-i b) e^{-2 i (e+f x)}}{a-i b}\right )+d \left (2 f (c+d x) \operatorname {PolyLog}\left (3,\frac {(-a-i b) e^{-2 i (e+f x)}}{a-i b}\right )-i d \operatorname {PolyLog}\left (4,\frac {(-a-i b) e^{-2 i (e+f x)}}{a-i b}\right )\right )\right )}{\left (a^2+b^2\right ) f^4}\right )+\frac {x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right ) \cos (e)}{4 (a \cos (e)+b \sin (e))} \]

[In]

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

[Out]

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1467 vs. \(2 (220 ) = 440\).

Time = 0.78 (sec) , antiderivative size = 1468, normalized size of antiderivative = 6.04

method	result	size
risch	\(\text {Expression too large to display}\)	\(1468\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1183 vs. \(2 (212) = 424\).

Time = 0.28 (sec) , antiderivative size = 1183, normalized size of antiderivative = 4.87 \[ \int \frac {(c+d x)^3}{a+b \tan (e+f x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/8*(2*a*d^3*f^4*x^4 + 8*a*c*d^2*f^4*x^3 + 12*a*c^2*d*f^4*x^2 + 8*a*c^3*f^4*x - 3*I*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*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)) - 6*(-I*b*d^3*f^2*x^2 - 2*I
*b*c*d^2*f^2*x - I*b*c^2*d*f^2)*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*b*d^3*f^2*x^2 + 2*I*b*c*d^2*f^2*x + I*b*c^2
*d*f^2)*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) + 4*(b*d^3*f^3*x^3 + 3*b*c*d^2*f^3*x^2 + 3*b*c^2*d*f^3*x + b*d^3*e^3 - 3
*b*c*d^2*e^2*f + 3*b*c^2*d*e*f^2)*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)) + 4*(b*d^3*f^3*x^3 + 3*b*c*d^2*f^3*x^2 + 3*b*c^2*d*f^
3*x + b*d^3*e^3 - 3*b*c*d^2*e^2*f + 3*b*c^2*d*e*f^2)*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)) - 4*(b*d^3*e^3 - 3*b*c*d^2*e^2*f
+ 3*b*c^2*d*e*f^2 - b*c^3*f^3)*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)) - 4*(b*d^3*e^3 - 3*b*c*d^2*e^2*f + 3*b*c^2*d*e*f^2 - b*c^3*f^3)*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*(b*d^3*f*x + b*c*d^2*f)*po
lylog(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)) + 6*(b*d^3*f*x + b*c*d^2*f)*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^2)*f^4)

Sympy [F]

\[ \int \frac {(c+d x)^3}{a+b \tan (e+f x)} \, dx=\int \frac {\left (c + d x\right )^{3}}{a + b \tan {\left (e + f x \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 983 vs. \(2 (212) = 424\).

Time = 0.57 (sec) , antiderivative size = 983, normalized size of antiderivative = 4.05 \[ \int \frac {(c+d x)^3}{a+b \tan (e+f x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/12*(18*c^2*d*e*(2*(f*x + e)*a/((a^2 + b^2)*f) + 2*b*log(b*tan(f*x + e) + a)/((a^2 + b^2)*f) - b*log(tan(f*x
+ e)^2 + 1)/((a^2 + b^2)*f)) - 6*(2*(f*x + e)*a/(a^2 + b^2) + 2*b*log(b*tan(f*x + e) + a)/(a^2 + b^2) - b*log
(tan(f*x + e)^2 + 1)/(a^2 + b^2))*c^3 - (3*(f*x + e)^4*(a - I*b)*d^3 + 12*I*b*d^3*polylog(4, (I*a + b)*e^(2*I*
f*x + 2*I*e)/(-I*a + b)) - 12*((a - I*b)*d^3*e - (a - I*b)*c*d^2*f)*(f*x + e)^3 + 18*((a - I*b)*d^3*e^2 - 2*(a
- I*b)*c*d^2*e*f + (a - I*b)*c^2*d*f^2)*(f*x + e)^2 - 12*((a - I*b)*d^3*e^3 - 3*(a - I*b)*c*d^2*e^2*f)*(f*x +
e) - 12*(I*b*d^3*e^3 - 3*I*b*c*d^2*e^2*f)*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*(4*I*(f*x + e)^3*b*d^3 + 9*(-I*b*d^3*e + I*b*c*d^2*f)*(f*x + e)^2 + 9*(I*b
*d^3*e^2 - 2*I*b*c*d^2*e*f + I*b*c^2*d*f^2)*(f*x + 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)) - 6*(4*
I*(f*x + e)^2*b*d^3 + 3*I*b*d^3*e^2 - 6*I*b*c*d^2*e*f + 3*I*b*c^2*d*f^2 + 6*(-I*b*d^3*e + I*b*c*d^2*f)*(f*x +
e))*dilog((I*a + b)*e^(2*I*f*x + 2*I*e)/(-I*a + b)) - 6*(b*d^3*e^3 - 3*b*c*d^2*e^2*f)*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*(4*(f*x + e)^3*b*d^3 - 9*(b*d^3*e - b*c*d^2*f)*(f*x + e)^2 + 9*(b*d^3*e^2 - 2*b*c*d^2*e*f + b*c^2*d*f
^2)*(f*x + 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)) + 6*(4*(f*x + e)*b*d^3 - 3*b*d^3*e + 3*b*c*d^2*f)*po
lylog(3, (I*a + b)*e^(2*I*f*x + 2*I*e)/(-I*a + b)))/((a^2 + b^2)*f^3))/f

Giac [F]

\[ \int \frac {(c+d x)^3}{a+b \tan (e+f x)} \, dx=\int { \frac {{\left (d x + c\right )}^{3}}{b \tan \left (f x + e\right ) + a} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^3}{a+b \tan (e+f x)} \, dx=\int \frac {{\left (c+d\,x\right )}^3}{a+b\,\mathrm {tan}\left (e+f\,x\right )} \,d x \]

[In]

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

[Out]

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