\(\int (c+d x)^3 \tan ^3(a+b x) \, dx\) [304]

Optimal result

Integrand size = 16, antiderivative size = 259 \[ \int (c+d x)^3 \tan ^3(a+b x) \, dx=\frac {3 i d (c+d x)^2}{2 b^2}+\frac {(c+d x)^3}{2 b}-\frac {i (c+d x)^4}{4 d}-\frac {3 d^2 (c+d x) \log \left (1+e^{2 i (a+b x)}\right )}{b^3}+\frac {(c+d x)^3 \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac {3 i d^3 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b^4}-\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b^2}+\frac {3 d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{2 b^3}+\frac {3 i d^3 \operatorname {PolyLog}\left (4,-e^{2 i (a+b x)}\right )}{4 b^4}-\frac {3 d (c+d x)^2 \tan (a+b x)}{2 b^2}+\frac {(c+d x)^3 \tan ^2(a+b x)}{2 b} \] Output:

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

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(814\) vs. \(2(259)=518\).

Time = 6.59 (sec) , antiderivative size = 814, normalized size of antiderivative = 3.14 \[ \int (c+d x)^3 \tan ^3(a+b x) \, dx =\text {Too large to display} \] Input:

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

Output:

((I/4)*c*d^2*(2*b^2*x^2*(2*b*x - (3*I)*(1 + E^((2*I)*a))*Log[1 + E^((-2*I) 
*(a + b*x))]) + 6*b*(1 + E^((2*I)*a))*x*PolyLog[2, -E^((-2*I)*(a + b*x))] 
- (3*I)*(1 + E^((2*I)*a))*PolyLog[3, -E^((-2*I)*(a + b*x))])*Sec[a])/(b^3* 
E^(I*a)) + ((I/8)*d^3*E^(I*a)*((2*b^4*x^4)/E^((2*I)*a) - (4*I)*b^3*(1 + E^ 
((-2*I)*a))*x^3*Log[1 + E^((-2*I)*(a + b*x))] + 6*b^2*(1 + E^((-2*I)*a))*x 
^2*PolyLog[2, -E^((-2*I)*(a + b*x))] - (6*I)*b*(1 + E^((-2*I)*a))*x*PolyLo 
g[3, -E^((-2*I)*(a + b*x))] - 3*(1 + E^((-2*I)*a))*PolyLog[4, -E^((-2*I)*( 
a + b*x))])*Sec[a])/b^4 + ((c + d*x)^3*Sec[a + b*x]^2)/(2*b) + (c^3*Sec[a] 
*(Cos[a]*Log[Cos[a]*Cos[b*x] - Sin[a]*Sin[b*x]] + b*x*Sin[a]))/(b*(Cos[a]^ 
2 + Sin[a]^2)) - (3*c*d^2*Sec[a]*(Cos[a]*Log[Cos[a]*Cos[b*x] - Sin[a]*Sin[ 
b*x]] + b*x*Sin[a]))/(b^3*(Cos[a]^2 + Sin[a]^2)) + (3*c^2*d*Csc[a]*((b^2*x 
^2)/E^(I*ArcTan[Cot[a]]) - (Cot[a]*(I*b*x*(-Pi - 2*ArcTan[Cot[a]]) - Pi*Lo 
g[1 + E^((-2*I)*b*x)] - 2*(b*x - ArcTan[Cot[a]])*Log[1 - E^((2*I)*(b*x - A 
rcTan[Cot[a]]))] + Pi*Log[Cos[b*x]] - 2*ArcTan[Cot[a]]*Log[Sin[b*x - ArcTa 
n[Cot[a]]]] + I*PolyLog[2, E^((2*I)*(b*x - ArcTan[Cot[a]]))]))/Sqrt[1 + Co 
t[a]^2])*Sec[a])/(2*b^2*Sqrt[Csc[a]^2*(Cos[a]^2 + Sin[a]^2)]) - (3*d^3*Csc 
[a]*((b^2*x^2)/E^(I*ArcTan[Cot[a]]) - (Cot[a]*(I*b*x*(-Pi - 2*ArcTan[Cot[a 
]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x - ArcTan[Cot[a]])*Log[1 - E^((2* 
I)*(b*x - ArcTan[Cot[a]]))] + Pi*Log[Cos[b*x]] - 2*ArcTan[Cot[a]]*Log[Sin[ 
b*x - ArcTan[Cot[a]]]] + I*PolyLog[2, E^((2*I)*(b*x - ArcTan[Cot[a]]))]...
 

Rubi [A] (verified)

Time = 1.51 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.14, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 4203, 3042, 4202, 2620, 3011, 4203, 17, 3042, 4202, 2620, 2715, 2838, 7163, 2720, 7143}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int (c+d x)^3 \tan ^3(a+b x) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (c+d x)^3 \tan (a+b x)^3dx\)

\(\Big \downarrow \) 4203

\(\displaystyle -\frac {3 d \int (c+d x)^2 \tan ^2(a+b x)dx}{2 b}-\int (c+d x)^3 \tan (a+b x)dx+\frac {(c+d x)^3 \tan ^2(a+b x)}{2 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\int (c+d x)^3 \tan (a+b x)dx-\frac {3 d \int (c+d x)^2 \tan (a+b x)^2dx}{2 b}+\frac {(c+d x)^3 \tan ^2(a+b x)}{2 b}\)

\(\Big \downarrow \) 4202

\(\displaystyle 2 i \int \frac {e^{2 i (a+b x)} (c+d x)^3}{1+e^{2 i (a+b x)}}dx-\frac {3 d \int (c+d x)^2 \tan (a+b x)^2dx}{2 b}+\frac {(c+d x)^3 \tan ^2(a+b x)}{2 b}-\frac {i (c+d x)^4}{4 d}\)

\(\Big \downarrow \) 2620

\(\displaystyle 2 i \left (\frac {3 i d \int (c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )dx}{2 b}-\frac {i (c+d x)^3 \log \left (1+e^{2 i (a+b x)}\right )}{2 b}\right )-\frac {3 d \int (c+d x)^2 \tan (a+b x)^2dx}{2 b}+\frac {(c+d x)^3 \tan ^2(a+b x)}{2 b}-\frac {i (c+d x)^4}{4 d}\)

\(\Big \downarrow \) 3011

\(\displaystyle 2 i \left (\frac {3 i d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b}-\frac {i d \int (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )dx}{b}\right )}{2 b}-\frac {i (c+d x)^3 \log \left (1+e^{2 i (a+b x)}\right )}{2 b}\right )-\frac {3 d \int (c+d x)^2 \tan (a+b x)^2dx}{2 b}+\frac {(c+d x)^3 \tan ^2(a+b x)}{2 b}-\frac {i (c+d x)^4}{4 d}\)

\(\Big \downarrow \) 4203

\(\displaystyle 2 i \left (\frac {3 i d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b}-\frac {i d \int (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )dx}{b}\right )}{2 b}-\frac {i (c+d x)^3 \log \left (1+e^{2 i (a+b x)}\right )}{2 b}\right )-\frac {3 d \left (-\frac {2 d \int (c+d x) \tan (a+b x)dx}{b}-\int (c+d x)^2dx+\frac {(c+d x)^2 \tan (a+b x)}{b}\right )}{2 b}+\frac {(c+d x)^3 \tan ^2(a+b x)}{2 b}-\frac {i (c+d x)^4}{4 d}\)

\(\Big \downarrow \) 17

\(\displaystyle 2 i \left (\frac {3 i d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b}-\frac {i d \int (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )dx}{b}\right )}{2 b}-\frac {i (c+d x)^3 \log \left (1+e^{2 i (a+b x)}\right )}{2 b}\right )-\frac {3 d \left (-\frac {2 d \int (c+d x) \tan (a+b x)dx}{b}+\frac {(c+d x)^2 \tan (a+b x)}{b}-\frac {(c+d x)^3}{3 d}\right )}{2 b}+\frac {(c+d x)^3 \tan ^2(a+b x)}{2 b}-\frac {i (c+d x)^4}{4 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 i \left (\frac {3 i d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b}-\frac {i d \int (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )dx}{b}\right )}{2 b}-\frac {i (c+d x)^3 \log \left (1+e^{2 i (a+b x)}\right )}{2 b}\right )-\frac {3 d \left (-\frac {2 d \int (c+d x) \tan (a+b x)dx}{b}+\frac {(c+d x)^2 \tan (a+b x)}{b}-\frac {(c+d x)^3}{3 d}\right )}{2 b}+\frac {(c+d x)^3 \tan ^2(a+b x)}{2 b}-\frac {i (c+d x)^4}{4 d}\)

\(\Big \downarrow \) 4202

\(\displaystyle 2 i \left (\frac {3 i d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b}-\frac {i d \int (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )dx}{b}\right )}{2 b}-\frac {i (c+d x)^3 \log \left (1+e^{2 i (a+b x)}\right )}{2 b}\right )-\frac {3 d \left (-\frac {2 d \left (\frac {i (c+d x)^2}{2 d}-2 i \int \frac {e^{2 i (a+b x)} (c+d x)}{1+e^{2 i (a+b x)}}dx\right )}{b}+\frac {(c+d x)^2 \tan (a+b x)}{b}-\frac {(c+d x)^3}{3 d}\right )}{2 b}+\frac {(c+d x)^3 \tan ^2(a+b x)}{2 b}-\frac {i (c+d x)^4}{4 d}\)

\(\Big \downarrow \) 2620

\(\displaystyle 2 i \left (\frac {3 i d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b}-\frac {i d \int (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )dx}{b}\right )}{2 b}-\frac {i (c+d x)^3 \log \left (1+e^{2 i (a+b x)}\right )}{2 b}\right )-\frac {3 d \left (-\frac {2 d \left (\frac {i (c+d x)^2}{2 d}-2 i \left (\frac {i d \int \log \left (1+e^{2 i (a+b x)}\right )dx}{2 b}-\frac {i (c+d x) \log \left (1+e^{2 i (a+b x)}\right )}{2 b}\right )\right )}{b}+\frac {(c+d x)^2 \tan (a+b x)}{b}-\frac {(c+d x)^3}{3 d}\right )}{2 b}+\frac {(c+d x)^3 \tan ^2(a+b x)}{2 b}-\frac {i (c+d x)^4}{4 d}\)

\(\Big \downarrow \) 2715

\(\displaystyle -\frac {3 d \left (-\frac {2 d \left (\frac {i (c+d x)^2}{2 d}-2 i \left (\frac {d \int e^{-2 i (a+b x)} \log \left (1+e^{2 i (a+b x)}\right )de^{2 i (a+b x)}}{4 b^2}-\frac {i (c+d x) \log \left (1+e^{2 i (a+b x)}\right )}{2 b}\right )\right )}{b}+\frac {(c+d x)^2 \tan (a+b x)}{b}-\frac {(c+d x)^3}{3 d}\right )}{2 b}+2 i \left (\frac {3 i d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b}-\frac {i d \int (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )dx}{b}\right )}{2 b}-\frac {i (c+d x)^3 \log \left (1+e^{2 i (a+b x)}\right )}{2 b}\right )+\frac {(c+d x)^3 \tan ^2(a+b x)}{2 b}-\frac {i (c+d x)^4}{4 d}\)

\(\Big \downarrow \) 2838

\(\displaystyle 2 i \left (\frac {3 i d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b}-\frac {i d \int (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )dx}{b}\right )}{2 b}-\frac {i (c+d x)^3 \log \left (1+e^{2 i (a+b x)}\right )}{2 b}\right )-\frac {3 d \left (-\frac {2 d \left (\frac {i (c+d x)^2}{2 d}-2 i \left (-\frac {d \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{4 b^2}-\frac {i (c+d x) \log \left (1+e^{2 i (a+b x)}\right )}{2 b}\right )\right )}{b}+\frac {(c+d x)^2 \tan (a+b x)}{b}-\frac {(c+d x)^3}{3 d}\right )}{2 b}+\frac {(c+d x)^3 \tan ^2(a+b x)}{2 b}-\frac {i (c+d x)^4}{4 d}\)

\(\Big \downarrow \) 7163

\(\displaystyle 2 i \left (\frac {3 i d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b}-\frac {i d \left (\frac {i d \int \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )dx}{2 b}-\frac {i (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{2 b}\right )}{b}\right )}{2 b}-\frac {i (c+d x)^3 \log \left (1+e^{2 i (a+b x)}\right )}{2 b}\right )-\frac {3 d \left (-\frac {2 d \left (\frac {i (c+d x)^2}{2 d}-2 i \left (-\frac {d \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{4 b^2}-\frac {i (c+d x) \log \left (1+e^{2 i (a+b x)}\right )}{2 b}\right )\right )}{b}+\frac {(c+d x)^2 \tan (a+b x)}{b}-\frac {(c+d x)^3}{3 d}\right )}{2 b}+\frac {(c+d x)^3 \tan ^2(a+b x)}{2 b}-\frac {i (c+d x)^4}{4 d}\)

\(\Big \downarrow \) 2720

\(\displaystyle 2 i \left (\frac {3 i d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b}-\frac {i d \left (\frac {d \int e^{-2 i (a+b x)} \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )de^{2 i (a+b x)}}{4 b^2}-\frac {i (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{2 b}\right )}{b}\right )}{2 b}-\frac {i (c+d x)^3 \log \left (1+e^{2 i (a+b x)}\right )}{2 b}\right )-\frac {3 d \left (-\frac {2 d \left (\frac {i (c+d x)^2}{2 d}-2 i \left (-\frac {d \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{4 b^2}-\frac {i (c+d x) \log \left (1+e^{2 i (a+b x)}\right )}{2 b}\right )\right )}{b}+\frac {(c+d x)^2 \tan (a+b x)}{b}-\frac {(c+d x)^3}{3 d}\right )}{2 b}+\frac {(c+d x)^3 \tan ^2(a+b x)}{2 b}-\frac {i (c+d x)^4}{4 d}\)

\(\Big \downarrow \) 7143

\(\displaystyle 2 i \left (\frac {3 i d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b}-\frac {i d \left (\frac {d \operatorname {PolyLog}\left (4,-e^{2 i (a+b x)}\right )}{4 b^2}-\frac {i (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{2 b}\right )}{b}\right )}{2 b}-\frac {i (c+d x)^3 \log \left (1+e^{2 i (a+b x)}\right )}{2 b}\right )-\frac {3 d \left (-\frac {2 d \left (\frac {i (c+d x)^2}{2 d}-2 i \left (-\frac {d \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{4 b^2}-\frac {i (c+d x) \log \left (1+e^{2 i (a+b x)}\right )}{2 b}\right )\right )}{b}+\frac {(c+d x)^2 \tan (a+b x)}{b}-\frac {(c+d x)^3}{3 d}\right )}{2 b}+\frac {(c+d x)^3 \tan ^2(a+b x)}{2 b}-\frac {i (c+d x)^4}{4 d}\)

Input:

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

Output:

((-1/4*I)*(c + d*x)^4)/d + (2*I)*(((-1/2*I)*(c + d*x)^3*Log[1 + E^((2*I)*( 
a + b*x))])/b + (((3*I)/2)*d*(((I/2)*(c + d*x)^2*PolyLog[2, -E^((2*I)*(a + 
 b*x))])/b - (I*d*(((-1/2*I)*(c + d*x)*PolyLog[3, -E^((2*I)*(a + b*x))])/b 
 + (d*PolyLog[4, -E^((2*I)*(a + b*x))])/(4*b^2)))/b))/b) + ((c + d*x)^3*Ta 
n[a + b*x]^2)/(2*b) - (3*d*(-1/3*(c + d*x)^3/d - (2*d*(((I/2)*(c + d*x)^2) 
/d - (2*I)*(((-1/2*I)*(c + d*x)*Log[1 + E^((2*I)*(a + b*x))])/b - (d*PolyL 
og[2, -E^((2*I)*(a + b*x))])/(4*b^2))))/b + ((c + d*x)^2*Tan[a + b*x])/b)) 
/(2*b)
 

Defintions of rubi rules used

rule 17

Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 
)/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
 

rule 2620

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] - Si 
mp[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 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[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 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{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 2838

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 3011

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] + Simp[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 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4202

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I 
*((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[2*I   Int[(c + d*x)^m*(E^(2*I*( 
e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] && IGt 
Q[m, 0]
 

rule 4203

Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symb 
ol] :> Simp[b*(c + d*x)^m*((b*Tan[e + f*x])^(n - 1)/(f*(n - 1))), x] + (-Si 
mp[b*d*(m/(f*(n - 1)))   Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1), x] 
, x] - Simp[b^2   Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; Free 
Q[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 0]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> 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 7163

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] - Simp[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]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 728 vs. \(2 (224 ) = 448\).

Time = 0.40 (sec) , antiderivative size = 729, normalized size of antiderivative = 2.81

method	result	size
risch	\(-\frac {3 i d \,c^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{2 i \left (b x +a \right )}\right )}{2 b^{2}}+\frac {6 i d^{3} a x}{b^{3}}+\frac {d^{3} \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right ) x^{3}}{b}+\frac {3 i d^{3} a^{2}}{b^{4}}+\frac {6 d^{2} c \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}-\frac {6 d^{3} a \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}+\frac {3 i d^{3} \operatorname {polylog}\left (4, -{\mathrm e}^{2 i \left (b x +a \right )}\right )}{4 b^{4}}+\frac {3 c \,d^{2} \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right ) x^{2}}{b}+\frac {3 d^{3} \operatorname {polylog}\left (3, -{\mathrm e}^{2 i \left (b x +a \right )}\right ) x}{2 b^{3}}+\frac {3 c \,d^{2} \operatorname {polylog}\left (3, -{\mathrm e}^{2 i \left (b x +a \right )}\right )}{2 b^{3}}-\frac {6 i d \,c^{2} a x}{b}+\frac {6 i c \,d^{2} a^{2} x}{b^{2}}-\frac {3 i c \,d^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{2 i \left (b x +a \right )}\right ) x}{b^{2}}+\frac {3 d \,c^{2} \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right ) x}{b}-\frac {3 d^{2} c \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )}{b^{3}}+\frac {3 i d^{3} x^{2}}{b^{2}}-\frac {3 d^{3} \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right ) x}{b^{3}}+\frac {3 i d^{3} \operatorname {polylog}\left (2, -{\mathrm e}^{2 i \left (b x +a \right )}\right )}{2 b^{4}}-\frac {3 i d^{3} a^{4}}{2 b^{4}}-i d^{2} c \,x^{3}-\frac {3 i d \,c^{2} x^{2}}{2}+\frac {2 d^{3} a^{3} \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}+\frac {c^{3} \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )}{b}+i c^{3} x +\frac {i c^{4}}{4 d}-\frac {3 i d^{3} \operatorname {polylog}\left (2, -{\mathrm e}^{2 i \left (b x +a \right )}\right ) x^{2}}{2 b^{2}}+\frac {2 b \,d^{3} x^{3} {\mathrm e}^{2 i \left (b x +a \right )}-3 i d^{3} x^{2} {\mathrm e}^{2 i \left (b x +a \right )}+6 b c \,d^{2} x^{2} {\mathrm e}^{2 i \left (b x +a \right )}-6 i c \,d^{2} x \,{\mathrm e}^{2 i \left (b x +a \right )}+6 b \,c^{2} d x \,{\mathrm e}^{2 i \left (b x +a \right )}-3 i c^{2} d \,{\mathrm e}^{2 i \left (b x +a \right )}-3 i d^{3} x^{2}+2 b \,c^{3} {\mathrm e}^{2 i \left (b x +a \right )}-6 i c \,d^{2} x -3 i c^{2} d}{b^{2} \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )^{2}}-\frac {6 c \,d^{2} a^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}+\frac {6 c^{2} d a \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2}}-\frac {3 i d \,c^{2} a^{2}}{b^{2}}-\frac {2 i d^{3} a^{3} x}{b^{3}}+\frac {4 i c \,d^{2} a^{3}}{b^{3}}-\frac {i d^{3} x^{4}}{4}-\frac {2 c^{3} \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b}\)	\(729\)

Input:

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

Output:

-6/b^3*c*d^2*a^2*ln(exp(I*(b*x+a)))-6*I/b*d*c^2*a*x+6*I/b^2*c*d^2*a^2*x+1/ 
b*d^3*ln(exp(2*I*(b*x+a))+1)*x^3+(2*b*d^3*x^3*exp(2*I*(b*x+a))-3*I*d^3*x^2 
*exp(2*I*(b*x+a))+6*b*c*d^2*x^2*exp(2*I*(b*x+a))-6*I*c*d^2*x*exp(2*I*(b*x+ 
a))+6*b*c^2*d*x*exp(2*I*(b*x+a))-3*I*c^2*d*exp(2*I*(b*x+a))-3*I*d^3*x^2+2* 
b*c^3*exp(2*I*(b*x+a))-6*I*c*d^2*x-3*I*c^2*d)/b^2/(exp(2*I*(b*x+a))+1)^2+3 
/2/b^3*d^3*polylog(3,-exp(2*I*(b*x+a)))*x+3/2/b^3*c*d^2*polylog(3,-exp(2*I 
*(b*x+a)))-3/b^3*d^2*c*ln(exp(2*I*(b*x+a))+1)-3/b^3*d^3*ln(exp(2*I*(b*x+a) 
)+1)*x+3*I/b^2*d^3*x^2+3*I/b^4*d^3*a^2-3*I/b^2*c*d^2*polylog(2,-exp(2*I*(b 
*x+a)))*x+6*I/b^3*d^3*a*x-3/2*I/b^2*d^3*polylog(2,-exp(2*I*(b*x+a)))*x^2-3 
/2*I/b^2*d*c^2*polylog(2,-exp(2*I*(b*x+a)))+6*d^2/b^3*c*ln(exp(I*(b*x+a))) 
-6*d^3/b^4*a*ln(exp(I*(b*x+a)))+I*c^3*x+1/4*I/d*c^4-3/2*I/b^4*d^3*a^4-I*d^ 
2*c*x^3-3/2*I*d*c^2*x^2+2/b^4*d^3*a^3*ln(exp(I*(b*x+a)))+3/b*d*c^2*ln(exp( 
2*I*(b*x+a))+1)*x+3/b*c*d^2*ln(exp(2*I*(b*x+a))+1)*x^2+6/b^2*c^2*d*a*ln(ex 
p(I*(b*x+a)))-3*I/b^2*d*c^2*a^2-2*I/b^3*d^3*a^3*x+4*I/b^3*c*d^2*a^3+1/b*c^ 
3*ln(exp(2*I*(b*x+a))+1)-1/4*I*d^3*x^4-2/b*c^3*ln(exp(I*(b*x+a)))+3/4*I*d^ 
3*polylog(4,-exp(2*I*(b*x+a)))/b^4+3/2*I*d^3*polylog(2,-exp(2*I*(b*x+a)))/ 
b^4
 

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. 592 vs. \(2 (217) = 434\).

Time = 0.09 (sec) , antiderivative size = 592, normalized size of antiderivative = 2.29 \[ \int (c+d x)^3 \tan ^3(a+b x) \, dx=\frac {4 \, b^{3} d^{3} x^{3} + 12 \, b^{3} c d^{2} x^{2} + 12 \, b^{3} c^{2} d x - 3 i \, d^{3} {\rm polylog}\left (4, \frac {\tan \left (b x + a\right )^{2} + 2 i \, \tan \left (b x + a\right ) - 1}{\tan \left (b x + a\right )^{2} + 1}\right ) + 3 i \, d^{3} {\rm polylog}\left (4, \frac {\tan \left (b x + a\right )^{2} - 2 i \, \tan \left (b x + a\right ) - 1}{\tan \left (b x + a\right )^{2} + 1}\right ) + 4 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \tan \left (b x + a\right )^{2} - 6 \, {\left (-i \, b^{2} d^{3} x^{2} - 2 i \, b^{2} c d^{2} x - i \, b^{2} c^{2} d + i \, d^{3}\right )} {\rm Li}_2\left (\frac {2 \, {\left (i \, \tan \left (b x + a\right ) - 1\right )}}{\tan \left (b x + a\right )^{2} + 1} + 1\right ) - 6 \, {\left (i \, b^{2} d^{3} x^{2} + 2 i \, b^{2} c d^{2} x + i \, b^{2} c^{2} d - i \, d^{3}\right )} {\rm Li}_2\left (\frac {2 \, {\left (-i \, \tan \left (b x + a\right ) - 1\right )}}{\tan \left (b x + a\right )^{2} + 1} + 1\right ) + 4 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + b^{3} c^{3} - 3 \, b c d^{2} + 3 \, {\left (b^{3} c^{2} d - b d^{3}\right )} x\right )} \log \left (-\frac {2 \, {\left (i \, \tan \left (b x + a\right ) - 1\right )}}{\tan \left (b x + a\right )^{2} + 1}\right ) + 4 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + b^{3} c^{3} - 3 \, b c d^{2} + 3 \, {\left (b^{3} c^{2} d - b d^{3}\right )} x\right )} \log \left (-\frac {2 \, {\left (-i \, \tan \left (b x + a\right ) - 1\right )}}{\tan \left (b x + a\right )^{2} + 1}\right ) + 6 \, {\left (b d^{3} x + b c d^{2}\right )} {\rm polylog}\left (3, \frac {\tan \left (b x + a\right )^{2} + 2 i \, \tan \left (b x + a\right ) - 1}{\tan \left (b x + a\right )^{2} + 1}\right ) + 6 \, {\left (b d^{3} x + b c d^{2}\right )} {\rm polylog}\left (3, \frac {\tan \left (b x + a\right )^{2} - 2 i \, \tan \left (b x + a\right ) - 1}{\tan \left (b x + a\right )^{2} + 1}\right ) - 12 \, {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} c^{2} d\right )} \tan \left (b x + a\right )}{8 \, b^{4}} \] Input:

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

Output:

1/8*(4*b^3*d^3*x^3 + 12*b^3*c*d^2*x^2 + 12*b^3*c^2*d*x - 3*I*d^3*polylog(4 
, (tan(b*x + a)^2 + 2*I*tan(b*x + a) - 1)/(tan(b*x + a)^2 + 1)) + 3*I*d^3* 
polylog(4, (tan(b*x + a)^2 - 2*I*tan(b*x + a) - 1)/(tan(b*x + a)^2 + 1)) + 
 4*(b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + b^3*c^3)*tan(b*x + a)^ 
2 - 6*(-I*b^2*d^3*x^2 - 2*I*b^2*c*d^2*x - I*b^2*c^2*d + I*d^3)*dilog(2*(I* 
tan(b*x + a) - 1)/(tan(b*x + a)^2 + 1) + 1) - 6*(I*b^2*d^3*x^2 + 2*I*b^2*c 
*d^2*x + I*b^2*c^2*d - I*d^3)*dilog(2*(-I*tan(b*x + a) - 1)/(tan(b*x + a)^ 
2 + 1) + 1) + 4*(b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + b^3*c^3 - 3*b*c*d^2 + 3*( 
b^3*c^2*d - b*d^3)*x)*log(-2*(I*tan(b*x + a) - 1)/(tan(b*x + a)^2 + 1)) + 
4*(b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + b^3*c^3 - 3*b*c*d^2 + 3*(b^3*c^2*d - b* 
d^3)*x)*log(-2*(-I*tan(b*x + a) - 1)/(tan(b*x + a)^2 + 1)) + 6*(b*d^3*x + 
b*c*d^2)*polylog(3, (tan(b*x + a)^2 + 2*I*tan(b*x + a) - 1)/(tan(b*x + a)^ 
2 + 1)) + 6*(b*d^3*x + b*c*d^2)*polylog(3, (tan(b*x + a)^2 - 2*I*tan(b*x + 
 a) - 1)/(tan(b*x + a)^2 + 1)) - 12*(b^2*d^3*x^2 + 2*b^2*c*d^2*x + b^2*c^2 
*d)*tan(b*x + a))/b^4
 

Sympy [F]

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

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

Output:

Integral((c + d*x)**3*tan(a + b*x)**3, 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. 2413 vs. \(2 (217) = 434\).

Time = 0.72 (sec) , antiderivative size = 2413, normalized size of antiderivative = 9.32 \[ \int (c+d x)^3 \tan ^3(a+b x) \, dx=\text {Too large to display} \] Input:

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (c+d x)^3 \tan ^3(a+b x) \, dx=\frac {2 \left (\int \tan \left (b x +a \right )^{3} x^{3}d x \right ) b \,d^{3}+6 \left (\int \tan \left (b x +a \right )^{3} x^{2}d x \right ) b c \,d^{2}+6 \left (\int \tan \left (b x +a \right )^{3} x d x \right ) b \,c^{2} d -\mathrm {log}\left (\tan \left (b x +a \right )^{2}+1\right ) c^{3}+\tan \left (b x +a \right )^{2} c^{3}}{2 b} \] Input:

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

Output:

(2*int(tan(a + b*x)**3*x**3,x)*b*d**3 + 6*int(tan(a + b*x)**3*x**2,x)*b*c* 
d**2 + 6*int(tan(a + b*x)**3*x,x)*b*c**2*d - log(tan(a + b*x)**2 + 1)*c**3 
 + tan(a + b*x)**2*c**3)/(2*b)