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

Optimal result

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

[Out]

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*polylog(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

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {3801, 3800, 2221, 2317, 2438, 32, 2611, 6744, 2320, 6724} \[ \int (c+d x)^3 \tan ^3(a+b x) \, dx=\frac {3 i d^3 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b^4}+\frac {3 i d^3 \operatorname {PolyLog}\left (4,-e^{2 i (a+b x)}\right )}{4 b^4}+\frac {3 d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{2 b^3}-\frac {3 d^2 (c+d x) \log \left (1+e^{2 i (a+b x)}\right )}{b^3}-\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b^2}-\frac {3 d (c+d x)^2 \tan (a+b x)}{2 b^2}+\frac {(c+d x)^3 \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac {(c+d x)^3 \tan ^2(a+b x)}{2 b}+\frac {3 i d (c+d x)^2}{2 b^2}+\frac {(c+d x)^3}{2 b}-\frac {i (c+d x)^4}{4 d} \]

[In]

Int[(c + d*x)^3*Tan[a + b*x]^3,x]

[Out]

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

Rule 32

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

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 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 3800

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I*((c + d*x)^(m + 1)/(d*(m + 1))), x
] - Dist[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] &&
IGtQ[m, 0]

Rule 3801

Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(c + d*x)^m*((b*Tan[e
+ f*x])^(n - 1)/(f*(n - 1))), x] + (-Dist[b*d*(m/(f*(n - 1))), Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1)
, x], x] - Dist[b^2, Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n,
1] && GtQ[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)^3 \tan ^2(a+b x)}{2 b}-\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 {i (c+d x)^4}{4 d}-\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}+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 \, dx}{2 b}+\frac {\left (3 d^2\right ) \int (c+d x) \tan (a+b x) \, dx}{b^2} \\ & = \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 {(c+d x)^3 \log \left (1+e^{2 i (a+b x)}\right )}{b}-\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}-\frac {(3 d) \int (c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right ) \, dx}{b}-\frac {\left (6 i d^2\right ) \int \frac {e^{2 i (a+b x)} (c+d x)}{1+e^{2 i (a+b x)}} \, dx}{b^2} \\ & = \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 (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b^2}-\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}+\frac {\left (3 i d^2\right ) \int (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right ) \, dx}{b^2}+\frac {\left (3 d^3\right ) \int \log \left (1+e^{2 i (a+b x)}\right ) \, dx}{b^3} \\ & = \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 (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 d (c+d x)^2 \tan (a+b x)}{2 b^2}+\frac {(c+d x)^3 \tan ^2(a+b x)}{2 b}-\frac {\left (3 i d^3\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i (a+b x)}\right )}{2 b^4}-\frac {\left (3 d^3\right ) \int \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right ) \, dx}{2 b^3} \\ & = \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 d (c+d x)^2 \tan (a+b x)}{2 b^2}+\frac {(c+d x)^3 \tan ^2(a+b x)}{2 b}+\frac {\left (3 i d^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{2 i (a+b x)}\right )}{4 b^4} \\ & = \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} \\ \end{align*}

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 = 6.92 (sec) , antiderivative size = 814, normalized size of antiderivative = 3.14 \[ \int (c+d x)^3 \tan ^3(a+b x) \, dx=\frac {i c d^2 e^{-i a} \left (2 b^2 x^2 \left (2 b x-3 i \left (1+e^{2 i a}\right ) \log \left (1+e^{-2 i (a+b x)}\right )\right )+6 b \left (1+e^{2 i a}\right ) x \operatorname {PolyLog}\left (2,-e^{-2 i (a+b x)}\right )-3 i \left (1+e^{2 i a}\right ) \operatorname {PolyLog}\left (3,-e^{-2 i (a+b x)}\right )\right ) \sec (a)}{4 b^3}+\frac {i d^3 e^{i a} \left (2 b^4 e^{-2 i a} x^4-4 i b^3 \left (1+e^{-2 i a}\right ) x^3 \log \left (1+e^{-2 i (a+b x)}\right )+6 b^2 \left (1+e^{-2 i a}\right ) x^2 \operatorname {PolyLog}\left (2,-e^{-2 i (a+b x)}\right )-6 i b \left (1+e^{-2 i a}\right ) x \operatorname {PolyLog}\left (3,-e^{-2 i (a+b x)}\right )-3 \left (1+e^{-2 i a}\right ) \operatorname {PolyLog}\left (4,-e^{-2 i (a+b x)}\right )\right ) \sec (a)}{8 b^4}+\frac {(c+d x)^3 \sec ^2(a+b x)}{2 b}+\frac {c^3 \sec (a) (\cos (a) \log (\cos (a) \cos (b x)-\sin (a) \sin (b x))+b x \sin (a))}{b \left (\cos ^2(a)+\sin ^2(a)\right )}-\frac {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 \left (\cos ^2(a)+\sin ^2(a)\right )}+\frac {3 c^2 d \csc (a) \left (b^2 e^{-i \arctan (\cot (a))} x^2-\frac {\cot (a) \left (i b x (-\pi -2 \arctan (\cot (a)))-\pi \log \left (1+e^{-2 i b x}\right )-2 (b x-\arctan (\cot (a))) \log \left (1-e^{2 i (b x-\arctan (\cot (a)))}\right )+\pi \log (\cos (b x))-2 \arctan (\cot (a)) \log (\sin (b x-\arctan (\cot (a))))+i \operatorname {PolyLog}\left (2,e^{2 i (b x-\arctan (\cot (a)))}\right )\right )}{\sqrt {1+\cot ^2(a)}}\right ) \sec (a)}{2 b^2 \sqrt {\csc ^2(a) \left (\cos ^2(a)+\sin ^2(a)\right )}}-\frac {3 d^3 \csc (a) \left (b^2 e^{-i \arctan (\cot (a))} x^2-\frac {\cot (a) \left (i b x (-\pi -2 \arctan (\cot (a)))-\pi \log \left (1+e^{-2 i b x}\right )-2 (b x-\arctan (\cot (a))) \log \left (1-e^{2 i (b x-\arctan (\cot (a)))}\right )+\pi \log (\cos (b x))-2 \arctan (\cot (a)) \log (\sin (b x-\arctan (\cot (a))))+i \operatorname {PolyLog}\left (2,e^{2 i (b x-\arctan (\cot (a)))}\right )\right )}{\sqrt {1+\cot ^2(a)}}\right ) \sec (a)}{2 b^4 \sqrt {\csc ^2(a) \left (\cos ^2(a)+\sin ^2(a)\right )}}-\frac {3 \sec (a) \sec (a+b x) \left (c^2 d \sin (b x)+2 c d^2 x \sin (b x)+d^3 x^2 \sin (b x)\right )}{2 b^2}-\frac {1}{4} x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right ) \tan (a) \]

[In]

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

[Out]

((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*Po
lyLog[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*(C
os[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*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[Co
s[b*x]] - 2*ArcTan[Cot[a]]*Log[Sin[b*x - ArcTan[Cot[a]]]] + I*PolyLog[2, E^((2*I)*(b*x - ArcTan[Cot[a]]))]))/S
qrt[1 + Cot[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]])*Lo
g[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]]))]))/Sqrt[1 + Cot[a]^2])*Sec[a])/(2*b^4*Sqrt[Csc[a]^2*(Cos[a]^2 +
Sin[a]^2)]) - (3*Sec[a]*Sec[a + b*x]*(c^2*d*Sin[b*x] + 2*c*d^2*x*Sin[b*x] + d^3*x^2*Sin[b*x]))/(2*b^2) - (x*(
4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3)*Tan[a])/4

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 = 0.72 (sec) , antiderivative size = 729, normalized size of antiderivative = 2.81

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

[In]

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

[Out]

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

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 = 0.27 (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}} \]

[In]

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

[Out]

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]

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

[In]

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

[Out]

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 = 0.90 (sec) , antiderivative size = 2413, normalized size of antiderivative = 9.32 \[ \int (c+d x)^3 \tan ^3(a+b x) \, dx=\text {Too large to display} \]

[In]

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

[Out]

-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*(b*c*d^2 - (a - I)*d^3)*(b*x + a)^3 + 6*(b^2*c^2*d - 2*(a - I)*b
*c*d^2 + (a^2 - 2*I*a - 1)*d^3)*(b*x + a)^2 + 12*(I*b^2*c^2*d + (-2*I*a - 1)*b*c*d^2 + (I*a^2 + a)*d^3)*(b*x +
a))*cos(2*b*x + 2*a) + 6*(3*b^2*c^2*d - 6*a*b*c*d^2 + 4*(b*x + a)^2*d^3 + 3*(a^2 - 1)*d^3 + 6*(b*c*d^2 - a*d^
3)*(b*x + a) + (3*b^2*c^2*d - 6*a*b*c*d^2 + 4*(b*x + a)^2*d^3 + 3*(a^2 - 1)*d^3 + 6*(b*c*d^2 - a*d^3)*(b*x + a
))*cos(4*b*x + 4*a) + 2*(3*b^2*c^2*d - 6*a*b*c*d^2 + 4*(b*x + a)^2*d^3 + 3*(a^2 - 1)*d^3 + 6*(b*c*d^2 - a*d^3)
*(b*x + a))*cos(2*b*x + 2*a) + (3*I*b^2*c^2*d - 6*I*a*b*c*d^2 + 4*I*(b*x + a)^2*d^3 + 3*(I*a^2 - I)*d^3 + 6*(I
*b*c*d^2 - I*a*d^3)*(b*x + a))*sin(4*b*x + 4*a) + 2*(3*I*b^2*c^2*d - 6*I*a*b*c*d^2 + 4*I*(b*x + a)^2*d^3 + 3*(
I*a^2 - I)*d^3 + 6*(I*b*c*d^2 - I*a*d^3)*(b*x + a))*sin(2*b*x + 2*a))*dilog(-e^(2*I*b*x + 2*I*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) + (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))*cos(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))*cos(2*b*x + 2*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))*sin(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))*sin(2*b*
x + 2*a))*log(cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a) + 1) - 12*(d^3*cos(4*b*x + 4*a) + 2
*d^3*cos(2*b*x + 2*a) + I*d^3*sin(4*b*x + 4*a) + 2*I*d^3*sin(2*b*x + 2*a) + d^3)*polylog(4, -e^(2*I*b*x + 2*I*
a)) + 6*(3*I*b*c*d^2 + 4*I*(b*x + a)*d^3 - 3*I*a*d^3 + (3*I*b*c*d^2 + 4*I*(b*x + a)*d^3 - 3*I*a*d^3)*cos(4*b*x
+ 4*a) + 2*(3*I*b*c*d^2 + 4*I*(b*x + a)*d^3 - 3*I*a*d^3)*cos(2*b*x + 2*a) - (3*b*c*d^2 + 4*(b*x + a)*d^3 - 3*
a*d^3)*sin(4*b*x + 4*a) - 2*(3*b*c*d^2 + 4*(b*x + a)*d^3 - 3*a*d^3)*sin(2*b*x + 2*a))*polylog(3, -e^(2*I*b*x +
2*I*a)) + 3*(I*(b*x + a)^4*d^3 + 4*(I*b*c*d^2 - I*a*d^3)*(b*x + a)^3 + 6*(I*b^2*c^2*d - 2*I*a*b*c*d^2 + (I*a^
2 - 2*I)*d^3)*(b*x + a)^2 + 24*(-I*b*c*d^2 + I*a*d^3)*(b*x + a))*sin(4*b*x + 4*a) + 6*(I*(b*x + a)^4*d^3 + 6*I
*b^2*c^2*d - 12*I*a*b*c*d^2 + 6*I*a^2*d^3 + 4*(I*b*c*d^2 + (-I*a - 1)*d^3)*(b*x + a)^3 + 6*(I*b^2*c^2*d + 2*(-
I*a - 1)*b*c*d^2 + (I*a^2 + 2*a - I)*d^3)*(b*x + a)^2 - 12*(b^2*c^2*d - (2*a - I)*b*c*d^2 + (a^2 - I*a)*d^3)*(
b*x + a))*sin(2*b*x + 2*a))/(-12*I*b^3*cos(4*b*x + 4*a) - 24*I*b^3*cos(2*b*x + 2*a) + 12*b^3*sin(4*b*x + 4*a)
+ 24*b^3*sin(2*b*x + 2*a) - 12*I*b^3))/b

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(tan(a + b*x)^3*(c + d*x)^3,x)

[Out]

int(tan(a + b*x)^3*(c + d*x)^3, x)