\(\int (c+d x)^3 \tan ^2(a+b x) \, dx\) [254]

Optimal result

Integrand size = 16, antiderivative size = 128 \[ \int (c+d x)^3 \tan ^2(a+b x) \, dx=-\frac {i (c+d x)^3}{b}-\frac {(c+d x)^4}{4 d}+\frac {3 d (c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{b^2}-\frac {3 i d^2 (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^3}+\frac {3 d^3 \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{2 b^4}+\frac {(c+d x)^3 \tan (a+b x)}{b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {3801, 3800, 2221, 2611, 2320, 6724, 32} \[ \int (c+d x)^3 \tan ^2(a+b x) \, dx=\frac {3 d^3 \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{2 b^4}-\frac {3 i d^2 (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^3}+\frac {3 d (c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{b^2}+\frac {(c+d x)^3 \tan (a+b x)}{b}-\frac {i (c+d x)^3}{b}-\frac {(c+d x)^4}{4 d} \]

[In]

Int[(c + d*x)^3*Tan[a + b*x]^2,x]

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \frac {(c+d x)^3 \tan (a+b x)}{b}-\frac {(3 d) \int (c+d x)^2 \tan (a+b x) \, dx}{b}-\int (c+d x)^3 \, dx \\ & = -\frac {i (c+d x)^3}{b}-\frac {(c+d x)^4}{4 d}+\frac {(c+d x)^3 \tan (a+b x)}{b}+\frac {(6 i d) \int \frac {e^{2 i (a+b x)} (c+d x)^2}{1+e^{2 i (a+b x)}} \, dx}{b} \\ & = -\frac {i (c+d x)^3}{b}-\frac {(c+d x)^4}{4 d}+\frac {3 d (c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{b^2}+\frac {(c+d x)^3 \tan (a+b x)}{b}-\frac {\left (6 d^2\right ) \int (c+d x) \log \left (1+e^{2 i (a+b x)}\right ) \, dx}{b^2} \\ & = -\frac {i (c+d x)^3}{b}-\frac {(c+d x)^4}{4 d}+\frac {3 d (c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{b^2}-\frac {3 i d^2 (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^3}+\frac {(c+d x)^3 \tan (a+b x)}{b}+\frac {\left (3 i d^3\right ) \int \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right ) \, dx}{b^3} \\ & = -\frac {i (c+d x)^3}{b}-\frac {(c+d x)^4}{4 d}+\frac {3 d (c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{b^2}-\frac {3 i d^2 (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^3}+\frac {(c+d x)^3 \tan (a+b x)}{b}+\frac {\left (3 d^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{2 i (a+b x)}\right )}{2 b^4} \\ & = -\frac {i (c+d x)^3}{b}-\frac {(c+d x)^4}{4 d}+\frac {3 d (c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{b^2}-\frac {3 i d^2 (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^3}+\frac {3 d^3 \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{2 b^4}+\frac {(c+d x)^3 \tan (a+b x)}{b} \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(424\) vs. \(2(128)=256\).

Time = 6.54 (sec) , antiderivative size = 424, normalized size of antiderivative = 3.31 \[ \int (c+d x)^3 \tan ^2(a+b x) \, dx=-\frac {1}{4} x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )+\frac {i d^3 e^{-i a} \left (2 b^2 x^2 \left (2 b x-3 i \left (1+e^{2 i a}\right ) \log \left (1+e^{-2 i (a+b x)}\right )\right )+6 b \left (1+e^{2 i a}\right ) x \operatorname {PolyLog}\left (2,-e^{-2 i (a+b x)}\right )-3 i \left (1+e^{2 i a}\right ) \operatorname {PolyLog}\left (3,-e^{-2 i (a+b x)}\right )\right ) \sec (a)}{4 b^4}+\frac {3 c^2 d \sec (a) (\cos (a) \log (\cos (a) \cos (b x)-\sin (a) \sin (b x))+b x \sin (a))}{b^2 \left (\cos ^2(a)+\sin ^2(a)\right )}+\frac {3 c d^2 \csc (a) \left (b^2 e^{-i \arctan (\cot (a))} x^2-\frac {\cot (a) \left (i b x (-\pi -2 \arctan (\cot (a)))-\pi \log \left (1+e^{-2 i b x}\right )-2 (b x-\arctan (\cot (a))) \log \left (1-e^{2 i (b x-\arctan (\cot (a)))}\right )+\pi \log (\cos (b x))-2 \arctan (\cot (a)) \log (\sin (b x-\arctan (\cot (a))))+i \operatorname {PolyLog}\left (2,e^{2 i (b x-\arctan (\cot (a)))}\right )\right )}{\sqrt {1+\cot ^2(a)}}\right ) \sec (a)}{b^3 \sqrt {\csc ^2(a) \left (\cos ^2(a)+\sin ^2(a)\right )}}+\frac {\sec (a) \sec (a+b x) \left (c^3 \sin (b x)+3 c^2 d x \sin (b x)+3 c d^2 x^2 \sin (b x)+d^3 x^3 \sin (b x)\right )}{b} \]

[In]

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

[Out]

-1/4*(x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3)) + ((I/4)*d^3*(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^4*E^(I*a)) + (3*c^2*d*Sec[a]*(Cos[a]*Log[Cos[a]*Cos[b*x]
- Sin[a]*Sin[b*x]] + b*x*Sin[a]))/(b^2*(Cos[a]^2 + Sin[a]^2)) + (3*c*d^2*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*PolyL
og[2, E^((2*I)*(b*x - ArcTan[Cot[a]]))]))/Sqrt[1 + Cot[a]^2])*Sec[a])/(b^3*Sqrt[Csc[a]^2*(Cos[a]^2 + Sin[a]^2)
]) + (Sec[a]*Sec[a + b*x]*(c^3*Sin[b*x] + 3*c^2*d*x*Sin[b*x] + 3*c*d^2*x^2*Sin[b*x] + d^3*x^3*Sin[b*x]))/b

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 355 vs. \(2 (116 ) = 232\).

Time = 1.70 (sec) , antiderivative size = 356, normalized size of antiderivative = 2.78

method	result	size
risch	\(-\frac {d^{3} x^{4}}{4}-d^{2} c \,x^{3}-\frac {3 d \,c^{2} x^{2}}{2}-c^{3} x -\frac {c^{4}}{4 d}-\frac {3 i d^{3} \operatorname {polylog}\left (2, -{\mathrm e}^{2 i \left (x b +a \right )}\right ) x}{b^{3}}+\frac {3 d^{3} \ln \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right ) x^{2}}{b^{2}}+\frac {3 d^{3} \operatorname {polylog}\left (3, -{\mathrm e}^{2 i \left (x b +a \right )}\right )}{2 b^{4}}+\frac {6 d^{2} c \ln \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right ) x}{b^{2}}-\frac {6 i d^{2} c \,x^{2}}{b}+\frac {6 i d^{3} a^{2} x}{b^{3}}+\frac {2 i \left (d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}\right )}{b \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right )}+\frac {12 d^{2} c a \ln \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}-\frac {3 i d^{2} c \operatorname {polylog}\left (2, -{\mathrm e}^{2 i \left (x b +a \right )}\right )}{b^{3}}-\frac {6 i d^{2} c \,a^{2}}{b^{3}}-\frac {6 d^{3} a^{2} \ln \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{4}}+\frac {3 d \,c^{2} \ln \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right )}{b^{2}}-\frac {6 d \,c^{2} \ln \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{2}}-\frac {12 i d^{2} c x a}{b^{2}}-\frac {2 i d^{3} x^{3}}{b}+\frac {4 i d^{3} a^{3}}{b^{4}}\)	\(356\)

[In]

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

[Out]

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

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. 373 vs. \(2 (113) = 226\).

Time = 0.26 (sec) , antiderivative size = 373, normalized size of antiderivative = 2.91 \[ \int (c+d x)^3 \tan ^2(a+b x) \, dx=-\frac {b^{4} d^{3} x^{4} + 4 \, b^{4} c d^{2} x^{3} + 6 \, b^{4} c^{2} d x^{2} + 4 \, b^{4} c^{3} x - 3 \, d^{3} {\rm polylog}\left (3, \frac {\tan \left (b x + a\right )^{2} + 2 i \, \tan \left (b x + a\right ) - 1}{\tan \left (b x + a\right )^{2} + 1}\right ) - 3 \, d^{3} {\rm polylog}\left (3, \frac {\tan \left (b x + a\right )^{2} - 2 i \, \tan \left (b x + a\right ) - 1}{\tan \left (b x + a\right )^{2} + 1}\right ) + 6 \, {\left (-i \, b d^{3} x - i \, b c d^{2}\right )} {\rm Li}_2\left (\frac {2 \, {\left (i \, \tan \left (b x + a\right ) - 1\right )}}{\tan \left (b x + a\right )^{2} + 1} + 1\right ) + 6 \, {\left (i \, b d^{3} x + i \, b c d^{2}\right )} {\rm Li}_2\left (\frac {2 \, {\left (-i \, \tan \left (b x + a\right ) - 1\right )}}{\tan \left (b x + a\right )^{2} + 1} + 1\right ) - 6 \, {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} c^{2} d\right )} \log \left (-\frac {2 \, {\left (i \, \tan \left (b x + a\right ) - 1\right )}}{\tan \left (b x + a\right )^{2} + 1}\right ) - 6 \, {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} c^{2} d\right )} \log \left (-\frac {2 \, {\left (-i \, \tan \left (b x + a\right ) - 1\right )}}{\tan \left (b x + a\right )^{2} + 1}\right ) - 4 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \tan \left (b x + a\right )}{4 \, b^{4}} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int (c+d x)^3 \tan ^2(a+b x) \, dx=\int \left (c + d x\right )^{3} \tan ^{2}{\left (a + b x \right )}\, dx \]

[In]

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

[Out]

Integral((c + d*x)**3*tan(a + b*x)**2, 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. 1363 vs. \(2 (113) = 226\).

Time = 0.40 (sec) , antiderivative size = 1363, normalized size of antiderivative = 10.65 \[ \int (c+d x)^3 \tan ^2(a+b x) \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Giac [F]

\[ \int (c+d x)^3 \tan ^2(a+b x) \, dx=\int { {\left (d x + c\right )}^{3} \tan \left (b x + a\right )^{2} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int (c+d x)^3 \tan ^2(a+b x) \, dx=\int {\mathrm {tan}\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^3 \,d x \]

[In]

int(tan(a + b*x)^2*(c + d*x)^3,x)

[Out]

int(tan(a + b*x)^2*(c + d*x)^3, x)