3.44 \(\int (c+d x)^3 (a+b \tan (e+f x))^2 \, dx\)

Optimal. Leaf size=300 \[ \frac {a^2 (c+d x)^4}{4 d}-\frac {3 a b d^2 (c+d x) \text {Li}_3\left (-e^{2 i (e+f x)}\right )}{f^3}+\frac {3 i a b d (c+d x)^2 \text {Li}_2\left (-e^{2 i (e+f x)}\right )}{f^2}-\frac {2 a b (c+d x)^3 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac {i a b (c+d x)^4}{2 d}-\frac {3 i a b d^3 \text {Li}_4\left (-e^{2 i (e+f x)}\right )}{2 f^4}-\frac {3 i b^2 d^2 (c+d x) \text {Li}_2\left (-e^{2 i (e+f x)}\right )}{f^3}+\frac {3 b^2 d (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f^2}+\frac {b^2 (c+d x)^3 \tan (e+f x)}{f}-\frac {i b^2 (c+d x)^3}{f}-\frac {b^2 (c+d x)^4}{4 d}+\frac {3 b^2 d^3 \text {Li}_3\left (-e^{2 i (e+f x)}\right )}{2 f^4} \]

[Out]

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

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Rubi [A]  time = 0.52, antiderivative size = 300, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {3722, 3719, 2190, 2531, 6609, 2282, 6589, 3720, 32} \[ \frac {a^2 (c+d x)^4}{4 d}-\frac {3 a b d^2 (c+d x) \text {Li}_3\left (-e^{2 i (e+f x)}\right )}{f^3}+\frac {3 i a b d (c+d x)^2 \text {Li}_2\left (-e^{2 i (e+f x)}\right )}{f^2}-\frac {2 a b (c+d x)^3 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac {i a b (c+d x)^4}{2 d}-\frac {3 i a b d^3 \text {Li}_4\left (-e^{2 i (e+f x)}\right )}{2 f^4}-\frac {3 i b^2 d^2 (c+d x) \text {Li}_2\left (-e^{2 i (e+f x)}\right )}{f^3}+\frac {3 b^2 d (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f^2}+\frac {b^2 (c+d x)^3 \tan (e+f x)}{f}-\frac {i b^2 (c+d x)^3}{f}-\frac {b^2 (c+d x)^4}{4 d}+\frac {3 b^2 d^3 \text {Li}_3\left (-e^{2 i (e+f x)}\right )}{2 f^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2282

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 2531

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 3719

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 3720

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 3722

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, (a + b*Tan[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[m, 0] && IGtQ[n, 0]

Rule 6589

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 6609

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*} \int (c+d x)^3 (a+b \tan (e+f x))^2 \, dx &=\int \left (a^2 (c+d x)^3+2 a b (c+d x)^3 \tan (e+f x)+b^2 (c+d x)^3 \tan ^2(e+f x)\right ) \, dx\\ &=\frac {a^2 (c+d x)^4}{4 d}+(2 a b) \int (c+d x)^3 \tan (e+f x) \, dx+b^2 \int (c+d x)^3 \tan ^2(e+f x) \, dx\\ &=\frac {a^2 (c+d x)^4}{4 d}+\frac {i a b (c+d x)^4}{2 d}+\frac {b^2 (c+d x)^3 \tan (e+f x)}{f}-(4 i a b) \int \frac {e^{2 i (e+f x)} (c+d x)^3}{1+e^{2 i (e+f x)}} \, dx-b^2 \int (c+d x)^3 \, dx-\frac {\left (3 b^2 d\right ) \int (c+d x)^2 \tan (e+f x) \, dx}{f}\\ &=-\frac {i b^2 (c+d x)^3}{f}+\frac {a^2 (c+d x)^4}{4 d}+\frac {i a b (c+d x)^4}{2 d}-\frac {b^2 (c+d x)^4}{4 d}-\frac {2 a b (c+d x)^3 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac {b^2 (c+d x)^3 \tan (e+f x)}{f}+\frac {(6 a b d) \int (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right ) \, dx}{f}+\frac {\left (6 i b^2 d\right ) \int \frac {e^{2 i (e+f x)} (c+d x)^2}{1+e^{2 i (e+f x)}} \, dx}{f}\\ &=-\frac {i b^2 (c+d x)^3}{f}+\frac {a^2 (c+d x)^4}{4 d}+\frac {i a b (c+d x)^4}{2 d}-\frac {b^2 (c+d x)^4}{4 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f^2}-\frac {2 a b (c+d x)^3 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac {3 i a b d (c+d x)^2 \text {Li}_2\left (-e^{2 i (e+f x)}\right )}{f^2}+\frac {b^2 (c+d x)^3 \tan (e+f x)}{f}-\frac {\left (6 i a b d^2\right ) \int (c+d x) \text {Li}_2\left (-e^{2 i (e+f x)}\right ) \, dx}{f^2}-\frac {\left (6 b^2 d^2\right ) \int (c+d x) \log \left (1+e^{2 i (e+f x)}\right ) \, dx}{f^2}\\ &=-\frac {i b^2 (c+d x)^3}{f}+\frac {a^2 (c+d x)^4}{4 d}+\frac {i a b (c+d x)^4}{2 d}-\frac {b^2 (c+d x)^4}{4 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f^2}-\frac {2 a b (c+d x)^3 \log \left (1+e^{2 i (e+f x)}\right )}{f}-\frac {3 i b^2 d^2 (c+d x) \text {Li}_2\left (-e^{2 i (e+f x)}\right )}{f^3}+\frac {3 i a b d (c+d x)^2 \text {Li}_2\left (-e^{2 i (e+f x)}\right )}{f^2}-\frac {3 a b d^2 (c+d x) \text {Li}_3\left (-e^{2 i (e+f x)}\right )}{f^3}+\frac {b^2 (c+d x)^3 \tan (e+f x)}{f}+\frac {\left (3 a b d^3\right ) \int \text {Li}_3\left (-e^{2 i (e+f x)}\right ) \, dx}{f^3}+\frac {\left (3 i b^2 d^3\right ) \int \text {Li}_2\left (-e^{2 i (e+f x)}\right ) \, dx}{f^3}\\ &=-\frac {i b^2 (c+d x)^3}{f}+\frac {a^2 (c+d x)^4}{4 d}+\frac {i a b (c+d x)^4}{2 d}-\frac {b^2 (c+d x)^4}{4 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f^2}-\frac {2 a b (c+d x)^3 \log \left (1+e^{2 i (e+f x)}\right )}{f}-\frac {3 i b^2 d^2 (c+d x) \text {Li}_2\left (-e^{2 i (e+f x)}\right )}{f^3}+\frac {3 i a b d (c+d x)^2 \text {Li}_2\left (-e^{2 i (e+f x)}\right )}{f^2}-\frac {3 a b d^2 (c+d x) \text {Li}_3\left (-e^{2 i (e+f x)}\right )}{f^3}+\frac {b^2 (c+d x)^3 \tan (e+f x)}{f}-\frac {\left (3 i a b d^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{2 i (e+f x)}\right )}{2 f^4}+\frac {\left (3 b^2 d^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 i (e+f x)}\right )}{2 f^4}\\ &=-\frac {i b^2 (c+d x)^3}{f}+\frac {a^2 (c+d x)^4}{4 d}+\frac {i a b (c+d x)^4}{2 d}-\frac {b^2 (c+d x)^4}{4 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f^2}-\frac {2 a b (c+d x)^3 \log \left (1+e^{2 i (e+f x)}\right )}{f}-\frac {3 i b^2 d^2 (c+d x) \text {Li}_2\left (-e^{2 i (e+f x)}\right )}{f^3}+\frac {3 i a b d (c+d x)^2 \text {Li}_2\left (-e^{2 i (e+f x)}\right )}{f^2}+\frac {3 b^2 d^3 \text {Li}_3\left (-e^{2 i (e+f x)}\right )}{2 f^4}-\frac {3 a b d^2 (c+d x) \text {Li}_3\left (-e^{2 i (e+f x)}\right )}{f^3}-\frac {3 i a b d^3 \text {Li}_4\left (-e^{2 i (e+f x)}\right )}{2 f^4}+\frac {b^2 (c+d x)^3 \tan (e+f x)}{f}\\ \end {align*}

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Mathematica [B]  time = 8.04, size = 1326, normalized size = 4.42 \[ \text {result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((I/4)*b^2*d^3*(2*f^2*x^2*(2*f*x - (3*I)*(1 + E^((2*I)*e))*Log[1 + E^((-2*I)*(e + f*x))]) + 6*(1 + E^((2*I)*e)
)*f*x*PolyLog[2, -E^((-2*I)*(e + f*x))] - (3*I)*(1 + E^((2*I)*e))*PolyLog[3, -E^((-2*I)*(e + f*x))])*Sec[e])/(
E^(I*e)*f^4) - ((I/2)*a*b*c*d^2*(2*f^2*x^2*(2*f*x - (3*I)*(1 + E^((2*I)*e))*Log[1 + E^((-2*I)*(e + f*x))]) + 6
*(1 + E^((2*I)*e))*f*x*PolyLog[2, -E^((-2*I)*(e + f*x))] - (3*I)*(1 + E^((2*I)*e))*PolyLog[3, -E^((-2*I)*(e +
f*x))])*Sec[e])/(E^(I*e)*f^3) - (I/4)*a*b*d^3*E^(I*e)*((2*x^4)/E^((2*I)*e) - ((4*I)*(1 + E^((-2*I)*e))*x^3*Log
[1 + E^((-2*I)*(e + f*x))])/f + (3*(1 + E^((2*I)*e))*(2*f^2*x^2*PolyLog[2, -E^((-2*I)*(e + f*x))] - (2*I)*f*x*
PolyLog[3, -E^((-2*I)*(e + f*x))] - PolyLog[4, -E^((-2*I)*(e + f*x))]))/(E^((2*I)*e)*f^4))*Sec[e] + (3*b^2*c^2
*d*Sec[e]*(Cos[e]*Log[Cos[e]*Cos[f*x] - Sin[e]*Sin[f*x]] + f*x*Sin[e]))/(f^2*(Cos[e]^2 + Sin[e]^2)) - (2*a*b*c
^3*Sec[e]*(Cos[e]*Log[Cos[e]*Cos[f*x] - Sin[e]*Sin[f*x]] + f*x*Sin[e]))/(f*(Cos[e]^2 + Sin[e]^2)) + (3*b^2*c*d
^2*Csc[e]*((f^2*x^2)/E^(I*ArcTan[Cot[e]]) - (Cot[e]*(I*f*x*(-Pi - 2*ArcTan[Cot[e]]) - Pi*Log[1 + E^((-2*I)*f*x
)] - 2*(f*x - ArcTan[Cot[e]])*Log[1 - E^((2*I)*(f*x - ArcTan[Cot[e]]))] + Pi*Log[Cos[f*x]] - 2*ArcTan[Cot[e]]*
Log[Sin[f*x - ArcTan[Cot[e]]]] + I*PolyLog[2, E^((2*I)*(f*x - ArcTan[Cot[e]]))]))/Sqrt[1 + Cot[e]^2])*Sec[e])/
(f^3*Sqrt[Csc[e]^2*(Cos[e]^2 + Sin[e]^2)]) - (3*a*b*c^2*d*Csc[e]*((f^2*x^2)/E^(I*ArcTan[Cot[e]]) - (Cot[e]*(I*
f*x*(-Pi - 2*ArcTan[Cot[e]]) - Pi*Log[1 + E^((-2*I)*f*x)] - 2*(f*x - ArcTan[Cot[e]])*Log[1 - E^((2*I)*(f*x - A
rcTan[Cot[e]]))] + Pi*Log[Cos[f*x]] - 2*ArcTan[Cot[e]]*Log[Sin[f*x - ArcTan[Cot[e]]]] + I*PolyLog[2, E^((2*I)*
(f*x - ArcTan[Cot[e]]))]))/Sqrt[1 + Cot[e]^2])*Sec[e])/(f^2*Sqrt[Csc[e]^2*(Cos[e]^2 + Sin[e]^2)]) + (Sec[e]*Se
c[e + f*x]*(4*a^2*c^3*f*x*Cos[f*x] - 4*b^2*c^3*f*x*Cos[f*x] + 6*a^2*c^2*d*f*x^2*Cos[f*x] - 6*b^2*c^2*d*f*x^2*C
os[f*x] + 4*a^2*c*d^2*f*x^3*Cos[f*x] - 4*b^2*c*d^2*f*x^3*Cos[f*x] + a^2*d^3*f*x^4*Cos[f*x] - b^2*d^3*f*x^4*Cos
[f*x] + 4*a^2*c^3*f*x*Cos[2*e + f*x] - 4*b^2*c^3*f*x*Cos[2*e + f*x] + 6*a^2*c^2*d*f*x^2*Cos[2*e + f*x] - 6*b^2
*c^2*d*f*x^2*Cos[2*e + f*x] + 4*a^2*c*d^2*f*x^3*Cos[2*e + f*x] - 4*b^2*c*d^2*f*x^3*Cos[2*e + f*x] + a^2*d^3*f*
x^4*Cos[2*e + f*x] - b^2*d^3*f*x^4*Cos[2*e + f*x] + 8*b^2*c^3*Sin[f*x] + 24*b^2*c^2*d*x*Sin[f*x] - 8*a*b*c^3*f
*x*Sin[f*x] + 24*b^2*c*d^2*x^2*Sin[f*x] - 12*a*b*c^2*d*f*x^2*Sin[f*x] + 8*b^2*d^3*x^3*Sin[f*x] - 8*a*b*c*d^2*f
*x^3*Sin[f*x] - 2*a*b*d^3*f*x^4*Sin[f*x] + 8*a*b*c^3*f*x*Sin[2*e + f*x] + 12*a*b*c^2*d*f*x^2*Sin[2*e + f*x] +
8*a*b*c*d^2*f*x^3*Sin[2*e + f*x] + 2*a*b*d^3*f*x^4*Sin[2*e + f*x]))/(8*f)

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fricas [C]  time = 0.66, size = 757, normalized size = 2.52 \[ \frac {{\left (a^{2} - b^{2}\right )} d^{3} f^{4} x^{4} + 4 \, {\left (a^{2} - b^{2}\right )} c d^{2} f^{4} x^{3} + 6 \, {\left (a^{2} - b^{2}\right )} c^{2} d f^{4} x^{2} + 4 \, {\left (a^{2} - b^{2}\right )} c^{3} f^{4} x + 3 i \, a b d^{3} {\rm polylog}\left (4, \frac {\tan \left (f x + e\right )^{2} + 2 i \, \tan \left (f x + e\right ) - 1}{\tan \left (f x + e\right )^{2} + 1}\right ) - 3 i \, a b d^{3} {\rm polylog}\left (4, \frac {\tan \left (f x + e\right )^{2} - 2 i \, \tan \left (f x + e\right ) - 1}{\tan \left (f x + e\right )^{2} + 1}\right ) + {\left (-6 i \, a b d^{3} f^{2} x^{2} - 6 i \, a b c^{2} d f^{2} + 6 i \, b^{2} c d^{2} f - 6 i \, {\left (2 \, a b c d^{2} f^{2} - b^{2} d^{3} f\right )} x\right )} {\rm Li}_2\left (\frac {2 \, {\left (i \, \tan \left (f x + e\right ) - 1\right )}}{\tan \left (f x + e\right )^{2} + 1} + 1\right ) + {\left (6 i \, a b d^{3} f^{2} x^{2} + 6 i \, a b c^{2} d f^{2} - 6 i \, b^{2} c d^{2} f + 6 i \, {\left (2 \, a b c d^{2} f^{2} - b^{2} d^{3} f\right )} x\right )} {\rm Li}_2\left (\frac {2 \, {\left (-i \, \tan \left (f x + e\right ) - 1\right )}}{\tan \left (f x + e\right )^{2} + 1} + 1\right ) - 2 \, {\left (2 \, a b d^{3} f^{3} x^{3} + 2 \, a b c^{3} f^{3} - 3 \, b^{2} c^{2} d f^{2} + 3 \, {\left (2 \, a b c d^{2} f^{3} - b^{2} d^{3} f^{2}\right )} x^{2} + 6 \, {\left (a b c^{2} d f^{3} - b^{2} c d^{2} f^{2}\right )} x\right )} \log \left (-\frac {2 \, {\left (i \, \tan \left (f x + e\right ) - 1\right )}}{\tan \left (f x + e\right )^{2} + 1}\right ) - 2 \, {\left (2 \, a b d^{3} f^{3} x^{3} + 2 \, a b c^{3} f^{3} - 3 \, b^{2} c^{2} d f^{2} + 3 \, {\left (2 \, a b c d^{2} f^{3} - b^{2} d^{3} f^{2}\right )} x^{2} + 6 \, {\left (a b c^{2} d f^{3} - b^{2} c d^{2} f^{2}\right )} x\right )} \log \left (-\frac {2 \, {\left (-i \, \tan \left (f x + e\right ) - 1\right )}}{\tan \left (f x + e\right )^{2} + 1}\right ) - 3 \, {\left (2 \, a b d^{3} f x + 2 \, a b c d^{2} f - b^{2} d^{3}\right )} {\rm polylog}\left (3, \frac {\tan \left (f x + e\right )^{2} + 2 i \, \tan \left (f x + e\right ) - 1}{\tan \left (f x + e\right )^{2} + 1}\right ) - 3 \, {\left (2 \, a b d^{3} f x + 2 \, a b c d^{2} f - b^{2} d^{3}\right )} {\rm polylog}\left (3, \frac {\tan \left (f x + e\right )^{2} - 2 i \, \tan \left (f x + e\right ) - 1}{\tan \left (f x + e\right )^{2} + 1}\right ) + 4 \, {\left (b^{2} d^{3} f^{3} x^{3} + 3 \, b^{2} c d^{2} f^{3} x^{2} + 3 \, b^{2} c^{2} d f^{3} x + b^{2} c^{3} f^{3}\right )} \tan \left (f x + e\right )}{4 \, f^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )}^{3} {\left (b \tan \left (f x + e\right ) + a\right )}^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.82, size = 919, normalized size = 3.06 \[ \frac {6 b^{2} c \,d^{2} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) x}{f^{2}}+\frac {12 b^{2} c \,d^{2} e \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f^{3}}-\frac {4 b a \,d^{3} e^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f^{4}}-\frac {3 b a \,d^{3} \polylog \left (3, -{\mathrm e}^{2 i \left (f x +e \right )}\right ) x}{f^{3}}-\frac {3 b a c \,d^{2} \polylog \left (3, -{\mathrm e}^{2 i \left (f x +e \right )}\right )}{f^{3}}-\frac {3 i b^{2} c \,d^{2} \polylog \left (2, -{\mathrm e}^{2 i \left (f x +e \right )}\right )}{f^{3}}-\frac {3 i b^{2} d^{3} \polylog \left (2, -{\mathrm e}^{2 i \left (f x +e \right )}\right ) x}{f^{3}}-\frac {3 i a b \,d^{3} \polylog \left (4, -{\mathrm e}^{2 i \left (f x +e \right )}\right )}{2 f^{4}}+a^{2} c^{3} x -b^{2} c^{3} x +\frac {12 i b a \,c^{2} d e x}{f}+\frac {6 i b a c \,d^{2} \polylog \left (2, -{\mathrm e}^{2 i \left (f x +e \right )}\right ) x}{f^{2}}-\frac {12 i b a c \,d^{2} e^{2} x}{f^{2}}+a^{2} c \,d^{2} x^{3}+\frac {3 b^{2} d^{3} \polylog \left (3, -{\mathrm e}^{2 i \left (f x +e \right )}\right )}{2 f^{4}}-b^{2} c \,d^{2} x^{3}+\frac {3 a^{2} c^{2} d \,x^{2}}{2}-\frac {3 b^{2} c^{2} d \,x^{2}}{2}+\frac {3 i b a \,c^{2} d \polylog \left (2, -{\mathrm e}^{2 i \left (f x +e \right )}\right )}{f^{2}}-\frac {2 b a \,d^{3} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) x^{3}}{f}+\frac {3 i b a \,d^{3} \polylog \left (2, -{\mathrm e}^{2 i \left (f x +e \right )}\right ) x^{2}}{f^{2}}-\frac {12 b a \,c^{2} d e \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f^{2}}+\frac {12 b a c \,d^{2} e^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f^{3}}-\frac {12 i b^{2} c \,d^{2} e x}{f^{2}}+\frac {4 i b a \,d^{3} e^{3} x}{f^{3}}-\frac {8 i b a c \,d^{2} e^{3}}{f^{3}}+\frac {6 i b a \,c^{2} d \,e^{2}}{f^{2}}-\frac {6 b a c \,d^{2} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) x^{2}}{f}-\frac {6 b \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) a \,c^{2} d x}{f}+\frac {2 i b^{2} \left (d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}\right )}{f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}-2 i a b \,c^{3} x +\frac {i a b \,d^{3} x^{4}}{2}+\frac {3 b^{2} d^{3} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) x^{2}}{f^{2}}+\frac {4 b a \,c^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f}-\frac {6 b^{2} c^{2} d \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f^{2}}+\frac {3 b^{2} c^{2} d \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}{f^{2}}-\frac {2 b a \,c^{3} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}{f}-\frac {6 b^{2} d^{3} e^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f^{4}}-\frac {2 i b^{2} d^{3} x^{3}}{f}+\frac {4 i b^{2} d^{3} e^{3}}{f^{4}}+\frac {a^{2} d^{3} x^{4}}{4}-\frac {b^{2} d^{3} x^{4}}{4}+\frac {3 i b a \,d^{3} e^{4}}{f^{4}}-\frac {6 i b^{2} c \,d^{2} x^{2}}{f}-\frac {6 i b^{2} c \,d^{2} e^{2}}{f^{3}}+\frac {6 i b^{2} d^{3} e^{2} x}{f^{3}}+3 i a b \,c^{2} d \,x^{2}+2 i a b c \,d^{2} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^3*(a+b*tan(f*x+e))^2,x)

[Out]

a^2*c^3*x-b^2*c^3*x+a^2*c*d^2*x^3+12*I/f*b*a*c^2*d*e*x+6*I/f^2*b*a*c*d^2*polylog(2,-exp(2*I*(f*x+e)))*x-12*I/f
^2*b*a*c*d^2*e^2*x+2*I*b^2*(d^3*x^3+3*c*d^2*x^2+3*c^2*d*x+c^3)/f/(exp(2*I*(f*x+e))+1)-b^2*c*d^2*x^3+3/2*a^2*c^
2*d*x^2-3/2*b^2*c^2*d*x^2-12/f^2*b*a*c^2*d*e*ln(exp(I*(f*x+e)))+12/f^3*b*a*c*d^2*e^2*ln(exp(I*(f*x+e)))-12*I/f
^2*b^2*c*d^2*e*x+4*I/f^3*b*a*d^3*e^3*x-8*I/f^3*b*a*c*d^2*e^3+3*I/f^2*b*a*c^2*d*polylog(2,-exp(2*I*(f*x+e)))+3*
I/f^2*b*a*d^3*polylog(2,-exp(2*I*(f*x+e)))*x^2+6*I/f^2*b*a*c^2*d*e^2-6/f*b*a*c*d^2*ln(exp(2*I*(f*x+e))+1)*x^2-
6/f*b*ln(exp(2*I*(f*x+e))+1)*a*c^2*d*x-2/f*b*a*d^3*ln(exp(2*I*(f*x+e))+1)*x^3-2*I*a*b*c^3*x+1/2*I*a*b*d^3*x^4+
1/4*a^2*d^3*x^4-1/4*b^2*d^3*x^4-3/f^3*b*a*d^3*polylog(3,-exp(2*I*(f*x+e)))*x+6/f^2*b^2*c*d^2*ln(exp(2*I*(f*x+e
))+1)*x+12/f^3*b^2*c*d^2*e*ln(exp(I*(f*x+e)))-4/f^4*b*a*d^3*e^3*ln(exp(I*(f*x+e)))-3/f^3*b*a*c*d^2*polylog(3,-
exp(2*I*(f*x+e)))+3/f^2*b^2*d^3*ln(exp(2*I*(f*x+e))+1)*x^2+4/f*b*a*c^3*ln(exp(I*(f*x+e)))-6/f^2*b^2*c^2*d*ln(e
xp(I*(f*x+e)))+3/f^2*b^2*c^2*d*ln(exp(2*I*(f*x+e))+1)-2/f*b*a*c^3*ln(exp(2*I*(f*x+e))+1)-6/f^4*b^2*d^3*e^2*ln(
exp(I*(f*x+e)))-2*I/f*b^2*d^3*x^3+4*I/f^4*b^2*d^3*e^3-3*I/f^3*b^2*c*d^2*polylog(2,-exp(2*I*(f*x+e)))-3*I/f^3*b
^2*d^3*polylog(2,-exp(2*I*(f*x+e)))*x+3*I/f^4*b*a*d^3*e^4-6*I/f*b^2*c*d^2*x^2-6*I/f^3*b^2*c*d^2*e^2+6*I/f^3*b^
2*d^3*e^2*x+3/2*b^2*d^3*polylog(3,-exp(2*I*(f*x+e)))/f^4+3*I*a*b*c^2*d*x^2+2*I*a*b*c*d^2*x^3-3/2*I*a*b*d^3*pol
ylog(4,-exp(2*I*(f*x+e)))/f^4

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maxima [B]  time = 2.79, size = 2521, normalized size = 8.40 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(4*(f*x + e)*a^2*c^3 + (f*x + e)^4*a^2*d^3/f^3 - 4*(f*x + e)^3*a^2*d^3*e/f^3 + 6*(f*x + e)^2*a^2*d^3*e^2/f
^3 - 4*(f*x + e)*a^2*d^3*e^3/f^3 + 4*(f*x + e)^3*a^2*c*d^2/f^2 - 12*(f*x + e)^2*a^2*c*d^2*e/f^2 + 12*(f*x + e)
*a^2*c*d^2*e^2/f^2 + 6*(f*x + e)^2*a^2*c^2*d/f - 12*(f*x + e)*a^2*c^2*d*e/f + 8*a*b*c^3*log(sec(f*x + e)) - 8*
a*b*d^3*e^3*log(sec(f*x + e))/f^3 + 24*a*b*c*d^2*e^2*log(sec(f*x + e))/f^2 - 24*a*b*c^2*d*e*log(sec(f*x + e))/
f + 4*(3*(2*a*b + I*b^2)*(f*x + e)^4*d^3 - 24*b^2*d^3*e^3 + 72*b^2*c*d^2*e^2*f - 72*b^2*c^2*d*e*f^2 + 24*b^2*c
^3*f^3 - 12*((2*a*b + I*b^2)*d^3*e - (2*a*b + I*b^2)*c*d^2*f)*(f*x + e)^3 + 18*((2*a*b + I*b^2)*d^3*e^2 - 2*(2
*a*b + I*b^2)*c*d^2*e*f + (2*a*b + I*b^2)*c^2*d*f^2)*(f*x + e)^2 - (12*I*b^2*d^3*e^3 - 36*I*b^2*c*d^2*e^2*f +
36*I*b^2*c^2*d*e*f^2 - 12*I*b^2*c^3*f^3)*(f*x + e) - (32*(f*x + e)^3*a*b*d^3 - 36*b^2*d^3*e^2 + 72*b^2*c*d^2*e
*f - 36*b^2*c^2*d*f^2 - 36*(2*a*b*d^3*e - 2*a*b*c*d^2*f + b^2*d^3)*(f*x + e)^2 + 72*(a*b*d^3*e^2 + a*b*c^2*d*f
^2 + b^2*d^3*e - (2*a*b*c*d^2*e + b^2*c*d^2)*f)*(f*x + e) + 4*(8*(f*x + e)^3*a*b*d^3 - 9*b^2*d^3*e^2 + 18*b^2*
c*d^2*e*f - 9*b^2*c^2*d*f^2 - 9*(2*a*b*d^3*e - 2*a*b*c*d^2*f + b^2*d^3)*(f*x + e)^2 + 18*(a*b*d^3*e^2 + a*b*c^
2*d*f^2 + b^2*d^3*e - (2*a*b*c*d^2*e + b^2*c*d^2)*f)*(f*x + e))*cos(2*f*x + 2*e) + (32*I*(f*x + e)^3*a*b*d^3 -
36*I*b^2*d^3*e^2 + 72*I*b^2*c*d^2*e*f - 36*I*b^2*c^2*d*f^2 + (-72*I*a*b*d^3*e + 72*I*a*b*c*d^2*f - 36*I*b^2*d
^3)*(f*x + e)^2 + (72*I*a*b*d^3*e^2 + 72*I*a*b*c^2*d*f^2 + 72*I*b^2*d^3*e + (-144*I*a*b*c*d^2*e - 72*I*b^2*c*d
^2)*f)*(f*x + e))*sin(2*f*x + 2*e))*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e) + 1) + (3*(2*a*b + I*b^2)*(f*x
+ e)^4*d^3 - 12*(2*b^2*d^3 + (2*a*b + I*b^2)*d^3*e - (2*a*b + I*b^2)*c*d^2*f)*(f*x + e)^3 + 18*(4*b^2*d^3*e +
(2*a*b + I*b^2)*d^3*e^2 + (2*a*b + I*b^2)*c^2*d*f^2 - 2*(2*b^2*c*d^2 + (2*a*b + I*b^2)*c*d^2*e)*f)*(f*x + e)^2
- (12*I*b^2*d^3*e^3 - 12*I*b^2*c^3*f^3 + 72*b^2*d^3*e^2 - 36*(-I*b^2*c^2*d*e - 2*b^2*c^2*d)*f^2 + (-36*I*b^2*
c*d^2*e^2 - 144*b^2*c*d^2*e)*f)*(f*x + e))*cos(2*f*x + 2*e) + (48*(f*x + e)^2*a*b*d^3 + 36*a*b*d^3*e^2 + 36*a*
b*c^2*d*f^2 + 36*b^2*d^3*e - 36*(2*a*b*d^3*e - 2*a*b*c*d^2*f + b^2*d^3)*(f*x + e) - 36*(2*a*b*c*d^2*e + b^2*c*
d^2)*f + 12*(4*(f*x + e)^2*a*b*d^3 + 3*a*b*d^3*e^2 + 3*a*b*c^2*d*f^2 + 3*b^2*d^3*e - 3*(2*a*b*d^3*e - 2*a*b*c*
d^2*f + b^2*d^3)*(f*x + e) - 3*(2*a*b*c*d^2*e + b^2*c*d^2)*f)*cos(2*f*x + 2*e) - (-48*I*(f*x + e)^2*a*b*d^3 -
36*I*a*b*d^3*e^2 - 36*I*a*b*c^2*d*f^2 - 36*I*b^2*d^3*e + (72*I*a*b*d^3*e - 72*I*a*b*c*d^2*f + 36*I*b^2*d^3)*(f
*x + e) + (72*I*a*b*c*d^2*e + 36*I*b^2*c*d^2)*f)*sin(2*f*x + 2*e))*dilog(-e^(2*I*f*x + 2*I*e)) - (-16*I*(f*x +
e)^3*a*b*d^3 + 18*I*b^2*d^3*e^2 - 36*I*b^2*c*d^2*e*f + 18*I*b^2*c^2*d*f^2 + (36*I*a*b*d^3*e - 36*I*a*b*c*d^2*
f + 18*I*b^2*d^3)*(f*x + e)^2 + (-36*I*a*b*d^3*e^2 - 36*I*a*b*c^2*d*f^2 - 36*I*b^2*d^3*e + (72*I*a*b*c*d^2*e +
36*I*b^2*c*d^2)*f)*(f*x + e) + (-16*I*(f*x + e)^3*a*b*d^3 + 18*I*b^2*d^3*e^2 - 36*I*b^2*c*d^2*e*f + 18*I*b^2*
c^2*d*f^2 + (36*I*a*b*d^3*e - 36*I*a*b*c*d^2*f + 18*I*b^2*d^3)*(f*x + e)^2 + (-36*I*a*b*d^3*e^2 - 36*I*a*b*c^2
*d*f^2 - 36*I*b^2*d^3*e + (72*I*a*b*c*d^2*e + 36*I*b^2*c*d^2)*f)*(f*x + e))*cos(2*f*x + 2*e) + 2*(8*(f*x + e)^
3*a*b*d^3 - 9*b^2*d^3*e^2 + 18*b^2*c*d^2*e*f - 9*b^2*c^2*d*f^2 - 9*(2*a*b*d^3*e - 2*a*b*c*d^2*f + b^2*d^3)*(f*
x + e)^2 + 18*(a*b*d^3*e^2 + a*b*c^2*d*f^2 + b^2*d^3*e - (2*a*b*c*d^2*e + b^2*c*d^2)*f)*(f*x + e))*sin(2*f*x +
2*e))*log(cos(2*f*x + 2*e)^2 + sin(2*f*x + 2*e)^2 + 2*cos(2*f*x + 2*e) + 1) - (24*a*b*d^3*cos(2*f*x + 2*e) +
24*I*a*b*d^3*sin(2*f*x + 2*e) + 24*a*b*d^3)*polylog(4, -e^(2*I*f*x + 2*I*e)) - (-48*I*(f*x + e)*a*b*d^3 + 36*I
*a*b*d^3*e - 36*I*a*b*c*d^2*f + 18*I*b^2*d^3 + (-48*I*(f*x + e)*a*b*d^3 + 36*I*a*b*d^3*e - 36*I*a*b*c*d^2*f +
18*I*b^2*d^3)*cos(2*f*x + 2*e) + 6*(8*(f*x + e)*a*b*d^3 - 6*a*b*d^3*e + 6*a*b*c*d^2*f - 3*b^2*d^3)*sin(2*f*x +
2*e))*polylog(3, -e^(2*I*f*x + 2*I*e)) - ((-6*I*a*b + 3*b^2)*(f*x + e)^4*d^3 + (24*I*b^2*d^3 + (24*I*a*b - 12
*b^2)*d^3*e + (-24*I*a*b + 12*b^2)*c*d^2*f)*(f*x + e)^3 + (-72*I*b^2*d^3*e + (-36*I*a*b + 18*b^2)*d^3*e^2 + (-
36*I*a*b + 18*b^2)*c^2*d*f^2 + (72*I*b^2*c*d^2 + (72*I*a*b - 36*b^2)*c*d^2*e)*f)*(f*x + e)^2 - (12*b^2*d^3*e^3
- 12*b^2*c^3*f^3 - 72*I*b^2*d^3*e^2 + (36*b^2*c^2*d*e - 72*I*b^2*c^2*d)*f^2 - 36*(b^2*c*d^2*e^2 - 4*I*b^2*c*d
^2*e)*f)*(f*x + e))*sin(2*f*x + 2*e))/(-12*I*f^3*cos(2*f*x + 2*e) + 12*f^3*sin(2*f*x + 2*e) - 12*I*f^3))/f

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^2\,{\left (c+d\,x\right )}^3 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*tan(e + f*x))^2*(c + d*x)^3,x)

[Out]

int((a + b*tan(e + f*x))^2*(c + d*x)^3, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \tan {\left (e + f x \right )}\right )^{2} \left (c + d x\right )^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*tan(e + f*x))**2*(c + d*x)**3, x)

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