3.1.51 \(\int (c+d x) (a+b \tan (e+f x))^3 \, dx\) [51]

Optimal. Leaf size=277 \[ -3 a b^2 c x+\frac {b^3 d x}{2 f}-\frac {3}{2} a b^2 d x^2+\frac {a^3 (c+d x)^2}{2 d}+\frac {3 i a^2 b (c+d x)^2}{2 d}-\frac {i b^3 (c+d x)^2}{2 d}-\frac {3 a^2 b (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac {b^3 (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac {3 a b^2 d \log (\cos (e+f x))}{f^2}+\frac {3 i a^2 b d \text {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{2 f^2}-\frac {i b^3 d \text {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{2 f^2}-\frac {b^3 d \tan (e+f x)}{2 f^2}+\frac {3 a b^2 (c+d x) \tan (e+f x)}{f}+\frac {b^3 (c+d x) \tan ^2(e+f x)}{2 f} \]

[Out]

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

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Rubi [A]
time = 0.23, antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 9, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3803, 3800, 2221, 2317, 2438, 3801, 3556, 3554, 8} \begin {gather*} \frac {a^3 (c+d x)^2}{2 d}-\frac {3 a^2 b (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac {3 i a^2 b (c+d x)^2}{2 d}+\frac {3 i a^2 b d \text {Li}_2\left (-e^{2 i (e+f x)}\right )}{2 f^2}+\frac {3 a b^2 (c+d x) \tan (e+f x)}{f}-3 a b^2 c x+\frac {3 a b^2 d \log (\cos (e+f x))}{f^2}-\frac {3}{2} a b^2 d x^2+\frac {b^3 (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac {b^3 (c+d x) \tan ^2(e+f x)}{2 f}-\frac {i b^3 (c+d x)^2}{2 d}-\frac {i b^3 d \text {Li}_2\left (-e^{2 i (e+f x)}\right )}{2 f^2}-\frac {b^3 d \tan (e+f x)}{2 f^2}+\frac {b^3 d x}{2 f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

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 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 3554

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d*x])^(n - 1)/(d*(n - 1))), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

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 3803

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]

Rubi steps

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

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Mathematica [A]
time = 3.51, size = 277, normalized size = 1.00 \begin {gather*} \frac {\cos (e+f x) \left (\cos ^2(e+f x) \left (-\left ((e+f x) \left (-3 i a^2 b d (e+f x)+i b^3 d (e+f x)+3 a b^2 (-d e+2 c f+d f x)+a^3 (-2 c f+d (e-f x))\right )\right )+2 b \left (-3 a^2+b^2\right ) d (e+f x) \log \left (1+e^{2 i (e+f x)}\right )+2 b \left (3 a b d+3 a^2 (d e-c f)+b^2 (-d e+c f)\right ) \log (\cos (e+f x))\right )-i b \left (-3 a^2+b^2\right ) d \cos ^2(e+f x) \text {PolyLog}\left (2,-e^{2 i (e+f x)}\right )+\frac {1}{2} b^2 (2 b f (c+d x)+(-b d+6 a f (c+d x)) \sin (2 (e+f x)))\right ) (a+b \tan (e+f x))^3}{2 f^2 (a \cos (e+f x)+b \sin (e+f x))^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.26, size = 493, normalized size = 1.78

method	result	size
risch	\(-\frac {i b^{3} d \polylog \left (2, -{\mathrm e}^{2 i \left (f x +e \right )}\right )}{2 f^{2}}+\frac {3 i a^{2} b d \,x^{2}}{2}-3 i a^{2} b c x +\frac {a^{3} d \,x^{2}}{2}+a^{3} c x -\frac {3 b \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) a^{2} d x}{f}-\frac {6 b \,a^{2} d e \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f^{2}}-\frac {2 i b^{3} d e x}{f}+\frac {3 i b \,a^{2} d \,e^{2}}{f^{2}}+\frac {3 i a^{2} b d \polylog \left (2, -{\mathrm e}^{2 i \left (f x +e \right )}\right )}{2 f^{2}}+\frac {6 i b \,a^{2} d e x}{f}+\frac {b^{3} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) d x}{f}+\frac {3 b^{2} a d \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}{f^{2}}-\frac {6 b^{2} a d \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f^{2}}-\frac {3 b \,a^{2} c \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}{f}+\frac {6 b \,a^{2} c \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f}+\frac {2 b^{3} d e \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f^{2}}-\frac {i b^{3} d \,e^{2}}{f^{2}}+\frac {b^{3} c \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}{f}-\frac {2 b^{3} c \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f}-\frac {i b^{3} d \,x^{2}}{2}+\frac {b^{2} \left (6 i a d f x \,{\mathrm e}^{2 i \left (f x +e \right )}+6 i a c f \,{\mathrm e}^{2 i \left (f x +e \right )}+2 b d f x \,{\mathrm e}^{2 i \left (f x +e \right )}+6 i a d f x -i b d \,{\mathrm e}^{2 i \left (f x +e \right )}+2 b c f \,{\mathrm e}^{2 i \left (f x +e \right )}+6 i a c f -i d b \right )}{f^{2} \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2}}+i b^{3} c x -3 a \,b^{2} c x -\frac {3 a \,b^{2} d \,x^{2}}{2}\)	\(493\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1384 vs. \(2 (249) = 498\).
time = 0.88, size = 1384, normalized size = 5.00 \begin {gather*} \frac {2 \, {\left (f x + e\right )} a^{3} c + \frac {{\left (f x + e\right )}^{2} a^{3} d}{f} - \frac {2 \, {\left (f x + e\right )} a^{3} d e}{f} + 6 \, a^{2} b c \log \left (\sec \left (f x + e\right )\right ) - \frac {6 \, a^{2} b d e \log \left (\sec \left (f x + e\right )\right )}{f} + \frac {2 \, {\left (12 \, a b^{2} c f + {\left (3 \, a^{2} b + 3 i \, a b^{2} - b^{3}\right )} {\left (f x + e\right )}^{2} d - 2 \, {\left ({\left (-3 i \, a b^{2} + b^{3}\right )} c f + {\left (3 i \, a b^{2} e - b^{3} e\right )} d\right )} {\left (f x + e\right )} - 2 \, {\left (6 \, a b^{2} e + b^{3}\right )} d + 2 \, {\left (b^{3} c f - {\left (3 \, a^{2} b - b^{3}\right )} {\left (f x + e\right )} d - {\left (b^{3} e - 3 \, a b^{2}\right )} d + {\left (b^{3} c f - {\left (3 \, a^{2} b - b^{3}\right )} {\left (f x + e\right )} d - {\left (b^{3} e - 3 \, a b^{2}\right )} d\right )} \cos \left (4 \, f x + 4 \, e\right ) + 2 \, {\left (b^{3} c f - {\left (3 \, a^{2} b - b^{3}\right )} {\left (f x + e\right )} d - {\left (b^{3} e - 3 \, a b^{2}\right )} d\right )} \cos \left (2 \, f x + 2 \, e\right ) - {\left (-i \, b^{3} c f + {\left (3 i \, a^{2} b - i \, b^{3}\right )} {\left (f x + e\right )} d + {\left (i \, b^{3} e - 3 i \, a b^{2}\right )} d\right )} \sin \left (4 \, f x + 4 \, e\right ) - 2 \, {\left (-i \, b^{3} c f + {\left (3 i \, a^{2} b - i \, b^{3}\right )} {\left (f x + e\right )} d + {\left (i \, b^{3} e - 3 i \, a b^{2}\right )} d\right )} \sin \left (2 \, f x + 2 \, e\right )\right )} \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right ) + {\left ({\left (3 \, a^{2} b + 3 i \, a b^{2} - b^{3}\right )} {\left (f x + e\right )}^{2} d - 2 \, {\left ({\left (-3 i \, a b^{2} + b^{3}\right )} c f + {\left (3 \, a b^{2} {\left (i \, e + 2\right )} - b^{3} e\right )} d\right )} {\left (f x + e\right )}\right )} \cos \left (4 \, f x + 4 \, e\right ) + 2 \, {\left ({\left (3 \, a^{2} b + 3 i \, a b^{2} - b^{3}\right )} {\left (f x + e\right )}^{2} d + 2 \, {\left (3 \, a b^{2} - i \, b^{3}\right )} c f - 2 \, {\left ({\left (-3 i \, a b^{2} + b^{3}\right )} c f - {\left (b^{3} {\left (e - i\right )} - 3 \, a b^{2} {\left (i \, e + 1\right )}\right )} d\right )} {\left (f x + e\right )} - {\left (b^{3} {\left (-2 i \, e + 1\right )} + 6 \, a b^{2} e\right )} d\right )} \cos \left (2 \, f x + 2 \, e\right ) + {\left ({\left (3 \, a^{2} b - b^{3}\right )} d \cos \left (4 \, f x + 4 \, e\right ) + 2 \, {\left (3 \, a^{2} b - b^{3}\right )} d \cos \left (2 \, f x + 2 \, e\right ) + {\left (3 i \, a^{2} b - i \, b^{3}\right )} d \sin \left (4 \, f x + 4 \, e\right ) - 2 \, {\left (-3 i \, a^{2} b + i \, b^{3}\right )} d \sin \left (2 \, f x + 2 \, e\right ) + {\left (3 \, a^{2} b - b^{3}\right )} d\right )} {\rm Li}_2\left (-e^{\left (2 i \, f x + 2 i \, e\right )}\right ) + {\left (-i \, b^{3} c f + {\left (3 i \, a^{2} b - i \, b^{3}\right )} {\left (f x + e\right )} d + {\left (i \, b^{3} e - 3 i \, a b^{2}\right )} d + {\left (-i \, b^{3} c f + {\left (3 i \, a^{2} b - i \, b^{3}\right )} {\left (f x + e\right )} d + {\left (i \, b^{3} e - 3 i \, a b^{2}\right )} d\right )} \cos \left (4 \, f x + 4 \, e\right ) - 2 \, {\left (i \, b^{3} c f + {\left (-3 i \, a^{2} b + i \, b^{3}\right )} {\left (f x + e\right )} d + {\left (-i \, b^{3} e + 3 i \, a b^{2}\right )} d\right )} \cos \left (2 \, f x + 2 \, e\right ) + {\left (b^{3} c f - {\left (3 \, a^{2} b - b^{3}\right )} {\left (f x + e\right )} d - {\left (b^{3} e - 3 \, a b^{2}\right )} d\right )} \sin \left (4 \, f x + 4 \, e\right ) + 2 \, {\left (b^{3} c f - {\left (3 \, a^{2} b - b^{3}\right )} {\left (f x + e\right )} d - {\left (b^{3} e - 3 \, a b^{2}\right )} d\right )} \sin \left (2 \, f x + 2 \, e\right )\right )} \log \left (\cos \left (2 \, f x + 2 \, e\right )^{2} + \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right ) + {\left ({\left (3 i \, a^{2} b - 3 \, a b^{2} - i \, b^{3}\right )} {\left (f x + e\right )}^{2} d - 2 \, {\left ({\left (3 \, a b^{2} + i \, b^{3}\right )} c f - {\left (3 \, a b^{2} {\left (e - 2 i\right )} + i \, b^{3} e\right )} d\right )} {\left (f x + e\right )}\right )} \sin \left (4 \, f x + 4 \, e\right ) - 2 \, {\left ({\left (-3 i \, a^{2} b + 3 \, a b^{2} + i \, b^{3}\right )} {\left (f x + e\right )}^{2} d + 2 \, {\left (-3 i \, a b^{2} - b^{3}\right )} c f + 2 \, {\left ({\left (3 \, a b^{2} + i \, b^{3}\right )} c f - {\left (3 \, a b^{2} {\left (e - i\right )} - b^{3} {\left (-i \, e - 1\right )}\right )} d\right )} {\left (f x + e\right )} + {\left (b^{3} {\left (2 \, e + i\right )} + 6 i \, a b^{2} e\right )} d\right )} \sin \left (2 \, f x + 2 \, e\right )\right )}}{-2 i \, f \cos \left (4 \, f x + 4 \, e\right ) - 4 i \, f \cos \left (2 \, f x + 2 \, e\right ) + 2 \, f \sin \left (4 \, f x + 4 \, e\right ) + 4 \, f \sin \left (2 \, f x + 2 \, e\right ) - 2 i \, f}}{2 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*(f*x + e)*a^3*c + (f*x + e)^2*a^3*d/f - 2*(f*x + e)*a^3*d*e/f + 6*a^2*b*c*log(sec(f*x + e)) - 6*a^2*b*d
*e*log(sec(f*x + e))/f + 2*(12*a*b^2*c*f + (3*a^2*b + 3*I*a*b^2 - b^3)*(f*x + e)^2*d - 2*((-3*I*a*b^2 + b^3)*c
*f + (3*I*a*b^2*e - b^3*e)*d)*(f*x + e) - 2*(6*a*b^2*e + b^3)*d + 2*(b^3*c*f - (3*a^2*b - b^3)*(f*x + e)*d - (
b^3*e - 3*a*b^2)*d + (b^3*c*f - (3*a^2*b - b^3)*(f*x + e)*d - (b^3*e - 3*a*b^2)*d)*cos(4*f*x + 4*e) + 2*(b^3*c
*f - (3*a^2*b - b^3)*(f*x + e)*d - (b^3*e - 3*a*b^2)*d)*cos(2*f*x + 2*e) - (-I*b^3*c*f + (3*I*a^2*b - I*b^3)*(
f*x + e)*d + (I*b^3*e - 3*I*a*b^2)*d)*sin(4*f*x + 4*e) - 2*(-I*b^3*c*f + (3*I*a^2*b - I*b^3)*(f*x + e)*d + (I*
b^3*e - 3*I*a*b^2)*d)*sin(2*f*x + 2*e))*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e) + 1) + ((3*a^2*b + 3*I*a*b^
2 - b^3)*(f*x + e)^2*d - 2*((-3*I*a*b^2 + b^3)*c*f + (3*a*b^2*(I*e + 2) - b^3*e)*d)*(f*x + e))*cos(4*f*x + 4*e
) + 2*((3*a^2*b + 3*I*a*b^2 - b^3)*(f*x + e)^2*d + 2*(3*a*b^2 - I*b^3)*c*f - 2*((-3*I*a*b^2 + b^3)*c*f - (b^3*
(e - I) - 3*a*b^2*(I*e + 1))*d)*(f*x + e) - (b^3*(-2*I*e + 1) + 6*a*b^2*e)*d)*cos(2*f*x + 2*e) + ((3*a^2*b - b
^3)*d*cos(4*f*x + 4*e) + 2*(3*a^2*b - b^3)*d*cos(2*f*x + 2*e) + (3*I*a^2*b - I*b^3)*d*sin(4*f*x + 4*e) - 2*(-3
*I*a^2*b + I*b^3)*d*sin(2*f*x + 2*e) + (3*a^2*b - b^3)*d)*dilog(-e^(2*I*f*x + 2*I*e)) + (-I*b^3*c*f + (3*I*a^2
*b - I*b^3)*(f*x + e)*d + (I*b^3*e - 3*I*a*b^2)*d + (-I*b^3*c*f + (3*I*a^2*b - I*b^3)*(f*x + e)*d + (I*b^3*e -
3*I*a*b^2)*d)*cos(4*f*x + 4*e) - 2*(I*b^3*c*f + (-3*I*a^2*b + I*b^3)*(f*x + e)*d + (-I*b^3*e + 3*I*a*b^2)*d)*
cos(2*f*x + 2*e) + (b^3*c*f - (3*a^2*b - b^3)*(f*x + e)*d - (b^3*e - 3*a*b^2)*d)*sin(4*f*x + 4*e) + 2*(b^3*c*f
- (3*a^2*b - b^3)*(f*x + e)*d - (b^3*e - 3*a*b^2)*d)*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) + ((3*I*a^2*b - 3*a*b^2 - I*b^3)*(f*x + e)^2*d - 2*((3*a*b^2 + I*b^3)*c*f - (3
*a*b^2*(e - 2*I) + I*b^3*e)*d)*(f*x + e))*sin(4*f*x + 4*e) - 2*((-3*I*a^2*b + 3*a*b^2 + I*b^3)*(f*x + e)^2*d +
2*(-3*I*a*b^2 - b^3)*c*f + 2*((3*a*b^2 + I*b^3)*c*f - (3*a*b^2*(e - I) - b^3*(-I*e - 1))*d)*(f*x + e) + (b^3*
(2*e + I) + 6*I*a*b^2*e)*d)*sin(2*f*x + 2*e))/(-2*I*f*cos(4*f*x + 4*e) - 4*I*f*cos(2*f*x + 2*e) + 2*f*sin(4*f*
x + 4*e) + 4*f*sin(2*f*x + 2*e) - 2*I*f))/f

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Fricas [A]
time = 0.37, size = 337, normalized size = 1.22 \begin {gather*} \frac {2 \, {\left (a^{3} - 3 \, a b^{2}\right )} d f^{2} x^{2} - i \, {\left (3 \, a^{2} b - b^{3}\right )} d {\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 ) + i \, {\left (3 \, a^{2} b - b^{3}\right )} d {\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 (b^{3} d f x + b^{3} c f\right )} \tan \left (f x + e\right )^{2} + 2 \, {\left (b^{3} d f + 2 \, {\left (a^{3} - 3 \, a b^{2}\right )} c f^{2}\right )} x + 2 \, {\left (3 \, a b^{2} d - {\left (3 \, a^{2} b - b^{3}\right )} d f x - {\left (3 \, a^{2} b - b^{3}\right )} c f\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 (3 \, a b^{2} d - {\left (3 \, a^{2} b - b^{3}\right )} d f x - {\left (3 \, a^{2} b - b^{3}\right )} c f\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 (6 \, a b^{2} d f x + 6 \, a b^{2} c f - b^{3} d\right )} \tan \left (f x + e\right )}{4 \, f^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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