\(\int (c+d x) (a+b \tanh (e+f x))^3 \, dx\) [65]

Optimal result

Integrand size = 18, antiderivative size = 261 \[ \int (c+d x) (a+b \tanh (e+f x))^3 \, dx=3 a b^2 c x+\frac {b^3 d x}{2 f}+\frac {3}{2} a b^2 d x^2+\frac {a^3 (c+d x)^2}{2 d}-\frac {3 a^2 b (c+d x)^2}{2 d}-\frac {b^3 (c+d x)^2}{2 d}+\frac {3 a^2 b (c+d x) \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {b^3 (c+d x) \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {3 a b^2 d \log (\cosh (e+f x))}{f^2}+\frac {3 a^2 b d \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{2 f^2}+\frac {b^3 d \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{2 f^2}-\frac {b^3 d \tanh (e+f x)}{2 f^2}-\frac {3 a b^2 (c+d x) \tanh (e+f x)}{f}-\frac {b^3 (c+d x) \tanh ^2(e+f x)}{2 f} \]

[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*a^2*b*(d*x+c)^2/d-1/2*b^3*(d*x+c)^2/d+3*a^2*
b*(d*x+c)*ln(1+exp(2*f*x+2*e))/f+b^3*(d*x+c)*ln(1+exp(2*f*x+2*e))/f+3*a*b^2*d*ln(cosh(f*x+e))/f^2+3/2*a^2*b*d*
polylog(2,-exp(2*f*x+2*e))/f^2+1/2*b^3*d*polylog(2,-exp(2*f*x+2*e))/f^2-1/2*b^3*d*tanh(f*x+e)/f^2-3*a*b^2*(d*x
+c)*tanh(f*x+e)/f-1/2*b^3*(d*x+c)*tanh(f*x+e)^2/f

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3803, 3799, 2221, 2317, 2438, 3801, 3556, 3554, 8} \[ \int (c+d x) (a+b \tanh (e+f x))^3 \, dx=\frac {a^3 (c+d x)^2}{2 d}+\frac {3 a^2 b (c+d x) \log \left (e^{2 (e+f x)}+1\right )}{f}-\frac {3 a^2 b (c+d x)^2}{2 d}+\frac {3 a^2 b d \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{2 f^2}-\frac {3 a b^2 (c+d x) \tanh (e+f x)}{f}+3 a b^2 c x+\frac {3 a b^2 d \log (\cosh (e+f x))}{f^2}+\frac {3}{2} a b^2 d x^2+\frac {b^3 (c+d x) \log \left (e^{2 (e+f x)}+1\right )}{f}-\frac {b^3 (c+d x) \tanh ^2(e+f x)}{2 f}-\frac {b^3 (c+d x)^2}{2 d}+\frac {b^3 d \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{2 f^2}-\frac {b^3 d \tanh (e+f x)}{2 f^2}+\frac {b^3 d x}{2 f} \]

[In]

Int[(c + d*x)*(a + b*Tanh[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*a^2*b*(c + d*x)^2)/(2*d) - (b
^3*(c + d*x)^2)/(2*d) + (3*a^2*b*(c + d*x)*Log[1 + E^(2*(e + f*x))])/f + (b^3*(c + d*x)*Log[1 + E^(2*(e + f*x)
)])/f + (3*a*b^2*d*Log[Cosh[e + f*x]])/f^2 + (3*a^2*b*d*PolyLog[2, -E^(2*(e + f*x))])/(2*f^2) + (b^3*d*PolyLog
[2, -E^(2*(e + f*x))])/(2*f^2) - (b^3*d*Tanh[e + f*x])/(2*f^2) - (3*a*b^2*(c + d*x)*Tanh[e + f*x])/f - (b^3*(c
+ d*x)*Tanh[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 3799

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(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*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x]
, x] /; FreeQ[{c, d, e, f, fz}, 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*} \text {integral}& = \int \left (a^3 (c+d x)+3 a^2 b (c+d x) \tanh (e+f x)+3 a b^2 (c+d x) \tanh ^2(e+f x)+b^3 (c+d x) \tanh ^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) \tanh (e+f x) \, dx+\left (3 a b^2\right ) \int (c+d x) \tanh ^2(e+f x) \, dx+b^3 \int (c+d x) \tanh ^3(e+f x) \, dx \\ & = \frac {a^3 (c+d x)^2}{2 d}-\frac {3 a^2 b (c+d x)^2}{2 d}-\frac {3 a b^2 (c+d x) \tanh (e+f x)}{f}-\frac {b^3 (c+d x) \tanh ^2(e+f x)}{2 f}+\left (6 a^2 b\right ) \int \frac {e^{2 (e+f x)} (c+d x)}{1+e^{2 (e+f x)}} \, dx+\left (3 a b^2\right ) \int (c+d x) \, dx+b^3 \int (c+d x) \tanh (e+f x) \, dx+\frac {\left (3 a b^2 d\right ) \int \tanh (e+f x) \, dx}{f}+\frac {\left (b^3 d\right ) \int \tanh ^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 a^2 b (c+d x)^2}{2 d}-\frac {b^3 (c+d x)^2}{2 d}+\frac {3 a^2 b (c+d x) \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {3 a b^2 d \log (\cosh (e+f x))}{f^2}-\frac {b^3 d \tanh (e+f x)}{2 f^2}-\frac {3 a b^2 (c+d x) \tanh (e+f x)}{f}-\frac {b^3 (c+d x) \tanh ^2(e+f x)}{2 f}+\left (2 b^3\right ) \int \frac {e^{2 (e+f x)} (c+d x)}{1+e^{2 (e+f x)}} \, dx-\frac {\left (3 a^2 b d\right ) \int \log \left (1+e^{2 (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 a^2 b (c+d x)^2}{2 d}-\frac {b^3 (c+d x)^2}{2 d}+\frac {3 a^2 b (c+d x) \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {b^3 (c+d x) \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {3 a b^2 d \log (\cosh (e+f x))}{f^2}-\frac {b^3 d \tanh (e+f x)}{2 f^2}-\frac {3 a b^2 (c+d x) \tanh (e+f x)}{f}-\frac {b^3 (c+d x) \tanh ^2(e+f x)}{2 f}-\frac {\left (3 a^2 b d\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{2 f^2}-\frac {\left (b^3 d\right ) \int \log \left (1+e^{2 (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 a^2 b (c+d x)^2}{2 d}-\frac {b^3 (c+d x)^2}{2 d}+\frac {3 a^2 b (c+d x) \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {b^3 (c+d x) \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {3 a b^2 d \log (\cosh (e+f x))}{f^2}+\frac {3 a^2 b d \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{2 f^2}-\frac {b^3 d \tanh (e+f x)}{2 f^2}-\frac {3 a b^2 (c+d x) \tanh (e+f x)}{f}-\frac {b^3 (c+d x) \tanh ^2(e+f x)}{2 f}-\frac {\left (b^3 d\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 (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 a^2 b (c+d x)^2}{2 d}-\frac {b^3 (c+d x)^2}{2 d}+\frac {3 a^2 b (c+d x) \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {b^3 (c+d x) \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {3 a b^2 d \log (\cosh (e+f x))}{f^2}+\frac {3 a^2 b d \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{2 f^2}+\frac {b^3 d \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{2 f^2}-\frac {b^3 d \tanh (e+f x)}{2 f^2}-\frac {3 a b^2 (c+d x) \tanh (e+f x)}{f}-\frac {b^3 (c+d x) \tanh ^2(e+f x)}{2 f} \\ \end{align*}

Mathematica [A] (verified)

Time = 7.47 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.16 \[ \int (c+d x) (a+b \tanh (e+f x))^3 \, dx=\frac {\cosh (e+f x) \left (b^3 f (c+d x)-a \left (a^2+3 b^2\right ) (e+f x) (-2 c f+d (e-f x)) \cosh ^2(e+f x)+b \cosh ^2(e+f x) \left (\frac {3 a^2 f^2 (c+d x)^2}{d}+\frac {b^2 f^2 (c+d x)^2}{d}-6 a b d (e+f x)+4 \left (3 a^2+b^2\right ) (d e-c f) (e+f x)+2 \left (3 a^2+b^2\right ) d (e+f x) \log \left (1+e^{-2 (e+f x)}\right )+6 a b d \log \left (1+e^{2 (e+f x)}\right )-2 \left (3 a^2+b^2\right ) (d e-c f) \log \left (1+e^{2 (e+f x)}\right )-\left (3 a^2+b^2\right ) d \operatorname {PolyLog}\left (2,-e^{-2 (e+f x)}\right )\right )-\frac {1}{2} b^2 (b d+6 a f (c+d x)) \sinh (2 (e+f x))\right ) (a+b \tanh (e+f x))^3}{2 f^2 (a \cosh (e+f x)+b \sinh (e+f x))^3} \]

[In]

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

[Out]

(Cosh[e + f*x]*(b^3*f*(c + d*x) - a*(a^2 + 3*b^2)*(e + f*x)*(-2*c*f + d*(e - f*x))*Cosh[e + f*x]^2 + b*Cosh[e
+ f*x]^2*((3*a^2*f^2*(c + d*x)^2)/d + (b^2*f^2*(c + d*x)^2)/d - 6*a*b*d*(e + f*x) + 4*(3*a^2 + b^2)*(d*e - c*f
)*(e + f*x) + 2*(3*a^2 + b^2)*d*(e + f*x)*Log[1 + E^(-2*(e + f*x))] + 6*a*b*d*Log[1 + E^(2*(e + f*x))] - 2*(3*
a^2 + b^2)*(d*e - c*f)*Log[1 + E^(2*(e + f*x))] - (3*a^2 + b^2)*d*PolyLog[2, -E^(-2*(e + f*x))]) - (b^2*(b*d +
6*a*f*(c + d*x))*Sinh[2*(e + f*x)])/2)*(a + b*Tanh[e + f*x])^3)/(2*f^2*(a*Cosh[e + f*x] + b*Sinh[e + f*x])^3)

Maple [A] (verified)

Time = 0.35 (sec) , antiderivative size = 459, normalized size of antiderivative = 1.76

method	result	size
risch	\(-\frac {3 a^{2} b d \,x^{2}}{2}+3 a^{2} b c x -\frac {6 b \,a^{2} d e x}{f}+\frac {6 b e \,a^{2} d \ln \left ({\mathrm e}^{f x +e}\right )}{f^{2}}-\frac {b^{3} d \,e^{2}}{f^{2}}+\frac {b^{3} c \ln \left (1+{\mathrm e}^{2 f x +2 e}\right )}{f}-\frac {2 b^{3} c \ln \left ({\mathrm e}^{f x +e}\right )}{f}+\frac {b^{2} \left (6 a d f x \,{\mathrm e}^{2 f x +2 e}+2 b d f x \,{\mathrm e}^{2 f x +2 e}+6 a c f \,{\mathrm e}^{2 f x +2 e}+2 b c f \,{\mathrm e}^{2 f x +2 e}+6 a d f x +{\mathrm e}^{2 f x +2 e} d b +6 a c f +b d \right )}{f^{2} \left (1+{\mathrm e}^{2 f x +2 e}\right )^{2}}+\frac {3 a^{2} b d \operatorname {polylog}\left (2, -{\mathrm e}^{2 f x +2 e}\right )}{2 f^{2}}+\frac {b^{3} d \operatorname {polylog}\left (2, -{\mathrm e}^{2 f x +2 e}\right )}{2 f^{2}}+3 a \,b^{2} c x +\frac {3 a \,b^{2} d \,x^{2}}{2}-\frac {2 b^{3} d e x}{f}-\frac {3 b \,a^{2} d \,e^{2}}{f^{2}}+\frac {b^{3} d \ln \left (1+{\mathrm e}^{2 f x +2 e}\right ) x}{f}+\frac {3 b \,a^{2} c \ln \left (1+{\mathrm e}^{2 f x +2 e}\right )}{f}-\frac {6 b \,a^{2} c \ln \left ({\mathrm e}^{f x +e}\right )}{f}+\frac {2 b^{3} e d \ln \left ({\mathrm e}^{f x +e}\right )}{f^{2}}+\frac {3 b^{2} d a \ln \left (1+{\mathrm e}^{2 f x +2 e}\right )}{f^{2}}-\frac {6 b^{2} d a \ln \left ({\mathrm e}^{f x +e}\right )}{f^{2}}+\frac {a^{3} d \,x^{2}}{2}-\frac {b^{3} d \,x^{2}}{2}+a^{3} c x +b^{3} c x +\frac {3 b \,a^{2} d \ln \left (1+{\mathrm e}^{2 f x +2 e}\right ) x}{f}\)	\(459\)

[In]

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

[Out]

-3/2*a^2*b*d*x^2+3*a^2*b*c*x-6/f*b*a^2*d*e*x+6/f^2*b*e*a^2*d*ln(exp(f*x+e))-1/f^2*b^3*d*e^2+1/f*b^3*c*ln(1+exp
(2*f*x+2*e))-2/f*b^3*c*ln(exp(f*x+e))+b^2*(6*a*d*f*x*exp(2*f*x+2*e)+2*b*d*f*x*exp(2*f*x+2*e)+6*a*c*f*exp(2*f*x
+2*e)+2*b*c*f*exp(2*f*x+2*e)+6*a*d*f*x+exp(2*f*x+2*e)*d*b+6*a*c*f+b*d)/f^2/(1+exp(2*f*x+2*e))^2+3/2*a^2*b*d*po
lylog(2,-exp(2*f*x+2*e))/f^2+1/2*b^3*d*polylog(2,-exp(2*f*x+2*e))/f^2+3*a*b^2*c*x+3/2*a*b^2*d*x^2-2/f*b^3*d*e*
x-3/f^2*b*a^2*d*e^2+1/f*b^3*d*ln(1+exp(2*f*x+2*e))*x+3/f*b*a^2*c*ln(1+exp(2*f*x+2*e))-6/f*b*a^2*c*ln(exp(f*x+e
))+2/f^2*b^3*e*d*ln(exp(f*x+e))+3/f^2*b^2*d*a*ln(1+exp(2*f*x+2*e))-6/f^2*b^2*d*a*ln(exp(f*x+e))+1/2*a^3*d*x^2-
1/2*b^3*d*x^2+a^3*c*x+b^3*c*x+3/f*b*a^2*d*ln(1+exp(2*f*x+2*e))*x

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.31 (sec) , antiderivative size = 3262, normalized size of antiderivative = 12.50 \[ \int (c+d x) (a+b \tanh (e+f x))^3 \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/2*((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*f^2*x^2 - 12*a*b^2*d*e + 2*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c*f^2*x + ((
a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*f^2*x^2 - 12*a*b^2*d*e + 2*(3*a^2*b + b^3)*d*e^2 - 4*(3*a^2*b + b^3)*c*e*f -
2*(6*a*b^2*d*f - (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c*f^2)*x)*cosh(f*x + e)^4 + 4*((a^3 - 3*a^2*b + 3*a*b^2 - b^3
)*d*f^2*x^2 - 12*a*b^2*d*e + 2*(3*a^2*b + b^3)*d*e^2 - 4*(3*a^2*b + b^3)*c*e*f - 2*(6*a*b^2*d*f - (a^3 - 3*a^2
*b + 3*a*b^2 - b^3)*c*f^2)*x)*cosh(f*x + e)*sinh(f*x + e)^3 + ((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*f^2*x^2 - 12*
a*b^2*d*e + 2*(3*a^2*b + b^3)*d*e^2 - 4*(3*a^2*b + b^3)*c*e*f - 2*(6*a*b^2*d*f - (a^3 - 3*a^2*b + 3*a*b^2 - b^
3)*c*f^2)*x)*sinh(f*x + e)^4 + 2*b^3*d + 2*(3*a^2*b + b^3)*d*e^2 + 2*((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*f^2*x^
2 - 12*a*b^2*d*e + b^3*d + 2*(3*a^2*b + b^3)*d*e^2 - 2*(2*(3*a^2*b + b^3)*c*e - (3*a*b^2 + b^3)*c)*f + 2*((a^3
- 3*a^2*b + 3*a*b^2 - b^3)*c*f^2 - (3*a*b^2 - b^3)*d*f)*x)*cosh(f*x + e)^2 + 2*((a^3 - 3*a^2*b + 3*a*b^2 - b^
3)*d*f^2*x^2 - 12*a*b^2*d*e + b^3*d + 2*(3*a^2*b + b^3)*d*e^2 + 3*((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*f^2*x^2 -
12*a*b^2*d*e + 2*(3*a^2*b + b^3)*d*e^2 - 4*(3*a^2*b + b^3)*c*e*f - 2*(6*a*b^2*d*f - (a^3 - 3*a^2*b + 3*a*b^2
- b^3)*c*f^2)*x)*cosh(f*x + e)^2 - 2*(2*(3*a^2*b + b^3)*c*e - (3*a*b^2 + b^3)*c)*f + 2*((a^3 - 3*a^2*b + 3*a*b
^2 - b^3)*c*f^2 - (3*a*b^2 - b^3)*d*f)*x)*sinh(f*x + e)^2 + 4*(3*a*b^2*c - (3*a^2*b + b^3)*c*e)*f + 2*((3*a^2*
b + b^3)*d*cosh(f*x + e)^4 + 4*(3*a^2*b + b^3)*d*cosh(f*x + e)*sinh(f*x + e)^3 + (3*a^2*b + b^3)*d*sinh(f*x +
e)^4 + 2*(3*a^2*b + b^3)*d*cosh(f*x + e)^2 + 2*(3*(3*a^2*b + b^3)*d*cosh(f*x + e)^2 + (3*a^2*b + b^3)*d)*sinh(
f*x + e)^2 + (3*a^2*b + b^3)*d + 4*((3*a^2*b + b^3)*d*cosh(f*x + e)^3 + (3*a^2*b + b^3)*d*cosh(f*x + e))*sinh(
f*x + e))*dilog(I*cosh(f*x + e) + I*sinh(f*x + e)) + 2*((3*a^2*b + b^3)*d*cosh(f*x + e)^4 + 4*(3*a^2*b + b^3)*
d*cosh(f*x + e)*sinh(f*x + e)^3 + (3*a^2*b + b^3)*d*sinh(f*x + e)^4 + 2*(3*a^2*b + b^3)*d*cosh(f*x + e)^2 + 2*
(3*(3*a^2*b + b^3)*d*cosh(f*x + e)^2 + (3*a^2*b + b^3)*d)*sinh(f*x + e)^2 + (3*a^2*b + b^3)*d + 4*((3*a^2*b +
b^3)*d*cosh(f*x + e)^3 + (3*a^2*b + b^3)*d*cosh(f*x + e))*sinh(f*x + e))*dilog(-I*cosh(f*x + e) - I*sinh(f*x +
e)) + 2*((3*a*b^2*d - (3*a^2*b + b^3)*d*e + (3*a^2*b + b^3)*c*f)*cosh(f*x + e)^4 + 4*(3*a*b^2*d - (3*a^2*b +
b^3)*d*e + (3*a^2*b + b^3)*c*f)*cosh(f*x + e)*sinh(f*x + e)^3 + (3*a*b^2*d - (3*a^2*b + b^3)*d*e + (3*a^2*b +
b^3)*c*f)*sinh(f*x + e)^4 + 3*a*b^2*d - (3*a^2*b + b^3)*d*e + (3*a^2*b + b^3)*c*f + 2*(3*a*b^2*d - (3*a^2*b +
b^3)*d*e + (3*a^2*b + b^3)*c*f)*cosh(f*x + e)^2 + 2*(3*a*b^2*d - (3*a^2*b + b^3)*d*e + (3*a^2*b + b^3)*c*f + 3
*(3*a*b^2*d - (3*a^2*b + b^3)*d*e + (3*a^2*b + b^3)*c*f)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 4*((3*a*b^2*d - (3
*a^2*b + b^3)*d*e + (3*a^2*b + b^3)*c*f)*cosh(f*x + e)^3 + (3*a*b^2*d - (3*a^2*b + b^3)*d*e + (3*a^2*b + b^3)*
c*f)*cosh(f*x + e))*sinh(f*x + e))*log(cosh(f*x + e) + sinh(f*x + e) + I) + 2*((3*a*b^2*d - (3*a^2*b + b^3)*d*
e + (3*a^2*b + b^3)*c*f)*cosh(f*x + e)^4 + 4*(3*a*b^2*d - (3*a^2*b + b^3)*d*e + (3*a^2*b + b^3)*c*f)*cosh(f*x
+ e)*sinh(f*x + e)^3 + (3*a*b^2*d - (3*a^2*b + b^3)*d*e + (3*a^2*b + b^3)*c*f)*sinh(f*x + e)^4 + 3*a*b^2*d - (
3*a^2*b + b^3)*d*e + (3*a^2*b + b^3)*c*f + 2*(3*a*b^2*d - (3*a^2*b + b^3)*d*e + (3*a^2*b + b^3)*c*f)*cosh(f*x
+ e)^2 + 2*(3*a*b^2*d - (3*a^2*b + b^3)*d*e + (3*a^2*b + b^3)*c*f + 3*(3*a*b^2*d - (3*a^2*b + b^3)*d*e + (3*a^
2*b + b^3)*c*f)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 4*((3*a*b^2*d - (3*a^2*b + b^3)*d*e + (3*a^2*b + b^3)*c*f)*
cosh(f*x + e)^3 + (3*a*b^2*d - (3*a^2*b + b^3)*d*e + (3*a^2*b + b^3)*c*f)*cosh(f*x + e))*sinh(f*x + e))*log(co
sh(f*x + e) + sinh(f*x + e) - I) + 2*(((3*a^2*b + b^3)*d*f*x + (3*a^2*b + b^3)*d*e)*cosh(f*x + e)^4 + 4*((3*a^
2*b + b^3)*d*f*x + (3*a^2*b + b^3)*d*e)*cosh(f*x + e)*sinh(f*x + e)^3 + ((3*a^2*b + b^3)*d*f*x + (3*a^2*b + b^
3)*d*e)*sinh(f*x + e)^4 + (3*a^2*b + b^3)*d*f*x + (3*a^2*b + b^3)*d*e + 2*((3*a^2*b + b^3)*d*f*x + (3*a^2*b +
b^3)*d*e)*cosh(f*x + e)^2 + 2*((3*a^2*b + b^3)*d*f*x + (3*a^2*b + b^3)*d*e + 3*((3*a^2*b + b^3)*d*f*x + (3*a^2
*b + b^3)*d*e)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 4*(((3*a^2*b + b^3)*d*f*x + (3*a^2*b + b^3)*d*e)*cosh(f*x +
e)^3 + ((3*a^2*b + b^3)*d*f*x + (3*a^2*b + b^3)*d*e)*cosh(f*x + e))*sinh(f*x + e))*log(I*cosh(f*x + e) + I*sin
h(f*x + e) + 1) + 2*(((3*a^2*b + b^3)*d*f*x + (3*a^2*b + b^3)*d*e)*cosh(f*x + e)^4 + 4*((3*a^2*b + b^3)*d*f*x
+ (3*a^2*b + b^3)*d*e)*cosh(f*x + e)*sinh(f*x + e)^3 + ((3*a^2*b + b^3)*d*f*x + (3*a^2*b + b^3)*d*e)*sinh(f*x
+ e)^4 + (3*a^2*b + b^3)*d*f*x + (3*a^2*b + b^3)*d*e + 2*((3*a^2*b + b^3)*d*f*x + (3*a^2*b + b^3)*d*e)*cosh(f*
x + e)^2 + 2*((3*a^2*b + b^3)*d*f*x + (3*a^2*b + b^3)*d*e + 3*((3*a^2*b + b^3)*d*f*x + (3*a^2*b + b^3)*d*e)*co
sh(f*x + e)^2)*sinh(f*x + e)^2 + 4*(((3*a^2*b + b^3)*d*f*x + (3*a^2*b + b^3)*d*e)*cosh(f*x + e)^3 + ((3*a^2*b
+ b^3)*d*f*x + (3*a^2*b + b^3)*d*e)*cosh(f*x + e))*sinh(f*x + e))*log(-I*cosh(f*x + e) - I*sinh(f*x + e) + 1)
+ 4*(((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*f^2*x^2 - 12*a*b^2*d*e + 2*(3*a^2*b + b^3)*d*e^2 - 4*(3*a^2*b + b^3)*c
*e*f - 2*(6*a*b^2*d*f - (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c*f^2)*x)*cosh(f*x + e)^3 + ((a^3 - 3*a^2*b + 3*a*b^2
- b^3)*d*f^2*x^2 - 12*a*b^2*d*e + b^3*d + 2*(3*a^2*b + b^3)*d*e^2 - 2*(2*(3*a^2*b + b^3)*c*e - (3*a*b^2 + b^3)
*c)*f + 2*((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c*f^2 - (3*a*b^2 - b^3)*d*f)*x)*cosh(f*x + e))*sinh(f*x + e))/(f^2*
cosh(f*x + e)^4 + 4*f^2*cosh(f*x + e)*sinh(f*x + e)^3 + f^2*sinh(f*x + e)^4 + 2*f^2*cosh(f*x + e)^2 + 2*(3*f^2
*cosh(f*x + e)^2 + f^2)*sinh(f*x + e)^2 + f^2 + 4*(f^2*cosh(f*x + e)^3 + f^2*cosh(f*x + e))*sinh(f*x + e))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 475, normalized size of antiderivative = 1.82 \[ \int (c+d x) (a+b \tanh (e+f x))^3 \, dx=\frac {1}{2} \, a^{3} d x^{2} + b^{3} c {\left (x + \frac {e}{f} + \frac {\log \left (e^{\left (-2 \, f x - 2 \, e\right )} + 1\right )}{f} + \frac {2 \, e^{\left (-2 \, f x - 2 \, e\right )}}{f {\left (2 \, e^{\left (-2 \, f x - 2 \, e\right )} + e^{\left (-4 \, f x - 4 \, e\right )} + 1\right )}}\right )} + a^{3} c x - \frac {6 \, a b^{2} d x}{f} + \frac {3 \, a^{2} b c \log \left (\cosh \left (f x + e\right )\right )}{f} - {\left (3 \, a^{2} b d + b^{3} d\right )} x^{2} + \frac {3 \, a b^{2} d \log \left (e^{\left (2 \, f x + 2 \, e\right )} + 1\right )}{f^{2}} + \frac {12 \, a b^{2} c f + 6 \, {\left (c f^{2} + 2 \, d f\right )} a b^{2} x + 2 \, b^{3} d + {\left (3 \, a^{2} b d f^{2} + 3 \, a b^{2} d f^{2} + b^{3} d f^{2}\right )} x^{2} + {\left (6 \, a b^{2} c f^{2} x e^{\left (4 \, e\right )} + {\left (3 \, a^{2} b d f^{2} e^{\left (4 \, e\right )} + 3 \, a b^{2} d f^{2} e^{\left (4 \, e\right )} + b^{3} d f^{2} e^{\left (4 \, e\right )}\right )} x^{2}\right )} e^{\left (4 \, f x\right )} + 2 \, {\left (6 \, a b^{2} c f e^{\left (2 \, e\right )} + b^{3} d e^{\left (2 \, e\right )} + {\left (3 \, a^{2} b d f^{2} e^{\left (2 \, e\right )} + 3 \, a b^{2} d f^{2} e^{\left (2 \, e\right )} + b^{3} d f^{2} e^{\left (2 \, e\right )}\right )} x^{2} + 2 \, {\left (b^{3} d f e^{\left (2 \, e\right )} + 3 \, {\left (c f^{2} e^{\left (2 \, e\right )} + d f e^{\left (2 \, e\right )}\right )} a b^{2}\right )} x\right )} e^{\left (2 \, f x\right )}}{2 \, {\left (f^{2} e^{\left (4 \, f x + 4 \, e\right )} + 2 \, f^{2} e^{\left (2 \, f x + 2 \, e\right )} + f^{2}\right )}} + \frac {{\left (3 \, a^{2} b d + b^{3} d\right )} {\left (2 \, f x \log \left (e^{\left (2 \, f x + 2 \, e\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (2 \, f x + 2 \, e\right )}\right )\right )}}{2 \, f^{2}} \]

[In]

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

[Out]

1/2*a^3*d*x^2 + b^3*c*(x + e/f + log(e^(-2*f*x - 2*e) + 1)/f + 2*e^(-2*f*x - 2*e)/(f*(2*e^(-2*f*x - 2*e) + e^(
-4*f*x - 4*e) + 1))) + a^3*c*x - 6*a*b^2*d*x/f + 3*a^2*b*c*log(cosh(f*x + e))/f - (3*a^2*b*d + b^3*d)*x^2 + 3*
a*b^2*d*log(e^(2*f*x + 2*e) + 1)/f^2 + 1/2*(12*a*b^2*c*f + 6*(c*f^2 + 2*d*f)*a*b^2*x + 2*b^3*d + (3*a^2*b*d*f^
2 + 3*a*b^2*d*f^2 + b^3*d*f^2)*x^2 + (6*a*b^2*c*f^2*x*e^(4*e) + (3*a^2*b*d*f^2*e^(4*e) + 3*a*b^2*d*f^2*e^(4*e)
+ b^3*d*f^2*e^(4*e))*x^2)*e^(4*f*x) + 2*(6*a*b^2*c*f*e^(2*e) + b^3*d*e^(2*e) + (3*a^2*b*d*f^2*e^(2*e) + 3*a*b
^2*d*f^2*e^(2*e) + b^3*d*f^2*e^(2*e))*x^2 + 2*(b^3*d*f*e^(2*e) + 3*(c*f^2*e^(2*e) + d*f*e^(2*e))*a*b^2)*x)*e^(
2*f*x))/(f^2*e^(4*f*x + 4*e) + 2*f^2*e^(2*f*x + 2*e) + f^2) + 1/2*(3*a^2*b*d + b^3*d)*(2*f*x*log(e^(2*f*x + 2*
e) + 1) + dilog(-e^(2*f*x + 2*e)))/f^2

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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