3.297 \(\int \sinh ^3(\frac{a+b x}{c+d x}) \, dx\)

Optimal. Leaf size=194 \[ -\frac{3 \cosh \left (\frac{b}{d}\right ) (b c-a d) \text{Chi}\left (\frac{b c-a d}{d (c+d x)}\right )}{4 d^2}+\frac{3 \cosh \left (\frac{3 b}{d}\right ) (b c-a d) \text{Chi}\left (\frac{3 (b c-a d)}{d (c+d x)}\right )}{4 d^2}+\frac{3 \sinh \left (\frac{b}{d}\right ) (b c-a d) \text{Shi}\left (\frac{b c-a d}{d (c+d x)}\right )}{4 d^2}-\frac{3 \sinh \left (\frac{3 b}{d}\right ) (b c-a d) \text{Shi}\left (\frac{3 (b c-a d)}{d (c+d x)}\right )}{4 d^2}+\frac{(c+d x) \sinh ^3\left (\frac{a+b x}{c+d x}\right )}{d} \]

[Out]

(-3*(b*c - a*d)*Cosh[b/d]*CoshIntegral[(b*c - a*d)/(d*(c + d*x))])/(4*d^2) + (3*(b*c - a*d)*Cosh[(3*b)/d]*Cosh
Integral[(3*(b*c - a*d))/(d*(c + d*x))])/(4*d^2) + ((c + d*x)*Sinh[(a + b*x)/(c + d*x)]^3)/d + (3*(b*c - a*d)*
Sinh[b/d]*SinhIntegral[(b*c - a*d)/(d*(c + d*x))])/(4*d^2) - (3*(b*c - a*d)*Sinh[(3*b)/d]*SinhIntegral[(3*(b*c
- a*d))/(d*(c + d*x))])/(4*d^2)

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Rubi [A]  time = 0.328356, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {5607, 3313, 3303, 3298, 3301} \[ -\frac{3 \cosh \left (\frac{b}{d}\right ) (b c-a d) \text{Chi}\left (\frac{b c-a d}{d (c+d x)}\right )}{4 d^2}+\frac{3 \cosh \left (\frac{3 b}{d}\right ) (b c-a d) \text{Chi}\left (\frac{3 (b c-a d)}{d (c+d x)}\right )}{4 d^2}+\frac{3 \sinh \left (\frac{b}{d}\right ) (b c-a d) \text{Shi}\left (\frac{b c-a d}{d (c+d x)}\right )}{4 d^2}-\frac{3 \sinh \left (\frac{3 b}{d}\right ) (b c-a d) \text{Shi}\left (\frac{3 (b c-a d)}{d (c+d x)}\right )}{4 d^2}+\frac{(c+d x) \sinh ^3\left (\frac{a+b x}{c+d x}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[(a + b*x)/(c + d*x)]^3,x]

[Out]

(-3*(b*c - a*d)*Cosh[b/d]*CoshIntegral[(b*c - a*d)/(d*(c + d*x))])/(4*d^2) + (3*(b*c - a*d)*Cosh[(3*b)/d]*Cosh
Integral[(3*(b*c - a*d))/(d*(c + d*x))])/(4*d^2) + ((c + d*x)*Sinh[(a + b*x)/(c + d*x)]^3)/d + (3*(b*c - a*d)*
Sinh[b/d]*SinhIntegral[(b*c - a*d)/(d*(c + d*x))])/(4*d^2) - (3*(b*c - a*d)*Sinh[(3*b)/d]*SinhIntegral[(3*(b*c
- a*d))/(d*(c + d*x))])/(4*d^2)

Rule 5607

Int[Sinh[((e_.)*((a_.) + (b_.)*(x_)))/((c_.) + (d_.)*(x_))]^(n_.), x_Symbol] :> -Dist[d^(-1), Subst[Int[Sinh[(
b*e)/d - (e*(b*c - a*d)*x)/d]^n/x^2, x], x, 1/(c + d*x)], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && NeQ[b*
c - a*d, 0]

Rule 3313

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Simp[((c + d*x)^(m + 1)*Sin[e + f*x]^
n)/(d*(m + 1)), x] - Dist[(f*n)/(d*(m + 1)), Int[ExpandTrigReduce[(c + d*x)^(m + 1), Cos[e + f*x]*Sin[e + f*x]
^(n - 1), x], x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && GeQ[m, -2] && LtQ[m, -1]

Rule 3303

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[(c*f)/d + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 3298

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(I*SinhIntegral[(c*f*fz)
/d + f*fz*x])/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rule 3301

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[(c*f*fz)/d
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rubi steps

\begin{align*} \int \sinh ^3\left (\frac{a+b x}{c+d x}\right ) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\sinh ^3\left (\frac{b}{d}-\frac{(b c-a d) x}{d}\right )}{x^2} \, dx,x,\frac{1}{c+d x}\right )}{d}\\ &=\frac{(c+d x) \sinh ^3\left (\frac{a+b x}{c+d x}\right )}{d}-\frac{(3 (b c-a d)) \operatorname{Subst}\left (\int \left (-\frac{\cosh \left (\frac{3 b}{d}-\frac{3 (b c-a d) x}{d}\right )}{4 x}+\frac{\cosh \left (\frac{b}{d}-\frac{(b c-a d) x}{d}\right )}{4 x}\right ) \, dx,x,\frac{1}{c+d x}\right )}{d^2}\\ &=\frac{(c+d x) \sinh ^3\left (\frac{a+b x}{c+d x}\right )}{d}+\frac{(3 (b c-a d)) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{3 b}{d}-\frac{3 (b c-a d) x}{d}\right )}{x} \, dx,x,\frac{1}{c+d x}\right )}{4 d^2}-\frac{(3 (b c-a d)) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{b}{d}-\frac{(b c-a d) x}{d}\right )}{x} \, dx,x,\frac{1}{c+d x}\right )}{4 d^2}\\ &=\frac{(c+d x) \sinh ^3\left (\frac{a+b x}{c+d x}\right )}{d}-\frac{\left (3 (b c-a d) \cosh \left (\frac{b}{d}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{(b c-a d) x}{d}\right )}{x} \, dx,x,\frac{1}{c+d x}\right )}{4 d^2}+\frac{\left (3 (b c-a d) \cosh \left (\frac{3 b}{d}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{3 (b c-a d) x}{d}\right )}{x} \, dx,x,\frac{1}{c+d x}\right )}{4 d^2}+\frac{\left (3 (b c-a d) \sinh \left (\frac{b}{d}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{(b c-a d) x}{d}\right )}{x} \, dx,x,\frac{1}{c+d x}\right )}{4 d^2}-\frac{\left (3 (b c-a d) \sinh \left (\frac{3 b}{d}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{3 (b c-a d) x}{d}\right )}{x} \, dx,x,\frac{1}{c+d x}\right )}{4 d^2}\\ &=-\frac{3 (b c-a d) \cosh \left (\frac{b}{d}\right ) \text{Chi}\left (\frac{b c-a d}{d (c+d x)}\right )}{4 d^2}+\frac{3 (b c-a d) \cosh \left (\frac{3 b}{d}\right ) \text{Chi}\left (\frac{3 (b c-a d)}{d (c+d x)}\right )}{4 d^2}+\frac{(c+d x) \sinh ^3\left (\frac{a+b x}{c+d x}\right )}{d}+\frac{3 (b c-a d) \sinh \left (\frac{b}{d}\right ) \text{Shi}\left (\frac{b c-a d}{d (c+d x)}\right )}{4 d^2}-\frac{3 (b c-a d) \sinh \left (\frac{3 b}{d}\right ) \text{Shi}\left (\frac{3 (b c-a d)}{d (c+d x)}\right )}{4 d^2}\\ \end{align*}

Mathematica [B]  time = 1.35566, size = 599, normalized size = 3.09 \[ \frac{-3 a d \sinh \left (\frac{b}{d}\right ) \text{Chi}\left (\frac{b c-a d}{x d^2+c d}\right )+3 b c \sinh \left (\frac{b}{d}\right ) \text{Chi}\left (\frac{b c-a d}{x d^2+c d}\right )+3 a d \cosh \left (\frac{b}{d}\right ) \text{Chi}\left (\frac{b c-a d}{x d^2+c d}\right )-3 b c \cosh \left (\frac{b}{d}\right ) \text{Chi}\left (\frac{b c-a d}{x d^2+c d}\right )+6 \cosh \left (\frac{3 b}{d}\right ) (b c-a d) \text{Chi}\left (\frac{3 (a d-b c)}{d (c+d x)}\right )-3 (b c-a d) \left (\sinh \left (\frac{b}{d}\right )+\cosh \left (\frac{b}{d}\right )\right ) \text{Chi}\left (\frac{a d-b c}{d (c+d x)}\right )-3 a d \sinh \left (\frac{b}{d}\right ) \text{Shi}\left (\frac{b c-a d}{x d^2+c d}\right )+3 b c \sinh \left (\frac{b}{d}\right ) \text{Shi}\left (\frac{b c-a d}{x d^2+c d}\right )+3 a d \cosh \left (\frac{b}{d}\right ) \text{Shi}\left (\frac{b c-a d}{x d^2+c d}\right )-3 b c \cosh \left (\frac{b}{d}\right ) \text{Shi}\left (\frac{b c-a d}{x d^2+c d}\right )-6 d^2 x \sinh \left (\frac{a+b x}{c+d x}\right )+2 d^2 x \sinh \left (\frac{3 (a+b x)}{c+d x}\right )+3 a d \sinh \left (\frac{b}{d}\right ) \text{Shi}\left (\frac{a d-b c}{d (c+d x)}\right )-6 a d \sinh \left (\frac{3 b}{d}\right ) \text{Shi}\left (\frac{3 (a d-b c)}{d (c+d x)}\right )-3 b c \sinh \left (\frac{b}{d}\right ) \text{Shi}\left (\frac{a d-b c}{d (c+d x)}\right )+6 b c \sinh \left (\frac{3 b}{d}\right ) \text{Shi}\left (\frac{3 (a d-b c)}{d (c+d x)}\right )+3 a d \cosh \left (\frac{b}{d}\right ) \text{Shi}\left (\frac{a d-b c}{d (c+d x)}\right )-3 b c \cosh \left (\frac{b}{d}\right ) \text{Shi}\left (\frac{a d-b c}{d (c+d x)}\right )-6 c d \sinh \left (\frac{a+b x}{c+d x}\right )+2 c d \sinh \left (\frac{3 (a+b x)}{c+d x}\right )}{8 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*(b*c - a*d)*Cosh[(3*b)/d]*CoshIntegral[(3*(-(b*c) + a*d))/(d*(c + d*x))] - 3*b*c*Cosh[b/d]*CoshIntegral[(b*
c - a*d)/(c*d + d^2*x)] + 3*a*d*Cosh[b/d]*CoshIntegral[(b*c - a*d)/(c*d + d^2*x)] + 3*b*c*CoshIntegral[(b*c -
a*d)/(c*d + d^2*x)]*Sinh[b/d] - 3*a*d*CoshIntegral[(b*c - a*d)/(c*d + d^2*x)]*Sinh[b/d] - 3*(b*c - a*d)*CoshIn
tegral[(-(b*c) + a*d)/(d*(c + d*x))]*(Cosh[b/d] + Sinh[b/d]) - 6*c*d*Sinh[(a + b*x)/(c + d*x)] - 6*d^2*x*Sinh[
(a + b*x)/(c + d*x)] + 2*c*d*Sinh[(3*(a + b*x))/(c + d*x)] + 2*d^2*x*Sinh[(3*(a + b*x))/(c + d*x)] - 3*b*c*Cos
h[b/d]*SinhIntegral[(-(b*c) + a*d)/(d*(c + d*x))] + 3*a*d*Cosh[b/d]*SinhIntegral[(-(b*c) + a*d)/(d*(c + d*x))]
- 3*b*c*Sinh[b/d]*SinhIntegral[(-(b*c) + a*d)/(d*(c + d*x))] + 3*a*d*Sinh[b/d]*SinhIntegral[(-(b*c) + a*d)/(d
*(c + d*x))] + 6*b*c*Sinh[(3*b)/d]*SinhIntegral[(3*(-(b*c) + a*d))/(d*(c + d*x))] - 6*a*d*Sinh[(3*b)/d]*SinhIn
tegral[(3*(-(b*c) + a*d))/(d*(c + d*x))] - 3*b*c*Cosh[b/d]*SinhIntegral[(b*c - a*d)/(c*d + d^2*x)] + 3*a*d*Cos
h[b/d]*SinhIntegral[(b*c - a*d)/(c*d + d^2*x)] + 3*b*c*Sinh[b/d]*SinhIntegral[(b*c - a*d)/(c*d + d^2*x)] - 3*a
*d*Sinh[b/d]*SinhIntegral[(b*c - a*d)/(c*d + d^2*x)])/(8*d^2)

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Maple [B]  time = 0.084, size = 700, normalized size = 3.6 \begin{align*} -{\frac{a}{8}{{\rm e}^{-3\,{\frac{bx+a}{dx+c}}}} \left ({\frac{da}{dx+c}}-{\frac{cb}{dx+c}} \right ) ^{-1}}+{\frac{cb}{8\,d}{{\rm e}^{-3\,{\frac{bx+a}{dx+c}}}} \left ({\frac{da}{dx+c}}-{\frac{cb}{dx+c}} \right ) ^{-1}}+{\frac{3\,a}{8\,d}{{\rm e}^{-3\,{\frac{b}{d}}}}{\it Ei} \left ( 1,3\,{\frac{da-cb}{d \left ( dx+c \right ) }} \right ) }-{\frac{3\,cb}{8\,{d}^{2}}{{\rm e}^{-3\,{\frac{b}{d}}}}{\it Ei} \left ( 1,3\,{\frac{da-cb}{d \left ( dx+c \right ) }} \right ) }+{\frac{3\,a}{8}{{\rm e}^{-{\frac{bx+a}{dx+c}}}} \left ({\frac{da}{dx+c}}-{\frac{cb}{dx+c}} \right ) ^{-1}}-{\frac{3\,cb}{8\,d}{{\rm e}^{-{\frac{bx+a}{dx+c}}}} \left ({\frac{da}{dx+c}}-{\frac{cb}{dx+c}} \right ) ^{-1}}-{\frac{3\,a}{8\,d}{{\rm e}^{-{\frac{b}{d}}}}{\it Ei} \left ( 1,{\frac{da-cb}{d \left ( dx+c \right ) }} \right ) }+{\frac{3\,cb}{8\,{d}^{2}}{{\rm e}^{-{\frac{b}{d}}}}{\it Ei} \left ( 1,{\frac{da-cb}{d \left ( dx+c \right ) }} \right ) }+{\frac{dxa}{8\,da-8\,cb}{{\rm e}^{3\,{\frac{bx+a}{dx+c}}}}}-{\frac{bcx}{8\,da-8\,cb}{{\rm e}^{3\,{\frac{bx+a}{dx+c}}}}}+{\frac{ac}{8\,da-8\,cb}{{\rm e}^{3\,{\frac{bx+a}{dx+c}}}}}-{\frac{b{c}^{2}}{8\,d \left ( da-cb \right ) }{{\rm e}^{3\,{\frac{bx+a}{dx+c}}}}}+{\frac{3\,a}{8\,d}{{\rm e}^{3\,{\frac{b}{d}}}}{\it Ei} \left ( 1,-3\,{\frac{da-cb}{d \left ( dx+c \right ) }} \right ) }-{\frac{3\,cb}{8\,{d}^{2}}{{\rm e}^{3\,{\frac{b}{d}}}}{\it Ei} \left ( 1,-3\,{\frac{da-cb}{d \left ( dx+c \right ) }} \right ) }-{\frac{3\,dxa}{8\,da-8\,cb}{{\rm e}^{{\frac{bx+a}{dx+c}}}}}+{\frac{3\,bcx}{8\,da-8\,cb}{{\rm e}^{{\frac{bx+a}{dx+c}}}}}-{\frac{3\,ac}{8\,da-8\,cb}{{\rm e}^{{\frac{bx+a}{dx+c}}}}}+{\frac{3\,b{c}^{2}}{8\,d \left ( da-cb \right ) }{{\rm e}^{{\frac{bx+a}{dx+c}}}}}-{\frac{3\,a}{8\,d}{{\rm e}^{{\frac{b}{d}}}}{\it Ei} \left ( 1,-{\frac{da-cb}{d \left ( dx+c \right ) }} \right ) }+{\frac{3\,cb}{8\,{d}^{2}}{{\rm e}^{{\frac{b}{d}}}}{\it Ei} \left ( 1,-{\frac{da-cb}{d \left ( dx+c \right ) }} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh((b*x+a)/(d*x+c))^3,x)

[Out]

-1/8*exp(-3*(b*x+a)/(d*x+c))/(d*a/(d*x+c)-b*c/(d*x+c))*a+1/8/d*exp(-3*(b*x+a)/(d*x+c))/(d*a/(d*x+c)-b*c/(d*x+c
))*c*b+3/8/d*exp(-3*b/d)*Ei(1,3*(a*d-b*c)/d/(d*x+c))*a-3/8/d^2*exp(-3*b/d)*Ei(1,3*(a*d-b*c)/d/(d*x+c))*c*b+3/8
*exp(-(b*x+a)/(d*x+c))/(d*a/(d*x+c)-b*c/(d*x+c))*a-3/8/d*exp(-(b*x+a)/(d*x+c))/(d*a/(d*x+c)-b*c/(d*x+c))*c*b-3
/8/d*exp(-b/d)*Ei(1,(a*d-b*c)/d/(d*x+c))*a+3/8/d^2*exp(-b/d)*Ei(1,(a*d-b*c)/d/(d*x+c))*c*b+1/8*d*exp(3*(b*x+a)
/(d*x+c))/(a*d-b*c)*x*a-1/8*exp(3*(b*x+a)/(d*x+c))/(a*d-b*c)*x*c*b+1/8*exp(3*(b*x+a)/(d*x+c))/(a*d-b*c)*c*a-1/
8/d*exp(3*(b*x+a)/(d*x+c))/(a*d-b*c)*c^2*b+3/8/d*exp(3*b/d)*Ei(1,-3*(a*d-b*c)/d/(d*x+c))*a-3/8/d^2*exp(3*b/d)*
Ei(1,-3*(a*d-b*c)/d/(d*x+c))*c*b-3/8*d*exp((b*x+a)/(d*x+c))/(a*d-b*c)*x*a+3/8*exp((b*x+a)/(d*x+c))/(a*d-b*c)*x
*c*b-3/8*exp((b*x+a)/(d*x+c))/(a*d-b*c)*c*a+3/8/d*exp((b*x+a)/(d*x+c))/(a*d-b*c)*c^2*b-3/8/d*exp(b/d)*Ei(1,-(a
*d-b*c)/d/(d*x+c))*a+3/8/d^2*exp(b/d)*Ei(1,-(a*d-b*c)/d/(d*x+c))*c*b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sinh \left (\frac{b x + a}{d x + c}\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.14485, size = 1536, normalized size = 7.92 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(6*(b*c - a*d)*Ei(-3*(b*c - a*d)/(d^2*x + c*d))*cosh((b*x + a)/(d*x + c))^2*cosh(3*b/d)*sinh((b*x + a)/(d
*x + c))^2 - 3*(b*c - a*d)*Ei(-3*(b*c - a*d)/(d^2*x + c*d))*cosh(3*b/d)*sinh((b*x + a)/(d*x + c))^4 - 2*(d^2*x
+ c*d)*sinh((b*x + a)/(d*x + c))^3 - 3*((b*c - a*d)*Ei(-3*(b*c - a*d)/(d^2*x + c*d))*cosh((b*x + a)/(d*x + c)
)^4 + (b*c - a*d)*Ei(3*(b*c - a*d)/(d^2*x + c*d)))*cosh(3*b/d) + 3*((b*c - a*d)*Ei((b*c - a*d)/(d^2*x + c*d))
+ (b*c - a*d)*Ei(-(b*c - a*d)/(d^2*x + c*d)))*cosh(b/d) + 6*(d^2*x - (d^2*x + c*d)*cosh((b*x + a)/(d*x + c))^2
+ c*d)*sinh((b*x + a)/(d*x + c)) - 3*((b*c - a*d)*Ei(-3*(b*c - a*d)/(d^2*x + c*d))*cosh((b*x + a)/(d*x + c))^
4 - 2*(b*c - a*d)*Ei(-3*(b*c - a*d)/(d^2*x + c*d))*cosh((b*x + a)/(d*x + c))^2*sinh((b*x + a)/(d*x + c))^2 + (
b*c - a*d)*Ei(-3*(b*c - a*d)/(d^2*x + c*d))*sinh((b*x + a)/(d*x + c))^4 - (b*c - a*d)*Ei(3*(b*c - a*d)/(d^2*x
+ c*d)))*sinh(3*b/d) - 3*((b*c - a*d)*Ei((b*c - a*d)/(d^2*x + c*d)) - (b*c - a*d)*Ei(-(b*c - a*d)/(d^2*x + c*d
)))*sinh(b/d))/(d^2*cosh((b*x + a)/(d*x + c))^4 - 2*d^2*cosh((b*x + a)/(d*x + c))^2*sinh((b*x + a)/(d*x + c))^
2 + d^2*sinh((b*x + a)/(d*x + c))^4)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sinh \left (\frac{b x + a}{d x + c}\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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