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

Optimal. Leaf size=143 \[ -\frac{3 b c \cosh \left (\frac{b}{d}\right ) \text{Chi}\left (\frac{b c}{d (c+d x)}\right )}{4 d^2}+\frac{3 b c \cosh \left (\frac{3 b}{d}\right ) \text{Chi}\left (\frac{3 b c}{d (c+d x)}\right )}{4 d^2}+\frac{3 b c \sinh \left (\frac{b}{d}\right ) \text{Shi}\left (\frac{b c}{d (c+d x)}\right )}{4 d^2}-\frac{3 b c \sinh \left (\frac{3 b}{d}\right ) \text{Shi}\left (\frac{3 b c}{d (c+d x)}\right )}{4 d^2}+\frac{(c+d x) \sinh ^3\left (\frac{b x}{c+d x}\right )}{d} \]

[Out]

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

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Rubi [A]  time = 0.246428, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {5607, 3313, 3303, 3298, 3301} \[ -\frac{3 b c \cosh \left (\frac{b}{d}\right ) \text{Chi}\left (\frac{b c}{d (c+d x)}\right )}{4 d^2}+\frac{3 b c \cosh \left (\frac{3 b}{d}\right ) \text{Chi}\left (\frac{3 b c}{d (c+d x)}\right )}{4 d^2}+\frac{3 b c \sinh \left (\frac{b}{d}\right ) \text{Shi}\left (\frac{b c}{d (c+d x)}\right )}{4 d^2}-\frac{3 b c \sinh \left (\frac{3 b}{d}\right ) \text{Shi}\left (\frac{3 b c}{d (c+d x)}\right )}{4 d^2}+\frac{(c+d x) \sinh ^3\left (\frac{b x}{c+d x}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[(b*x)/(c + d*x)]^3,x]

[Out]

(-3*b*c*Cosh[b/d]*CoshIntegral[(b*c)/(d*(c + d*x))])/(4*d^2) + (3*b*c*Cosh[(3*b)/d]*CoshIntegral[(3*b*c)/(d*(c
+ d*x))])/(4*d^2) + ((c + d*x)*Sinh[(b*x)/(c + d*x)]^3)/d + (3*b*c*Sinh[b/d]*SinhIntegral[(b*c)/(d*(c + d*x))
])/(4*d^2) - (3*b*c*Sinh[(3*b)/d]*SinhIntegral[(3*b*c)/(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{b x}{c+d x}\right ) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\sinh ^3\left (\frac{b}{d}-\frac{b c x}{d}\right )}{x^2} \, dx,x,\frac{1}{c+d x}\right )}{d}\\ &=\frac{(c+d x) \sinh ^3\left (\frac{b x}{c+d x}\right )}{d}-\frac{(3 b c) \operatorname{Subst}\left (\int \left (-\frac{\cosh \left (\frac{3 b}{d}-\frac{3 b c x}{d}\right )}{4 x}+\frac{\cosh \left (\frac{b}{d}-\frac{b c x}{d}\right )}{4 x}\right ) \, dx,x,\frac{1}{c+d x}\right )}{d^2}\\ &=\frac{(c+d x) \sinh ^3\left (\frac{b x}{c+d x}\right )}{d}+\frac{(3 b c) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{3 b}{d}-\frac{3 b c x}{d}\right )}{x} \, dx,x,\frac{1}{c+d x}\right )}{4 d^2}-\frac{(3 b c) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{b}{d}-\frac{b c x}{d}\right )}{x} \, dx,x,\frac{1}{c+d x}\right )}{4 d^2}\\ &=\frac{(c+d x) \sinh ^3\left (\frac{b x}{c+d x}\right )}{d}-\frac{\left (3 b c \cosh \left (\frac{b}{d}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{b c x}{d}\right )}{x} \, dx,x,\frac{1}{c+d x}\right )}{4 d^2}+\frac{\left (3 b c \cosh \left (\frac{3 b}{d}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{3 b c x}{d}\right )}{x} \, dx,x,\frac{1}{c+d x}\right )}{4 d^2}+\frac{\left (3 b c \sinh \left (\frac{b}{d}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{b c x}{d}\right )}{x} \, dx,x,\frac{1}{c+d x}\right )}{4 d^2}-\frac{\left (3 b c \sinh \left (\frac{3 b}{d}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{3 b c x}{d}\right )}{x} \, dx,x,\frac{1}{c+d x}\right )}{4 d^2}\\ &=-\frac{3 b c \cosh \left (\frac{b}{d}\right ) \text{Chi}\left (\frac{b c}{d (c+d x)}\right )}{4 d^2}+\frac{3 b c \cosh \left (\frac{3 b}{d}\right ) \text{Chi}\left (\frac{3 b c}{d (c+d x)}\right )}{4 d^2}+\frac{(c+d x) \sinh ^3\left (\frac{b x}{c+d x}\right )}{d}+\frac{3 b c \sinh \left (\frac{b}{d}\right ) \text{Shi}\left (\frac{b c}{d (c+d x)}\right )}{4 d^2}-\frac{3 b c \sinh \left (\frac{3 b}{d}\right ) \text{Shi}\left (\frac{3 b c}{d (c+d x)}\right )}{4 d^2}\\ \end{align*}

Mathematica [A]  time = 0.458828, size = 172, normalized size = 1.2 \[ \frac{-3 b c \cosh \left (\frac{b}{d}\right ) \text{Chi}\left (\frac{b c}{d (c+d x)}\right )+3 b c \cosh \left (\frac{3 b}{d}\right ) \text{Chi}\left (\frac{3 b c}{d (c+d x)}\right )-3 d^2 x \sinh \left (\frac{b x}{c+d x}\right )+d^2 x \sinh \left (\frac{3 b x}{c+d x}\right )+3 b c \sinh \left (\frac{b}{d}\right ) \text{Shi}\left (\frac{b c}{d (c+d x)}\right )-3 b c \sinh \left (\frac{3 b}{d}\right ) \text{Shi}\left (\frac{3 b c}{d (c+d x)}\right )-3 c d \sinh \left (\frac{b x}{c+d x}\right )+c d \sinh \left (\frac{3 b x}{c+d x}\right )}{4 d^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[(b*x)/(c + d*x)]^3,x]

[Out]

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

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Maple [A]  time = 0.088, size = 228, normalized size = 1.6 \begin{align*} -{\frac{dx+c}{8\,d}{{\rm e}^{-3\,{\frac{bx}{dx+c}}}}}-{\frac{3\,cb}{8\,{d}^{2}}{{\rm e}^{-3\,{\frac{b}{d}}}}{\it Ei} \left ( 1,-3\,{\frac{cb}{d \left ( dx+c \right ) }} \right ) }+{\frac{3\,dx+3\,c}{8\,d}{{\rm e}^{-{\frac{bx}{dx+c}}}}}+{\frac{3\,cb}{8\,{d}^{2}}{{\rm e}^{-{\frac{b}{d}}}}{\it Ei} \left ( 1,-{\frac{cb}{d \left ( dx+c \right ) }} \right ) }+{\frac{x}{8}{{\rm e}^{3\,{\frac{bx}{dx+c}}}}}+{\frac{c}{8\,d}{{\rm e}^{3\,{\frac{bx}{dx+c}}}}}-{\frac{3\,cb}{8\,{d}^{2}}{{\rm e}^{3\,{\frac{b}{d}}}}{\it Ei} \left ( 1,3\,{\frac{cb}{d \left ( dx+c \right ) }} \right ) }-{\frac{3\,x}{8}{{\rm e}^{{\frac{bx}{dx+c}}}}}-{\frac{3\,c}{8\,d}{{\rm e}^{{\frac{bx}{dx+c}}}}}+{\frac{3\,cb}{8\,{d}^{2}}{{\rm e}^{{\frac{b}{d}}}}{\it Ei} \left ( 1,{\frac{cb}{d \left ( dx+c \right ) }} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{3}{8} \, b c \int \frac{x e^{\left (\frac{3 \, b c}{d^{2} x + c d}\right )}}{d^{2} x^{2} e^{\left (\frac{3 \, b}{d}\right )} + 2 \, c d x e^{\left (\frac{3 \, b}{d}\right )} + c^{2} e^{\left (\frac{3 \, b}{d}\right )}}\,{d x} + \frac{3}{8} \, b c \int \frac{x e^{\left (\frac{b c}{d^{2} x + c d}\right )}}{d^{2} x^{2} e^{\frac{b}{d}} + 2 \, c d x e^{\frac{b}{d}} + c^{2} e^{\frac{b}{d}}}\,{d x} + \frac{3}{8} \, b c \int \frac{x e^{\left (-\frac{b c}{d^{2} x + c d} + \frac{b}{d}\right )}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\,{d x} - \frac{3}{8} \, b c \int \frac{x e^{\left (-\frac{3 \, b c}{d^{2} x + c d} + \frac{3 \, b}{d}\right )}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\,{d x} - \frac{1}{8} \,{\left (x e^{\left (\frac{3 \, b c}{d^{2} x + c d}\right )} - 3 \, x e^{\left (\frac{b c}{d^{2} x + c d} + \frac{2 \, b}{d}\right )} + 3 \, x e^{\left (-\frac{b c}{d^{2} x + c d} + \frac{4 \, b}{d}\right )} - x e^{\left (-\frac{3 \, b c}{d^{2} x + c d} + \frac{6 \, b}{d}\right )}\right )} e^{\left (-\frac{3 \, b}{d}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/8*b*c*integrate(x*e^(3*b*c/(d^2*x + c*d))/(d^2*x^2*e^(3*b/d) + 2*c*d*x*e^(3*b/d) + c^2*e^(3*b/d)), x) + 3/8
*b*c*integrate(x*e^(b*c/(d^2*x + c*d))/(d^2*x^2*e^(b/d) + 2*c*d*x*e^(b/d) + c^2*e^(b/d)), x) + 3/8*b*c*integra
te(x*e^(-b*c/(d^2*x + c*d) + b/d)/(d^2*x^2 + 2*c*d*x + c^2), x) - 3/8*b*c*integrate(x*e^(-3*b*c/(d^2*x + c*d)
+ 3*b/d)/(d^2*x^2 + 2*c*d*x + c^2), x) - 1/8*(x*e^(3*b*c/(d^2*x + c*d)) - 3*x*e^(b*c/(d^2*x + c*d) + 2*b/d) +
3*x*e^(-b*c/(d^2*x + c*d) + 4*b/d) - x*e^(-3*b*c/(d^2*x + c*d) + 6*b/d))*e^(-3*b/d)

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Fricas [B]  time = 2.18035, size = 1524, normalized size = 10.66 \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/(d*x+c))^3,x, algorithm="fricas")

[Out]

1/8*(3*(b*c*Ei(-3*b*c/(d^2*x + c*d))*cosh(3*b/d) - b*c*Ei(-b*c/(d^2*x + c*d))*cosh(b/d))*sinh(b*x/(d*x + c))^4
+ 2*(d^2*x + c*d)*sinh(b*x/(d*x + c))^3 - 6*(b*c*Ei(-3*b*c/(d^2*x + c*d))*cosh(b*x/(d*x + c))^2*cosh(3*b/d) -
b*c*Ei(-b*c/(d^2*x + c*d))*cosh(b*x/(d*x + c))^2*cosh(b/d))*sinh(b*x/(d*x + c))^2 + 3*(b*c*Ei(-3*b*c/(d^2*x +
c*d))*cosh(b*x/(d*x + c))^4 + b*c*Ei(3*b*c/(d^2*x + c*d)))*cosh(3*b/d) - 3*(b*c*Ei(-b*c/(d^2*x + c*d))*cosh(b
*x/(d*x + c))^4 + b*c*Ei(b*c/(d^2*x + c*d)))*cosh(b/d) - 6*(d^2*x - (d^2*x + c*d)*cosh(b*x/(d*x + c))^2 + c*d)
*sinh(b*x/(d*x + c)) + 3*(b*c*Ei(-3*b*c/(d^2*x + c*d))*cosh(b*x/(d*x + c))^4 - 2*b*c*Ei(-3*b*c/(d^2*x + c*d))*
cosh(b*x/(d*x + c))^2*sinh(b*x/(d*x + c))^2 + b*c*Ei(-3*b*c/(d^2*x + c*d))*sinh(b*x/(d*x + c))^4 - b*c*Ei(3*b*
c/(d^2*x + c*d)))*sinh(3*b/d) - 3*(b*c*Ei(-b*c/(d^2*x + c*d))*cosh(b*x/(d*x + c))^4 - 2*b*c*Ei(-b*c/(d^2*x + c
*d))*cosh(b*x/(d*x + c))^2*sinh(b*x/(d*x + c))^2 + b*c*Ei(-b*c/(d^2*x + c*d))*sinh(b*x/(d*x + c))^4 - b*c*Ei(b
*c/(d^2*x + c*d)))*sinh(b/d))/(d^2*cosh(b*x/(d*x + c))^4 - 2*d^2*cosh(b*x/(d*x + c))^2*sinh(b*x/(d*x + c))^2 +
d^2*sinh(b*x/(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/(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}{d x + c}\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sinh(b*x/(d*x + c))^3, x)