3.21 \(\int \frac{\sinh ^3(a+b x)}{(c+d x)^2} \, dx\)

Optimal. Leaf size=145 \[ -\frac{3 b \cosh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (\frac{b c}{d}+b x\right )}{4 d^2}+\frac{3 b \cosh \left (3 a-\frac{3 b c}{d}\right ) \text{Chi}\left (\frac{3 b c}{d}+3 b x\right )}{4 d^2}-\frac{3 b \sinh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (\frac{b c}{d}+b x\right )}{4 d^2}+\frac{3 b \sinh \left (3 a-\frac{3 b c}{d}\right ) \text{Shi}\left (\frac{3 b c}{d}+3 b x\right )}{4 d^2}-\frac{\sinh ^3(a+b x)}{d (c+d x)} \]

[Out]

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

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Rubi [A]  time = 0.262214, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3313, 3303, 3298, 3301} \[ -\frac{3 b \cosh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (\frac{b c}{d}+b x\right )}{4 d^2}+\frac{3 b \cosh \left (3 a-\frac{3 b c}{d}\right ) \text{Chi}\left (\frac{3 b c}{d}+3 b x\right )}{4 d^2}-\frac{3 b \sinh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (\frac{b c}{d}+b x\right )}{4 d^2}+\frac{3 b \sinh \left (3 a-\frac{3 b c}{d}\right ) \text{Shi}\left (\frac{3 b c}{d}+3 b x\right )}{4 d^2}-\frac{\sinh ^3(a+b x)}{d (c+d x)} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[a + b*x]^3/(c + d*x)^2,x]

[Out]

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

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

Mathematica [A]  time = 1.34078, size = 160, normalized size = 1.1 \[ \frac{6 b (c+d x) \left (-\cosh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (b \left (\frac{c}{d}+x\right )\right )+\cosh \left (3 a-\frac{3 b c}{d}\right ) \text{Chi}\left (\frac{3 b (c+d x)}{d}\right )-\sinh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (b \left (\frac{c}{d}+x\right )\right )+\sinh \left (3 a-\frac{3 b c}{d}\right ) \text{Shi}\left (\frac{3 b (c+d x)}{d}\right )\right )+6 d \sinh (a) \cosh (b x)-2 d \sinh (3 a) \cosh (3 b x)+6 d \cosh (a) \sinh (b x)-2 d \cosh (3 a) \sinh (3 b x)}{8 d^2 (c+d x)} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[a + b*x]^3/(c + d*x)^2,x]

[Out]

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

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Maple [A]  time = 0.091, size = 271, normalized size = 1.9 \begin{align*}{\frac{b{{\rm e}^{-3\,bx-3\,a}}}{ \left ( 8\,bdx+8\,cb \right ) d}}-{\frac{3\,b}{8\,{d}^{2}}{{\rm e}^{-3\,{\frac{da-cb}{d}}}}{\it Ei} \left ( 1,3\,bx+3\,a-3\,{\frac{da-cb}{d}} \right ) }-{\frac{3\,b{{\rm e}^{-bx-a}}}{8\,d \left ( bdx+cb \right ) }}+{\frac{3\,b}{8\,{d}^{2}}{{\rm e}^{-{\frac{da-cb}{d}}}}{\it Ei} \left ( 1,bx+a-{\frac{da-cb}{d}} \right ) }+{\frac{3\,b{{\rm e}^{bx+a}}}{8\,{d}^{2}} \left ({\frac{cb}{d}}+bx \right ) ^{-1}}+{\frac{3\,b}{8\,{d}^{2}}{{\rm e}^{{\frac{da-cb}{d}}}}{\it Ei} \left ( 1,-bx-a-{\frac{-da+cb}{d}} \right ) }-{\frac{b{{\rm e}^{3\,bx+3\,a}}}{8\,{d}^{2}} \left ({\frac{cb}{d}}+bx \right ) ^{-1}}-{\frac{3\,b}{8\,{d}^{2}}{{\rm e}^{3\,{\frac{da-cb}{d}}}}{\it Ei} \left ( 1,-3\,bx-3\,a-3\,{\frac{-da+cb}{d}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(b*x+a)^3/(d*x+c)^2,x)

[Out]

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

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Maxima [A]  time = 1.38717, size = 196, normalized size = 1.35 \begin{align*} \frac{e^{\left (-3 \, a + \frac{3 \, b c}{d}\right )} E_{2}\left (\frac{3 \,{\left (d x + c\right )} b}{d}\right )}{8 \,{\left (d x + c\right )} d} - \frac{3 \, e^{\left (-a + \frac{b c}{d}\right )} E_{2}\left (\frac{{\left (d x + c\right )} b}{d}\right )}{8 \,{\left (d x + c\right )} d} + \frac{3 \, e^{\left (a - \frac{b c}{d}\right )} E_{2}\left (-\frac{{\left (d x + c\right )} b}{d}\right )}{8 \,{\left (d x + c\right )} d} - \frac{e^{\left (3 \, a - \frac{3 \, b c}{d}\right )} E_{2}\left (-\frac{3 \,{\left (d x + c\right )} b}{d}\right )}{8 \,{\left (d x + c\right )} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(b*x+a)^3/(d*x+c)^2,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.69591, size = 662, normalized size = 4.57 \begin{align*} -\frac{2 \, d \sinh \left (b x + a\right )^{3} + 3 \,{\left ({\left (b d x + b c\right )}{\rm Ei}\left (\frac{b d x + b c}{d}\right ) +{\left (b d x + b c\right )}{\rm Ei}\left (-\frac{b d x + b c}{d}\right )\right )} \cosh \left (-\frac{b c - a d}{d}\right ) - 3 \,{\left ({\left (b d x + b c\right )}{\rm Ei}\left (\frac{3 \,{\left (b d x + b c\right )}}{d}\right ) +{\left (b d x + b c\right )}{\rm Ei}\left (-\frac{3 \,{\left (b d x + b c\right )}}{d}\right )\right )} \cosh \left (-\frac{3 \,{\left (b c - a d\right )}}{d}\right ) + 6 \,{\left (d \cosh \left (b x + a\right )^{2} - d\right )} \sinh \left (b x + a\right ) + 3 \,{\left ({\left (b d x + b c\right )}{\rm Ei}\left (\frac{b d x + b c}{d}\right ) -{\left (b d x + b c\right )}{\rm Ei}\left (-\frac{b d x + b c}{d}\right )\right )} \sinh \left (-\frac{b c - a d}{d}\right ) - 3 \,{\left ({\left (b d x + b c\right )}{\rm Ei}\left (\frac{3 \,{\left (b d x + b c\right )}}{d}\right ) -{\left (b d x + b c\right )}{\rm Ei}\left (-\frac{3 \,{\left (b d x + b c\right )}}{d}\right )\right )} \sinh \left (-\frac{3 \,{\left (b c - a d\right )}}{d}\right )}{8 \,{\left (d^{3} x + c d^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(b*x+a)^3/(d*x+c)^2,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh ^{3}{\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(b*x+a)**3/(d*x+c)**2,x)

[Out]

Integral(sinh(a + b*x)**3/(c + d*x)**2, x)

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Giac [B]  time = 1.24623, size = 400, normalized size = 2.76 \begin{align*} \frac{3 \, b d x{\rm Ei}\left (\frac{3 \,{\left (b d x + b c\right )}}{d}\right ) e^{\left (3 \, a - \frac{3 \, b c}{d}\right )} - 3 \, b d x{\rm Ei}\left (\frac{b d x + b c}{d}\right ) e^{\left (a - \frac{b c}{d}\right )} - 3 \, b d x{\rm Ei}\left (-\frac{b d x + b c}{d}\right ) e^{\left (-a + \frac{b c}{d}\right )} + 3 \, b d x{\rm Ei}\left (-\frac{3 \,{\left (b d x + b c\right )}}{d}\right ) e^{\left (-3 \, a + \frac{3 \, b c}{d}\right )} + 3 \, b c{\rm Ei}\left (\frac{3 \,{\left (b d x + b c\right )}}{d}\right ) e^{\left (3 \, a - \frac{3 \, b c}{d}\right )} - 3 \, b c{\rm Ei}\left (\frac{b d x + b c}{d}\right ) e^{\left (a - \frac{b c}{d}\right )} - 3 \, b c{\rm Ei}\left (-\frac{b d x + b c}{d}\right ) e^{\left (-a + \frac{b c}{d}\right )} + 3 \, b c{\rm Ei}\left (-\frac{3 \,{\left (b d x + b c\right )}}{d}\right ) e^{\left (-3 \, a + \frac{3 \, b c}{d}\right )} - d e^{\left (3 \, b x + 3 \, a\right )} + 3 \, d e^{\left (b x + a\right )} - 3 \, d e^{\left (-b x - a\right )} + d e^{\left (-3 \, b x - 3 \, a\right )}}{8 \,{\left (d^{3} x + c d^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(b*x+a)^3/(d*x+c)^2,x, algorithm="giac")

[Out]

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