3.321 \(\int \frac{\cosh ^2(a+b x) \sinh ^3(a+b x)}{x} \, dx\)

Optimal. Leaf size=73 \[ -\frac{1}{8} \sinh (a) \text{Chi}(b x)-\frac{1}{16} \sinh (3 a) \text{Chi}(3 b x)+\frac{1}{16} \sinh (5 a) \text{Chi}(5 b x)-\frac{1}{8} \cosh (a) \text{Shi}(b x)-\frac{1}{16} \cosh (3 a) \text{Shi}(3 b x)+\frac{1}{16} \cosh (5 a) \text{Shi}(5 b x) \]

[Out]

-(CoshIntegral[b*x]*Sinh[a])/8 - (CoshIntegral[3*b*x]*Sinh[3*a])/16 + (CoshIntegral[5*b*x]*Sinh[5*a])/16 - (Co
sh[a]*SinhIntegral[b*x])/8 - (Cosh[3*a]*SinhIntegral[3*b*x])/16 + (Cosh[5*a]*SinhIntegral[5*b*x])/16

________________________________________________________________________________________

Rubi [A]  time = 0.182328, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {5448, 3303, 3298, 3301} \[ -\frac{1}{8} \sinh (a) \text{Chi}(b x)-\frac{1}{16} \sinh (3 a) \text{Chi}(3 b x)+\frac{1}{16} \sinh (5 a) \text{Chi}(5 b x)-\frac{1}{8} \cosh (a) \text{Shi}(b x)-\frac{1}{16} \cosh (3 a) \text{Shi}(3 b x)+\frac{1}{16} \cosh (5 a) \text{Shi}(5 b x) \]

Antiderivative was successfully verified.

[In]

Int[(Cosh[a + b*x]^2*Sinh[a + b*x]^3)/x,x]

[Out]

-(CoshIntegral[b*x]*Sinh[a])/8 - (CoshIntegral[3*b*x]*Sinh[3*a])/16 + (CoshIntegral[5*b*x]*Sinh[5*a])/16 - (Co
sh[a]*SinhIntegral[b*x])/8 - (Cosh[3*a]*SinhIntegral[3*b*x])/16 + (Cosh[5*a]*SinhIntegral[5*b*x])/16

Rule 5448

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

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{\cosh ^2(a+b x) \sinh ^3(a+b x)}{x} \, dx &=\int \left (-\frac{\sinh (a+b x)}{8 x}-\frac{\sinh (3 a+3 b x)}{16 x}+\frac{\sinh (5 a+5 b x)}{16 x}\right ) \, dx\\ &=-\left (\frac{1}{16} \int \frac{\sinh (3 a+3 b x)}{x} \, dx\right )+\frac{1}{16} \int \frac{\sinh (5 a+5 b x)}{x} \, dx-\frac{1}{8} \int \frac{\sinh (a+b x)}{x} \, dx\\ &=-\left (\frac{1}{8} \cosh (a) \int \frac{\sinh (b x)}{x} \, dx\right )-\frac{1}{16} \cosh (3 a) \int \frac{\sinh (3 b x)}{x} \, dx+\frac{1}{16} \cosh (5 a) \int \frac{\sinh (5 b x)}{x} \, dx-\frac{1}{8} \sinh (a) \int \frac{\cosh (b x)}{x} \, dx-\frac{1}{16} \sinh (3 a) \int \frac{\cosh (3 b x)}{x} \, dx+\frac{1}{16} \sinh (5 a) \int \frac{\cosh (5 b x)}{x} \, dx\\ &=-\frac{1}{8} \text{Chi}(b x) \sinh (a)-\frac{1}{16} \text{Chi}(3 b x) \sinh (3 a)+\frac{1}{16} \text{Chi}(5 b x) \sinh (5 a)-\frac{1}{8} \cosh (a) \text{Shi}(b x)-\frac{1}{16} \cosh (3 a) \text{Shi}(3 b x)+\frac{1}{16} \cosh (5 a) \text{Shi}(5 b x)\\ \end{align*}

Mathematica [A]  time = 0.10901, size = 63, normalized size = 0.86 \[ \frac{1}{16} (-2 \sinh (a) \text{Chi}(b x)-\sinh (3 a) \text{Chi}(3 b x)+\sinh (5 a) \text{Chi}(5 b x)-2 \cosh (a) \text{Shi}(b x)-\cosh (3 a) \text{Shi}(3 b x)+\cosh (5 a) \text{Shi}(5 b x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*CoshIntegral[b*x]*Sinh[a] - CoshIntegral[3*b*x]*Sinh[3*a] + CoshIntegral[5*b*x]*Sinh[5*a] - 2*Cosh[a]*Sinh
Integral[b*x] - Cosh[3*a]*SinhIntegral[3*b*x] + Cosh[5*a]*SinhIntegral[5*b*x])/16

________________________________________________________________________________________

Maple [A]  time = 0.091, size = 71, normalized size = 1. \begin{align*}{\frac{{{\rm e}^{-5\,a}}{\it Ei} \left ( 1,5\,bx \right ) }{32}}-{\frac{{{\rm e}^{-3\,a}}{\it Ei} \left ( 1,3\,bx \right ) }{32}}-{\frac{{{\rm e}^{-a}}{\it Ei} \left ( 1,bx \right ) }{16}}+{\frac{{{\rm e}^{a}}{\it Ei} \left ( 1,-bx \right ) }{16}}+{\frac{{{\rm e}^{3\,a}}{\it Ei} \left ( 1,-3\,bx \right ) }{32}}-{\frac{{{\rm e}^{5\,a}}{\it Ei} \left ( 1,-5\,bx \right ) }{32}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*exp(-5*a)*Ei(1,5*b*x)-1/32*exp(-3*a)*Ei(1,3*b*x)-1/16*exp(-a)*Ei(1,b*x)+1/16*exp(a)*Ei(1,-b*x)+1/32*exp(3
*a)*Ei(1,-3*b*x)-1/32*exp(5*a)*Ei(1,-5*b*x)

________________________________________________________________________________________

Maxima [A]  time = 1.30887, size = 86, normalized size = 1.18 \begin{align*} \frac{1}{32} \,{\rm Ei}\left (5 \, b x\right ) e^{\left (5 \, a\right )} - \frac{1}{32} \,{\rm Ei}\left (3 \, b x\right ) e^{\left (3 \, a\right )} + \frac{1}{16} \,{\rm Ei}\left (-b x\right ) e^{\left (-a\right )} + \frac{1}{32} \,{\rm Ei}\left (-3 \, b x\right ) e^{\left (-3 \, a\right )} - \frac{1}{32} \,{\rm Ei}\left (-5 \, b x\right ) e^{\left (-5 \, a\right )} - \frac{1}{16} \,{\rm Ei}\left (b x\right ) e^{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*Ei(5*b*x)*e^(5*a) - 1/32*Ei(3*b*x)*e^(3*a) + 1/16*Ei(-b*x)*e^(-a) + 1/32*Ei(-3*b*x)*e^(-3*a) - 1/32*Ei(-5
*b*x)*e^(-5*a) - 1/16*Ei(b*x)*e^a

________________________________________________________________________________________

Fricas [A]  time = 1.80776, size = 323, normalized size = 4.42 \begin{align*} \frac{1}{32} \,{\left ({\rm Ei}\left (5 \, b x\right ) -{\rm Ei}\left (-5 \, b x\right )\right )} \cosh \left (5 \, a\right ) - \frac{1}{32} \,{\left ({\rm Ei}\left (3 \, b x\right ) -{\rm Ei}\left (-3 \, b x\right )\right )} \cosh \left (3 \, a\right ) - \frac{1}{16} \,{\left ({\rm Ei}\left (b x\right ) -{\rm Ei}\left (-b x\right )\right )} \cosh \left (a\right ) + \frac{1}{32} \,{\left ({\rm Ei}\left (5 \, b x\right ) +{\rm Ei}\left (-5 \, b x\right )\right )} \sinh \left (5 \, a\right ) - \frac{1}{32} \,{\left ({\rm Ei}\left (3 \, b x\right ) +{\rm Ei}\left (-3 \, b x\right )\right )} \sinh \left (3 \, a\right ) - \frac{1}{16} \,{\left ({\rm Ei}\left (b x\right ) +{\rm Ei}\left (-b x\right )\right )} \sinh \left (a\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*(Ei(5*b*x) - Ei(-5*b*x))*cosh(5*a) - 1/32*(Ei(3*b*x) - Ei(-3*b*x))*cosh(3*a) - 1/16*(Ei(b*x) - Ei(-b*x))*
cosh(a) + 1/32*(Ei(5*b*x) + Ei(-5*b*x))*sinh(5*a) - 1/32*(Ei(3*b*x) + Ei(-3*b*x))*sinh(3*a) - 1/16*(Ei(b*x) +
Ei(-b*x))*sinh(a)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.17892, size = 86, normalized size = 1.18 \begin{align*} \frac{1}{32} \,{\rm Ei}\left (5 \, b x\right ) e^{\left (5 \, a\right )} - \frac{1}{32} \,{\rm Ei}\left (3 \, b x\right ) e^{\left (3 \, a\right )} + \frac{1}{16} \,{\rm Ei}\left (-b x\right ) e^{\left (-a\right )} + \frac{1}{32} \,{\rm Ei}\left (-3 \, b x\right ) e^{\left (-3 \, a\right )} - \frac{1}{32} \,{\rm Ei}\left (-5 \, b x\right ) e^{\left (-5 \, a\right )} - \frac{1}{16} \,{\rm Ei}\left (b x\right ) e^{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*Ei(5*b*x)*e^(5*a) - 1/32*Ei(3*b*x)*e^(3*a) + 1/16*Ei(-b*x)*e^(-a) + 1/32*Ei(-3*b*x)*e^(-3*a) - 1/32*Ei(-5
*b*x)*e^(-5*a) - 1/16*Ei(b*x)*e^a