### 3.314 $$\int \frac{\cosh (a+b x) \sinh ^3(a+b x)}{x^3} \, dx$$

Optimal. Leaf size=125 $-\frac{1}{2} b^2 \sinh (2 a) \text{Chi}(2 b x)+b^2 \sinh (4 a) \text{Chi}(4 b x)-\frac{1}{2} b^2 \cosh (2 a) \text{Shi}(2 b x)+b^2 \cosh (4 a) \text{Shi}(4 b x)+\frac{\sinh (2 a+2 b x)}{8 x^2}-\frac{\sinh (4 a+4 b x)}{16 x^2}+\frac{b \cosh (2 a+2 b x)}{4 x}-\frac{b \cosh (4 a+4 b x)}{4 x}$

[Out]

(b*Cosh[2*a + 2*b*x])/(4*x) - (b*Cosh[4*a + 4*b*x])/(4*x) - (b^2*CoshIntegral[2*b*x]*Sinh[2*a])/2 + b^2*CoshIn
tegral[4*b*x]*Sinh[4*a] + Sinh[2*a + 2*b*x]/(8*x^2) - Sinh[4*a + 4*b*x]/(16*x^2) - (b^2*Cosh[2*a]*SinhIntegral
[2*b*x])/2 + b^2*Cosh[4*a]*SinhIntegral[4*b*x]

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Rubi [A]  time = 0.229893, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 5, integrand size = 18, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.278, Rules used = {5448, 3297, 3303, 3298, 3301} $-\frac{1}{2} b^2 \sinh (2 a) \text{Chi}(2 b x)+b^2 \sinh (4 a) \text{Chi}(4 b x)-\frac{1}{2} b^2 \cosh (2 a) \text{Shi}(2 b x)+b^2 \cosh (4 a) \text{Shi}(4 b x)+\frac{\sinh (2 a+2 b x)}{8 x^2}-\frac{\sinh (4 a+4 b x)}{16 x^2}+\frac{b \cosh (2 a+2 b x)}{4 x}-\frac{b \cosh (4 a+4 b x)}{4 x}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(b*Cosh[2*a + 2*b*x])/(4*x) - (b*Cosh[4*a + 4*b*x])/(4*x) - (b^2*CoshIntegral[2*b*x]*Sinh[2*a])/2 + b^2*CoshIn
tegral[4*b*x]*Sinh[4*a] + Sinh[2*a + 2*b*x]/(8*x^2) - Sinh[4*a + 4*b*x]/(16*x^2) - (b^2*Cosh[2*a]*SinhIntegral
[2*b*x])/2 + b^2*Cosh[4*a]*SinhIntegral[4*b*x]

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 3297

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[((c + d*x)^(m + 1)*Sin[e + f*x])/(d*(
m + 1)), x] - Dist[f/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && 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{\cosh (a+b x) \sinh ^3(a+b x)}{x^3} \, dx &=\int \left (-\frac{\sinh (2 a+2 b x)}{4 x^3}+\frac{\sinh (4 a+4 b x)}{8 x^3}\right ) \, dx\\ &=\frac{1}{8} \int \frac{\sinh (4 a+4 b x)}{x^3} \, dx-\frac{1}{4} \int \frac{\sinh (2 a+2 b x)}{x^3} \, dx\\ &=\frac{\sinh (2 a+2 b x)}{8 x^2}-\frac{\sinh (4 a+4 b x)}{16 x^2}-\frac{1}{4} b \int \frac{\cosh (2 a+2 b x)}{x^2} \, dx+\frac{1}{4} b \int \frac{\cosh (4 a+4 b x)}{x^2} \, dx\\ &=\frac{b \cosh (2 a+2 b x)}{4 x}-\frac{b \cosh (4 a+4 b x)}{4 x}+\frac{\sinh (2 a+2 b x)}{8 x^2}-\frac{\sinh (4 a+4 b x)}{16 x^2}-\frac{1}{2} b^2 \int \frac{\sinh (2 a+2 b x)}{x} \, dx+b^2 \int \frac{\sinh (4 a+4 b x)}{x} \, dx\\ &=\frac{b \cosh (2 a+2 b x)}{4 x}-\frac{b \cosh (4 a+4 b x)}{4 x}+\frac{\sinh (2 a+2 b x)}{8 x^2}-\frac{\sinh (4 a+4 b x)}{16 x^2}-\frac{1}{2} \left (b^2 \cosh (2 a)\right ) \int \frac{\sinh (2 b x)}{x} \, dx+\left (b^2 \cosh (4 a)\right ) \int \frac{\sinh (4 b x)}{x} \, dx-\frac{1}{2} \left (b^2 \sinh (2 a)\right ) \int \frac{\cosh (2 b x)}{x} \, dx+\left (b^2 \sinh (4 a)\right ) \int \frac{\cosh (4 b x)}{x} \, dx\\ &=\frac{b \cosh (2 a+2 b x)}{4 x}-\frac{b \cosh (4 a+4 b x)}{4 x}-\frac{1}{2} b^2 \text{Chi}(2 b x) \sinh (2 a)+b^2 \text{Chi}(4 b x) \sinh (4 a)+\frac{\sinh (2 a+2 b x)}{8 x^2}-\frac{\sinh (4 a+4 b x)}{16 x^2}-\frac{1}{2} b^2 \cosh (2 a) \text{Shi}(2 b x)+b^2 \cosh (4 a) \text{Shi}(4 b x)\\ \end{align*}

Mathematica [A]  time = 0.545119, size = 113, normalized size = 0.9 $b^2 \sinh (4 a) \text{Chi}(4 b x)+b^2 \sinh (a) (-\cosh (a)) \text{Chi}(2 b x)-\frac{1}{2} b^2 \cosh (2 a) \text{Shi}(2 b x)+b^2 \cosh (4 a) \text{Shi}(4 b x)+\frac{\sinh (2 (a+b x))+2 b x \cosh (2 (a+b x))}{8 x^2}-\frac{\sinh (4 (a+b x))+4 b x \cosh (4 (a+b x))}{16 x^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(b^2*Cosh[a]*CoshIntegral[2*b*x]*Sinh[a]) + b^2*CoshIntegral[4*b*x]*Sinh[4*a] + (2*b*x*Cosh[2*(a + b*x)] + Si
nh[2*(a + b*x)])/(8*x^2) - (4*b*x*Cosh[4*(a + b*x)] + Sinh[4*(a + b*x)])/(16*x^2) - (b^2*Cosh[2*a]*SinhIntegra
l[2*b*x])/2 + b^2*Cosh[4*a]*SinhIntegral[4*b*x]

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Maple [A]  time = 0.069, size = 178, normalized size = 1.4 \begin{align*} -{\frac{b{{\rm e}^{-4\,bx-4\,a}}}{8\,x}}+{\frac{{{\rm e}^{-4\,bx-4\,a}}}{32\,{x}^{2}}}+{\frac{{b}^{2}{{\rm e}^{-4\,a}}{\it Ei} \left ( 1,4\,bx \right ) }{2}}+{\frac{b{{\rm e}^{-2\,bx-2\,a}}}{8\,x}}-{\frac{{{\rm e}^{-2\,bx-2\,a}}}{16\,{x}^{2}}}-{\frac{{b}^{2}{{\rm e}^{-2\,a}}{\it Ei} \left ( 1,2\,bx \right ) }{4}}-{\frac{{{\rm e}^{4\,bx+4\,a}}}{32\,{x}^{2}}}-{\frac{b{{\rm e}^{4\,bx+4\,a}}}{8\,x}}-{\frac{{b}^{2}{{\rm e}^{4\,a}}{\it Ei} \left ( 1,-4\,bx \right ) }{2}}+{\frac{{{\rm e}^{2\,bx+2\,a}}}{16\,{x}^{2}}}+{\frac{b{{\rm e}^{2\,bx+2\,a}}}{8\,x}}+{\frac{{b}^{2}{{\rm e}^{2\,a}}{\it Ei} \left ( 1,-2\,bx \right ) }{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*b*exp(-4*b*x-4*a)/x+1/32*exp(-4*b*x-4*a)/x^2+1/2*b^2*exp(-4*a)*Ei(1,4*b*x)+1/8*b*exp(-2*b*x-2*a)/x-1/16*e
xp(-2*b*x-2*a)/x^2-1/4*b^2*exp(-2*a)*Ei(1,2*b*x)-1/32/x^2*exp(4*b*x+4*a)-1/8*b/x*exp(4*b*x+4*a)-1/2*b^2*exp(4*
a)*Ei(1,-4*b*x)+1/16*exp(2*b*x+2*a)/x^2+1/8*b*exp(2*b*x+2*a)/x+1/4*b^2*exp(2*a)*Ei(1,-2*b*x)

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Maxima [A]  time = 1.22558, size = 81, normalized size = 0.65 \begin{align*} b^{2} e^{\left (-4 \, a\right )} \Gamma \left (-2, 4 \, b x\right ) - \frac{1}{2} \, b^{2} e^{\left (-2 \, a\right )} \Gamma \left (-2, 2 \, b x\right ) + \frac{1}{2} \, b^{2} e^{\left (2 \, a\right )} \Gamma \left (-2, -2 \, b x\right ) - b^{2} e^{\left (4 \, a\right )} \Gamma \left (-2, -4 \, b x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^2*e^(-4*a)*gamma(-2, 4*b*x) - 1/2*b^2*e^(-2*a)*gamma(-2, 2*b*x) + 1/2*b^2*e^(2*a)*gamma(-2, -2*b*x) - b^2*e^
(4*a)*gamma(-2, -4*b*x)

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Fricas [B]  time = 1.85163, size = 570, normalized size = 4.56 \begin{align*} -\frac{b x \cosh \left (b x + a\right )^{4} + b x \sinh \left (b x + a\right )^{4} - b x \cosh \left (b x + a\right )^{2} + \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} +{\left (6 \, b x \cosh \left (b x + a\right )^{2} - b x\right )} \sinh \left (b x + a\right )^{2} - 2 \,{\left (b^{2} x^{2}{\rm Ei}\left (4 \, b x\right ) - b^{2} x^{2}{\rm Ei}\left (-4 \, b x\right )\right )} \cosh \left (4 \, a\right ) +{\left (b^{2} x^{2}{\rm Ei}\left (2 \, b x\right ) - b^{2} x^{2}{\rm Ei}\left (-2 \, b x\right )\right )} \cosh \left (2 \, a\right ) +{\left (\cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) - 2 \,{\left (b^{2} x^{2}{\rm Ei}\left (4 \, b x\right ) + b^{2} x^{2}{\rm Ei}\left (-4 \, b x\right )\right )} \sinh \left (4 \, a\right ) +{\left (b^{2} x^{2}{\rm Ei}\left (2 \, b x\right ) + b^{2} x^{2}{\rm Ei}\left (-2 \, b x\right )\right )} \sinh \left (2 \, a\right )}{4 \, x^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.19316, size = 227, normalized size = 1.82 \begin{align*} \frac{16 \, b^{2} x^{2}{\rm Ei}\left (4 \, b x\right ) e^{\left (4 \, a\right )} - 8 \, b^{2} x^{2}{\rm Ei}\left (2 \, b x\right ) e^{\left (2 \, a\right )} + 8 \, b^{2} x^{2}{\rm Ei}\left (-2 \, b x\right ) e^{\left (-2 \, a\right )} - 16 \, b^{2} x^{2}{\rm Ei}\left (-4 \, b x\right ) e^{\left (-4 \, a\right )} - 4 \, b x e^{\left (4 \, b x + 4 \, a\right )} + 4 \, b x e^{\left (2 \, b x + 2 \, a\right )} + 4 \, b x e^{\left (-2 \, b x - 2 \, a\right )} - 4 \, b x e^{\left (-4 \, b x - 4 \, a\right )} - e^{\left (4 \, b x + 4 \, a\right )} + 2 \, e^{\left (2 \, b x + 2 \, a\right )} - 2 \, e^{\left (-2 \, b x - 2 \, a\right )} + e^{\left (-4 \, b x - 4 \, a\right )}}{32 \, x^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*(16*b^2*x^2*Ei(4*b*x)*e^(4*a) - 8*b^2*x^2*Ei(2*b*x)*e^(2*a) + 8*b^2*x^2*Ei(-2*b*x)*e^(-2*a) - 16*b^2*x^2*
Ei(-4*b*x)*e^(-4*a) - 4*b*x*e^(4*b*x + 4*a) + 4*b*x*e^(2*b*x + 2*a) + 4*b*x*e^(-2*b*x - 2*a) - 4*b*x*e^(-4*b*x
- 4*a) - e^(4*b*x + 4*a) + 2*e^(2*b*x + 2*a) - 2*e^(-2*b*x - 2*a) + e^(-4*b*x - 4*a))/x^2