### 3.288 $$\int \frac{\cosh (a+b x) \sinh ^2(a+b x)}{x^4} \, dx$$

Optimal. Leaf size=154 $-\frac{1}{24} b^3 \sinh (a) \text{Chi}(b x)+\frac{9}{8} b^3 \sinh (3 a) \text{Chi}(3 b x)-\frac{1}{24} b^3 \cosh (a) \text{Shi}(b x)+\frac{9}{8} b^3 \cosh (3 a) \text{Shi}(3 b x)+\frac{b^2 \cosh (a+b x)}{24 x}-\frac{3 b^2 \cosh (3 a+3 b x)}{8 x}+\frac{b \sinh (a+b x)}{24 x^2}-\frac{b \sinh (3 a+3 b x)}{8 x^2}+\frac{\cosh (a+b x)}{12 x^3}-\frac{\cosh (3 a+3 b x)}{12 x^3}$

[Out]

Cosh[a + b*x]/(12*x^3) + (b^2*Cosh[a + b*x])/(24*x) - Cosh[3*a + 3*b*x]/(12*x^3) - (3*b^2*Cosh[3*a + 3*b*x])/(
8*x) - (b^3*CoshIntegral[b*x]*Sinh[a])/24 + (9*b^3*CoshIntegral[3*b*x]*Sinh[3*a])/8 + (b*Sinh[a + b*x])/(24*x^
2) - (b*Sinh[3*a + 3*b*x])/(8*x^2) - (b^3*Cosh[a]*SinhIntegral[b*x])/24 + (9*b^3*Cosh[3*a]*SinhIntegral[3*b*x]
)/8

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Rubi [A]  time = 0.281335, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 14, 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} $-\frac{1}{24} b^3 \sinh (a) \text{Chi}(b x)+\frac{9}{8} b^3 \sinh (3 a) \text{Chi}(3 b x)-\frac{1}{24} b^3 \cosh (a) \text{Shi}(b x)+\frac{9}{8} b^3 \cosh (3 a) \text{Shi}(3 b x)+\frac{b^2 \cosh (a+b x)}{24 x}-\frac{3 b^2 \cosh (3 a+3 b x)}{8 x}+\frac{b \sinh (a+b x)}{24 x^2}-\frac{b \sinh (3 a+3 b x)}{8 x^2}+\frac{\cosh (a+b x)}{12 x^3}-\frac{\cosh (3 a+3 b x)}{12 x^3}$

Antiderivative was successfully veriﬁed.

[In]

Int[(Cosh[a + b*x]*Sinh[a + b*x]^2)/x^4,x]

[Out]

Cosh[a + b*x]/(12*x^3) + (b^2*Cosh[a + b*x])/(24*x) - Cosh[3*a + 3*b*x]/(12*x^3) - (3*b^2*Cosh[3*a + 3*b*x])/(
8*x) - (b^3*CoshIntegral[b*x]*Sinh[a])/24 + (9*b^3*CoshIntegral[3*b*x]*Sinh[3*a])/8 + (b*Sinh[a + b*x])/(24*x^
2) - (b*Sinh[3*a + 3*b*x])/(8*x^2) - (b^3*Cosh[a]*SinhIntegral[b*x])/24 + (9*b^3*Cosh[3*a]*SinhIntegral[3*b*x]
)/8

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 ^2(a+b x)}{x^4} \, dx &=\int \left (-\frac{\cosh (a+b x)}{4 x^4}+\frac{\cosh (3 a+3 b x)}{4 x^4}\right ) \, dx\\ &=-\left (\frac{1}{4} \int \frac{\cosh (a+b x)}{x^4} \, dx\right )+\frac{1}{4} \int \frac{\cosh (3 a+3 b x)}{x^4} \, dx\\ &=\frac{\cosh (a+b x)}{12 x^3}-\frac{\cosh (3 a+3 b x)}{12 x^3}-\frac{1}{12} b \int \frac{\sinh (a+b x)}{x^3} \, dx+\frac{1}{4} b \int \frac{\sinh (3 a+3 b x)}{x^3} \, dx\\ &=\frac{\cosh (a+b x)}{12 x^3}-\frac{\cosh (3 a+3 b x)}{12 x^3}+\frac{b \sinh (a+b x)}{24 x^2}-\frac{b \sinh (3 a+3 b x)}{8 x^2}-\frac{1}{24} b^2 \int \frac{\cosh (a+b x)}{x^2} \, dx+\frac{1}{8} \left (3 b^2\right ) \int \frac{\cosh (3 a+3 b x)}{x^2} \, dx\\ &=\frac{\cosh (a+b x)}{12 x^3}+\frac{b^2 \cosh (a+b x)}{24 x}-\frac{\cosh (3 a+3 b x)}{12 x^3}-\frac{3 b^2 \cosh (3 a+3 b x)}{8 x}+\frac{b \sinh (a+b x)}{24 x^2}-\frac{b \sinh (3 a+3 b x)}{8 x^2}-\frac{1}{24} b^3 \int \frac{\sinh (a+b x)}{x} \, dx+\frac{1}{8} \left (9 b^3\right ) \int \frac{\sinh (3 a+3 b x)}{x} \, dx\\ &=\frac{\cosh (a+b x)}{12 x^3}+\frac{b^2 \cosh (a+b x)}{24 x}-\frac{\cosh (3 a+3 b x)}{12 x^3}-\frac{3 b^2 \cosh (3 a+3 b x)}{8 x}+\frac{b \sinh (a+b x)}{24 x^2}-\frac{b \sinh (3 a+3 b x)}{8 x^2}-\frac{1}{24} \left (b^3 \cosh (a)\right ) \int \frac{\sinh (b x)}{x} \, dx+\frac{1}{8} \left (9 b^3 \cosh (3 a)\right ) \int \frac{\sinh (3 b x)}{x} \, dx-\frac{1}{24} \left (b^3 \sinh (a)\right ) \int \frac{\cosh (b x)}{x} \, dx+\frac{1}{8} \left (9 b^3 \sinh (3 a)\right ) \int \frac{\cosh (3 b x)}{x} \, dx\\ &=\frac{\cosh (a+b x)}{12 x^3}+\frac{b^2 \cosh (a+b x)}{24 x}-\frac{\cosh (3 a+3 b x)}{12 x^3}-\frac{3 b^2 \cosh (3 a+3 b x)}{8 x}-\frac{1}{24} b^3 \text{Chi}(b x) \sinh (a)+\frac{9}{8} b^3 \text{Chi}(3 b x) \sinh (3 a)+\frac{b \sinh (a+b x)}{24 x^2}-\frac{b \sinh (3 a+3 b x)}{8 x^2}-\frac{1}{24} b^3 \cosh (a) \text{Shi}(b x)+\frac{9}{8} b^3 \cosh (3 a) \text{Shi}(3 b x)\\ \end{align*}

Mathematica [A]  time = 0.305473, size = 138, normalized size = 0.9 $\frac{-b^3 x^3 \sinh (a) \text{Chi}(b x)+27 b^3 x^3 \sinh (3 a) \text{Chi}(3 b x)-b^3 x^3 \cosh (a) \text{Shi}(b x)+27 b^3 x^3 \cosh (3 a) \text{Shi}(3 b x)+b^2 x^2 \cosh (a+b x)-9 b^2 x^2 \cosh (3 (a+b x))+b x \sinh (a+b x)-3 b x \sinh (3 (a+b x))+2 \cosh (a+b x)-2 \cosh (3 (a+b x))}{24 x^3}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(Cosh[a + b*x]*Sinh[a + b*x]^2)/x^4,x]

[Out]

(2*Cosh[a + b*x] + b^2*x^2*Cosh[a + b*x] - 2*Cosh[3*(a + b*x)] - 9*b^2*x^2*Cosh[3*(a + b*x)] - b^3*x^3*CoshInt
egral[b*x]*Sinh[a] + 27*b^3*x^3*CoshIntegral[3*b*x]*Sinh[3*a] + b*x*Sinh[a + b*x] - 3*b*x*Sinh[3*(a + b*x)] -
b^3*x^3*Cosh[a]*SinhIntegral[b*x] + 27*b^3*x^3*Cosh[3*a]*SinhIntegral[3*b*x])/(24*x^3)

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Maple [A]  time = 0.076, size = 234, normalized size = 1.5 \begin{align*} -{\frac{3\,{b}^{2}{{\rm e}^{-3\,bx-3\,a}}}{16\,x}}+{\frac{b{{\rm e}^{-3\,bx-3\,a}}}{16\,{x}^{2}}}-{\frac{{{\rm e}^{-3\,bx-3\,a}}}{24\,{x}^{3}}}+{\frac{9\,{b}^{3}{{\rm e}^{-3\,a}}{\it Ei} \left ( 1,3\,bx \right ) }{16}}+{\frac{{b}^{2}{{\rm e}^{-bx-a}}}{48\,x}}-{\frac{b{{\rm e}^{-bx-a}}}{48\,{x}^{2}}}+{\frac{{{\rm e}^{-bx-a}}}{24\,{x}^{3}}}-{\frac{{b}^{3}{{\rm e}^{-a}}{\it Ei} \left ( 1,bx \right ) }{48}}+{\frac{{{\rm e}^{bx+a}}}{24\,{x}^{3}}}+{\frac{b{{\rm e}^{bx+a}}}{48\,{x}^{2}}}+{\frac{{b}^{2}{{\rm e}^{bx+a}}}{48\,x}}+{\frac{{b}^{3}{{\rm e}^{a}}{\it Ei} \left ( 1,-bx \right ) }{48}}-{\frac{{{\rm e}^{3\,bx+3\,a}}}{24\,{x}^{3}}}-{\frac{b{{\rm e}^{3\,bx+3\,a}}}{16\,{x}^{2}}}-{\frac{3\,{b}^{2}{{\rm e}^{3\,bx+3\,a}}}{16\,x}}-{\frac{9\,{b}^{3}{{\rm e}^{3\,a}}{\it Ei} \left ( 1,-3\,bx \right ) }{16}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(b*x+a)*sinh(b*x+a)^2/x^4,x)

[Out]

-3/16*b^2*exp(-3*b*x-3*a)/x+1/16*b*exp(-3*b*x-3*a)/x^2-1/24*exp(-3*b*x-3*a)/x^3+9/16*b^3*exp(-3*a)*Ei(1,3*b*x)
+1/48*b^2*exp(-b*x-a)/x-1/48*b*exp(-b*x-a)/x^2+1/24*exp(-b*x-a)/x^3-1/48*b^3*exp(-a)*Ei(1,b*x)+1/24/x^3*exp(b*
x+a)+1/48*b/x^2*exp(b*x+a)+1/48*b^2/x*exp(b*x+a)+1/48*b^3*exp(a)*Ei(1,-b*x)-1/24/x^3*exp(3*b*x+3*a)-1/16*b/x^2
*exp(3*b*x+3*a)-3/16*b^2/x*exp(3*b*x+3*a)-9/16*b^3*exp(3*a)*Ei(1,-3*b*x)

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Maxima [A]  time = 1.34099, size = 78, normalized size = 0.51 \begin{align*} -\frac{27}{8} \, b^{3} e^{\left (-3 \, a\right )} \Gamma \left (-3, 3 \, b x\right ) + \frac{1}{8} \, b^{3} e^{\left (-a\right )} \Gamma \left (-3, b x\right ) - \frac{1}{8} \, b^{3} e^{a} \Gamma \left (-3, -b x\right ) + \frac{27}{8} \, b^{3} e^{\left (3 \, a\right )} \Gamma \left (-3, -3 \, 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)^2/x^4,x, algorithm="maxima")

[Out]

-27/8*b^3*e^(-3*a)*gamma(-3, 3*b*x) + 1/8*b^3*e^(-a)*gamma(-3, b*x) - 1/8*b^3*e^a*gamma(-3, -b*x) + 27/8*b^3*e
^(3*a)*gamma(-3, -3*b*x)

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Fricas [A]  time = 1.63779, size = 549, normalized size = 3.56 \begin{align*} -\frac{6 \, b x \sinh \left (b x + a\right )^{3} + 2 \,{\left (9 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )^{3} + 6 \,{\left (9 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - 2 \,{\left (b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right ) - 27 \,{\left (b^{3} x^{3}{\rm Ei}\left (3 \, b x\right ) - b^{3} x^{3}{\rm Ei}\left (-3 \, b x\right )\right )} \cosh \left (3 \, a\right ) +{\left (b^{3} x^{3}{\rm Ei}\left (b x\right ) - b^{3} x^{3}{\rm Ei}\left (-b x\right )\right )} \cosh \left (a\right ) + 2 \,{\left (9 \, b x \cosh \left (b x + a\right )^{2} - b x\right )} \sinh \left (b x + a\right ) - 27 \,{\left (b^{3} x^{3}{\rm Ei}\left (3 \, b x\right ) + b^{3} x^{3}{\rm Ei}\left (-3 \, b x\right )\right )} \sinh \left (3 \, a\right ) +{\left (b^{3} x^{3}{\rm Ei}\left (b x\right ) + b^{3} x^{3}{\rm Ei}\left (-b x\right )\right )} \sinh \left (a\right )}{48 \, x^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)*sinh(b*x+a)^2/x^4,x, algorithm="fricas")

[Out]

-1/48*(6*b*x*sinh(b*x + a)^3 + 2*(9*b^2*x^2 + 2)*cosh(b*x + a)^3 + 6*(9*b^2*x^2 + 2)*cosh(b*x + a)*sinh(b*x +
a)^2 - 2*(b^2*x^2 + 2)*cosh(b*x + a) - 27*(b^3*x^3*Ei(3*b*x) - b^3*x^3*Ei(-3*b*x))*cosh(3*a) + (b^3*x^3*Ei(b*x
) - b^3*x^3*Ei(-b*x))*cosh(a) + 2*(9*b*x*cosh(b*x + a)^2 - b*x)*sinh(b*x + a) - 27*(b^3*x^3*Ei(3*b*x) + b^3*x^
3*Ei(-3*b*x))*sinh(3*a) + (b^3*x^3*Ei(b*x) + b^3*x^3*Ei(-b*x))*sinh(a))/x^3

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh ^{2}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{x^{4}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)*sinh(b*x+a)**2/x**4,x)

[Out]

Integral(sinh(a + b*x)**2*cosh(a + b*x)/x**4, x)

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Giac [A]  time = 1.1941, size = 300, normalized size = 1.95 \begin{align*} \frac{27 \, b^{3} x^{3}{\rm Ei}\left (3 \, b x\right ) e^{\left (3 \, a\right )} + b^{3} x^{3}{\rm Ei}\left (-b x\right ) e^{\left (-a\right )} - 27 \, b^{3} x^{3}{\rm Ei}\left (-3 \, b x\right ) e^{\left (-3 \, a\right )} - b^{3} x^{3}{\rm Ei}\left (b x\right ) e^{a} - 9 \, b^{2} x^{2} e^{\left (3 \, b x + 3 \, a\right )} + b^{2} x^{2} e^{\left (b x + a\right )} + b^{2} x^{2} e^{\left (-b x - a\right )} - 9 \, b^{2} x^{2} e^{\left (-3 \, b x - 3 \, a\right )} - 3 \, b x e^{\left (3 \, b x + 3 \, a\right )} + b x e^{\left (b x + a\right )} - b x e^{\left (-b x - a\right )} + 3 \, b x e^{\left (-3 \, b x - 3 \, a\right )} - 2 \, e^{\left (3 \, b x + 3 \, a\right )} + 2 \, e^{\left (b x + a\right )} + 2 \, e^{\left (-b x - a\right )} - 2 \, e^{\left (-3 \, b x - 3 \, a\right )}}{48 \, x^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)*sinh(b*x+a)^2/x^4,x, algorithm="giac")

[Out]

1/48*(27*b^3*x^3*Ei(3*b*x)*e^(3*a) + b^3*x^3*Ei(-b*x)*e^(-a) - 27*b^3*x^3*Ei(-3*b*x)*e^(-3*a) - b^3*x^3*Ei(b*x
)*e^a - 9*b^2*x^2*e^(3*b*x + 3*a) + b^2*x^2*e^(b*x + a) + b^2*x^2*e^(-b*x - a) - 9*b^2*x^2*e^(-3*b*x - 3*a) -
3*b*x*e^(3*b*x + 3*a) + b*x*e^(b*x + a) - b*x*e^(-b*x - a) + 3*b*x*e^(-3*b*x - 3*a) - 2*e^(3*b*x + 3*a) + 2*e^
(b*x + a) + 2*e^(-b*x - a) - 2*e^(-3*b*x - 3*a))/x^3