3.1.46 \(\int \frac {\cosh (b x) \text {Shi}(b x)}{x^3} \, dx\) [46]

Optimal. Leaf size=102 \[ -\frac {b \cosh (2 b x)}{4 x}-\frac {b \sinh ^2(b x)}{2 x}-\frac {\sinh (2 b x)}{8 x^2}-\frac {\cosh (b x) \text {Shi}(b x)}{2 x^2}-\frac {b \sinh (b x) \text {Shi}(b x)}{2 x}+b^2 \text {Shi}(2 b x)+\frac {1}{2} b^2 \text {Int}\left (\frac {\cosh (b x) \text {Shi}(b x)}{x},x\right ) \]

[Out]

1/2*b^2*CannotIntegrate(cosh(b*x)*Shi(b*x)/x,x)-1/4*b*cosh(2*b*x)/x-1/2*cosh(b*x)*Shi(b*x)/x^2+b^2*Shi(2*b*x)-
1/2*b*Shi(b*x)*sinh(b*x)/x-1/2*b*sinh(b*x)^2/x-1/8*sinh(2*b*x)/x^2

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Rubi [A]
time = 0.14, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\cosh (b x) \text {Shi}(b x)}{x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(Cosh[b*x]*SinhIntegral[b*x])/x^3,x]

[Out]

-1/4*(b*Cosh[2*b*x])/x - (b*Sinh[b*x]^2)/(2*x) - Sinh[2*b*x]/(8*x^2) - (Cosh[b*x]*SinhIntegral[b*x])/(2*x^2) -
(b*Sinh[b*x]*SinhIntegral[b*x])/(2*x) + b^2*SinhIntegral[2*b*x] + (b^2*Defer[Int][(Cosh[b*x]*SinhIntegral[b*x
])/x, x])/2

Rubi steps

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

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Mathematica [A]
time = 0.32, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cosh (b x) \text {Shi}(b x)}{x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(Cosh[b*x]*SinhIntegral[b*x])/x^3,x]

[Out]

Integrate[(Cosh[b*x]*SinhIntegral[b*x])/x^3, x]

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Maple [A]
time = 0.18, size = 0, normalized size = 0.00 \[\int \frac {\cosh \left (b x \right ) \hyperbolicSineIntegral \left (b x \right )}{x^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(b*x)*Shi(b*x)/x^3,x)

[Out]

int(cosh(b*x)*Shi(b*x)/x^3,x)

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x)*Shi(b*x)/x^3,x, algorithm="maxima")

[Out]

integrate(Shi(b*x)*cosh(b*x)/x^3, x)

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x)*Shi(b*x)/x^3,x, algorithm="fricas")

[Out]

integral(cosh(b*x)*sinh_integral(b*x)/x^3, x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cosh {\left (b x \right )} \operatorname {Shi}{\left (b x \right )}}{x^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x)*Shi(b*x)/x**3,x)

[Out]

Integral(cosh(b*x)*Shi(b*x)/x**3, x)

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x)*Shi(b*x)/x^3,x, algorithm="giac")

[Out]

integrate(Shi(b*x)*cosh(b*x)/x^3, x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\mathrm {sinhint}\left (b\,x\right )\,\mathrm {cosh}\left (b\,x\right )}{x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((sinhint(b*x)*cosh(b*x))/x^3,x)

[Out]

int((sinhint(b*x)*cosh(b*x))/x^3, x)

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