3.114 \(\int \frac{\text{Chi}(b x) \sinh (b x)}{x^3} \, dx\)

Optimal. Leaf size=101 \[ \frac{1}{2} b^2 \text{CannotIntegrate}\left (\frac{\text{Chi}(b x) \sinh (b x)}{x},x\right )+b^2 \text{Shi}(2 b x)-\frac{\text{Chi}(b x) \sinh (b x)}{2 x^2}-\frac{b \text{Chi}(b x) \cosh (b x)}{2 x}-\frac{\sinh (2 b x)}{8 x^2}-\frac{b \cosh ^2(b x)}{2 x}-\frac{b \cosh (2 b x)}{4 x} \]

[Out]

(b^2*CannotIntegrate[(CoshIntegral[b*x]*Sinh[b*x])/x, x])/2 - (b*Cosh[b*x]^2)/(2*x) - (b*Cosh[2*b*x])/(4*x) -
(b*Cosh[b*x]*CoshIntegral[b*x])/(2*x) - (CoshIntegral[b*x]*Sinh[b*x])/(2*x^2) - Sinh[2*b*x]/(8*x^2) + b^2*Sinh
Integral[2*b*x]

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Rubi [A]  time = 0.206122, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\text{Chi}(b x) \sinh (b x)}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(CoshIntegral[b*x]*Sinh[b*x])/x^3,x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.428396, size = 0, normalized size = 0. \[ \int \frac{\text{Chi}(b x) \sinh (b x)}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(CoshIntegral[b*x]*Sinh[b*x])/x^3,x]

[Out]

Integrate[(CoshIntegral[b*x]*Sinh[b*x])/x^3, x]

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Maple [A]  time = 0.056, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it Chi} \left ( bx \right ) \sinh \left ( bx \right ) }{{x}^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(Chi(b*x)*sinh(b*x)/x^3,x)

[Out]

int(Chi(b*x)*sinh(b*x)/x^3,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Chi}\left (b x\right ) \sinh \left (b x\right )}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Chi(b*x)*sinh(b*x)/x^3,x, algorithm="maxima")

[Out]

integrate(Chi(b*x)*sinh(b*x)/x^3, x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{Chi}\left (b x\right ) \sinh \left (b x\right )}{x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Chi(b*x)*sinh(b*x)/x^3,x, algorithm="fricas")

[Out]

integral(cosh_integral(b*x)*sinh(b*x)/x^3, x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh{\left (b x \right )} \operatorname{Chi}\left (b x\right )}{x^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Chi(b*x)*sinh(b*x)/x**3,x)

[Out]

Integral(sinh(b*x)*Chi(b*x)/x**3, x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Chi}\left (b x\right ) \sinh \left (b x\right )}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Chi(b*x)*sinh(b*x)/x^3,x, algorithm="giac")

[Out]

integrate(Chi(b*x)*sinh(b*x)/x^3, x)