Optimal. Leaf size=102 \[ \frac {1}{2} b^2 \text {Int}\left (\frac {\text {Shi}(b x) \cosh (b x)}{x},x\right )+b^2 \text {Shi}(2 b x)-\frac {\text {Shi}(b x) \cosh (b x)}{2 x^2}-\frac {b \text {Shi}(b x) \sinh (b x)}{2 x}-\frac {\sinh (2 b x)}{8 x^2}-\frac {b \sinh ^2(b x)}{2 x}-\frac {b \cosh (2 b x)}{4 x} \]
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Rubi [A] time = 0.20, 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 = {} \[ \int \frac {\cosh (b x) \text {Shi}(b x)}{x^3} \, dx \]
Verification is Not applicable to the result.
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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 = 0.44, size = 0, normalized size = 0.00 \[ \int \frac {\cosh (b x) \text {Shi}(b x)}{x^3} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cosh \left (b x\right ) \operatorname {Shi}\left (b x\right )}{x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm Shi}\left (b x\right ) \cosh \left (b x\right )}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {\cosh \left (b x \right ) \Shi \left (b x \right )}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm Shi}\left (b x\right ) \cosh \left (b x\right )}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {sinhint}\left (b\,x\right )\,\mathrm {cosh}\left (b\,x\right )}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh {\left (b x \right )} \operatorname {Shi}{\left (b x \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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