Optimal. Leaf size=62 \[ \frac {\text {Shi}(2 b x)}{2 b^2}-\frac {\text {Shi}(b x) \cosh (b x)}{b^2}-\frac {\sinh (b x) \cosh (b x)}{2 b^2}+\frac {x \text {Shi}(b x) \sinh (b x)}{b}+\frac {x}{2 b} \]
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Rubi [A] time = 0.07, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {6548, 12, 2635, 8, 6540, 5448, 3298} \[ \frac {\text {Shi}(2 b x)}{2 b^2}-\frac {\text {Shi}(b x) \cosh (b x)}{b^2}-\frac {\sinh (b x) \cosh (b x)}{2 b^2}+\frac {x \text {Shi}(b x) \sinh (b x)}{b}+\frac {x}{2 b} \]
Antiderivative was successfully verified.
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Rule 8
Rule 12
Rule 2635
Rule 3298
Rule 5448
Rule 6540
Rule 6548
Rubi steps
\begin {align*} \int x \cosh (b x) \text {Shi}(b x) \, dx &=\frac {x \sinh (b x) \text {Shi}(b x)}{b}-\frac {\int \sinh (b x) \text {Shi}(b x) \, dx}{b}-\int \frac {\sinh ^2(b x)}{b} \, dx\\ &=-\frac {\cosh (b x) \text {Shi}(b x)}{b^2}+\frac {x \sinh (b x) \text {Shi}(b x)}{b}+\frac {\int \frac {\cosh (b x) \sinh (b x)}{b x} \, dx}{b}-\frac {\int \sinh ^2(b x) \, dx}{b}\\ &=-\frac {\cosh (b x) \sinh (b x)}{2 b^2}-\frac {\cosh (b x) \text {Shi}(b x)}{b^2}+\frac {x \sinh (b x) \text {Shi}(b x)}{b}+\frac {\int \frac {\cosh (b x) \sinh (b x)}{x} \, dx}{b^2}+\frac {\int 1 \, dx}{2 b}\\ &=\frac {x}{2 b}-\frac {\cosh (b x) \sinh (b x)}{2 b^2}-\frac {\cosh (b x) \text {Shi}(b x)}{b^2}+\frac {x \sinh (b x) \text {Shi}(b x)}{b}+\frac {\int \frac {\sinh (2 b x)}{2 x} \, dx}{b^2}\\ &=\frac {x}{2 b}-\frac {\cosh (b x) \sinh (b x)}{2 b^2}-\frac {\cosh (b x) \text {Shi}(b x)}{b^2}+\frac {x \sinh (b x) \text {Shi}(b x)}{b}+\frac {\int \frac {\sinh (2 b x)}{x} \, dx}{2 b^2}\\ &=\frac {x}{2 b}-\frac {\cosh (b x) \sinh (b x)}{2 b^2}-\frac {\cosh (b x) \text {Shi}(b x)}{b^2}+\frac {x \sinh (b x) \text {Shi}(b x)}{b}+\frac {\text {Shi}(2 b x)}{2 b^2}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 46, normalized size = 0.74 \[ \frac {2 \text {Shi}(2 b x)+4 \text {Shi}(b x) (b x \sinh (b x)-\cosh (b x))+2 b x-\sinh (2 b x)}{4 b^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x \cosh \left (b x\right ) \operatorname {Shi}\left (b x\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x {\rm Shi}\left (b x\right ) \cosh \left (b x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 46, normalized size = 0.74 \[ \frac {\Shi \left (b x \right ) \left (b x \sinh \left (b x \right )-\cosh \left (b x \right )\right )-\frac {\sinh \left (b x \right ) \cosh \left (b x \right )}{2}+\frac {b x}{2}+\frac {\Shi \left (2 b x \right )}{2}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x {\rm Shi}\left (b x\right ) \cosh \left (b x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x\,\mathrm {sinhint}\left (b\,x\right )\,\mathrm {cosh}\left (b\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \cosh {\left (b x \right )} \operatorname {Shi}{\left (b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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