Optimal. Leaf size=33 \[ \frac {\text {Shi}(a+b x) \cosh (a+b x)}{b}-\frac {\text {Shi}(2 a+2 b x)}{2 b} \]
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Rubi [A] time = 0.06, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {6540, 5448, 12, 3298} \[ \frac {\text {Shi}(a+b x) \cosh (a+b x)}{b}-\frac {\text {Shi}(2 a+2 b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3298
Rule 5448
Rule 6540
Rubi steps
\begin {align*} \int \sinh (a+b x) \text {Shi}(a+b x) \, dx &=\frac {\cosh (a+b x) \text {Shi}(a+b x)}{b}-\int \frac {\cosh (a+b x) \sinh (a+b x)}{a+b x} \, dx\\ &=\frac {\cosh (a+b x) \text {Shi}(a+b x)}{b}-\int \frac {\sinh (2 a+2 b x)}{2 (a+b x)} \, dx\\ &=\frac {\cosh (a+b x) \text {Shi}(a+b x)}{b}-\frac {1}{2} \int \frac {\sinh (2 a+2 b x)}{a+b x} \, dx\\ &=\frac {\cosh (a+b x) \text {Shi}(a+b x)}{b}-\frac {\text {Shi}(2 a+2 b x)}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 32, normalized size = 0.97 \[ \frac {\text {Shi}(a+b x) \cosh (a+b x)}{b}-\frac {\text {Shi}(2 (a+b x))}{2 b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sinh \left (b x + a\right ) \operatorname {Shi}\left (b x + a\right ), x\right ) \]
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 \[ \int {\rm Shi}\left (b x + a\right ) \sinh \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 30, normalized size = 0.91 \[ \frac {\cosh \left (b x +a \right ) \Shi \left (b x +a \right )-\frac {\Shi \left (2 b x +2 a \right )}{2}}{b} \]
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 \[ \int {\rm Shi}\left (b x + a\right ) \sinh \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \mathrm {sinhint}\left (a+b\,x\right )\,\mathrm {sinh}\left (a+b\,x\right ) \,d x \]
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 \[ \int \sinh {\left (a + b x \right )} \operatorname {Shi}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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