Optimal. Leaf size=153 \[ \frac{\sinh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (x (b-d)+\frac{c (b-d)}{d}\right )}{2 b}-\frac{\sinh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (x (b+d)+\frac{c (b+d)}{d}\right )}{2 b}+\frac{\cosh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (x (b-d)+\frac{c (b-d)}{d}\right )}{2 b}+\frac{\cosh (a+b x) \text{Shi}(c+d x)}{b}-\frac{\cosh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (x (b+d)+\frac{c (b+d)}{d}\right )}{2 b} \]
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Rubi [A] time = 0.251837, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {6540, 5472, 3303, 3298, 3301} \[ \frac{\sinh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (x (b-d)+\frac{c (b-d)}{d}\right )}{2 b}-\frac{\sinh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (x (b+d)+\frac{c (b+d)}{d}\right )}{2 b}+\frac{\cosh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (x (b-d)+\frac{c (b-d)}{d}\right )}{2 b}+\frac{\cosh (a+b x) \text{Shi}(c+d x)}{b}-\frac{\cosh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (x (b+d)+\frac{c (b+d)}{d}\right )}{2 b} \]
Antiderivative was successfully verified.
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Rule 6540
Rule 5472
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \sinh (a+b x) \text{Shi}(c+d x) \, dx &=\frac{\cosh (a+b x) \text{Shi}(c+d x)}{b}-\frac{d \int \frac{\cosh (a+b x) \sinh (c+d x)}{c+d x} \, dx}{b}\\ &=\frac{\cosh (a+b x) \text{Shi}(c+d x)}{b}-\frac{d \int \left (-\frac{\sinh (a-c+(b-d) x)}{2 (c+d x)}+\frac{\sinh (a+c+(b+d) x)}{2 (c+d x)}\right ) \, dx}{b}\\ &=\frac{\cosh (a+b x) \text{Shi}(c+d x)}{b}+\frac{d \int \frac{\sinh (a-c+(b-d) x)}{c+d x} \, dx}{2 b}-\frac{d \int \frac{\sinh (a+c+(b+d) x)}{c+d x} \, dx}{2 b}\\ &=\frac{\cosh (a+b x) \text{Shi}(c+d x)}{b}+\frac{\left (d \cosh \left (a-\frac{b c}{d}\right )\right ) \int \frac{\sinh \left (\frac{c (b-d)}{d}+(b-d) x\right )}{c+d x} \, dx}{2 b}-\frac{\left (d \cosh \left (a-\frac{b c}{d}\right )\right ) \int \frac{\sinh \left (\frac{c (b+d)}{d}+(b+d) x\right )}{c+d x} \, dx}{2 b}+\frac{\left (d \sinh \left (a-\frac{b c}{d}\right )\right ) \int \frac{\cosh \left (\frac{c (b-d)}{d}+(b-d) x\right )}{c+d x} \, dx}{2 b}-\frac{\left (d \sinh \left (a-\frac{b c}{d}\right )\right ) \int \frac{\cosh \left (\frac{c (b+d)}{d}+(b+d) x\right )}{c+d x} \, dx}{2 b}\\ &=\frac{\text{Chi}\left (\frac{c (b-d)}{d}+(b-d) x\right ) \sinh \left (a-\frac{b c}{d}\right )}{2 b}-\frac{\text{Chi}\left (\frac{c (b+d)}{d}+(b+d) x\right ) \sinh \left (a-\frac{b c}{d}\right )}{2 b}+\frac{\cosh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (\frac{c (b-d)}{d}+(b-d) x\right )}{2 b}+\frac{\cosh (a+b x) \text{Shi}(c+d x)}{b}-\frac{\cosh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (\frac{c (b+d)}{d}+(b+d) x\right )}{2 b}\\ \end{align*}
Mathematica [A] time = 2.27803, size = 209, normalized size = 1.37 \[ \frac{2 \sinh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (-\frac{(b-d) (c+d x)}{d}\right )-2 \sinh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (\frac{(b+d) (c+d x)}{d}\right )+\sinh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (\frac{(b-d) (c+d x)}{d}\right )+\sinh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (-\frac{b c}{d}+c-b x+d x\right )+4 \cosh (a+b x) \text{Shi}(c+d x)+\cosh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (\frac{(b-d) (c+d x)}{d}\right )-2 \cosh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (\frac{(b+d) (c+d x)}{d}\right )-\cosh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (-\frac{b c}{d}+c-b x+d x\right )}{4 b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.132, size = 0, normalized size = 0. \begin{align*} \int{\it Shi} \left ( dx+c \right ) \sinh \left ( bx+a \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\rm Shi}\left (d x + c\right ) \sinh \left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sinh \left (b x + a\right ) \operatorname{Shi}\left (d x + c\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sinh{\left (a + b x \right )} \operatorname{Shi}{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\rm Shi}\left (d x + c\right ) \sinh \left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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