Integrand size = 13, antiderivative size = 153 \[ \int \sinh (a+b x) \text {Shi}(c+d x) \, dx=\frac {\text {Chi}\left (\frac {c (b-d)}{d}+(b-d) x\right ) \sinh \left (a-\frac {b c}{d}\right )}{2 b}-\frac {\text {Chi}\left (\frac {c (b+d)}{d}+(b+d) x\right ) \sinh \left (a-\frac {b c}{d}\right )}{2 b}+\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b}+\frac {\cosh (a+b x) \text {Shi}(c+d x)}{b}-\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {c (b+d)}{d}+(b+d) x\right )}{2 b} \]
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Time = 0.21 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {6675, 5580, 3384, 3379, 3382} \[ \int \sinh (a+b x) \text {Shi}(c+d x) \, dx=\frac {\sinh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 b}-\frac {\sinh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 b}+\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 b}+\frac {\cosh (a+b x) \text {Shi}(c+d x)}{b}-\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 b} \]
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Rule 3379
Rule 3382
Rule 3384
Rule 5580
Rule 6675
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.68 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.90 \[ \int \sinh (a+b x) \text {Shi}(c+d x) \, dx=\frac {e^{-a-\frac {b c}{d}} \left (-e^{\frac {2 b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {(b-d) (c+d x)}{d}\right )+e^{2 a} \operatorname {ExpIntegralEi}\left (\frac {(b-d) (c+d x)}{d}\right )+e^{\frac {2 b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {(b+d) (c+d x)}{d}\right )-e^{2 a} \operatorname {ExpIntegralEi}\left (\frac {(b+d) (c+d x)}{d}\right )+4 e^{a+\frac {b c}{d}} \cosh (a+b x) \text {Shi}(c+d x)\right )}{4 b} \]
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\[\int \operatorname {Shi}\left (d x +c \right ) \sinh \left (b x +a \right )d x\]
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\[ \int \sinh (a+b x) \text {Shi}(c+d x) \, dx=\int { {\rm Shi}\left (d x + c\right ) \sinh \left (b x + a\right ) \,d x } \]
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\[ \int \sinh (a+b x) \text {Shi}(c+d x) \, dx=\int \sinh {\left (a + b x \right )} \operatorname {Shi}{\left (c + d x \right )}\, dx \]
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\[ \int \sinh (a+b x) \text {Shi}(c+d x) \, dx=\int { {\rm Shi}\left (d x + c\right ) \sinh \left (b x + a\right ) \,d x } \]
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\[ \int \sinh (a+b x) \text {Shi}(c+d x) \, dx=\int { {\rm Shi}\left (d x + c\right ) \sinh \left (b x + a\right ) \,d x } \]
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Timed out. \[ \int \sinh (a+b x) \text {Shi}(c+d x) \, dx=\int \mathrm {sinhint}\left (c+d\,x\right )\,\mathrm {sinh}\left (a+b\,x\right ) \,d x \]
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