Optimal. Leaf size=46 \[ -\frac {\text {Chi}(2 a+2 b x)}{2 b}+\frac {\text {Chi}(a+b x) \cosh (a+b x)}{b}-\frac {\log (a+b x)}{2 b} \]
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Rubi [A] time = 0.08, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {6547, 3312, 3301} \[ -\frac {\text {Chi}(2 a+2 b x)}{2 b}+\frac {\text {Chi}(a+b x) \cosh (a+b x)}{b}-\frac {\log (a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 3301
Rule 3312
Rule 6547
Rubi steps
\begin {align*} \int \text {Chi}(a+b x) \sinh (a+b x) \, dx &=\frac {\cosh (a+b x) \text {Chi}(a+b x)}{b}-\int \frac {\cosh ^2(a+b x)}{a+b x} \, dx\\ &=\frac {\cosh (a+b x) \text {Chi}(a+b x)}{b}-\int \left (\frac {1}{2 (a+b x)}+\frac {\cosh (2 a+2 b x)}{2 (a+b x)}\right ) \, dx\\ &=\frac {\cosh (a+b x) \text {Chi}(a+b x)}{b}-\frac {\log (a+b x)}{2 b}-\frac {1}{2} \int \frac {\cosh (2 a+2 b x)}{a+b x} \, dx\\ &=\frac {\cosh (a+b x) \text {Chi}(a+b x)}{b}-\frac {\text {Chi}(2 a+2 b x)}{2 b}-\frac {\log (a+b x)}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 45, normalized size = 0.98 \[ -\frac {\text {Chi}(2 (a+b x))}{2 b}+\frac {\text {Chi}(a+b x) \cosh (a+b x)}{b}-\frac {\log (a+b x)}{2 b} \]
Antiderivative was successfully verified.
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fricas [F] time = 4.05, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\operatorname {Chi}\left (b x + a\right ) \sinh \left (b x + a\right ), x\right ) \]
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 \[ \int {\rm Chi}\left (b x + a\right ) \sinh \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 43, normalized size = 0.93 \[ -\frac {\Chi \left (2 b x +2 a \right )}{2 b}+\frac {\Chi \left (b x +a \right ) \cosh \left (b x +a \right )}{b}-\frac {\ln \left (b x +a \right )}{2 b} \]
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 \[ \int {\rm Chi}\left (b x + a\right ) \sinh \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \mathrm {coshint}\left (a+b\,x\right )\,\mathrm {sinh}\left (a+b\,x\right ) \,d x \]
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 \[ \int \sinh {\left (a + b x \right )} \operatorname {Chi}\left (a + b x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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