Optimal. Leaf size=34 \[ \frac {\cosh (b x) \text {Chi}(b x)}{b}-\frac {\text {Chi}(2 b x)}{2 b}-\frac {\log (x)}{2 b} \]
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Rubi [A]
time = 0.04, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {6682, 12, 3393,
3382} \begin {gather*} -\frac {\text {Chi}(2 b x)}{2 b}+\frac {\text {Chi}(b x) \cosh (b x)}{b}-\frac {\log (x)}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 3382
Rule 3393
Rule 6682
Rubi steps
\begin {align*} \int \text {Chi}(b x) \sinh (b x) \, dx &=\frac {\cosh (b x) \text {Chi}(b x)}{b}-\int \frac {\cosh ^2(b x)}{b x} \, dx\\ &=\frac {\cosh (b x) \text {Chi}(b x)}{b}-\frac {\int \frac {\cosh ^2(b x)}{x} \, dx}{b}\\ &=\frac {\cosh (b x) \text {Chi}(b x)}{b}-\frac {\int \left (\frac {1}{2 x}+\frac {\cosh (2 b x)}{2 x}\right ) \, dx}{b}\\ &=\frac {\cosh (b x) \text {Chi}(b x)}{b}-\frac {\log (x)}{2 b}-\frac {\int \frac {\cosh (2 b x)}{x} \, dx}{2 b}\\ &=\frac {\cosh (b x) \text {Chi}(b x)}{b}-\frac {\text {Chi}(2 b x)}{2 b}-\frac {\log (x)}{2 b}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 36, normalized size = 1.06 \begin {gather*} \frac {\cosh (b x) \text {Chi}(b x)}{b}-\frac {\text {Chi}(2 b x)}{2 b}-\frac {\log (b x)}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.28, size = 28, normalized size = 0.82
method | result | size |
derivativedivides | \(\frac {\hyperbolicCosineIntegral \left (b x \right ) \cosh \left (b x \right )-\frac {\ln \left (b x \right )}{2}-\frac {\hyperbolicCosineIntegral \left (2 b x \right )}{2}}{b}\) | \(28\) |
default | \(\frac {\hyperbolicCosineIntegral \left (b x \right ) \cosh \left (b x \right )-\frac {\ln \left (b x \right )}{2}-\frac {\hyperbolicCosineIntegral \left (2 b x \right )}{2}}{b}\) | \(28\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \sinh {\left (b x \right )} \operatorname {Chi}\left (b x\right )\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \mathrm {coshint}\left (b\,x\right )\,\mathrm {sinh}\left (b\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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