Optimal. Leaf size=46 \[ -\frac {\text {Chi}(2 a+2 b x)}{2 b}+\frac {\log (a+b x)}{2 b}+\frac {\sinh (a+b x) \text {Shi}(a+b x)}{b} \]
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Rubi [A]
time = 0.05, 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 = {6681, 3393,
3382} \begin {gather*} -\frac {\text {Chi}(2 a+2 b x)}{2 b}+\frac {\text {Shi}(a+b x) \sinh (a+b x)}{b}+\frac {\log (a+b x)}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 3382
Rule 3393
Rule 6681
Rubi steps
\begin {align*} \int \cosh (a+b x) \text {Shi}(a+b x) \, dx &=\frac {\sinh (a+b x) \text {Shi}(a+b x)}{b}-\int \frac {\sinh ^2(a+b x)}{a+b x} \, dx\\ &=\frac {\sinh (a+b x) \text {Shi}(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 {\log (a+b x)}{2 b}+\frac {\sinh (a+b x) \text {Shi}(a+b x)}{b}-\frac {1}{2} \int \frac {\cosh (2 a+2 b x)}{a+b x} \, dx\\ &=-\frac {\text {Chi}(2 a+2 b x)}{2 b}+\frac {\log (a+b x)}{2 b}+\frac {\sinh (a+b x) \text {Shi}(a+b x)}{b}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 45, normalized size = 0.98 \begin {gather*} -\frac {\text {Chi}(2 (a+b x))}{2 b}+\frac {\log (a+b x)}{2 b}+\frac {\sinh (a+b x) \text {Shi}(a+b x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.37, size = 38, normalized size = 0.83
method | result | size |
derivativedivides | \(\frac {\hyperbolicSineIntegral \left (b x +a \right ) \sinh \left (b x +a \right )+\frac {\ln \left (b x +a \right )}{2}-\frac {\hyperbolicCosineIntegral \left (2 b x +2 a \right )}{2}}{b}\) | \(38\) |
default | \(\frac {\hyperbolicSineIntegral \left (b x +a \right ) \sinh \left (b x +a \right )+\frac {\ln \left (b x +a \right )}{2}-\frac {\hyperbolicCosineIntegral \left (2 b x +2 a \right )}{2}}{b}\) | \(38\) |
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 \cosh {\left (a + b x \right )} \operatorname {Shi}{\left (a + 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 = -0.02 \begin {gather*} \int \mathrm {sinhint}\left (a+b\,x\right )\,\mathrm {cosh}\left (a+b\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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