Optimal. Leaf size=44 \[ b \text {Chi}(2 b x)-\frac {\sinh (2 b x)}{2 x}-\frac {\cosh (b x) \text {Shi}(b x)}{x}+\frac {1}{2} b \text {Shi}(b x)^2 \]
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Rubi [A]
time = 0.07, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6685, 6818, 12,
5556, 3378, 3382} \begin {gather*} b \text {Chi}(2 b x)+\frac {1}{2} b \text {Shi}(b x)^2-\frac {\text {Shi}(b x) \cosh (b x)}{x}-\frac {\sinh (2 b x)}{2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 3378
Rule 3382
Rule 5556
Rule 6685
Rule 6818
Rubi steps
\begin {align*} \int \frac {\cosh (b x) \text {Shi}(b x)}{x^2} \, dx &=-\frac {\cosh (b x) \text {Shi}(b x)}{x}+b \int \frac {\cosh (b x) \sinh (b x)}{b x^2} \, dx+b \int \frac {\sinh (b x) \text {Shi}(b x)}{x} \, dx\\ &=-\frac {\cosh (b x) \text {Shi}(b x)}{x}+\frac {1}{2} b \text {Shi}(b x)^2+\int \frac {\cosh (b x) \sinh (b x)}{x^2} \, dx\\ &=-\frac {\cosh (b x) \text {Shi}(b x)}{x}+\frac {1}{2} b \text {Shi}(b x)^2+\int \frac {\sinh (2 b x)}{2 x^2} \, dx\\ &=-\frac {\cosh (b x) \text {Shi}(b x)}{x}+\frac {1}{2} b \text {Shi}(b x)^2+\frac {1}{2} \int \frac {\sinh (2 b x)}{x^2} \, dx\\ &=-\frac {\sinh (2 b x)}{2 x}-\frac {\cosh (b x) \text {Shi}(b x)}{x}+\frac {1}{2} b \text {Shi}(b x)^2+b \int \frac {\cosh (2 b x)}{x} \, dx\\ &=b \text {Chi}(2 b x)-\frac {\sinh (2 b x)}{2 x}-\frac {\cosh (b x) \text {Shi}(b x)}{x}+\frac {1}{2} b \text {Shi}(b x)^2\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 44, normalized size = 1.00 \begin {gather*} b \text {Chi}(2 b x)-\frac {\sinh (2 b x)}{2 x}-\frac {\cosh (b x) \text {Shi}(b x)}{x}+\frac {1}{2} b \text {Shi}(b x)^2 \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.17, size = 0, normalized size = 0.00 \[\int \frac {\cosh \left (b x \right ) \hyperbolicSineIntegral \left (b x \right )}{x^{2}}\, dx\]
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 \frac {\cosh {\left (b x \right )} \operatorname {Shi}{\left (b x \right )}}{x^{2}}\, 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 \frac {\mathrm {sinhint}\left (b\,x\right )\,\mathrm {cosh}\left (b\,x\right )}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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