Optimal. Leaf size=29 \[ \frac {\text {Chi}(3 x)}{10}-\frac {\text {Chi}(7 x)}{10}+\frac {1}{5} \sinh (5 x) \text {Shi}(2 x) \]
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Rubi [A]
time = 0.04, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {6681, 12, 5578,
3382} \begin {gather*} \frac {\text {Chi}(3 x)}{10}-\frac {\text {Chi}(7 x)}{10}+\frac {1}{5} \text {Shi}(2 x) \sinh (5 x) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 3382
Rule 5578
Rule 6681
Rubi steps
\begin {align*} \int \cosh (5 x) \text {Shi}(2 x) \, dx &=\frac {1}{5} \sinh (5 x) \text {Shi}(2 x)-\frac {2}{5} \int \frac {\sinh (2 x) \sinh (5 x)}{2 x} \, dx\\ &=\frac {1}{5} \sinh (5 x) \text {Shi}(2 x)-\frac {1}{5} \int \frac {\sinh (2 x) \sinh (5 x)}{x} \, dx\\ &=\frac {1}{5} \sinh (5 x) \text {Shi}(2 x)-\frac {1}{5} \int \left (-\frac {\cosh (3 x)}{2 x}+\frac {\cosh (7 x)}{2 x}\right ) \, dx\\ &=\frac {1}{5} \sinh (5 x) \text {Shi}(2 x)+\frac {1}{10} \int \frac {\cosh (3 x)}{x} \, dx-\frac {1}{10} \int \frac {\cosh (7 x)}{x} \, dx\\ &=\frac {\text {Chi}(3 x)}{10}-\frac {\text {Chi}(7 x)}{10}+\frac {1}{5} \sinh (5 x) \text {Shi}(2 x)\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 25, normalized size = 0.86 \begin {gather*} \frac {1}{10} (\text {Chi}(3 x)-\text {Chi}(7 x)+2 \sinh (5 x) \text {Shi}(2 x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.93, size = 24, normalized size = 0.83
method | result | size |
default | \(\frac {\hyperbolicCosineIntegral \left (3 x \right )}{10}-\frac {\hyperbolicCosineIntegral \left (7 x \right )}{10}+\frac {\hyperbolicSineIntegral \left (2 x \right ) \sinh \left (5 x \right )}{5}\) | \(24\) |
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 (5 x \right )} \operatorname {Shi}{\left (2 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 {sinhint}\left (2\,x\right )\,\mathrm {cosh}\left (5\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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