Optimal. Leaf size=33 \[ \frac {\text {Chi}(a+b x) \sinh (a+b x)}{b}-\frac {\text {Shi}(2 a+2 b x)}{2 b} \]
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Rubi [A]
time = 0.04, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {6676, 5556, 12,
3379} \begin {gather*} \frac {\text {Chi}(a+b x) \sinh (a+b x)}{b}-\frac {\text {Shi}(2 a+2 b x)}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 3379
Rule 5556
Rule 6676
Rubi steps
\begin {align*} \int \cosh (a+b x) \text {Chi}(a+b x) \, dx &=\frac {\text {Chi}(a+b x) \sinh (a+b x)}{b}-\int \frac {\cosh (a+b x) \sinh (a+b x)}{a+b x} \, dx\\ &=\frac {\text {Chi}(a+b x) \sinh (a+b x)}{b}-\int \frac {\sinh (2 a+2 b x)}{2 (a+b x)} \, dx\\ &=\frac {\text {Chi}(a+b x) \sinh (a+b x)}{b}-\frac {1}{2} \int \frac {\sinh (2 a+2 b x)}{a+b x} \, dx\\ &=\frac {\text {Chi}(a+b x) \sinh (a+b x)}{b}-\frac {\text {Shi}(2 a+2 b x)}{2 b}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 32, normalized size = 0.97 \begin {gather*} \frac {\text {Chi}(a+b x) \sinh (a+b x)}{b}-\frac {\text {Shi}(2 (a+b x))}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.25, size = 30, normalized size = 0.91
method | result | size |
derivativedivides | \(\frac {\hyperbolicCosineIntegral \left (b x +a \right ) \sinh \left (b x +a \right )-\frac {\hyperbolicSineIntegral \left (2 b x +2 a \right )}{2}}{b}\) | \(30\) |
default | \(\frac {\hyperbolicCosineIntegral \left (b x +a \right ) \sinh \left (b x +a \right )-\frac {\hyperbolicSineIntegral \left (2 b x +2 a \right )}{2}}{b}\) | \(30\) |
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 {Chi}\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 = -0.03 \begin {gather*} \int \mathrm {coshint}\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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