Optimal. Leaf size=78 \[ \frac {\cosh \left (2 a-\frac {2 b c}{d}\right ) \text {Chi}\left (\frac {2 b c}{d}+2 b x\right )}{2 d}+\frac {\sinh \left (2 a-\frac {2 b c}{d}\right ) \text {Shi}\left (\frac {2 b c}{d}+2 b x\right )}{2 d}+\frac {\log (c+d x)}{2 d} \]
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Rubi [A] time = 0.15, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3312, 3303, 3298, 3301} \[ \frac {\cosh \left (2 a-\frac {2 b c}{d}\right ) \text {Chi}\left (\frac {2 b c}{d}+2 b x\right )}{2 d}+\frac {\sinh \left (2 a-\frac {2 b c}{d}\right ) \text {Shi}\left (\frac {2 b c}{d}+2 b x\right )}{2 d}+\frac {\log (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 3298
Rule 3301
Rule 3303
Rule 3312
Rubi steps
\begin {align*} \int \frac {\cosh ^2(a+b x)}{c+d x} \, dx &=\int \left (\frac {1}{2 (c+d x)}+\frac {\cosh (2 a+2 b x)}{2 (c+d x)}\right ) \, dx\\ &=\frac {\log (c+d x)}{2 d}+\frac {1}{2} \int \frac {\cosh (2 a+2 b x)}{c+d x} \, dx\\ &=\frac {\log (c+d x)}{2 d}+\frac {1}{2} \cosh \left (2 a-\frac {2 b c}{d}\right ) \int \frac {\cosh \left (\frac {2 b c}{d}+2 b x\right )}{c+d x} \, dx+\frac {1}{2} \sinh \left (2 a-\frac {2 b c}{d}\right ) \int \frac {\sinh \left (\frac {2 b c}{d}+2 b x\right )}{c+d x} \, dx\\ &=\frac {\cosh \left (2 a-\frac {2 b c}{d}\right ) \text {Chi}\left (\frac {2 b c}{d}+2 b x\right )}{2 d}+\frac {\log (c+d x)}{2 d}+\frac {\sinh \left (2 a-\frac {2 b c}{d}\right ) \text {Shi}\left (\frac {2 b c}{d}+2 b x\right )}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 64, normalized size = 0.82 \[ \frac {\cosh \left (2 a-\frac {2 b c}{d}\right ) \text {Chi}\left (\frac {2 b (c+d x)}{d}\right )+\sinh \left (2 a-\frac {2 b c}{d}\right ) \text {Shi}\left (\frac {2 b (c+d x)}{d}\right )+\log (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 104, normalized size = 1.33 \[ \frac {{\left ({\rm Ei}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) + {\rm Ei}\left (-\frac {2 \, {\left (b d x + b c\right )}}{d}\right )\right )} \cosh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + {\left ({\rm Ei}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) - {\rm Ei}\left (-\frac {2 \, {\left (b d x + b c\right )}}{d}\right )\right )} \sinh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + 2 \, \log \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 68, normalized size = 0.87 \[ \frac {{\rm Ei}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (2 \, a - \frac {2 \, b c}{d}\right )} + {\rm Ei}\left (-\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (-2 \, a + \frac {2 \, b c}{d}\right )} + 2 \, \log \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 97, normalized size = 1.24 \[ \frac {\ln \left (d x +c \right )}{2 d}-\frac {{\mathrm e}^{-\frac {2 \left (d a -c b \right )}{d}} \Ei \left (1, 2 b x +2 a -\frac {2 \left (d a -c b \right )}{d}\right )}{4 d}-\frac {{\mathrm e}^{\frac {2 d a -2 c b}{d}} \Ei \left (1, -2 b x -2 a -\frac {2 \left (-d a +c b \right )}{d}\right )}{4 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 72, normalized size = 0.92 \[ -\frac {e^{\left (-2 \, a + \frac {2 \, b c}{d}\right )} E_{1}\left (\frac {2 \, {\left (d x + c\right )} b}{d}\right )}{4 \, d} - \frac {e^{\left (2 \, a - \frac {2 \, b c}{d}\right )} E_{1}\left (-\frac {2 \, {\left (d x + c\right )} b}{d}\right )}{4 \, d} + \frac {\log \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^2}{c+d\,x} \,d x \]
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 \[ \int \frac {\cosh ^{2}{\left (a + b x \right )}}{c + d x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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