Optimal. Leaf size=73 \[ -\frac {\cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{a}-\frac {\sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{a}+\frac {\cosh (c) \text {Chi}(d x)}{a}+\frac {\sinh (c) \text {Shi}(d x)}{a} \]
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Rubi [A] time = 0.26, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {6742, 3303, 3298, 3301} \[ -\frac {\cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{a}-\frac {\sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{a}+\frac {\cosh (c) \text {Chi}(d x)}{a}+\frac {\sinh (c) \text {Shi}(d x)}{a} \]
Antiderivative was successfully verified.
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Rule 3298
Rule 3301
Rule 3303
Rule 6742
Rubi steps
\begin {align*} \int \frac {\cosh (c+d x)}{x (a+b x)} \, dx &=\int \left (\frac {\cosh (c+d x)}{a x}-\frac {b \cosh (c+d x)}{a (a+b x)}\right ) \, dx\\ &=\frac {\int \frac {\cosh (c+d x)}{x} \, dx}{a}-\frac {b \int \frac {\cosh (c+d x)}{a+b x} \, dx}{a}\\ &=\frac {\cosh (c) \int \frac {\cosh (d x)}{x} \, dx}{a}-\frac {\left (b \cosh \left (c-\frac {a d}{b}\right )\right ) \int \frac {\cosh \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{a}+\frac {\sinh (c) \int \frac {\sinh (d x)}{x} \, dx}{a}-\frac {\left (b \sinh \left (c-\frac {a d}{b}\right )\right ) \int \frac {\sinh \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{a}\\ &=\frac {\cosh (c) \text {Chi}(d x)}{a}-\frac {\cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{a}+\frac {\sinh (c) \text {Shi}(d x)}{a}-\frac {\sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{a}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 63, normalized size = 0.86 \[ \frac {-\cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (d \left (\frac {a}{b}+x\right )\right )-\sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )+\cosh (c) \text {Chi}(d x)+\sinh (c) \text {Shi}(d x)}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 123, normalized size = 1.68 \[ \frac {{\left ({\rm Ei}\left (d x\right ) + {\rm Ei}\left (-d x\right )\right )} \cosh \relax (c) - {\left ({\rm Ei}\left (\frac {b d x + a d}{b}\right ) + {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \cosh \left (-\frac {b c - a d}{b}\right ) + {\left ({\rm Ei}\left (d x\right ) - {\rm Ei}\left (-d x\right )\right )} \sinh \relax (c) + {\left ({\rm Ei}\left (\frac {b d x + a d}{b}\right ) - {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \sinh \left (-\frac {b c - a d}{b}\right )}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 75, normalized size = 1.03 \[ \frac {{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + {\rm Ei}\left (d x\right ) e^{c} - {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )}}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 108, normalized size = 1.48 \[ \frac {{\mathrm e}^{\frac {d a -c b}{b}} \Ei \left (1, d x +c +\frac {d a -c b}{b}\right )}{2 a}-\frac {{\mathrm e}^{-c} \Ei \left (1, d x \right )}{2 a}+\frac {{\mathrm e}^{-\frac {d a -c b}{b}} \Ei \left (1, -d x -c -\frac {d a -c b}{b}\right )}{2 a}-\frac {{\mathrm e}^{c} \Ei \left (1, -d x \right )}{2 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.41, size = 155, normalized size = 2.12 \[ \frac {1}{2} \, d {\left (\frac {b {\left (\frac {e^{\left (-c + \frac {a d}{b}\right )} E_{1}\left (\frac {{\left (b x + a\right )} d}{b}\right )}{b} + \frac {e^{\left (c - \frac {a d}{b}\right )} E_{1}\left (-\frac {{\left (b x + a\right )} d}{b}\right )}{b}\right )}}{a d} + \frac {2 \, \cosh \left (d x + c\right ) \log \left (b x + a\right )}{a d} - \frac {2 \, \cosh \left (d x + c\right ) \log \relax (x)}{a d} + \frac {{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + {\rm Ei}\left (d x\right ) e^{c}}{a d}\right )} - {\left (\frac {\log \left (b x + a\right )}{a} - \frac {\log \relax (x)}{a}\right )} \cosh \left (d x + c\right ) \]
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 \[ \int \frac {\mathrm {cosh}\left (c+d\,x\right )}{x\,\left (a+b\,x\right )} \,d x \]
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 \[ \int \frac {\cosh {\left (c + d x \right )}}{x \left (a + b x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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