Optimal. Leaf size=68 \[ -\frac {a \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{b^2}-\frac {a \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{b^2}+\frac {\sinh (c+d x)}{b d} \]
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Rubi [A] time = 0.18, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6742, 2637, 3303, 3298, 3301} \[ -\frac {a \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{b^2}-\frac {a \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{b^2}+\frac {\sinh (c+d x)}{b d} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3298
Rule 3301
Rule 3303
Rule 6742
Rubi steps
\begin {align*} \int \frac {x \cosh (c+d x)}{a+b x} \, dx &=\int \left (\frac {\cosh (c+d x)}{b}-\frac {a \cosh (c+d x)}{b (a+b x)}\right ) \, dx\\ &=\frac {\int \cosh (c+d x) \, dx}{b}-\frac {a \int \frac {\cosh (c+d x)}{a+b x} \, dx}{b}\\ &=\frac {\sinh (c+d x)}{b d}-\frac {\left (a \cosh \left (c-\frac {a d}{b}\right )\right ) \int \frac {\cosh \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{b}-\frac {\left (a \sinh \left (c-\frac {a d}{b}\right )\right ) \int \frac {\sinh \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{b}\\ &=-\frac {a \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{b^2}+\frac {\sinh (c+d x)}{b d}-\frac {a \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^2}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 64, normalized size = 0.94 \[ \frac {-a d \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (d \left (\frac {a}{b}+x\right )\right )-a d \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )+b \sinh (c+d x)}{b^2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 118, normalized size = 1.74 \[ -\frac {{\left (a d {\rm Ei}\left (\frac {b d x + a d}{b}\right ) + a d {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \cosh \left (-\frac {b c - a d}{b}\right ) - 2 \, b \sinh \left (d x + c\right ) - {\left (a d {\rm Ei}\left (\frac {b d x + a d}{b}\right ) - a d {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \sinh \left (-\frac {b c - a d}{b}\right )}{2 \, b^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 83, normalized size = 1.22 \[ -\frac {a d {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + a d {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} - b e^{\left (d x + c\right )} + b e^{\left (-d x - c\right )}}{2 \, b^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 114, normalized size = 1.68 \[ -\frac {{\mathrm e}^{-d x -c}}{2 d b}+\frac {{\mathrm e}^{\frac {d a -c b}{b}} \Ei \left (1, d x +c +\frac {d a -c b}{b}\right ) a}{2 b^{2}}+\frac {{\mathrm e}^{d x +c}}{2 d b}+\frac {{\mathrm e}^{-\frac {d a -c b}{b}} \Ei \left (1, -d x -c -\frac {d a -c b}{b}\right ) a}{2 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.38, size = 156, normalized size = 2.29 \[ \frac {1}{2} \, d {\left (\frac {a {\left (\frac {e^{\left (-c + \frac {a d}{b}\right )} E_{1}\left (\frac {{\left (b x + a\right )} d}{b}\right )}{b} + \frac {e^{\left (c - \frac {a d}{b}\right )} E_{1}\left (-\frac {{\left (b x + a\right )} d}{b}\right )}{b}\right )}}{b d} - \frac {\frac {{\left (d x e^{c} - e^{c}\right )} e^{\left (d x\right )}}{d^{2}} + \frac {{\left (d x + 1\right )} e^{\left (-d x - c\right )}}{d^{2}}}{b} + \frac {2 \, a \cosh \left (d x + c\right ) \log \left (b x + a\right )}{b^{2} d}\right )} + {\left (\frac {x}{b} - \frac {a \log \left (b x + a\right )}{b^{2}}\right )} \cosh \left (d x + c\right ) \]
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 \[ \int \frac {x\,\mathrm {cosh}\left (c+d\,x\right )}{a+b\,x} \,d x \]
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 \[ \int \frac {x \cosh {\left (c + d x \right )}}{a + b x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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