Optimal. Leaf size=28 \[ a \cosh (c) \text {Chi}(d x)+a \sinh (c) \text {Shi}(d x)+\frac {b \sinh (c+d x)}{d} \]
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Rubi [A] time = 0.15, antiderivative size = 28, 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} \[ a \cosh (c) \text {Chi}(d x)+a \sinh (c) \text {Shi}(d x)+\frac {b \sinh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3298
Rule 3301
Rule 3303
Rule 6742
Rubi steps
\begin {align*} \int \frac {(a+b x) \cosh (c+d x)}{x} \, dx &=\int \left (b \cosh (c+d x)+\frac {a \cosh (c+d x)}{x}\right ) \, dx\\ &=a \int \frac {\cosh (c+d x)}{x} \, dx+b \int \cosh (c+d x) \, dx\\ &=\frac {b \sinh (c+d x)}{d}+(a \cosh (c)) \int \frac {\cosh (d x)}{x} \, dx+(a \sinh (c)) \int \frac {\sinh (d x)}{x} \, dx\\ &=a \cosh (c) \text {Chi}(d x)+\frac {b \sinh (c+d x)}{d}+a \sinh (c) \text {Shi}(d x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 39, normalized size = 1.39 \[ a \cosh (c) \text {Chi}(d x)+a \sinh (c) \text {Shi}(d x)+\frac {b \sinh (c) \cosh (d x)}{d}+\frac {b \cosh (c) \sinh (d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 54, normalized size = 1.93 \[ \frac {{\left (a d {\rm Ei}\left (d x\right ) + a d {\rm Ei}\left (-d x\right )\right )} \cosh \relax (c) + 2 \, b \sinh \left (d x + c\right ) + {\left (a d {\rm Ei}\left (d x\right ) - a d {\rm Ei}\left (-d x\right )\right )} \sinh \relax (c)}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 47, normalized size = 1.68 \[ \frac {a d {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a d {\rm Ei}\left (d x\right ) e^{c} + b e^{\left (d x + c\right )} - b e^{\left (-d x - c\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 52, normalized size = 1.86 \[ -\frac {a \,{\mathrm e}^{-c} \Ei \left (1, d x \right )}{2}-\frac {b \,{\mathrm e}^{-d x -c}}{2 d}-\frac {a \,{\mathrm e}^{c} \Ei \left (1, -d x \right )}{2}+\frac {b \,{\mathrm e}^{d x +c}}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.53, size = 97, normalized size = 3.46 \[ -\frac {1}{2} \, {\left (b {\left (\frac {{\left (d x e^{c} - e^{c}\right )} e^{\left (d x\right )}}{d^{2}} + \frac {{\left (d x + 1\right )} e^{\left (-d x - c\right )}}{d^{2}}\right )} + \frac {2 \, a \cosh \left (d x + c\right ) \log \relax (x)}{d} - \frac {{\left ({\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + {\rm Ei}\left (d x\right ) e^{c}\right )} a}{d}\right )} d + {\left (b x + a \log \relax (x)\right )} \cosh \left (d x + c\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ a\,\mathrm {coshint}\left (d\,x\right )\,\mathrm {cosh}\relax (c)+a\,\mathrm {sinhint}\left (d\,x\right )\,\mathrm {sinh}\relax (c)+\frac {b\,\mathrm {sinh}\left (c+d\,x\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.19, size = 34, normalized size = 1.21 \[ a \sinh {\relax (c )} \operatorname {Shi}{\left (d x \right )} + a \cosh {\relax (c )} \operatorname {Chi}\left (d x\right ) + b \left (\begin {cases} \frac {\sinh {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \cosh {\relax (c )} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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