Optimal. Leaf size=41 \[ a \cosh (c) \text {Chi}(d x)+a \sinh (c) \text {Shi}(d x)-\frac {b \cosh (c+d x)}{d^2}+\frac {b x \sinh (c+d x)}{d} \]
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Rubi [A] time = 0.10, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {5287, 3303, 3298, 3301, 3296, 2638} \[ a \cosh (c) \text {Chi}(d x)+a \sinh (c) \text {Shi}(d x)-\frac {b \cosh (c+d x)}{d^2}+\frac {b x \sinh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3296
Rule 3298
Rule 3301
Rule 3303
Rule 5287
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right ) \cosh (c+d x)}{x} \, dx &=\int \left (\frac {a \cosh (c+d x)}{x}+b x \cosh (c+d x)\right ) \, dx\\ &=a \int \frac {\cosh (c+d x)}{x} \, dx+b \int x \cosh (c+d x) \, dx\\ &=\frac {b x \sinh (c+d x)}{d}-\frac {b \int \sinh (c+d x) \, dx}{d}+(a \cosh (c)) \int \frac {\cosh (d x)}{x} \, dx+(a \sinh (c)) \int \frac {\sinh (d x)}{x} \, dx\\ &=-\frac {b \cosh (c+d x)}{d^2}+a \cosh (c) \text {Chi}(d x)+\frac {b x \sinh (c+d x)}{d}+a \sinh (c) \text {Shi}(d x)\\ \end {align*}
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Mathematica [A] time = 0.12, size = 55, normalized size = 1.34 \[ a \cosh (c) \text {Chi}(d x)+a \sinh (c) \text {Shi}(d x)+\frac {b \cosh (d x) (d x \sinh (c)-\cosh (c))}{d^2}+\frac {b \sinh (d x) (d x \cosh (c)-\sinh (c))}{d^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 73, normalized size = 1.78 \[ \frac {2 \, b d x \sinh \left (d x + c\right ) - 2 \, b \cosh \left (d x + c\right ) + {\left (a d^{2} {\rm Ei}\left (d x\right ) + a d^{2} {\rm Ei}\left (-d x\right )\right )} \cosh \relax (c) + {\left (a d^{2} {\rm Ei}\left (d x\right ) - a d^{2} {\rm Ei}\left (-d x\right )\right )} \sinh \relax (c)}{2 \, d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 76, normalized size = 1.85 \[ \frac {a d^{2} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a d^{2} {\rm Ei}\left (d x\right ) e^{c} + b d x e^{\left (d x + c\right )} - b d x e^{\left (-d x - c\right )} - b e^{\left (d x + c\right )} - b e^{\left (-d x - c\right )}}{2 \, d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 81, normalized size = 1.98 \[ -\frac {a \,{\mathrm e}^{-c} \Ei \left (1, d x \right )}{2}-\frac {b \,{\mathrm e}^{-d x -c} x}{2 d}-\frac {b \,{\mathrm e}^{-d x -c}}{2 d^{2}}-\frac {a \,{\mathrm e}^{c} \Ei \left (1, -d x \right )}{2}+\frac {b \,{\mathrm e}^{d x +c} x}{2 d}-\frac {b \,{\mathrm e}^{d x +c}}{2 d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.39, size = 122, normalized size = 2.98 \[ -\frac {1}{4} \, {\left (b {\left (\frac {{\left (d^{2} x^{2} e^{c} - 2 \, d x e^{c} + 2 \, e^{c}\right )} e^{\left (d x\right )}}{d^{3}} + \frac {{\left (d^{2} x^{2} + 2 \, d x + 2\right )} e^{\left (-d x - c\right )}}{d^{3}}\right )} + \frac {2 \, a \cosh \left (d x + c\right ) \log \left (x^{2}\right )}{d} - \frac {2 \, {\left ({\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + {\rm Ei}\left (d x\right ) e^{c}\right )} a}{d}\right )} d + \frac {1}{2} \, {\left (b x^{2} + a \log \left (x^{2}\right )\right )} \cosh \left (d x + c\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ a\,\mathrm {coshint}\left (d\,x\right )\,\mathrm {cosh}\relax (c)+a\,\mathrm {sinhint}\left (d\,x\right )\,\mathrm {sinh}\relax (c)-\frac {b\,\left (\mathrm {cosh}\left (c+d\,x\right )-d\,x\,\mathrm {sinh}\left (c+d\,x\right )\right )}{d^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.41, size = 49, normalized size = 1.20 \[ a \sinh {\relax (c )} \operatorname {Shi}{\left (d x \right )} + a \cosh {\relax (c )} \operatorname {Chi}\left (d x\right ) + b \left (\begin {cases} \frac {x \sinh {\left (c + d x \right )}}{d} - \frac {\cosh {\left (c + d x \right )}}{d^{2}} & \text {for}\: d \neq 0 \\\frac {x^{2} \cosh {\relax (c )}}{2} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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