Integrand size = 17, antiderivative size = 56 \[ \int \frac {\left (a+b x^3\right ) \cosh (c+d x)}{x} \, dx=-\frac {2 b x \cosh (c+d x)}{d^2}+a \cosh (c) \text {Chi}(d x)+\frac {2 b \sinh (c+d x)}{d^3}+\frac {b x^2 \sinh (c+d x)}{d}+a \sinh (c) \text {Shi}(d x) \]
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Time = 0.09 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {5395, 3384, 3379, 3382, 3377, 2717} \[ \int \frac {\left (a+b x^3\right ) \cosh (c+d x)}{x} \, dx=a \cosh (c) \text {Chi}(d x)+a \sinh (c) \text {Shi}(d x)+\frac {2 b \sinh (c+d x)}{d^3}-\frac {2 b x \cosh (c+d x)}{d^2}+\frac {b x^2 \sinh (c+d x)}{d} \]
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Rule 2717
Rule 3377
Rule 3379
Rule 3382
Rule 3384
Rule 5395
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a \cosh (c+d x)}{x}+b x^2 \cosh (c+d x)\right ) \, dx \\ & = a \int \frac {\cosh (c+d x)}{x} \, dx+b \int x^2 \cosh (c+d x) \, dx \\ & = \frac {b x^2 \sinh (c+d x)}{d}-\frac {(2 b) \int x \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 {2 b x \cosh (c+d x)}{d^2}+a \cosh (c) \text {Chi}(d x)+\frac {b x^2 \sinh (c+d x)}{d}+a \sinh (c) \text {Shi}(d x)+\frac {(2 b) \int \cosh (c+d x) \, dx}{d^2} \\ & = -\frac {2 b x \cosh (c+d x)}{d^2}+a \cosh (c) \text {Chi}(d x)+\frac {2 b \sinh (c+d x)}{d^3}+\frac {b x^2 \sinh (c+d x)}{d}+a \sinh (c) \text {Shi}(d x) \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+b x^3\right ) \cosh (c+d x)}{x} \, dx=a \cosh (c) \text {Chi}(d x)+\frac {b \left (-2 d x \cosh (c+d x)+\left (2+d^2 x^2\right ) \sinh (c+d x)\right )}{d^3}+a \sinh (c) \text {Shi}(d x) \]
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Time = 0.07 (sec) , antiderivative size = 113, normalized size of antiderivative = 2.02
method | result | size |
risch | \(-\frac {a \,{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right )}{2}-\frac {{\mathrm e}^{-d x -c} b \,x^{2}}{2 d}-\frac {a \,{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right )}{2}+\frac {{\mathrm e}^{d x +c} b \,x^{2}}{2 d}-\frac {{\mathrm e}^{-d x -c} b x}{d^{2}}-\frac {{\mathrm e}^{d x +c} b x}{d^{2}}-\frac {{\mathrm e}^{-d x -c} b}{d^{3}}+\frac {{\mathrm e}^{d x +c} b}{d^{3}}\) | \(113\) |
meijerg | \(\frac {4 i b \cosh \left (c \right ) \sqrt {\pi }\, \left (\frac {i x d \cosh \left (d x \right )}{2 \sqrt {\pi }}-\frac {i \left (\frac {3 x^{2} d^{2}}{2}+3\right ) \sinh \left (d x \right )}{6 \sqrt {\pi }}\right )}{d^{3}}+\frac {4 b \sinh \left (c \right ) \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\left (\frac {x^{2} d^{2}}{2}+1\right ) \cosh \left (d x \right )}{2 \sqrt {\pi }}-\frac {d x \sinh \left (d x \right )}{2 \sqrt {\pi }}\right )}{d^{3}}+\frac {a \cosh \left (c \right ) \sqrt {\pi }\, \left (\frac {2 \gamma +2 \ln \left (x \right )+2 \ln \left (i d \right )}{\sqrt {\pi }}+\frac {2 \,\operatorname {Chi}\left (d x \right )-2 \ln \left (d x \right )-2 \gamma }{\sqrt {\pi }}\right )}{2}+a \,\operatorname {Shi}\left (d x \right ) \sinh \left (c \right )\) | \(149\) |
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Time = 0.26 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.54 \[ \int \frac {\left (a+b x^3\right ) \cosh (c+d x)}{x} \, dx=-\frac {4 \, b d x \cosh \left (d x + c\right ) - {\left (a d^{3} {\rm Ei}\left (d x\right ) + a d^{3} {\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) - 2 \, {\left (b d^{2} x^{2} + 2 \, b\right )} \sinh \left (d x + c\right ) - {\left (a d^{3} {\rm Ei}\left (d x\right ) - a d^{3} {\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right )}{2 \, d^{3}} \]
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Time = 1.67 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.18 \[ \int \frac {\left (a+b x^3\right ) \cosh (c+d x)}{x} \, dx=a \sinh {\left (c \right )} \operatorname {Shi}{\left (d x \right )} + a \cosh {\left (c \right )} \operatorname {Chi}\left (d x\right ) + b \left (\begin {cases} \frac {x^{2} \sinh {\left (c + d x \right )}}{d} - \frac {2 x \cosh {\left (c + d x \right )}}{d^{2}} + \frac {2 \sinh {\left (c + d x \right )}}{d^{3}} & \text {for}\: d \neq 0 \\\frac {x^{3} \cosh {\left (c \right )}}{3} & \text {otherwise} \end {cases}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 140 vs. \(2 (56) = 112\).
Time = 0.23 (sec) , antiderivative size = 140, normalized size of antiderivative = 2.50 \[ \int \frac {\left (a+b x^3\right ) \cosh (c+d x)}{x} \, dx=-\frac {1}{6} \, {\left (b {\left (\frac {{\left (d^{3} x^{3} e^{c} - 3 \, d^{2} x^{2} e^{c} + 6 \, d x e^{c} - 6 \, e^{c}\right )} e^{\left (d x\right )}}{d^{4}} + \frac {{\left (d^{3} x^{3} + 3 \, d^{2} x^{2} + 6 \, d x + 6\right )} e^{\left (-d x - c\right )}}{d^{4}}\right )} + \frac {2 \, a \cosh \left (d x + c\right ) \log \left (x^{3}\right )}{d} - \frac {3 \, {\left ({\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + {\rm Ei}\left (d x\right ) e^{c}\right )} a}{d}\right )} d + \frac {1}{3} \, {\left (b x^{3} + a \log \left (x^{3}\right )\right )} \cosh \left (d x + c\right ) \]
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Time = 0.26 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.95 \[ \int \frac {\left (a+b x^3\right ) \cosh (c+d x)}{x} \, dx=\frac {b d^{2} x^{2} e^{\left (d x + c\right )} - b d^{2} x^{2} e^{\left (-d x - c\right )} + a d^{3} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a d^{3} {\rm Ei}\left (d x\right ) e^{c} - 2 \, b d x e^{\left (d x + c\right )} - 2 \, b d x e^{\left (-d x - c\right )} + 2 \, b e^{\left (d x + c\right )} - 2 \, b e^{\left (-d x - c\right )}}{2 \, d^{3}} \]
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Timed out. \[ \int \frac {\left (a+b x^3\right ) \cosh (c+d x)}{x} \, dx=a\,\mathrm {coshint}\left (d\,x\right )\,\mathrm {cosh}\left (c\right )+a\,\mathrm {sinhint}\left (d\,x\right )\,\mathrm {sinh}\left (c\right )+\frac {b\,\left (2\,\mathrm {sinh}\left (c+d\,x\right )+d^2\,x^2\,\mathrm {sinh}\left (c+d\,x\right )-2\,d\,x\,\mathrm {cosh}\left (c+d\,x\right )\right )}{d^3} \]
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