Integrand size = 17, antiderivative size = 62 \[ \int \frac {(a+b x)^2 \cosh (c+d x)}{x} \, dx=-\frac {b^2 \cosh (c+d x)}{d^2}+a^2 \cosh (c) \text {Chi}(d x)+\frac {2 a b \sinh (c+d x)}{d}+\frac {b^2 x \sinh (c+d x)}{d}+a^2 \sinh (c) \text {Shi}(d x) \]
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Time = 0.14 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {6874, 2717, 3384, 3379, 3382, 3377, 2718} \[ \int \frac {(a+b x)^2 \cosh (c+d x)}{x} \, dx=a^2 \cosh (c) \text {Chi}(d x)+a^2 \sinh (c) \text {Shi}(d x)+\frac {2 a b \sinh (c+d x)}{d}-\frac {b^2 \cosh (c+d x)}{d^2}+\frac {b^2 x \sinh (c+d x)}{d} \]
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Rule 2717
Rule 2718
Rule 3377
Rule 3379
Rule 3382
Rule 3384
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (2 a b \cosh (c+d x)+\frac {a^2 \cosh (c+d x)}{x}+b^2 x \cosh (c+d x)\right ) \, dx \\ & = a^2 \int \frac {\cosh (c+d x)}{x} \, dx+(2 a b) \int \cosh (c+d x) \, dx+b^2 \int x \cosh (c+d x) \, dx \\ & = \frac {2 a b \sinh (c+d x)}{d}+\frac {b^2 x \sinh (c+d x)}{d}-\frac {b^2 \int \sinh (c+d x) \, dx}{d}+\left (a^2 \cosh (c)\right ) \int \frac {\cosh (d x)}{x} \, dx+\left (a^2 \sinh (c)\right ) \int \frac {\sinh (d x)}{x} \, dx \\ & = -\frac {b^2 \cosh (c+d x)}{d^2}+a^2 \cosh (c) \text {Chi}(d x)+\frac {2 a b \sinh (c+d x)}{d}+\frac {b^2 x \sinh (c+d x)}{d}+a^2 \sinh (c) \text {Shi}(d x) \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.82 \[ \int \frac {(a+b x)^2 \cosh (c+d x)}{x} \, dx=a^2 \cosh (c) \text {Chi}(d x)+\frac {b (-b \cosh (c+d x)+d (2 a+b x) \sinh (c+d x))}{d^2}+a^2 \sinh (c) \text {Shi}(d x) \]
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Time = 0.22 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.95
method | result | size |
risch | \(-\frac {a^{2} {\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right )}{2}-\frac {a^{2} {\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right )}{2}-\frac {{\mathrm e}^{-d x -c} b^{2} x}{2 d}+\frac {{\mathrm e}^{d x +c} b^{2} x}{2 d}-\frac {{\mathrm e}^{-d x -c} a b}{d}+\frac {{\mathrm e}^{d x +c} a b}{d}-\frac {{\mathrm e}^{-d x -c} b^{2}}{2 d^{2}}-\frac {{\mathrm e}^{d x +c} b^{2}}{2 d^{2}}\) | \(121\) |
meijerg | \(-\frac {2 b^{2} \cosh \left (c \right ) \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\cosh \left (d x \right )}{2 \sqrt {\pi }}-\frac {d x \sinh \left (d x \right )}{2 \sqrt {\pi }}\right )}{d^{2}}+\frac {b^{2} \sinh \left (c \right ) \left (\cosh \left (d x \right ) x d -\sinh \left (d x \right )\right )}{d^{2}}+\frac {2 a b \cosh \left (c \right ) \sinh \left (d x \right )}{d}-\frac {2 b a \sinh \left (c \right ) \sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cosh \left (d x \right )}{\sqrt {\pi }}\right )}{d}+\frac {a^{2} \cosh \left (c \right ) \sqrt {\pi }\, \left (\frac {2 \gamma +2 \ln \left (x \right )+2 \ln \left (i d \right )}{\sqrt {\pi }}+\frac {2 \,\operatorname {Chi}\left (d x \right )-2 \ln \left (d x \right )-2 \gamma }{\sqrt {\pi }}\right )}{2}+a^{2} \operatorname {Shi}\left (d x \right ) \sinh \left (c \right )\) | \(161\) |
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Time = 0.25 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.52 \[ \int \frac {(a+b x)^2 \cosh (c+d x)}{x} \, dx=-\frac {2 \, b^{2} \cosh \left (d x + c\right ) - {\left (a^{2} d^{2} {\rm Ei}\left (d x\right ) + a^{2} d^{2} {\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) - 2 \, {\left (b^{2} d x + 2 \, a b d\right )} \sinh \left (d x + c\right ) - {\left (a^{2} d^{2} {\rm Ei}\left (d x\right ) - a^{2} d^{2} {\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right )}{2 \, d^{2}} \]
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Time = 1.66 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.18 \[ \int \frac {(a+b x)^2 \cosh (c+d x)}{x} \, dx=a^{2} \sinh {\left (c \right )} \operatorname {Shi}{\left (d x \right )} + a^{2} \cosh {\left (c \right )} \operatorname {Chi}\left (d x\right ) + 2 a b \left (\begin {cases} x \cosh {\left (c \right )} & \text {for}\: d = 0 \\\frac {\sinh {\left (c + d x \right )}}{d} & \text {otherwise} \end {cases}\right ) + b^{2} \left (\begin {cases} \frac {x \sinh {\left (c + d x \right )}}{d} - \frac {\cosh {\left (c + d x \right )}}{d^{2}} & \text {for}\: d \neq 0 \\\frac {x^{2} \cosh {\left (c \right )}}{2} & \text {otherwise} \end {cases}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 175 vs. \(2 (62) = 124\).
Time = 0.28 (sec) , antiderivative size = 175, normalized size of antiderivative = 2.82 \[ \int \frac {(a+b x)^2 \cosh (c+d x)}{x} \, dx=-\frac {1}{4} \, {\left (4 \, a b {\left (\frac {{\left (d x e^{c} - e^{c}\right )} e^{\left (d x\right )}}{d^{2}} + \frac {{\left (d x + 1\right )} e^{\left (-d x - c\right )}}{d^{2}}\right )} + b^{2} {\left (\frac {{\left (d^{2} x^{2} e^{c} - 2 \, d x e^{c} + 2 \, e^{c}\right )} e^{\left (d x\right )}}{d^{3}} + \frac {{\left (d^{2} x^{2} + 2 \, d x + 2\right )} e^{\left (-d x - c\right )}}{d^{3}}\right )} + \frac {4 \, a^{2} \cosh \left (d x + c\right ) \log \left (x\right )}{d} - \frac {2 \, {\left ({\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + {\rm Ei}\left (d x\right ) e^{c}\right )} a^{2}}{d}\right )} d + \frac {1}{2} \, {\left (b^{2} x^{2} + 4 \, a b x + 2 \, a^{2} \log \left (x\right )\right )} \cosh \left (d x + c\right ) \]
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Time = 0.28 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.82 \[ \int \frac {(a+b x)^2 \cosh (c+d x)}{x} \, dx=\frac {a^{2} d^{2} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a^{2} d^{2} {\rm Ei}\left (d x\right ) e^{c} + b^{2} d x e^{\left (d x + c\right )} - b^{2} d x e^{\left (-d x - c\right )} + 2 \, a b d e^{\left (d x + c\right )} - 2 \, a b d e^{\left (-d x - c\right )} - b^{2} e^{\left (d x + c\right )} - b^{2} e^{\left (-d x - c\right )}}{2 \, d^{2}} \]
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Timed out. \[ \int \frac {(a+b x)^2 \cosh (c+d x)}{x} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )\,{\left (a+b\,x\right )}^2}{x} \,d x \]
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