Integrand size = 17, antiderivative size = 100 \[ \int \frac {x^2 \cosh (c+d x)}{a+b x} \, dx=-\frac {\cosh (c+d x)}{b d^2}+\frac {a^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{b^3}-\frac {a \sinh (c+d x)}{b^2 d}+\frac {x \sinh (c+d x)}{b d}+\frac {a^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^3} \]
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Time = 0.19 (sec) , antiderivative size = 100, 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, 3377, 2718, 3384, 3379, 3382} \[ \int \frac {x^2 \cosh (c+d x)}{a+b x} \, dx=\frac {a^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{b^3}+\frac {a^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{b^3}-\frac {a \sinh (c+d x)}{b^2 d}-\frac {\cosh (c+d x)}{b d^2}+\frac {x \sinh (c+d x)}{b d} \]
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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 (-\frac {a \cosh (c+d x)}{b^2}+\frac {x \cosh (c+d x)}{b}+\frac {a^2 \cosh (c+d x)}{b^2 (a+b x)}\right ) \, dx \\ & = -\frac {a \int \cosh (c+d x) \, dx}{b^2}+\frac {a^2 \int \frac {\cosh (c+d x)}{a+b x} \, dx}{b^2}+\frac {\int x \cosh (c+d x) \, dx}{b} \\ & = -\frac {a \sinh (c+d x)}{b^2 d}+\frac {x \sinh (c+d x)}{b d}-\frac {\int \sinh (c+d x) \, dx}{b d}+\frac {\left (a^2 \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^2}+\frac {\left (a^2 \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^2} \\ & = -\frac {\cosh (c+d x)}{b d^2}+\frac {a^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{b^3}-\frac {a \sinh (c+d x)}{b^2 d}+\frac {x \sinh (c+d x)}{b d}+\frac {a^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^3} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.89 \[ \int \frac {x^2 \cosh (c+d x)}{a+b x} \, dx=\frac {a^2 d^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (d \left (\frac {a}{b}+x\right )\right )+b (-b \cosh (c+d x)+d (-a+b x) \sinh (c+d x))+a^2 d^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )}{b^3 d^2} \]
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Time = 0.22 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.84
method | result | size |
risch | \(-\frac {{\mathrm e}^{-\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (-d x -c -\frac {d a -c b}{b}\right ) a^{2}}{2 b^{3}}-\frac {{\mathrm e}^{\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (d x +c +\frac {d a -c b}{b}\right ) a^{2}}{2 b^{3}}-\frac {{\mathrm e}^{-d x -c} x}{2 d b}+\frac {{\mathrm e}^{d x +c} x}{2 d b}+\frac {{\mathrm e}^{-d x -c} a}{2 d \,b^{2}}-\frac {a \,{\mathrm e}^{d x +c}}{2 d \,b^{2}}-\frac {{\mathrm e}^{-d x -c}}{2 d^{2} b}-\frac {{\mathrm e}^{d x +c}}{2 d^{2} b}\) | \(184\) |
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Time = 0.25 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.56 \[ \int \frac {x^2 \cosh (c+d x)}{a+b x} \, dx=-\frac {2 \, b^{2} \cosh \left (d x + c\right ) - {\left (a^{2} d^{2} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) + a^{2} d^{2} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \cosh \left (-\frac {b c - a d}{b}\right ) - 2 \, {\left (b^{2} d x - a b d\right )} \sinh \left (d x + c\right ) + {\left (a^{2} d^{2} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) - a^{2} d^{2} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \sinh \left (-\frac {b c - a d}{b}\right )}{2 \, b^{3} d^{2}} \]
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\[ \int \frac {x^2 \cosh (c+d x)}{a+b x} \, dx=\int \frac {x^{2} \cosh {\left (c + d x \right )}}{a + b x}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 233 vs. \(2 (103) = 206\).
Time = 0.23 (sec) , antiderivative size = 233, normalized size of antiderivative = 2.33 \[ \int \frac {x^2 \cosh (c+d x)}{a+b x} \, dx=-\frac {1}{4} \, d {\left (\frac {2 \, a^{2} {\left (\frac {e^{\left (-c + \frac {a d}{b}\right )} E_{1}\left (\frac {{\left (b x + a\right )} d}{b}\right )}{b} + \frac {e^{\left (c - \frac {a d}{b}\right )} E_{1}\left (-\frac {{\left (b x + a\right )} d}{b}\right )}{b}\right )}}{b^{2} d} - \frac {2 \, a {\left (\frac {{\left (d x e^{c} - e^{c}\right )} e^{\left (d x\right )}}{d^{2}} + \frac {{\left (d x + 1\right )} e^{\left (-d x - c\right )}}{d^{2}}\right )}}{b^{2}} + \frac {\frac {{\left (d^{2} x^{2} e^{c} - 2 \, d x e^{c} + 2 \, e^{c}\right )} e^{\left (d x\right )}}{d^{3}} + \frac {{\left (d^{2} x^{2} + 2 \, d x + 2\right )} e^{\left (-d x - c\right )}}{d^{3}}}{b} + \frac {4 \, a^{2} \cosh \left (d x + c\right ) \log \left (b x + a\right )}{b^{3} d}\right )} + \frac {1}{2} \, {\left (\frac {2 \, a^{2} \log \left (b x + a\right )}{b^{3}} + \frac {b x^{2} - 2 \, a x}{b^{2}}\right )} \cosh \left (d x + c\right ) \]
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Time = 0.46 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.48 \[ \int \frac {x^2 \cosh (c+d x)}{a+b x} \, dx=\frac {a^{2} d^{2} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + a^{2} d^{2} {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} + b^{2} d x e^{\left (d x + c\right )} - b^{2} d x e^{\left (-d x - c\right )} - a b d e^{\left (d x + c\right )} + a b d e^{\left (-d x - c\right )} - b^{2} e^{\left (d x + c\right )} - b^{2} e^{\left (-d x - c\right )}}{2 \, b^{3} d^{2}} \]
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Timed out. \[ \int \frac {x^2 \cosh (c+d x)}{a+b x} \, dx=\int \frac {x^2\,\mathrm {cosh}\left (c+d\,x\right )}{a+b\,x} \,d x \]
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