Integrand size = 16, antiderivative size = 81 \[ \int \frac {\cosh ^2(a+b x)}{(c+d x)^2} \, dx=-\frac {\cosh ^2(a+b x)}{d (c+d x)}+\frac {b \text {Chi}\left (\frac {2 b c}{d}+2 b x\right ) \sinh \left (2 a-\frac {2 b c}{d}\right )}{d^2}+\frac {b \cosh \left (2 a-\frac {2 b c}{d}\right ) \text {Shi}\left (\frac {2 b c}{d}+2 b x\right )}{d^2} \]
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Time = 0.11 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {3394, 12, 3384, 3379, 3382} \[ \int \frac {\cosh ^2(a+b x)}{(c+d x)^2} \, dx=\frac {b \sinh \left (2 a-\frac {2 b c}{d}\right ) \text {Chi}\left (\frac {2 b c}{d}+2 b x\right )}{d^2}+\frac {b \cosh \left (2 a-\frac {2 b c}{d}\right ) \text {Shi}\left (\frac {2 b c}{d}+2 b x\right )}{d^2}-\frac {\cosh ^2(a+b x)}{d (c+d x)} \]
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Rule 12
Rule 3379
Rule 3382
Rule 3384
Rule 3394
Rubi steps \begin{align*} \text {integral}& = -\frac {\cosh ^2(a+b x)}{d (c+d x)}+\frac {(2 i b) \int -\frac {i \sinh (2 a+2 b x)}{2 (c+d x)} \, dx}{d} \\ & = -\frac {\cosh ^2(a+b x)}{d (c+d x)}+\frac {b \int \frac {\sinh (2 a+2 b x)}{c+d x} \, dx}{d} \\ & = -\frac {\cosh ^2(a+b x)}{d (c+d x)}+\frac {\left (b \cosh \left (2 a-\frac {2 b c}{d}\right )\right ) \int \frac {\sinh \left (\frac {2 b c}{d}+2 b x\right )}{c+d x} \, dx}{d}+\frac {\left (b \sinh \left (2 a-\frac {2 b c}{d}\right )\right ) \int \frac {\cosh \left (\frac {2 b c}{d}+2 b x\right )}{c+d x} \, dx}{d} \\ & = -\frac {\cosh ^2(a+b x)}{d (c+d x)}+\frac {b \text {Chi}\left (\frac {2 b c}{d}+2 b x\right ) \sinh \left (2 a-\frac {2 b c}{d}\right )}{d^2}+\frac {b \cosh \left (2 a-\frac {2 b c}{d}\right ) \text {Shi}\left (\frac {2 b c}{d}+2 b x\right )}{d^2} \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.93 \[ \int \frac {\cosh ^2(a+b x)}{(c+d x)^2} \, dx=\frac {-\frac {d \cosh ^2(a+b x)}{c+d x}+b \text {Chi}\left (\frac {2 b (c+d x)}{d}\right ) \sinh \left (2 a-\frac {2 b c}{d}\right )+b \cosh \left (2 a-\frac {2 b c}{d}\right ) \text {Shi}\left (\frac {2 b (c+d x)}{d}\right )}{d^2} \]
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Time = 0.36 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.88
method | result | size |
risch | \(-\frac {1}{2 \left (d x +c \right ) d}-\frac {b \,{\mathrm e}^{-2 b x -2 a}}{4 d \left (d x b +c b \right )}+\frac {b \,{\mathrm e}^{-\frac {2 \left (d a -c b \right )}{d}} \operatorname {Ei}_{1}\left (2 b x +2 a -\frac {2 \left (d a -c b \right )}{d}\right )}{2 d^{2}}-\frac {b \,{\mathrm e}^{2 b x +2 a}}{4 d^{2} \left (\frac {b c}{d}+b x \right )}-\frac {b \,{\mathrm e}^{\frac {2 d a -2 c b}{d}} \operatorname {Ei}_{1}\left (-2 b x -2 a -\frac {2 \left (-d a +c b \right )}{d}\right )}{2 d^{2}}\) | \(152\) |
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Leaf count of result is larger than twice the leaf count of optimal. 164 vs. \(2 (81) = 162\).
Time = 0.26 (sec) , antiderivative size = 164, normalized size of antiderivative = 2.02 \[ \int \frac {\cosh ^2(a+b x)}{(c+d x)^2} \, dx=-\frac {d \cosh \left (b x + a\right )^{2} + d \sinh \left (b x + a\right )^{2} - {\left ({\left (b d x + b c\right )} {\rm Ei}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) - {\left (b d x + b c\right )} {\rm Ei}\left (-\frac {2 \, {\left (b d x + b c\right )}}{d}\right )\right )} \cosh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - {\left ({\left (b d x + b c\right )} {\rm Ei}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) + {\left (b d x + b c\right )} {\rm Ei}\left (-\frac {2 \, {\left (b d x + b c\right )}}{d}\right )\right )} \sinh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + d}{2 \, {\left (d^{3} x + c d^{2}\right )}} \]
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\[ \int \frac {\cosh ^2(a+b x)}{(c+d x)^2} \, dx=\int \frac {\cosh ^{2}{\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \]
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Time = 0.23 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.09 \[ \int \frac {\cosh ^2(a+b x)}{(c+d x)^2} \, dx=-\frac {e^{\left (-2 \, a + \frac {2 \, b c}{d}\right )} E_{2}\left (\frac {2 \, {\left (d x + c\right )} b}{d}\right )}{4 \, {\left (d x + c\right )} d} - \frac {e^{\left (2 \, a - \frac {2 \, b c}{d}\right )} E_{2}\left (-\frac {2 \, {\left (d x + c\right )} b}{d}\right )}{4 \, {\left (d x + c\right )} d} - \frac {1}{2 \, {\left (d^{2} x + c d\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 574 vs. \(2 (81) = 162\).
Time = 0.30 (sec) , antiderivative size = 574, normalized size of antiderivative = 7.09 \[ \int \frac {\cosh ^2(a+b x)}{(c+d x)^2} \, dx=-\frac {{\left (2 \, {\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} b^{2} {\rm Ei}\left (-\frac {2 \, {\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d\right )}}{d}\right ) e^{\left (\frac {2 \, {\left (b c - a d\right )}}{d}\right )} + 2 \, b^{3} c {\rm Ei}\left (-\frac {2 \, {\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d\right )}}{d}\right ) e^{\left (\frac {2 \, {\left (b c - a d\right )}}{d}\right )} - 2 \, a b^{2} d {\rm Ei}\left (-\frac {2 \, {\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d\right )}}{d}\right ) e^{\left (\frac {2 \, {\left (b c - a d\right )}}{d}\right )} - 2 \, {\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} b^{2} {\rm Ei}\left (\frac {2 \, {\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )} - 2 \, b^{3} c {\rm Ei}\left (\frac {2 \, {\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )} + 2 \, a b^{2} d {\rm Ei}\left (\frac {2 \, {\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )} + b^{2} d e^{\left (\frac {2 \, {\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )}}{d}\right )} + b^{2} d e^{\left (-\frac {2 \, {\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )}}{d}\right )} + 2 \, b^{2} d\right )} d^{2}}{4 \, {\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} d^{4} + b c d^{4} - a d^{5}\right )} b} \]
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Timed out. \[ \int \frac {\cosh ^2(a+b x)}{(c+d x)^2} \, dx=\int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^2} \,d x \]
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