Integrand size = 16, antiderivative size = 145 \[ \int \frac {\cosh ^3(a+b x)}{(c+d x)^2} \, dx=-\frac {\cosh ^3(a+b x)}{d (c+d x)}+\frac {3 b \text {Chi}\left (\frac {3 b c}{d}+3 b x\right ) \sinh \left (3 a-\frac {3 b c}{d}\right )}{4 d^2}+\frac {3 b \text {Chi}\left (\frac {b c}{d}+b x\right ) \sinh \left (a-\frac {b c}{d}\right )}{4 d^2}+\frac {3 b \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{4 d^2}+\frac {3 b \cosh \left (3 a-\frac {3 b c}{d}\right ) \text {Shi}\left (\frac {3 b c}{d}+3 b x\right )}{4 d^2} \]
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Time = 0.21 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3394, 3384, 3379, 3382} \[ \int \frac {\cosh ^3(a+b x)}{(c+d x)^2} \, dx=\frac {3 b \sinh \left (3 a-\frac {3 b c}{d}\right ) \text {Chi}\left (\frac {3 b c}{d}+3 b x\right )}{4 d^2}+\frac {3 b \sinh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {b c}{d}+b x\right )}{4 d^2}+\frac {3 b \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{4 d^2}+\frac {3 b \cosh \left (3 a-\frac {3 b c}{d}\right ) \text {Shi}\left (\frac {3 b c}{d}+3 b x\right )}{4 d^2}-\frac {\cosh ^3(a+b x)}{d (c+d x)} \]
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Rule 3379
Rule 3382
Rule 3384
Rule 3394
Rubi steps \begin{align*} \text {integral}& = -\frac {\cosh ^3(a+b x)}{d (c+d x)}+\frac {(3 i b) \int \left (-\frac {i \sinh (a+b x)}{4 (c+d x)}-\frac {i \sinh (3 a+3 b x)}{4 (c+d x)}\right ) \, dx}{d} \\ & = -\frac {\cosh ^3(a+b x)}{d (c+d x)}+\frac {(3 b) \int \frac {\sinh (a+b x)}{c+d x} \, dx}{4 d}+\frac {(3 b) \int \frac {\sinh (3 a+3 b x)}{c+d x} \, dx}{4 d} \\ & = -\frac {\cosh ^3(a+b x)}{d (c+d x)}+\frac {\left (3 b \cosh \left (3 a-\frac {3 b c}{d}\right )\right ) \int \frac {\sinh \left (\frac {3 b c}{d}+3 b x\right )}{c+d x} \, dx}{4 d}+\frac {\left (3 b \cosh \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sinh \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx}{4 d}+\frac {\left (3 b \sinh \left (3 a-\frac {3 b c}{d}\right )\right ) \int \frac {\cosh \left (\frac {3 b c}{d}+3 b x\right )}{c+d x} \, dx}{4 d}+\frac {\left (3 b \sinh \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cosh \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx}{4 d} \\ & = -\frac {\cosh ^3(a+b x)}{d (c+d x)}+\frac {3 b \text {Chi}\left (\frac {3 b c}{d}+3 b x\right ) \sinh \left (3 a-\frac {3 b c}{d}\right )}{4 d^2}+\frac {3 b \text {Chi}\left (\frac {b c}{d}+b x\right ) \sinh \left (a-\frac {b c}{d}\right )}{4 d^2}+\frac {3 b \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{4 d^2}+\frac {3 b \cosh \left (3 a-\frac {3 b c}{d}\right ) \text {Shi}\left (\frac {3 b c}{d}+3 b x\right )}{4 d^2} \\ \end{align*}
Time = 0.47 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.35 \[ \int \frac {\cosh ^3(a+b x)}{(c+d x)^2} \, dx=-\frac {3 \cosh (a) \cosh (b x)}{4 d (c+d x)}-\frac {\cosh (3 a) \cosh (3 b x)}{4 d (c+d x)}-\frac {3 \sinh (a) \sinh (b x)}{4 d (c+d x)}-\frac {\sinh (3 a) \sinh (3 b x)}{4 d (c+d x)}-\frac {3 b \left (-2 \text {Chi}\left (\frac {3 b c}{d}+3 b x\right ) \sinh \left (3 a-\frac {3 b c}{d}\right )-2 \text {Chi}\left (\frac {b c}{d}+b x\right ) \sinh \left (a-\frac {b c}{d}\right )-2 \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )-2 \cosh \left (3 a-\frac {3 b c}{d}\right ) \text {Shi}\left (\frac {3 b c}{d}+3 b x\right )\right )}{8 d^2} \]
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Time = 0.42 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.87
method | result | size |
risch | \(-\frac {b \,{\mathrm e}^{-3 b x -3 a}}{8 d \left (d x b +c b \right )}+\frac {3 b \,{\mathrm e}^{-\frac {3 \left (d a -c b \right )}{d}} \operatorname {Ei}_{1}\left (3 b x +3 a -\frac {3 \left (d a -c b \right )}{d}\right )}{8 d^{2}}-\frac {3 b \,{\mathrm e}^{-b x -a}}{8 d \left (d x b +c b \right )}+\frac {3 b \,{\mathrm e}^{-\frac {d a -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {d a -c b}{d}\right )}{8 d^{2}}-\frac {3 b \,{\mathrm e}^{b x +a}}{8 d^{2} \left (\frac {b c}{d}+b x \right )}-\frac {3 b \,{\mathrm e}^{\frac {d a -c b}{d}} \operatorname {Ei}_{1}\left (-b x -a -\frac {-d a +c b}{d}\right )}{8 d^{2}}-\frac {b \,{\mathrm e}^{3 b x +3 a}}{8 d^{2} \left (\frac {b c}{d}+b x \right )}-\frac {3 b \,{\mathrm e}^{\frac {3 d a -3 c b}{d}} \operatorname {Ei}_{1}\left (-3 b x -3 a -\frac {3 \left (-d a +c b \right )}{d}\right )}{8 d^{2}}\) | \(271\) |
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Leaf count of result is larger than twice the leaf count of optimal. 305 vs. \(2 (137) = 274\).
Time = 0.25 (sec) , antiderivative size = 305, normalized size of antiderivative = 2.10 \[ \int \frac {\cosh ^3(a+b x)}{(c+d x)^2} \, dx=-\frac {2 \, d \cosh \left (b x + a\right )^{3} + 6 \, d \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + 6 \, d \cosh \left (b x + a\right ) - 3 \, {\left ({\left (b d x + b c\right )} {\rm Ei}\left (\frac {b d x + b c}{d}\right ) - {\left (b d x + b c\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right )\right )} \cosh \left (-\frac {b c - a d}{d}\right ) - 3 \, {\left ({\left (b d x + b c\right )} {\rm Ei}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) - {\left (b d x + b c\right )} {\rm Ei}\left (-\frac {3 \, {\left (b d x + b c\right )}}{d}\right )\right )} \cosh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) - 3 \, {\left ({\left (b d x + b c\right )} {\rm Ei}\left (\frac {b d x + b c}{d}\right ) + {\left (b d x + b c\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right )\right )} \sinh \left (-\frac {b c - a d}{d}\right ) - 3 \, {\left ({\left (b d x + b c\right )} {\rm Ei}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) + {\left (b d x + b c\right )} {\rm Ei}\left (-\frac {3 \, {\left (b d x + b c\right )}}{d}\right )\right )} \sinh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )}{8 \, {\left (d^{3} x + c d^{2}\right )}} \]
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\[ \int \frac {\cosh ^3(a+b x)}{(c+d x)^2} \, dx=\int \frac {\cosh ^{3}{\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \]
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Time = 0.24 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00 \[ \int \frac {\cosh ^3(a+b x)}{(c+d x)^2} \, dx=-\frac {e^{\left (-3 \, a + \frac {3 \, b c}{d}\right )} E_{2}\left (\frac {3 \, {\left (d x + c\right )} b}{d}\right )}{8 \, {\left (d x + c\right )} d} - \frac {3 \, e^{\left (-a + \frac {b c}{d}\right )} E_{2}\left (\frac {{\left (d x + c\right )} b}{d}\right )}{8 \, {\left (d x + c\right )} d} - \frac {3 \, e^{\left (a - \frac {b c}{d}\right )} E_{2}\left (-\frac {{\left (d x + c\right )} b}{d}\right )}{8 \, {\left (d x + c\right )} d} - \frac {e^{\left (3 \, a - \frac {3 \, b c}{d}\right )} E_{2}\left (-\frac {3 \, {\left (d x + c\right )} b}{d}\right )}{8 \, {\left (d x + c\right )} d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1075 vs. \(2 (137) = 274\).
Time = 0.32 (sec) , antiderivative size = 1075, normalized size of antiderivative = 7.41 \[ \int \frac {\cosh ^3(a+b x)}{(c+d x)^2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\cosh ^3(a+b x)}{(c+d x)^2} \, dx=\int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^3}{{\left (c+d\,x\right )}^2} \,d x \]
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