Integrand size = 20, antiderivative size = 145 \[ \int \frac {(a+a \cosh (e+f x))^2}{c+d x} \, dx=\frac {2 a^2 \cosh \left (e-\frac {c f}{d}\right ) \text {Chi}\left (\frac {c f}{d}+f x\right )}{d}+\frac {a^2 \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Chi}\left (\frac {2 c f}{d}+2 f x\right )}{2 d}+\frac {3 a^2 \log (c+d x)}{2 d}+\frac {2 a^2 \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (\frac {c f}{d}+f x\right )}{d}+\frac {a^2 \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{2 d} \]
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Time = 0.27 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3399, 3393, 3384, 3379, 3382} \[ \int \frac {(a+a \cosh (e+f x))^2}{c+d x} \, dx=\frac {2 a^2 \text {Chi}\left (x f+\frac {c f}{d}\right ) \cosh \left (e-\frac {c f}{d}\right )}{d}+\frac {a^2 \text {Chi}\left (2 x f+\frac {2 c f}{d}\right ) \cosh \left (2 e-\frac {2 c f}{d}\right )}{2 d}+\frac {2 a^2 \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (x f+\frac {c f}{d}\right )}{d}+\frac {a^2 \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{2 d}+\frac {3 a^2 \log (c+d x)}{2 d} \]
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Rule 3379
Rule 3382
Rule 3384
Rule 3393
Rule 3399
Rubi steps \begin{align*} \text {integral}& = \left (4 a^2\right ) \int \frac {\sin ^4\left (\frac {1}{2} (i e+\pi )+\frac {i f x}{2}\right )}{c+d x} \, dx \\ & = \left (4 a^2\right ) \int \left (\frac {3}{8 (c+d x)}+\frac {\cosh (e+f x)}{2 (c+d x)}+\frac {\cosh (2 e+2 f x)}{8 (c+d x)}\right ) \, dx \\ & = \frac {3 a^2 \log (c+d x)}{2 d}+\frac {1}{2} a^2 \int \frac {\cosh (2 e+2 f x)}{c+d x} \, dx+\left (2 a^2\right ) \int \frac {\cosh (e+f x)}{c+d x} \, dx \\ & = \frac {3 a^2 \log (c+d x)}{2 d}+\frac {1}{2} \left (a^2 \cosh \left (2 e-\frac {2 c f}{d}\right )\right ) \int \frac {\cosh \left (\frac {2 c f}{d}+2 f x\right )}{c+d x} \, dx+\left (2 a^2 \cosh \left (e-\frac {c f}{d}\right )\right ) \int \frac {\cosh \left (\frac {c f}{d}+f x\right )}{c+d x} \, dx+\frac {1}{2} \left (a^2 \sinh \left (2 e-\frac {2 c f}{d}\right )\right ) \int \frac {\sinh \left (\frac {2 c f}{d}+2 f x\right )}{c+d x} \, dx+\left (2 a^2 \sinh \left (e-\frac {c f}{d}\right )\right ) \int \frac {\sinh \left (\frac {c f}{d}+f x\right )}{c+d x} \, dx \\ & = \frac {2 a^2 \cosh \left (e-\frac {c f}{d}\right ) \text {Chi}\left (\frac {c f}{d}+f x\right )}{d}+\frac {a^2 \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Chi}\left (\frac {2 c f}{d}+2 f x\right )}{2 d}+\frac {3 a^2 \log (c+d x)}{2 d}+\frac {2 a^2 \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (\frac {c f}{d}+f x\right )}{d}+\frac {a^2 \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{2 d} \\ \end{align*}
Time = 0.53 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.78 \[ \int \frac {(a+a \cosh (e+f x))^2}{c+d x} \, dx=\frac {a^2 \left (4 \cosh \left (e-\frac {c f}{d}\right ) \text {Chi}\left (f \left (\frac {c}{d}+x\right )\right )+\cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Chi}\left (\frac {2 f (c+d x)}{d}\right )+3 \log (c+d x)+4 \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (f \left (\frac {c}{d}+x\right )\right )+\sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )\right )}{2 d} \]
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Time = 0.47 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.32
method | result | size |
risch | \(-\frac {a^{2} {\mathrm e}^{\frac {c f -d e}{d}} \operatorname {Ei}_{1}\left (f x +e +\frac {c f -d e}{d}\right )}{d}-\frac {a^{2} {\mathrm e}^{-\frac {c f -d e}{d}} \operatorname {Ei}_{1}\left (-f x -e -\frac {c f -d e}{d}\right )}{d}+\frac {3 a^{2} \ln \left (d x +c \right )}{2 d}-\frac {a^{2} {\mathrm e}^{-\frac {2 \left (c f -d e \right )}{d}} \operatorname {Ei}_{1}\left (-2 f x -2 e -\frac {2 \left (c f -d e \right )}{d}\right )}{4 d}-\frac {a^{2} {\mathrm e}^{\frac {2 c f -2 d e}{d}} \operatorname {Ei}_{1}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right )}{4 d}\) | \(191\) |
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Time = 0.25 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.57 \[ \int \frac {(a+a \cosh (e+f x))^2}{c+d x} \, dx=\frac {6 \, a^{2} \log \left (d x + c\right ) + 4 \, {\left (a^{2} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) + a^{2} {\rm Ei}\left (-\frac {d f x + c f}{d}\right )\right )} \cosh \left (-\frac {d e - c f}{d}\right ) + {\left (a^{2} {\rm Ei}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) + a^{2} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right )\right )} \cosh \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) - 4 \, {\left (a^{2} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) - a^{2} {\rm Ei}\left (-\frac {d f x + c f}{d}\right )\right )} \sinh \left (-\frac {d e - c f}{d}\right ) - {\left (a^{2} {\rm Ei}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) - a^{2} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right )\right )} \sinh \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right )}{4 \, d} \]
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\[ \int \frac {(a+a \cosh (e+f x))^2}{c+d x} \, dx=a^{2} \left (\int \frac {2 \cosh {\left (e + f x \right )}}{c + d x}\, dx + \int \frac {\cosh ^{2}{\left (e + f x \right )}}{c + d x}\, dx + \int \frac {1}{c + d x}\, dx\right ) \]
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Time = 0.24 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.03 \[ \int \frac {(a+a \cosh (e+f x))^2}{c+d x} \, dx=-\frac {1}{4} \, a^{2} {\left (\frac {e^{\left (-2 \, e + \frac {2 \, c f}{d}\right )} E_{1}\left (\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{d} + \frac {e^{\left (2 \, e - \frac {2 \, c f}{d}\right )} E_{1}\left (-\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{d} - \frac {2 \, \log \left (d x + c\right )}{d}\right )} - a^{2} {\left (\frac {e^{\left (-e + \frac {c f}{d}\right )} E_{1}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{d} + \frac {e^{\left (e - \frac {c f}{d}\right )} E_{1}\left (-\frac {{\left (d x + c\right )} f}{d}\right )}{d}\right )} + \frac {a^{2} \log \left (d x + c\right )}{d} \]
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Time = 0.31 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.93 \[ \int \frac {(a+a \cosh (e+f x))^2}{c+d x} \, dx=\frac {a^{2} {\rm Ei}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) e^{\left (2 \, e - \frac {2 \, c f}{d}\right )} + 4 \, a^{2} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) e^{\left (e - \frac {c f}{d}\right )} + 4 \, a^{2} {\rm Ei}\left (-\frac {d f x + c f}{d}\right ) e^{\left (-e + \frac {c f}{d}\right )} + a^{2} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) e^{\left (-2 \, e + \frac {2 \, c f}{d}\right )} + 6 \, a^{2} \log \left (d x + c\right )}{4 \, d} \]
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Timed out. \[ \int \frac {(a+a \cosh (e+f x))^2}{c+d x} \, dx=\int \frac {{\left (a+a\,\mathrm {cosh}\left (e+f\,x\right )\right )}^2}{c+d\,x} \,d x \]
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