Integrand size = 20, antiderivative size = 88 \[ \int \frac {(c+d x)^2}{a+a \cosh (e+f x)} \, dx=\frac {(c+d x)^2}{a f}-\frac {4 d (c+d x) \log \left (1+e^{e+f x}\right )}{a f^2}-\frac {4 d^2 \operatorname {PolyLog}\left (2,-e^{e+f x}\right )}{a f^3}+\frac {(c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f} \]
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Time = 0.14 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3399, 4269, 3799, 2221, 2317, 2438} \[ \int \frac {(c+d x)^2}{a+a \cosh (e+f x)} \, dx=-\frac {4 d (c+d x) \log \left (e^{e+f x}+1\right )}{a f^2}+\frac {(c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}+\frac {(c+d x)^2}{a f}-\frac {4 d^2 \operatorname {PolyLog}\left (2,-e^{e+f x}\right )}{a f^3} \]
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Rule 2221
Rule 2317
Rule 2438
Rule 3399
Rule 3799
Rule 4269
Rubi steps \begin{align*} \text {integral}& = \frac {\int (c+d x)^2 \csc ^2\left (\frac {1}{2} (i e+\pi )+\frac {i f x}{2}\right ) \, dx}{2 a} \\ & = \frac {(c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}-\frac {(2 d) \int (c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right ) \, dx}{a f} \\ & = \frac {(c+d x)^2}{a f}+\frac {(c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}-\frac {(4 d) \int \frac {e^{2 \left (\frac {e}{2}+\frac {f x}{2}\right )} (c+d x)}{1+e^{2 \left (\frac {e}{2}+\frac {f x}{2}\right )}} \, dx}{a f} \\ & = \frac {(c+d x)^2}{a f}-\frac {4 d (c+d x) \log \left (1+e^{e+f x}\right )}{a f^2}+\frac {(c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}+\frac {\left (4 d^2\right ) \int \log \left (1+e^{2 \left (\frac {e}{2}+\frac {f x}{2}\right )}\right ) \, dx}{a f^2} \\ & = \frac {(c+d x)^2}{a f}-\frac {4 d (c+d x) \log \left (1+e^{e+f x}\right )}{a f^2}+\frac {(c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}+\frac {\left (4 d^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \left (\frac {e}{2}+\frac {f x}{2}\right )}\right )}{a f^3} \\ & = \frac {(c+d x)^2}{a f}-\frac {4 d (c+d x) \log \left (1+e^{e+f x}\right )}{a f^2}-\frac {4 d^2 \operatorname {PolyLog}\left (2,-e^{e+f x}\right )}{a f^3}+\frac {(c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f} \\ \end{align*}
Time = 0.67 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.51 \[ \int \frac {(c+d x)^2}{a+a \cosh (e+f x)} \, dx=\frac {2 \cosh \left (\frac {1}{2} (e+f x)\right ) \left (-\frac {2 \cosh \left (\frac {1}{2} (e+f x)\right ) \left (f (c+d x) \left (f (c+d x)+2 d \left (1+e^e\right ) \log \left (1+e^{-e-f x}\right )\right )-2 d^2 \left (1+e^e\right ) \operatorname {PolyLog}\left (2,-e^{-e-f x}\right )\right )}{\left (1+e^e\right ) f^2}+(c+d x)^2 \text {sech}\left (\frac {e}{2}\right ) \sinh \left (\frac {f x}{2}\right )\right )}{a f (1+\cosh (e+f x))} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(173\) vs. \(2(82)=164\).
Time = 0.15 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.98
method | result | size |
risch | \(-\frac {2 \left (x^{2} d^{2}+2 c d x +c^{2}\right )}{f a \left (1+{\mathrm e}^{f x +e}\right )}+\frac {4 d c \ln \left ({\mathrm e}^{f x +e}\right )}{a \,f^{2}}-\frac {4 d c \ln \left (1+{\mathrm e}^{f x +e}\right )}{a \,f^{2}}+\frac {2 d^{2} x^{2}}{a f}+\frac {4 d^{2} e x}{a \,f^{2}}+\frac {2 d^{2} e^{2}}{a \,f^{3}}-\frac {4 d^{2} \ln \left (1+{\mathrm e}^{f x +e}\right ) x}{a \,f^{2}}-\frac {4 d^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{f x +e}\right )}{a \,f^{3}}-\frac {4 d^{2} e \ln \left ({\mathrm e}^{f x +e}\right )}{a \,f^{3}}\) | \(174\) |
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Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (81) = 162\).
Time = 0.25 (sec) , antiderivative size = 243, normalized size of antiderivative = 2.76 \[ \int \frac {(c+d x)^2}{a+a \cosh (e+f x)} \, dx=-\frac {2 \, {\left (d^{2} e^{2} - 2 \, c d e f + c^{2} f^{2} - {\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x - d^{2} e^{2} + 2 \, c d e f\right )} \cosh \left (f x + e\right ) + 2 \, {\left (d^{2} \cosh \left (f x + e\right ) + d^{2} \sinh \left (f x + e\right ) + d^{2}\right )} {\rm Li}_2\left (-\cosh \left (f x + e\right ) - \sinh \left (f x + e\right )\right ) + 2 \, {\left (d^{2} f x + c d f + {\left (d^{2} f x + c d f\right )} \cosh \left (f x + e\right ) + {\left (d^{2} f x + c d f\right )} \sinh \left (f x + e\right )\right )} \log \left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right ) + 1\right ) - {\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x - d^{2} e^{2} + 2 \, c d e f\right )} \sinh \left (f x + e\right )\right )}}{a f^{3} \cosh \left (f x + e\right ) + a f^{3} \sinh \left (f x + e\right ) + a f^{3}} \]
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\[ \int \frac {(c+d x)^2}{a+a \cosh (e+f x)} \, dx=\frac {\int \frac {c^{2}}{\cosh {\left (e + f x \right )} + 1}\, dx + \int \frac {d^{2} x^{2}}{\cosh {\left (e + f x \right )} + 1}\, dx + \int \frac {2 c d x}{\cosh {\left (e + f x \right )} + 1}\, dx}{a} \]
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\[ \int \frac {(c+d x)^2}{a+a \cosh (e+f x)} \, dx=\int { \frac {{\left (d x + c\right )}^{2}}{a \cosh \left (f x + e\right ) + a} \,d x } \]
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\[ \int \frac {(c+d x)^2}{a+a \cosh (e+f x)} \, dx=\int { \frac {{\left (d x + c\right )}^{2}}{a \cosh \left (f x + e\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {(c+d x)^2}{a+a \cosh (e+f x)} \, dx=\int \frac {{\left (c+d\,x\right )}^2}{a+a\,\mathrm {cosh}\left (e+f\,x\right )} \,d x \]
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