Integrand size = 20, antiderivative size = 200 \[ \int \frac {(c+d x)^2}{(a+a \cosh (e+f x))^2} \, dx=\frac {(c+d x)^2}{3 a^2 f}-\frac {4 d (c+d x) \log \left (1+e^{e+f x}\right )}{3 a^2 f^2}-\frac {4 d^2 \operatorname {PolyLog}\left (2,-e^{e+f x}\right )}{3 a^2 f^3}+\frac {d (c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^2}-\frac {2 d^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^3}+\frac {(c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f} \]
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Time = 0.19 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {3399, 4271, 3852, 8, 4269, 3799, 2221, 2317, 2438} \[ \int \frac {(c+d x)^2}{(a+a \cosh (e+f x))^2} \, dx=-\frac {4 d (c+d x) \log \left (e^{e+f x}+1\right )}{3 a^2 f^2}+\frac {d (c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^2}+\frac {(c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {(c+d x)^2}{3 a^2 f}-\frac {4 d^2 \operatorname {PolyLog}\left (2,-e^{e+f x}\right )}{3 a^2 f^3}-\frac {2 d^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^3} \]
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Rule 8
Rule 2221
Rule 2317
Rule 2438
Rule 3399
Rule 3799
Rule 3852
Rule 4269
Rule 4271
Rubi steps \begin{align*} \text {integral}& = \frac {\int (c+d x)^2 \csc ^4\left (\frac {1}{2} (i e+\pi )+\frac {i f x}{2}\right ) \, dx}{4 a^2} \\ & = \frac {d (c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^2}+\frac {(c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {\int (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \, dx}{6 a^2}-\frac {d^2 \int \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \, dx}{3 a^2 f^2} \\ & = \frac {d (c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^2}+\frac {(c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}-\frac {\left (2 i d^2\right ) \text {Subst}\left (\int 1 \, dx,x,-i \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{3 a^2 f^3}-\frac {(2 d) \int (c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right ) \, dx}{3 a^2 f} \\ & = \frac {(c+d x)^2}{3 a^2 f}+\frac {d (c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^2}-\frac {2 d^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^3}+\frac {(c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 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}{3 a^2 f} \\ & = \frac {(c+d x)^2}{3 a^2 f}-\frac {4 d (c+d x) \log \left (1+e^{e+f x}\right )}{3 a^2 f^2}+\frac {d (c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^2}-\frac {2 d^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^3}+\frac {(c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {\left (4 d^2\right ) \int \log \left (1+e^{2 \left (\frac {e}{2}+\frac {f x}{2}\right )}\right ) \, dx}{3 a^2 f^2} \\ & = \frac {(c+d x)^2}{3 a^2 f}-\frac {4 d (c+d x) \log \left (1+e^{e+f x}\right )}{3 a^2 f^2}+\frac {d (c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^2}-\frac {2 d^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^3}+\frac {(c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 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 )}{3 a^2 f^3} \\ & = \frac {(c+d x)^2}{3 a^2 f}-\frac {4 d (c+d x) \log \left (1+e^{e+f x}\right )}{3 a^2 f^2}-\frac {4 d^2 \operatorname {PolyLog}\left (2,-e^{e+f x}\right )}{3 a^2 f^3}+\frac {d (c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^2}-\frac {2 d^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^3}+\frac {(c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f} \\ \end{align*}
Time = 1.05 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.48 \[ \int \frac {(c+d x)^2}{(a+a \cosh (e+f x))^2} \, dx=\frac {\cosh \left (\frac {1}{2} (e+f x)\right ) \left (-\frac {8 \cosh ^3\left (\frac {1}{2} (e+f x)\right ) \left (f (c+d x) \left (f (c+d x)+2 d \left (1+e^e\right ) \log \left (1+e^{-e-f x}\right )\right )-2 d^2 \left (1+e^e\right ) \operatorname {PolyLog}\left (2,-e^{-e-f x}\right )\right )}{1+e^e}+\text {sech}\left (\frac {e}{2}\right ) \left (2 d f (c+d x) \cosh \left (\frac {f x}{2}\right )+2 d f (c+d x) \cosh \left (e+\frac {f x}{2}\right )-4 d^2 \sinh \left (\frac {f x}{2}\right )+3 c^2 f^2 \sinh \left (\frac {f x}{2}\right )+6 c d f^2 x \sinh \left (\frac {f x}{2}\right )+3 d^2 f^2 x^2 \sinh \left (\frac {f x}{2}\right )+2 d^2 \sinh \left (e+\frac {f x}{2}\right )-2 d^2 \sinh \left (e+\frac {3 f x}{2}\right )+c^2 f^2 \sinh \left (e+\frac {3 f x}{2}\right )+2 c d f^2 x \sinh \left (e+\frac {3 f x}{2}\right )+d^2 f^2 x^2 \sinh \left (e+\frac {3 f x}{2}\right )\right )\right )}{3 a^2 f^3 (1+\cosh (e+f x))^2} \]
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Time = 0.21 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.56
method | result | size |
risch | \(-\frac {2 \left (3 \,{\mathrm e}^{f x +e} d^{2} f^{2} x^{2}+6 \,{\mathrm e}^{f x +e} c d \,f^{2} x +d^{2} x^{2} f^{2}-2 d^{2} f x \,{\mathrm e}^{2 f x +2 e}+3 \,{\mathrm e}^{f x +e} c^{2} f^{2}+2 c d \,f^{2} x -2 c d f \,{\mathrm e}^{2 f x +2 e}-2 d^{2} f x \,{\mathrm e}^{f x +e}+c^{2} f^{2}-2 c d f \,{\mathrm e}^{f x +e}-2 \,{\mathrm e}^{2 f x +2 e} d^{2}-4 \,{\mathrm e}^{f x +e} d^{2}-2 d^{2}\right )}{3 f^{3} a^{2} \left (1+{\mathrm e}^{f x +e}\right )^{3}}+\frac {4 d c \ln \left ({\mathrm e}^{f x +e}\right )}{3 a^{2} f^{2}}-\frac {4 d c \ln \left (1+{\mathrm e}^{f x +e}\right )}{3 a^{2} f^{2}}+\frac {2 d^{2} x^{2}}{3 a^{2} f}+\frac {4 d^{2} e x}{3 a^{2} f^{2}}+\frac {2 d^{2} e^{2}}{3 a^{2} f^{3}}-\frac {4 d^{2} \ln \left (1+{\mathrm e}^{f x +e}\right ) x}{3 a^{2} f^{2}}-\frac {4 d^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{f x +e}\right )}{3 a^{2} f^{3}}-\frac {4 d^{2} e \ln \left ({\mathrm e}^{f x +e}\right )}{3 a^{2} f^{3}}\) | \(313\) |
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Leaf count of result is larger than twice the leaf count of optimal. 963 vs. \(2 (163) = 326\).
Time = 0.25 (sec) , antiderivative size = 963, normalized size of antiderivative = 4.82 \[ \int \frac {(c+d x)^2}{(a+a \cosh (e+f x))^2} \, dx=\text {Too large to display} \]
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\[ \int \frac {(c+d x)^2}{(a+a \cosh (e+f x))^2} \, dx=\frac {\int \frac {c^{2}}{\cosh ^{2}{\left (e + f x \right )} + 2 \cosh {\left (e + f x \right )} + 1}\, dx + \int \frac {d^{2} x^{2}}{\cosh ^{2}{\left (e + f x \right )} + 2 \cosh {\left (e + f x \right )} + 1}\, dx + \int \frac {2 c d x}{\cosh ^{2}{\left (e + f x \right )} + 2 \cosh {\left (e + f x \right )} + 1}\, dx}{a^{2}} \]
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\[ \int \frac {(c+d x)^2}{(a+a \cosh (e+f x))^2} \, dx=\int { \frac {{\left (d x + c\right )}^{2}}{{\left (a \cosh \left (f x + e\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {(c+d x)^2}{(a+a \cosh (e+f x))^2} \, dx=\int { \frac {{\left (d x + c\right )}^{2}}{{\left (a \cosh \left (f x + e\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {(c+d x)^2}{(a+a \cosh (e+f x))^2} \, dx=\int \frac {{\left (c+d\,x\right )}^2}{{\left (a+a\,\mathrm {cosh}\left (e+f\,x\right )\right )}^2} \,d x \]
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