Optimal. Leaf size=88 \[ -\frac {4 d (c+d x) \log \left (e^{e+f x}+1\right )}{a f^2}+\frac {(c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}+\frac {(c+d x)^2}{a f}-\frac {4 d^2 \text {Li}_2\left (-e^{e+f x}\right )}{a f^3} \]
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Rubi [A] time = 0.20, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3318, 4184, 3718, 2190, 2279, 2391} \[ -\frac {4 d^2 \text {PolyLog}\left (2,-e^{e+f x}\right )}{a f^3}-\frac {4 d (c+d x) \log \left (e^{e+f x}+1\right )}{a f^2}+\frac {(c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}+\frac {(c+d x)^2}{a f} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 3318
Rule 3718
Rule 4184
Rubi steps
\begin {align*} \int \frac {(c+d x)^2}{a+a \cosh (e+f x)} \, dx &=\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 ) \operatorname {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 \text {Li}_2\left (-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*}
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Mathematica [C] time = 6.38, size = 472, normalized size = 5.36 \[ \frac {2 \text {sech}\left (\frac {e}{2}\right ) \cosh \left (\frac {e}{2}+\frac {f x}{2}\right ) \left (c^2 \sinh \left (\frac {f x}{2}\right )+2 c d x \sinh \left (\frac {f x}{2}\right )+d^2 x^2 \sinh \left (\frac {f x}{2}\right )\right )}{f (a \cosh (e+f x)+a)}-\frac {8 c d \text {sech}\left (\frac {e}{2}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \left (\cosh \left (\frac {e}{2}\right ) \log \left (\sinh \left (\frac {e}{2}\right ) \sinh \left (\frac {f x}{2}\right )+\cosh \left (\frac {e}{2}\right ) \cosh \left (\frac {f x}{2}\right )\right )-\frac {1}{2} f x \sinh \left (\frac {e}{2}\right )\right )}{f^2 \left (\cosh ^2\left (\frac {e}{2}\right )-\sinh ^2\left (\frac {e}{2}\right )\right ) (a \cosh (e+f x)+a)}-\frac {8 d^2 \text {csch}\left (\frac {e}{2}\right ) \text {sech}\left (\frac {e}{2}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \left (\frac {1}{4} f^2 x^2 e^{-\tanh ^{-1}\left (\coth \left (\frac {e}{2}\right )\right )}-\frac {i \coth \left (\frac {e}{2}\right ) \left (i \text {Li}_2\left (e^{2 i \left (\frac {i f x}{2}+i \tanh ^{-1}\left (\coth \left (\frac {e}{2}\right )\right )\right )}\right )-\frac {1}{2} f x \left (-\pi +2 i \tanh ^{-1}\left (\coth \left (\frac {e}{2}\right )\right )\right )-2 \left (i \tanh ^{-1}\left (\coth \left (\frac {e}{2}\right )\right )+\frac {i f x}{2}\right ) \log \left (1-e^{2 i \left (i \tanh ^{-1}\left (\coth \left (\frac {e}{2}\right )\right )+\frac {i f x}{2}\right )}\right )+2 i \tanh ^{-1}\left (\coth \left (\frac {e}{2}\right )\right ) \log \left (i \sinh \left (\tanh ^{-1}\left (\coth \left (\frac {e}{2}\right )\right )+\frac {f x}{2}\right )\right )-\pi \log \left (e^{f x}+1\right )+\pi \log \left (\cosh \left (\frac {f x}{2}\right )\right )\right )}{\sqrt {1-\coth ^2\left (\frac {e}{2}\right )}}\right )}{f^3 \sqrt {\text {csch}^2\left (\frac {e}{2}\right ) \left (\sinh ^2\left (\frac {e}{2}\right )-\cosh ^2\left (\frac {e}{2}\right )\right )} (a \cosh (e+f x)+a)} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.41, size = 243, normalized size = 2.76 \[ -\frac {2 \, {\left (d^{2} e^{2} - 2 \, c d e f + c^{2} f^{2} - {\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x - d^{2} e^{2} + 2 \, c d e f\right )} \cosh \left (f x + e\right ) + 2 \, {\left (d^{2} \cosh \left (f x + e\right ) + d^{2} \sinh \left (f x + e\right ) + d^{2}\right )} {\rm Li}_2\left (-\cosh \left (f x + e\right ) - \sinh \left (f x + e\right )\right ) + 2 \, {\left (d^{2} f x + c d f + {\left (d^{2} f x + c d f\right )} \cosh \left (f x + e\right ) + {\left (d^{2} f x + c d f\right )} \sinh \left (f x + e\right )\right )} \log \left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right ) + 1\right ) - {\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x - d^{2} e^{2} + 2 \, c d e f\right )} \sinh \left (f x + e\right )\right )}}{a f^{3} \cosh \left (f x + e\right ) + a f^{3} \sinh \left (f x + e\right ) + a f^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (d x + c\right )}^{2}}{a \cosh \left (f x + e\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.21, size = 174, normalized size = 1.98 \[ -\frac {2 \left (d^{2} x^{2}+2 c d x +c^{2}\right )}{f a \left ({\mathrm e}^{f x +e}+1\right )}-\frac {4 d c \ln \left ({\mathrm e}^{f x +e}+1\right )}{a \,f^{2}}+\frac {4 d \ln \left ({\mathrm e}^{f x +e}\right ) c}{a \,f^{2}}+\frac {2 d^{2} x^{2}}{a f}+\frac {4 d^{2} e x}{a \,f^{2}}+\frac {2 d^{2} e^{2}}{a \,f^{3}}-\frac {4 d^{2} \ln \left ({\mathrm e}^{f x +e}+1\right ) x}{a \,f^{2}}-\frac {4 d^{2} \polylog \left (2, -{\mathrm e}^{f x +e}\right )}{a \,f^{3}}-\frac {4 d^{2} e \ln \left ({\mathrm e}^{f x +e}\right )}{a \,f^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -2 \, d^{2} {\left (\frac {x^{2}}{a f e^{\left (f x + e\right )} + a f} - 2 \, \int \frac {x}{a f e^{\left (f x + e\right )} + a f}\,{d x}\right )} + 4 \, c d {\left (\frac {x e^{\left (f x + e\right )}}{a f e^{\left (f x + e\right )} + a f} - \frac {\log \left ({\left (e^{\left (f x + e\right )} + 1\right )} e^{\left (-e\right )}\right )}{a f^{2}}\right )} + \frac {2 \, c^{2}}{{\left (a e^{\left (-f x - e\right )} + a\right )} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c+d\,x\right )}^2}{a+a\,\mathrm {cosh}\left (e+f\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {c^{2}}{\cosh {\left (e + f x \right )} + 1}\, dx + \int \frac {d^{2} x^{2}}{\cosh {\left (e + f x \right )} + 1}\, dx + \int \frac {2 c d x}{\cosh {\left (e + f x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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