Optimal. Leaf size=200 \[ -\frac {4 d (c+d x) \log \left (e^{e+f x}+1\right )}{3 a^2 f^2}+\frac {d (c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^2}+\frac {(c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {(c+d x)^2}{3 a^2 f}-\frac {4 d^2 \text {Li}_2\left (-e^{e+f x}\right )}{3 a^2 f^3}-\frac {2 d^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^3} \]
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Rubi [A] time = 0.25, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {3318, 4186, 3767, 8, 4184, 3718, 2190, 2279, 2391} \[ -\frac {4 d^2 \text {PolyLog}\left (2,-e^{e+f x}\right )}{3 a^2 f^3}-\frac {4 d (c+d x) \log \left (e^{e+f x}+1\right )}{3 a^2 f^2}+\frac {d (c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^2}+\frac {(c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {(c+d x)^2}{3 a^2 f}-\frac {2 d^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^3} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2190
Rule 2279
Rule 2391
Rule 3318
Rule 3718
Rule 3767
Rule 4184
Rule 4186
Rubi steps
\begin {align*} \int \frac {(c+d x)^2}{(a+a \cosh (e+f x))^2} \, dx &=\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 ) \operatorname {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 ) \operatorname {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 \text {Li}_2\left (-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*}
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Mathematica [C] time = 6.48, size = 637, normalized size = 3.18 \[ \frac {\text {sech}\left (\frac {e}{2}\right ) \cosh \left (\frac {e}{2}+\frac {f x}{2}\right ) \left (c^2 f^2 \sinh \left (e+\frac {3 f x}{2}\right )+3 c^2 f^2 \sinh \left (\frac {f x}{2}\right )+2 c d f^2 x \sinh \left (e+\frac {3 f x}{2}\right )+2 c d f \cosh \left (e+\frac {f x}{2}\right )+6 c d f^2 x \sinh \left (\frac {f x}{2}\right )+2 c d f \cosh \left (\frac {f x}{2}\right )+d^2 f^2 x^2 \sinh \left (e+\frac {3 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 )+2 d^2 f x \cosh \left (e+\frac {f x}{2}\right )+3 d^2 f^2 x^2 \sinh \left (\frac {f x}{2}\right )-4 d^2 \sinh \left (\frac {f x}{2}\right )+2 d^2 f x \cosh \left (\frac {f x}{2}\right )\right )}{3 f^3 (a \cosh (e+f x)+a)^2}-\frac {16 c d \text {sech}\left (\frac {e}{2}\right ) \cosh ^4\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 )}{3 f^2 \left (\cosh ^2\left (\frac {e}{2}\right )-\sinh ^2\left (\frac {e}{2}\right )\right ) (a \cosh (e+f x)+a)^2}-\frac {16 d^2 \text {csch}\left (\frac {e}{2}\right ) \text {sech}\left (\frac {e}{2}\right ) \cosh ^4\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 )}{3 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)^2} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.69, size = 963, normalized size = 4.82 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (d x + c\right )}^{2}}{{\left (a \cosh \left (f x + e\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 313, normalized size = 1.56 \[ -\frac {2 \left (3 f^{2} d^{2} x^{2} {\mathrm e}^{f x +e}+6 f^{2} c d x \,{\mathrm e}^{f x +e}+d^{2} f^{2} x^{2}-2 d^{2} f x \,{\mathrm e}^{2 f x +2 e}+3 f^{2} c^{2} {\mathrm e}^{f x +e}+2 c d \,f^{2} x -2 c d f \,{\mathrm e}^{2 f x +2 e}-2 f \,d^{2} x \,{\mathrm e}^{f x +e}+c^{2} f^{2}-2 f c d \,{\mathrm e}^{f x +e}-2 d^{2} {\mathrm e}^{2 f x +2 e}-4 d^{2} {\mathrm e}^{f x +e}-2 d^{2}\right )}{3 f^{3} a^{2} \left ({\mathrm e}^{f x +e}+1\right )^{3}}-\frac {4 d c \ln \left ({\mathrm e}^{f x +e}+1\right )}{3 a^{2} f^{2}}+\frac {4 d c \ln \left ({\mathrm e}^{f x +e}\right )}{3 a^{2} f^{2}}+\frac {2 d^{2} x^{2}}{3 a^{2} f}+\frac {4 d^{2} e x}{3 a^{2} f^{2}}+\frac {2 d^{2} e^{2}}{3 a^{2} f^{3}}-\frac {4 d^{2} \ln \left ({\mathrm e}^{f x +e}+1\right ) x}{3 a^{2} f^{2}}-\frac {4 d^{2} \polylog \left (2, -{\mathrm e}^{f x +e}\right )}{3 a^{2} f^{3}}-\frac {4 d^{2} e \ln \left ({\mathrm e}^{f x +e}\right )}{3 a^{2} f^{3}} \]
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 \[ -\frac {2}{3} \, d^{2} {\left (\frac {f^{2} x^{2} - 2 \, {\left (f x e^{\left (2 \, e\right )} + e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )} + {\left (3 \, f^{2} x^{2} e^{e} - 2 \, f x e^{e} - 4 \, e^{e}\right )} e^{\left (f x\right )} - 2}{a^{2} f^{3} e^{\left (3 \, f x + 3 \, e\right )} + 3 \, a^{2} f^{3} e^{\left (2 \, f x + 2 \, e\right )} + 3 \, a^{2} f^{3} e^{\left (f x + e\right )} + a^{2} f^{3}} - 6 \, \int \frac {x}{3 \, {\left (a^{2} f e^{\left (f x + e\right )} + a^{2} f\right )}}\,{d x}\right )} + \frac {4}{3} \, c d {\left (\frac {f x e^{\left (3 \, f x + 3 \, e\right )} + {\left (3 \, f x e^{\left (2 \, e\right )} + e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )} + e^{\left (f x + e\right )}}{a^{2} f^{2} e^{\left (3 \, f x + 3 \, e\right )} + 3 \, a^{2} f^{2} e^{\left (2 \, f x + 2 \, e\right )} + 3 \, a^{2} f^{2} e^{\left (f x + e\right )} + a^{2} f^{2}} - \frac {\log \left ({\left (e^{\left (f x + e\right )} + 1\right )} e^{\left (-e\right )}\right )}{a^{2} f^{2}}\right )} + \frac {2}{3} \, c^{2} {\left (\frac {3 \, e^{\left (-f x - e\right )}}{{\left (3 \, a^{2} e^{\left (-f x - e\right )} + 3 \, a^{2} e^{\left (-2 \, f x - 2 \, e\right )} + a^{2} e^{\left (-3 \, f x - 3 \, e\right )} + a^{2}\right )} f} + \frac {1}{{\left (3 \, a^{2} e^{\left (-f x - e\right )} + 3 \, a^{2} e^{\left (-2 \, f x - 2 \, e\right )} + a^{2} e^{\left (-3 \, f x - 3 \, e\right )} + a^{2}\right )} f}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c+d\,x\right )}^2}{{\left (a+a\,\mathrm {cosh}\left (e+f\,x\right )\right )}^2} \,d x \]
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 \[ \frac {\int \frac {c^{2}}{\cosh ^{2}{\left (e + f x \right )} + 2 \cosh {\left (e + f x \right )} + 1}\, dx + \int \frac {d^{2} x^{2}}{\cosh ^{2}{\left (e + f x \right )} + 2 \cosh {\left (e + f x \right )} + 1}\, dx + \int \frac {2 c d x}{\cosh ^{2}{\left (e + f x \right )} + 2 \cosh {\left (e + f x \right )} + 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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