Optimal. Leaf size=255 \[ -\frac {4 d^2 (c+d x) \text {Li}_2\left (-e^{e+f x}\right )}{a^2 f^3}-\frac {2 d^2 (c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{a^2 f^3}-\frac {2 d (c+d x)^2 \log \left (e^{e+f x}+1\right )}{a^2 f^2}+\frac {d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{2 a^2 f^2}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {(c+d x)^3}{3 a^2 f}+\frac {4 d^3 \text {Li}_3\left (-e^{e+f x}\right )}{a^2 f^4}+\frac {4 d^3 \log \left (\cosh \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{a^2 f^4} \]
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Rubi [A] time = 0.36, antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {3318, 4186, 4184, 3475, 3718, 2190, 2531, 2282, 6589} \[ -\frac {4 d^2 (c+d x) \text {PolyLog}\left (2,-e^{e+f x}\right )}{a^2 f^3}+\frac {4 d^3 \text {PolyLog}\left (3,-e^{e+f x}\right )}{a^2 f^4}-\frac {2 d^2 (c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{a^2 f^3}-\frac {2 d (c+d x)^2 \log \left (e^{e+f x}+1\right )}{a^2 f^2}+\frac {d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{2 a^2 f^2}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {(c+d x)^3}{3 a^2 f}+\frac {4 d^3 \log \left (\cosh \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{a^2 f^4} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2282
Rule 2531
Rule 3318
Rule 3475
Rule 3718
Rule 4184
Rule 4186
Rule 6589
Rubi steps
\begin {align*} \int \frac {(c+d x)^3}{(a+a \cosh (e+f x))^2} \, dx &=\frac {\int (c+d x)^3 \csc ^4\left (\frac {1}{2} (i e+\pi )+\frac {i f x}{2}\right ) \, dx}{4 a^2}\\ &=\frac {d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{2 a^2 f^2}+\frac {(c+d x)^3 \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)^3 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \, dx}{6 a^2}-\frac {d^2 \int (c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \, dx}{a^2 f^2}\\ &=\frac {d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{2 a^2 f^2}-\frac {2 d^2 (c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{a^2 f^3}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^3 \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 d^3\right ) \int \tanh \left (\frac {e}{2}+\frac {f x}{2}\right ) \, dx}{a^2 f^3}-\frac {d \int (c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right ) \, dx}{a^2 f}\\ &=\frac {(c+d x)^3}{3 a^2 f}+\frac {4 d^3 \log \left (\cosh \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{a^2 f^4}+\frac {d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{2 a^2 f^2}-\frac {2 d^2 (c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{a^2 f^3}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^3 \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 {(2 d) \int \frac {e^{2 \left (\frac {e}{2}+\frac {f x}{2}\right )} (c+d x)^2}{1+e^{2 \left (\frac {e}{2}+\frac {f x}{2}\right )}} \, dx}{a^2 f}\\ &=\frac {(c+d x)^3}{3 a^2 f}-\frac {2 d (c+d x)^2 \log \left (1+e^{e+f x}\right )}{a^2 f^2}+\frac {4 d^3 \log \left (\cosh \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{a^2 f^4}+\frac {d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{2 a^2 f^2}-\frac {2 d^2 (c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{a^2 f^3}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^3 \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 (c+d x) \log \left (1+e^{2 \left (\frac {e}{2}+\frac {f x}{2}\right )}\right ) \, dx}{a^2 f^2}\\ &=\frac {(c+d x)^3}{3 a^2 f}-\frac {2 d (c+d x)^2 \log \left (1+e^{e+f x}\right )}{a^2 f^2}+\frac {4 d^3 \log \left (\cosh \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{a^2 f^4}-\frac {4 d^2 (c+d x) \text {Li}_2\left (-e^{e+f x}\right )}{a^2 f^3}+\frac {d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{2 a^2 f^2}-\frac {2 d^2 (c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{a^2 f^3}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^3 \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^3\right ) \int \text {Li}_2\left (-e^{2 \left (\frac {e}{2}+\frac {f x}{2}\right )}\right ) \, dx}{a^2 f^3}\\ &=\frac {(c+d x)^3}{3 a^2 f}-\frac {2 d (c+d x)^2 \log \left (1+e^{e+f x}\right )}{a^2 f^2}+\frac {4 d^3 \log \left (\cosh \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{a^2 f^4}-\frac {4 d^2 (c+d x) \text {Li}_2\left (-e^{e+f x}\right )}{a^2 f^3}+\frac {d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{2 a^2 f^2}-\frac {2 d^2 (c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{a^2 f^3}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^3 \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^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 \left (\frac {e}{2}+\frac {f x}{2}\right )}\right )}{a^2 f^4}\\ &=\frac {(c+d x)^3}{3 a^2 f}-\frac {2 d (c+d x)^2 \log \left (1+e^{e+f x}\right )}{a^2 f^2}+\frac {4 d^3 \log \left (\cosh \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{a^2 f^4}-\frac {4 d^2 (c+d x) \text {Li}_2\left (-e^{e+f x}\right )}{a^2 f^3}+\frac {4 d^3 \text {Li}_3\left (-e^{e+f x}\right )}{a^2 f^4}+\frac {d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{2 a^2 f^2}-\frac {2 d^2 (c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{a^2 f^3}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^3 \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 [A] time = 3.49, size = 462, normalized size = 1.81 \[ \frac {\cosh \left (\frac {1}{2} (e+f x)\right ) \left (\text {sech}\left (\frac {e}{2}\right ) (c+d x) \left (c^2 f^2 \sinh \left (e+\frac {3 f x}{2}\right )+3 c^2 f^2 \sinh \left (\frac {f x}{2}\right )+2 c d f^2 x \sinh \left (e+\frac {3 f x}{2}\right )+3 d f (c+d x) \cosh \left (e+\frac {f x}{2}\right )+6 c d f^2 x \sinh \left (\frac {f x}{2}\right )+3 d f (c+d x) \cosh \left (\frac {f x}{2}\right )+d^2 f^2 x^2 \sinh \left (e+\frac {3 f x}{2}\right )+6 d^2 \sinh \left (e+\frac {f x}{2}\right )-6 d^2 \sinh \left (e+\frac {3 f x}{2}\right )+3 d^2 f^2 x^2 \sinh \left (\frac {f x}{2}\right )-12 d^2 \sinh \left (\frac {f x}{2}\right )\right )-\frac {8 d \cosh ^3\left (\frac {1}{2} (e+f x)\right ) \left (-\frac {3 \left (c^2 f^2-2 d^2\right ) (\sinh (e)+\cosh (e)+1) (f x-\log (\sinh (e+f x)+\cosh (e+f x)+1))}{f}+3 c^2 f^2 x-6 c d (\sinh (e)+\cosh (e)+1) \text {Li}_2(\sinh (e+f x)-\cosh (e+f x))+6 c d f x (\sinh (e)+\cosh (e)+1) \log (-\sinh (e+f x)+\cosh (e+f x)+1)+3 c d f^2 x^2-\frac {6 d^2 (\sinh (e)+\cosh (e)+1) (f x \text {Li}_2(\sinh (e+f x)-\cosh (e+f x))+\text {Li}_3(\sinh (e+f x)-\cosh (e+f x)))}{f}+3 d^2 f x^2 (\sinh (e)+\cosh (e)+1) \log (-\sinh (e+f x)+\cosh (e+f x)+1)+d^2 f^2 x^3-6 d^2 x\right )}{\sinh (e)+\cosh (e)+1}\right )}{3 a^2 f^3 (\cosh (e+f x)+1)^2} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.58, size = 1863, normalized size = 7.31 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (d x + c\right )}^{3}}{{\left (a \cosh \left (f x + e\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.30, size = 600, normalized size = 2.35 \[ -\frac {2 \left (3 f^{2} d^{3} x^{3} {\mathrm e}^{f x +e}+9 f^{2} c \,d^{2} x^{2} {\mathrm e}^{f x +e}+d^{3} f^{2} x^{3}-3 d^{3} f \,x^{2} {\mathrm e}^{2 f x +2 e}+9 f^{2} c^{2} d x \,{\mathrm e}^{f x +e}+3 c \,d^{2} f^{2} x^{2}-6 c \,d^{2} f x \,{\mathrm e}^{2 f x +2 e}-3 f \,d^{3} x^{2} {\mathrm e}^{f x +e}+3 f^{2} c^{3} {\mathrm e}^{f x +e}+3 c^{2} d \,f^{2} x -3 c^{2} d f \,{\mathrm e}^{2 f x +2 e}-6 f c \,d^{2} x \,{\mathrm e}^{f x +e}-6 d^{3} x \,{\mathrm e}^{2 f x +2 e}+c^{3} f^{2}-3 f \,c^{2} d \,{\mathrm e}^{f x +e}-6 c \,d^{2} {\mathrm e}^{2 f x +2 e}-12 d^{3} x \,{\mathrm e}^{f x +e}-12 c \,d^{2} {\mathrm e}^{f x +e}-6 d^{3} x -6 c \,d^{2}\right )}{3 f^{3} a^{2} \left ({\mathrm e}^{f x +e}+1\right )^{3}}-\frac {4 d^{2} \ln \left ({\mathrm e}^{f x +e}+1\right ) c x}{a^{2} f^{2}}-\frac {4 d^{3} e^{3}}{3 a^{2} f^{4}}-\frac {2 d^{3} \ln \left ({\mathrm e}^{f x +e}+1\right ) x^{2}}{a^{2} f^{2}}-\frac {4 d^{3} \polylog \left (2, -{\mathrm e}^{f x +e}\right ) x}{a^{2} f^{3}}+\frac {2 d^{3} x^{3}}{3 a^{2} f}-\frac {2 d \,c^{2} \ln \left ({\mathrm e}^{f x +e}+1\right )}{a^{2} f^{2}}+\frac {2 d \,c^{2} \ln \left ({\mathrm e}^{f x +e}\right )}{a^{2} f^{2}}+\frac {2 d^{3} e^{2} \ln \left ({\mathrm e}^{f x +e}\right )}{a^{2} f^{4}}-\frac {4 d^{2} c \polylog \left (2, -{\mathrm e}^{f x +e}\right )}{a^{2} f^{3}}-\frac {4 d^{2} c e \ln \left ({\mathrm e}^{f x +e}\right )}{a^{2} f^{3}}+\frac {4 d^{2} c e x}{a^{2} f^{2}}+\frac {4 d^{3} \polylog \left (3, -{\mathrm e}^{f x +e}\right )}{a^{2} f^{4}}+\frac {4 d^{3} \ln \left ({\mathrm e}^{f x +e}+1\right )}{a^{2} f^{4}}-\frac {4 d^{3} \ln \left ({\mathrm e}^{f x +e}\right )}{a^{2} f^{4}}-\frac {2 d^{3} e^{2} x}{a^{2} f^{3}}+\frac {2 d^{2} c \,x^{2}}{a^{2} f}+\frac {2 d^{2} c \,e^{2}}{a^{2} f^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.62, size = 610, normalized size = 2.39 \[ 2 \, c^{2} 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^{3} {\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 )} - \frac {2 \, {\left (d^{3} f^{2} x^{3} + 3 \, c d^{2} f^{2} x^{2} - 6 \, d^{3} x - 6 \, c d^{2} - 3 \, {\left (d^{3} f x^{2} e^{\left (2 \, e\right )} + 2 \, c d^{2} e^{\left (2 \, e\right )} + 2 \, {\left (c d^{2} f e^{\left (2 \, e\right )} + d^{3} e^{\left (2 \, e\right )}\right )} x\right )} e^{\left (2 \, f x\right )} + 3 \, {\left (d^{3} f^{2} x^{3} e^{e} - 4 \, c d^{2} e^{e} + {\left (3 \, c d^{2} f^{2} e^{e} - d^{3} f e^{e}\right )} x^{2} - 2 \, {\left (c d^{2} f e^{e} + 2 \, d^{3} e^{e}\right )} x\right )} e^{\left (f x\right )}\right )}}{3 \, {\left (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}\right )}} - \frac {4 \, {\left (f x \log \left (e^{\left (f x + e\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (f x + e\right )}\right )\right )} c d^{2}}{a^{2} f^{3}} - \frac {4 \, d^{3} x}{a^{2} f^{3}} - \frac {2 \, {\left (f^{2} x^{2} \log \left (e^{\left (f x + e\right )} + 1\right ) + 2 \, f x {\rm Li}_2\left (-e^{\left (f x + e\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (f x + e\right )})\right )} d^{3}}{a^{2} f^{4}} + \frac {4 \, d^{3} \log \left (e^{\left (f x + e\right )} + 1\right )}{a^{2} f^{4}} + \frac {2 \, {\left (d^{3} f^{3} x^{3} + 3 \, c d^{2} f^{3} x^{2}\right )}}{3 \, a^{2} f^{4}} \]
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 \[ \int \frac {{\left (c+d\,x\right )}^3}{{\left (a+a\,\mathrm {cosh}\left (e+f\,x\right )\right )}^2} \,d x \]
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 \[ \frac {\int \frac {c^{3}}{\cosh ^{2}{\left (e + f x \right )} + 2 \cosh {\left (e + f x \right )} + 1}\, dx + \int \frac {d^{3} x^{3}}{\cosh ^{2}{\left (e + f x \right )} + 2 \cosh {\left (e + f x \right )} + 1}\, dx + \int \frac {3 c d^{2} x^{2}}{\cosh ^{2}{\left (e + f x \right )} + 2 \cosh {\left (e + f x \right )} + 1}\, dx + \int \frac {3 c^{2} d x}{\cosh ^{2}{\left (e + f x \right )} + 2 \cosh {\left (e + f x \right )} + 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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