Optimal. Leaf size=200 \[ \frac {(c+d x)^2}{3 a^2 f}-\frac {4 d (c+d x) \log \left (1+e^{e+f x}\right )}{3 a^2 f^2}-\frac {4 d^2 \text {PolyLog}\left (2,-e^{e+f x}\right )}{3 a^2 f^3}+\frac {d (c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^2}-\frac {2 d^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^3}+\frac {(c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f} \]
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Rubi [A]
time = 0.18, 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 = {3399, 4271,
3852, 8, 4269, 3799, 2221, 2317, 2438} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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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*} \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 ) \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 \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] Result contains complex when optimal does not.
time = 6.27, size = 637, normalized size = 3.18 \begin {gather*} -\frac {16 c d \cosh ^4\left (\frac {e}{2}+\frac {f x}{2}\right ) \text {sech}\left (\frac {e}{2}\right ) \left (\cosh \left (\frac {e}{2}\right ) \log \left (\cosh \left (\frac {e}{2}\right ) \cosh \left (\frac {f x}{2}\right )+\sinh \left (\frac {e}{2}\right ) \sinh \left (\frac {f x}{2}\right )\right )-\frac {1}{2} f x \sinh \left (\frac {e}{2}\right )\right )}{3 f^2 (a+a \cosh (e+f x))^2 \left (\cosh ^2\left (\frac {e}{2}\right )-\sinh ^2\left (\frac {e}{2}\right )\right )}-\frac {16 d^2 \cosh ^4\left (\frac {e}{2}+\frac {f x}{2}\right ) \text {csch}\left (\frac {e}{2}\right ) \left (\frac {1}{4} e^{-\tanh ^{-1}\left (\coth \left (\frac {e}{2}\right )\right )} f^2 x^2-\frac {i \coth \left (\frac {e}{2}\right ) \left (-\frac {1}{2} f x \left (-\pi +2 i \tanh ^{-1}\left (\coth \left (\frac {e}{2}\right )\right )\right )-\pi \log \left (1+e^{f x}\right )-2 \left (\frac {i f x}{2}+i \tanh ^{-1}\left (\coth \left (\frac {e}{2}\right )\right )\right ) \log \left (1-e^{2 i \left (\frac {i f x}{2}+i \tanh ^{-1}\left (\coth \left (\frac {e}{2}\right )\right )\right )}\right )+\pi \log \left (\cosh \left (\frac {f x}{2}\right )\right )+2 i \tanh ^{-1}\left (\coth \left (\frac {e}{2}\right )\right ) \log \left (i \sinh \left (\frac {f x}{2}+\tanh ^{-1}\left (\coth \left (\frac {e}{2}\right )\right )\right )\right )+i \text {PolyLog}\left (2,e^{2 i \left (\frac {i f x}{2}+i \tanh ^{-1}\left (\coth \left (\frac {e}{2}\right )\right )\right )}\right )\right )}{\sqrt {1-\coth ^2\left (\frac {e}{2}\right )}}\right ) \text {sech}\left (\frac {e}{2}\right )}{3 f^3 (a+a \cosh (e+f x))^2 \sqrt {\text {csch}^2\left (\frac {e}{2}\right ) \left (-\cosh ^2\left (\frac {e}{2}\right )+\sinh ^2\left (\frac {e}{2}\right )\right )}}+\frac {\cosh \left (\frac {e}{2}+\frac {f x}{2}\right ) \text {sech}\left (\frac {e}{2}\right ) \left (2 c d f \cosh \left (\frac {f x}{2}\right )+2 d^2 f x \cosh \left (\frac {f x}{2}\right )+2 c d f \cosh \left (e+\frac {f x}{2}\right )+2 d^2 f 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 )}{3 f^3 (a+a \cosh (e+f x))^2} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 1.60, size = 313, normalized size = 1.56
method | result | size |
risch | \(-\frac {2 \left (3 f^{2} d^{2} x^{2} {\mathrm e}^{f x +e}+6 f^{2} c d x \,{\mathrm e}^{f x +e}+d^{2} x^{2} f^{2}-2 d^{2} f x \,{\mathrm e}^{2 f x +2 e}+3 f^{2} c^{2} {\mathrm e}^{f x +e}+2 c d \,f^{2} x -2 c d f \,{\mathrm e}^{2 f x +2 e}-2 f \,d^{2} x \,{\mathrm e}^{f x +e}+c^{2} f^{2}-2 f c d \,{\mathrm e}^{f x +e}-2 d^{2} {\mathrm e}^{2 f x +2 e}-4 d^{2} {\mathrm e}^{f x +e}-2 d^{2}\right )}{3 f^{3} a^{2} \left ({\mathrm e}^{f x +e}+1\right )^{3}}-\frac {4 d c \ln \left ({\mathrm e}^{f x +e}+1\right )}{3 a^{2} f^{2}}+\frac {4 d c \ln \left ({\mathrm e}^{f x +e}\right )}{3 a^{2} f^{2}}+\frac {2 d^{2} x^{2}}{3 a^{2} f}+\frac {4 d^{2} e x}{3 a^{2} f^{2}}+\frac {2 d^{2} e^{2}}{3 a^{2} f^{3}}-\frac {4 d^{2} \ln \left ({\mathrm e}^{f x +e}+1\right ) x}{3 a^{2} f^{2}}-\frac {4 d^{2} \polylog \left (2, -{\mathrm e}^{f x +e}\right )}{3 a^{2} f^{3}}-\frac {4 d^{2} e \ln \left ({\mathrm e}^{f x +e}\right )}{3 a^{2} f^{3}}\) | \(313\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1347 vs.
\(2 (170) = 340\).
time = 0.45, size = 1347, normalized size = 6.74 \begin {gather*} \text {Too large to display} \end {gather*}
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 \begin {gather*} \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}} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (c+d\,x\right )}^2}{{\left (a+a\,\mathrm {cosh}\left (e+f\,x\right )\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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