Optimal. Leaf size=405 \[ \frac {b^3 c d x}{f}+\frac {b^3 d^2 x^2}{2 f}-\frac {3 a b^2 (c+d x)^2}{f}+\frac {a^3 (c+d x)^3}{3 d}-\frac {a^2 b (c+d x)^3}{d}+\frac {a b^2 (c+d x)^3}{d}-\frac {b^3 (c+d x)^3}{3 d}+\frac {6 a b^2 d (c+d x) \log \left (1+e^{2 (e+f x)}\right )}{f^2}+\frac {3 a^2 b (c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {b^3 (c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {b^3 d^2 \log (\cosh (e+f x))}{f^3}+\frac {3 a b^2 d^2 \text {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{f^3}+\frac {3 a^2 b d (c+d x) \text {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{f^2}+\frac {b^3 d (c+d x) \text {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{f^2}-\frac {3 a^2 b d^2 \text {PolyLog}\left (3,-e^{2 (e+f x)}\right )}{2 f^3}-\frac {b^3 d^2 \text {PolyLog}\left (3,-e^{2 (e+f x)}\right )}{2 f^3}-\frac {b^3 d (c+d x) \tanh (e+f x)}{f^2}-\frac {3 a b^2 (c+d x)^2 \tanh (e+f x)}{f}-\frac {b^3 (c+d x)^2 \tanh ^2(e+f x)}{2 f} \]
[Out]
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Rubi [A]
time = 0.48, antiderivative size = 405, normalized size of antiderivative = 1.00, number of steps
used = 22, number of rules used = 11, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {3803, 3799,
2221, 2611, 2320, 6724, 3801, 2317, 2438, 32, 3556} \begin {gather*} \frac {a^3 (c+d x)^3}{3 d}+\frac {3 a^2 b d (c+d x) \text {Li}_2\left (-e^{2 (e+f x)}\right )}{f^2}+\frac {3 a^2 b (c+d x)^2 \log \left (e^{2 (e+f x)}+1\right )}{f}-\frac {a^2 b (c+d x)^3}{d}-\frac {3 a^2 b d^2 \text {Li}_3\left (-e^{2 (e+f x)}\right )}{2 f^3}+\frac {6 a b^2 d (c+d x) \log \left (e^{2 (e+f x)}+1\right )}{f^2}-\frac {3 a b^2 (c+d x)^2 \tanh (e+f x)}{f}-\frac {3 a b^2 (c+d x)^2}{f}+\frac {a b^2 (c+d x)^3}{d}+\frac {3 a b^2 d^2 \text {Li}_2\left (-e^{2 (e+f x)}\right )}{f^3}+\frac {b^3 d (c+d x) \text {Li}_2\left (-e^{2 (e+f x)}\right )}{f^2}-\frac {b^3 d (c+d x) \tanh (e+f x)}{f^2}+\frac {b^3 (c+d x)^2 \log \left (e^{2 (e+f x)}+1\right )}{f}-\frac {b^3 (c+d x)^2 \tanh ^2(e+f x)}{2 f}+\frac {b^3 c d x}{f}-\frac {b^3 (c+d x)^3}{3 d}-\frac {b^3 d^2 \text {Li}_3\left (-e^{2 (e+f x)}\right )}{2 f^3}+\frac {b^3 d^2 \log (\cosh (e+f x))}{f^3}+\frac {b^3 d^2 x^2}{2 f} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 32
Rule 2221
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 3556
Rule 3799
Rule 3801
Rule 3803
Rule 6724
Rubi steps
\begin {align*} \int (c+d x)^2 (a+b \tanh (e+f x))^3 \, dx &=\int \left (a^3 (c+d x)^2+3 a^2 b (c+d x)^2 \tanh (e+f x)+3 a b^2 (c+d x)^2 \tanh ^2(e+f x)+b^3 (c+d x)^2 \tanh ^3(e+f x)\right ) \, dx\\ &=\frac {a^3 (c+d x)^3}{3 d}+\left (3 a^2 b\right ) \int (c+d x)^2 \tanh (e+f x) \, dx+\left (3 a b^2\right ) \int (c+d x)^2 \tanh ^2(e+f x) \, dx+b^3 \int (c+d x)^2 \tanh ^3(e+f x) \, dx\\ &=\frac {a^3 (c+d x)^3}{3 d}-\frac {a^2 b (c+d x)^3}{d}-\frac {3 a b^2 (c+d x)^2 \tanh (e+f x)}{f}-\frac {b^3 (c+d x)^2 \tanh ^2(e+f x)}{2 f}+\left (6 a^2 b\right ) \int \frac {e^{2 (e+f x)} (c+d x)^2}{1+e^{2 (e+f x)}} \, dx+\left (3 a b^2\right ) \int (c+d x)^2 \, dx+b^3 \int (c+d x)^2 \tanh (e+f x) \, dx+\frac {\left (6 a b^2 d\right ) \int (c+d x) \tanh (e+f x) \, dx}{f}+\frac {\left (b^3 d\right ) \int (c+d x) \tanh ^2(e+f x) \, dx}{f}\\ &=-\frac {3 a b^2 (c+d x)^2}{f}+\frac {a^3 (c+d x)^3}{3 d}-\frac {a^2 b (c+d x)^3}{d}+\frac {a b^2 (c+d x)^3}{d}-\frac {b^3 (c+d x)^3}{3 d}+\frac {3 a^2 b (c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f}-\frac {b^3 d (c+d x) \tanh (e+f x)}{f^2}-\frac {3 a b^2 (c+d x)^2 \tanh (e+f x)}{f}-\frac {b^3 (c+d x)^2 \tanh ^2(e+f x)}{2 f}+\left (2 b^3\right ) \int \frac {e^{2 (e+f x)} (c+d x)^2}{1+e^{2 (e+f x)}} \, dx+\frac {\left (b^3 d^2\right ) \int \tanh (e+f x) \, dx}{f^2}-\frac {\left (6 a^2 b d\right ) \int (c+d x) \log \left (1+e^{2 (e+f x)}\right ) \, dx}{f}+\frac {\left (12 a b^2 d\right ) \int \frac {e^{2 (e+f x)} (c+d x)}{1+e^{2 (e+f x)}} \, dx}{f}+\frac {\left (b^3 d\right ) \int (c+d x) \, dx}{f}\\ &=\frac {b^3 c d x}{f}+\frac {b^3 d^2 x^2}{2 f}-\frac {3 a b^2 (c+d x)^2}{f}+\frac {a^3 (c+d x)^3}{3 d}-\frac {a^2 b (c+d x)^3}{d}+\frac {a b^2 (c+d x)^3}{d}-\frac {b^3 (c+d x)^3}{3 d}+\frac {6 a b^2 d (c+d x) \log \left (1+e^{2 (e+f x)}\right )}{f^2}+\frac {3 a^2 b (c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {b^3 (c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {b^3 d^2 \log (\cosh (e+f x))}{f^3}+\frac {3 a^2 b d (c+d x) \text {Li}_2\left (-e^{2 (e+f x)}\right )}{f^2}-\frac {b^3 d (c+d x) \tanh (e+f x)}{f^2}-\frac {3 a b^2 (c+d x)^2 \tanh (e+f x)}{f}-\frac {b^3 (c+d x)^2 \tanh ^2(e+f x)}{2 f}-\frac {\left (3 a^2 b d^2\right ) \int \text {Li}_2\left (-e^{2 (e+f x)}\right ) \, dx}{f^2}-\frac {\left (6 a b^2 d^2\right ) \int \log \left (1+e^{2 (e+f x)}\right ) \, dx}{f^2}-\frac {\left (2 b^3 d\right ) \int (c+d x) \log \left (1+e^{2 (e+f x)}\right ) \, dx}{f}\\ &=\frac {b^3 c d x}{f}+\frac {b^3 d^2 x^2}{2 f}-\frac {3 a b^2 (c+d x)^2}{f}+\frac {a^3 (c+d x)^3}{3 d}-\frac {a^2 b (c+d x)^3}{d}+\frac {a b^2 (c+d x)^3}{d}-\frac {b^3 (c+d x)^3}{3 d}+\frac {6 a b^2 d (c+d x) \log \left (1+e^{2 (e+f x)}\right )}{f^2}+\frac {3 a^2 b (c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {b^3 (c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {b^3 d^2 \log (\cosh (e+f x))}{f^3}+\frac {3 a^2 b d (c+d x) \text {Li}_2\left (-e^{2 (e+f x)}\right )}{f^2}+\frac {b^3 d (c+d x) \text {Li}_2\left (-e^{2 (e+f x)}\right )}{f^2}-\frac {b^3 d (c+d x) \tanh (e+f x)}{f^2}-\frac {3 a b^2 (c+d x)^2 \tanh (e+f x)}{f}-\frac {b^3 (c+d x)^2 \tanh ^2(e+f x)}{2 f}-\frac {\left (3 a^2 b d^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{2 f^3}-\frac {\left (3 a b^2 d^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{f^3}-\frac {\left (b^3 d^2\right ) \int \text {Li}_2\left (-e^{2 (e+f x)}\right ) \, dx}{f^2}\\ &=\frac {b^3 c d x}{f}+\frac {b^3 d^2 x^2}{2 f}-\frac {3 a b^2 (c+d x)^2}{f}+\frac {a^3 (c+d x)^3}{3 d}-\frac {a^2 b (c+d x)^3}{d}+\frac {a b^2 (c+d x)^3}{d}-\frac {b^3 (c+d x)^3}{3 d}+\frac {6 a b^2 d (c+d x) \log \left (1+e^{2 (e+f x)}\right )}{f^2}+\frac {3 a^2 b (c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {b^3 (c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {b^3 d^2 \log (\cosh (e+f x))}{f^3}+\frac {3 a b^2 d^2 \text {Li}_2\left (-e^{2 (e+f x)}\right )}{f^3}+\frac {3 a^2 b d (c+d x) \text {Li}_2\left (-e^{2 (e+f x)}\right )}{f^2}+\frac {b^3 d (c+d x) \text {Li}_2\left (-e^{2 (e+f x)}\right )}{f^2}-\frac {3 a^2 b d^2 \text {Li}_3\left (-e^{2 (e+f x)}\right )}{2 f^3}-\frac {b^3 d (c+d x) \tanh (e+f x)}{f^2}-\frac {3 a b^2 (c+d x)^2 \tanh (e+f x)}{f}-\frac {b^3 (c+d x)^2 \tanh ^2(e+f x)}{2 f}-\frac {\left (b^3 d^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{2 f^3}\\ &=\frac {b^3 c d x}{f}+\frac {b^3 d^2 x^2}{2 f}-\frac {3 a b^2 (c+d x)^2}{f}+\frac {a^3 (c+d x)^3}{3 d}-\frac {a^2 b (c+d x)^3}{d}+\frac {a b^2 (c+d x)^3}{d}-\frac {b^3 (c+d x)^3}{3 d}+\frac {6 a b^2 d (c+d x) \log \left (1+e^{2 (e+f x)}\right )}{f^2}+\frac {3 a^2 b (c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {b^3 (c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {b^3 d^2 \log (\cosh (e+f x))}{f^3}+\frac {3 a b^2 d^2 \text {Li}_2\left (-e^{2 (e+f x)}\right )}{f^3}+\frac {3 a^2 b d (c+d x) \text {Li}_2\left (-e^{2 (e+f x)}\right )}{f^2}+\frac {b^3 d (c+d x) \text {Li}_2\left (-e^{2 (e+f x)}\right )}{f^2}-\frac {3 a^2 b d^2 \text {Li}_3\left (-e^{2 (e+f x)}\right )}{2 f^3}-\frac {b^3 d^2 \text {Li}_3\left (-e^{2 (e+f x)}\right )}{2 f^3}-\frac {b^3 d (c+d x) \tanh (e+f x)}{f^2}-\frac {3 a b^2 (c+d x)^2 \tanh (e+f x)}{f}-\frac {b^3 (c+d x)^2 \tanh ^2(e+f x)}{2 f}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1142\) vs. \(2(405)=810\).
time = 9.63, size = 1142, normalized size = 2.82 \begin {gather*} \frac {b \left (-\frac {4 e^{2 e} f x \left (9 a b d f (2 c+d x)+3 a^2 f^2 \left (3 c^2+3 c d x+d^2 x^2\right )+b^2 \left (3 c^2 f^2+3 c d f^2 x+d^2 \left (3+f^2 x^2\right )\right )\right )}{1+e^{2 e}}+6 \left (6 a b d f (c+d x)+3 a^2 f^2 (c+d x)^2+b^2 \left (c^2 f^2+2 c d f^2 x+d^2 \left (1+f^2 x^2\right )\right )\right ) \log \left (1+e^{2 (e+f x)}\right )+6 d \left (3 a b d+3 a^2 f (c+d x)+b^2 f (c+d x)\right ) \text {PolyLog}\left (2,-e^{2 (e+f x)}\right )-3 \left (3 a^2+b^2\right ) d^2 \text {PolyLog}\left (3,-e^{2 (e+f x)}\right )\right )}{6 f^3}+\frac {\text {sech}(e) \text {sech}^2(e+f x) \left (6 b^3 c^2 f \cosh (e)+12 b^3 c d f x \cosh (e)+6 a^3 c^2 f^2 x \cosh (e)+18 a b^2 c^2 f^2 x \cosh (e)+6 b^3 d^2 f x^2 \cosh (e)+6 a^3 c d f^2 x^2 \cosh (e)+18 a b^2 c d f^2 x^2 \cosh (e)+2 a^3 d^2 f^2 x^3 \cosh (e)+6 a b^2 d^2 f^2 x^3 \cosh (e)+3 a^3 c^2 f^2 x \cosh (e+2 f x)+9 a b^2 c^2 f^2 x \cosh (e+2 f x)+3 a^3 c d f^2 x^2 \cosh (e+2 f x)+9 a b^2 c d f^2 x^2 \cosh (e+2 f x)+a^3 d^2 f^2 x^3 \cosh (e+2 f x)+3 a b^2 d^2 f^2 x^3 \cosh (e+2 f x)+3 a^3 c^2 f^2 x \cosh (3 e+2 f x)+9 a b^2 c^2 f^2 x \cosh (3 e+2 f x)+3 a^3 c d f^2 x^2 \cosh (3 e+2 f x)+9 a b^2 c d f^2 x^2 \cosh (3 e+2 f x)+a^3 d^2 f^2 x^3 \cosh (3 e+2 f x)+3 a b^2 d^2 f^2 x^3 \cosh (3 e+2 f x)+6 b^3 c d \sinh (e)+18 a b^2 c^2 f \sinh (e)+6 b^3 d^2 x \sinh (e)+36 a b^2 c d f x \sinh (e)+18 a^2 b c^2 f^2 x \sinh (e)+6 b^3 c^2 f^2 x \sinh (e)+18 a b^2 d^2 f x^2 \sinh (e)+18 a^2 b c d f^2 x^2 \sinh (e)+6 b^3 c d f^2 x^2 \sinh (e)+6 a^2 b d^2 f^2 x^3 \sinh (e)+2 b^3 d^2 f^2 x^3 \sinh (e)-6 b^3 c d \sinh (e+2 f x)-18 a b^2 c^2 f \sinh (e+2 f x)-6 b^3 d^2 x \sinh (e+2 f x)-36 a b^2 c d f x \sinh (e+2 f x)-9 a^2 b c^2 f^2 x \sinh (e+2 f x)-3 b^3 c^2 f^2 x \sinh (e+2 f x)-18 a b^2 d^2 f x^2 \sinh (e+2 f x)-9 a^2 b c d f^2 x^2 \sinh (e+2 f x)-3 b^3 c d f^2 x^2 \sinh (e+2 f x)-3 a^2 b d^2 f^2 x^3 \sinh (e+2 f x)-b^3 d^2 f^2 x^3 \sinh (e+2 f x)+9 a^2 b c^2 f^2 x \sinh (3 e+2 f x)+3 b^3 c^2 f^2 x \sinh (3 e+2 f x)+9 a^2 b c d f^2 x^2 \sinh (3 e+2 f x)+3 b^3 c d f^2 x^2 \sinh (3 e+2 f x)+3 a^2 b d^2 f^2 x^3 \sinh (3 e+2 f x)+b^3 d^2 f^2 x^3 \sinh (3 e+2 f x)\right )}{12 f^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1065\) vs.
\(2(393)=786\).
time = 3.34, size = 1066, normalized size = 2.63
method | result | size |
risch | \(\frac {b^{3} \polylog \left (2, -{\mathrm e}^{2 f x +2 e}\right ) d^{2} x}{f^{2}}-\frac {12 b \,a^{2} c d e x}{f}+\frac {6 b \ln \left (1+{\mathrm e}^{2 f x +2 e}\right ) a^{2} c d x}{f}+\frac {12 b \,a^{2} c d e \ln \left ({\mathrm e}^{f x +e}\right )}{f^{2}}+\frac {2 b^{3} \ln \left (1+{\mathrm e}^{2 f x +2 e}\right ) c d x}{f}+\frac {6 b^{2} \ln \left (1+{\mathrm e}^{2 f x +2 e}\right ) a \,d^{2} x}{f^{2}}+\frac {6 b^{2} a c d \ln \left (1+{\mathrm e}^{2 f x +2 e}\right )}{f^{2}}-\frac {12 b^{2} a c d \ln \left ({\mathrm e}^{f x +e}\right )}{f^{2}}+\frac {4 b^{3} c d e \ln \left ({\mathrm e}^{f x +e}\right )}{f^{2}}+\frac {3 b \polylog \left (2, -{\mathrm e}^{2 f x +2 e}\right ) a^{2} d^{2} x}{f^{2}}+\frac {3 b \,a^{2} c d \polylog \left (2, -{\mathrm e}^{2 f x +2 e}\right )}{f^{2}}+\frac {3 b \ln \left (1+{\mathrm e}^{2 f x +2 e}\right ) a^{2} d^{2} x^{2}}{f}+\frac {12 b^{2} a \,d^{2} e \ln \left ({\mathrm e}^{f x +e}\right )}{f^{3}}-\frac {6 b \,a^{2} d^{2} e^{2} \ln \left ({\mathrm e}^{f x +e}\right )}{f^{3}}-\frac {12 b^{2} a \,d^{2} e x}{f^{2}}-\frac {4 b^{3} c d e x}{f}+\frac {6 b \,a^{2} d^{2} e^{2} x}{f^{2}}-\frac {6 b \,a^{2} c d \,e^{2}}{f^{2}}-\frac {6 b^{2} a \,d^{2} x^{2}}{f}+\frac {4 b^{3} d^{2} e^{3}}{3 f^{3}}-\frac {2 b^{3} d^{2} \ln \left ({\mathrm e}^{f x +e}\right )}{f^{3}}+\frac {b^{3} d^{2} \ln \left (1+{\mathrm e}^{2 f x +2 e}\right )}{f^{3}}-\frac {2 b^{3} c^{2} \ln \left ({\mathrm e}^{f x +e}\right )}{f}+\frac {b^{3} c^{2} \ln \left (1+{\mathrm e}^{2 f x +2 e}\right )}{f}+\frac {b^{3} c d \polylog \left (2, -{\mathrm e}^{2 f x +2 e}\right )}{f^{2}}+\frac {3 a \,b^{2} d^{2} \polylog \left (2, -{\mathrm e}^{2 f x +2 e}\right )}{f^{3}}+\frac {3 b \,a^{2} c^{2} \ln \left (1+{\mathrm e}^{2 f x +2 e}\right )}{f}-\frac {2 b^{3} d^{2} e^{2} \ln \left ({\mathrm e}^{f x +e}\right )}{f^{3}}-\frac {6 b \,a^{2} c^{2} \ln \left ({\mathrm e}^{f x +e}\right )}{f}+\frac {b^{3} \ln \left (1+{\mathrm e}^{2 f x +2 e}\right ) d^{2} x^{2}}{f}-\frac {3 a^{2} b \,d^{2} \polylog \left (3, -{\mathrm e}^{2 f x +2 e}\right )}{2 f^{3}}+\frac {d^{2} a^{3} x^{3}}{3}-\frac {d^{2} b^{3} x^{3}}{3}+\frac {a^{3} c^{3}}{3 d}+\frac {b^{3} c^{3}}{3 d}-3 d \,a^{2} b c \,x^{2}+3 d a \,b^{2} c \,x^{2}+3 a^{2} b \,c^{2} x +3 a \,b^{2} c^{2} x -d^{2} a^{2} b \,x^{3}-\frac {2 b^{3} c d \,e^{2}}{f^{2}}-\frac {6 b^{2} a \,d^{2} e^{2}}{f^{3}}+\frac {4 b \,a^{2} d^{2} e^{3}}{f^{3}}+\frac {2 b^{3} d^{2} e^{2} x}{f^{2}}+\frac {2 b^{2} \left (3 a \,d^{2} f \,x^{2} {\mathrm e}^{2 f x +2 e}+b \,d^{2} f \,x^{2} {\mathrm e}^{2 f x +2 e}+6 a c d f x \,{\mathrm e}^{2 f x +2 e}+2 b c d f x \,{\mathrm e}^{2 f x +2 e}+3 a \,c^{2} f \,{\mathrm e}^{2 f x +2 e}+3 a \,d^{2} f \,x^{2}+b \,c^{2} f \,{\mathrm e}^{2 f x +2 e}+b \,d^{2} x \,{\mathrm e}^{2 f x +2 e}+6 a c d f x +b c d \,{\mathrm e}^{2 f x +2 e}+3 a \,c^{2} f +b \,d^{2} x +b c d \right )}{f^{2} \left (1+{\mathrm e}^{2 f x +2 e}\right )^{2}}+d^{2} a \,b^{2} x^{3}+d \,a^{3} c \,x^{2}-d \,b^{3} c \,x^{2}+a^{3} c^{2} x +b^{3} c^{2} x +\frac {a^{2} b \,c^{3}}{d}+\frac {a \,b^{2} c^{3}}{d}-\frac {b^{3} d^{2} \polylog \left (3, -{\mathrm e}^{2 f x +2 e}\right )}{2 f^{3}}\) | \(1066\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 835 vs.
\(2 (402) = 804\).
time = 0.57, size = 835, normalized size = 2.06 \begin {gather*} \frac {1}{3} \, a^{3} d^{2} x^{3} + a^{3} c d x^{2} + b^{3} c^{2} {\left (x + \frac {e}{f} + \frac {\log \left (e^{\left (-2 \, f x - 2 \, e\right )} + 1\right )}{f} + \frac {2 \, e^{\left (-2 \, f x - 2 \, e\right )}}{f {\left (2 \, e^{\left (-2 \, f x - 2 \, e\right )} + e^{\left (-4 \, f x - 4 \, e\right )} + 1\right )}}\right )} + a^{3} c^{2} x + \frac {3 \, a^{2} b c^{2} \log \left (\cosh \left (f x + e\right )\right )}{f} + \frac {18 \, a b^{2} c^{2} f + 6 \, b^{3} c d + {\left (3 \, a^{2} b d^{2} f^{2} + 3 \, a b^{2} d^{2} f^{2} + b^{3} d^{2} f^{2}\right )} x^{3} + 3 \, {\left (3 \, a^{2} b c d f^{2} + b^{3} c d f^{2} + 3 \, {\left (c d f^{2} + 2 \, d^{2} f\right )} a b^{2}\right )} x^{2} + 3 \, {\left (2 \, b^{3} d^{2} + 3 \, {\left (c^{2} f^{2} + 4 \, c d f\right )} a b^{2}\right )} x + {\left (9 \, a b^{2} c^{2} f^{2} x e^{\left (4 \, e\right )} + {\left (3 \, a^{2} b d^{2} f^{2} + 3 \, a b^{2} d^{2} f^{2} + b^{3} d^{2} f^{2}\right )} x^{3} e^{\left (4 \, e\right )} + 3 \, {\left (3 \, a^{2} b c d f^{2} + 3 \, a b^{2} c d f^{2} + b^{3} c d f^{2}\right )} x^{2} e^{\left (4 \, e\right )}\right )} e^{\left (4 \, f x\right )} + 2 \, {\left ({\left (3 \, a^{2} b d^{2} f^{2} + 3 \, a b^{2} d^{2} f^{2} + b^{3} d^{2} f^{2}\right )} x^{3} e^{\left (2 \, e\right )} + 3 \, {\left (3 \, a^{2} b c d f^{2} + 3 \, {\left (c d f^{2} + d^{2} f\right )} a b^{2} + {\left (c d f^{2} + d^{2} f\right )} b^{3}\right )} x^{2} e^{\left (2 \, e\right )} + 3 \, {\left (3 \, {\left (c^{2} f^{2} + 2 \, c d f\right )} a b^{2} + {\left (2 \, c d f + d^{2}\right )} b^{3}\right )} x e^{\left (2 \, e\right )} + 3 \, {\left (3 \, a b^{2} c^{2} f + b^{3} c d\right )} e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{3 \, {\left (f^{2} e^{\left (4 \, f x + 4 \, e\right )} + 2 \, f^{2} e^{\left (2 \, f x + 2 \, e\right )} + f^{2}\right )}} - \frac {2 \, {\left (6 \, a b^{2} c d f + b^{3} d^{2}\right )} x}{f^{2}} + \frac {{\left (3 \, a^{2} b d^{2} + b^{3} d^{2}\right )} {\left (2 \, f^{2} x^{2} \log \left (e^{\left (2 \, f x + 2 \, e\right )} + 1\right ) + 2 \, f x {\rm Li}_2\left (-e^{\left (2 \, f x + 2 \, e\right )}\right ) - {\rm Li}_{3}(-e^{\left (2 \, f x + 2 \, e\right )})\right )}}{2 \, f^{3}} + \frac {{\left (3 \, a^{2} b c d f + b^{3} c d f + 3 \, a b^{2} d^{2}\right )} {\left (2 \, f x \log \left (e^{\left (2 \, f x + 2 \, e\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (2 \, f x + 2 \, e\right )}\right )\right )}}{f^{3}} + \frac {{\left (6 \, a b^{2} c d f + b^{3} d^{2}\right )} \log \left (e^{\left (2 \, f x + 2 \, e\right )} + 1\right )}{f^{3}} - \frac {2 \, {\left ({\left (3 \, a^{2} b d^{2} + b^{3} d^{2}\right )} f^{3} x^{3} + 3 \, {\left (3 \, a^{2} b c d f + b^{3} c d f + 3 \, a b^{2} d^{2}\right )} f^{2} x^{2}\right )}}{3 \, f^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains complex when optimal does not.
time = 0.61, size = 11295, normalized size = 27.89 \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*} \int \left (a + b \tanh {\left (e + f x \right )}\right )^{3} \left (c + d x\right )^{2}\, dx \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 {\left (a+b\,\mathrm {tanh}\left (e+f\,x\right )\right )}^3\,{\left (c+d\,x\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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