Optimal. Leaf size=117 \[ -\frac {(c+d x)^4}{4 d}+\frac {(c+d x)^3 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {3 d (c+d x)^2 \text {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{2 f^2}-\frac {3 d^2 (c+d x) \text {PolyLog}\left (3,-e^{2 (e+f x)}\right )}{2 f^3}+\frac {3 d^3 \text {PolyLog}\left (4,-e^{2 (e+f x)}\right )}{4 f^4} \]
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Rubi [A]
time = 0.13, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3799, 2221,
2611, 6744, 2320, 6724} \begin {gather*} -\frac {3 d^2 (c+d x) \text {Li}_3\left (-e^{2 (e+f x)}\right )}{2 f^3}+\frac {3 d (c+d x)^2 \text {Li}_2\left (-e^{2 (e+f x)}\right )}{2 f^2}+\frac {(c+d x)^3 \log \left (e^{2 (e+f x)}+1\right )}{f}-\frac {(c+d x)^4}{4 d}+\frac {3 d^3 \text {Li}_4\left (-e^{2 (e+f x)}\right )}{4 f^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2320
Rule 2611
Rule 3799
Rule 6724
Rule 6744
Rubi steps
\begin {align*} \int (c+d x)^3 \tanh (e+f x) \, dx &=-\frac {(c+d x)^4}{4 d}+2 \int \frac {e^{2 (e+f x)} (c+d x)^3}{1+e^{2 (e+f x)}} \, dx\\ &=-\frac {(c+d x)^4}{4 d}+\frac {(c+d x)^3 \log \left (1+e^{2 (e+f x)}\right )}{f}-\frac {(3 d) \int (c+d x)^2 \log \left (1+e^{2 (e+f x)}\right ) \, dx}{f}\\ &=-\frac {(c+d x)^4}{4 d}+\frac {(c+d x)^3 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {3 d (c+d x)^2 \text {Li}_2\left (-e^{2 (e+f x)}\right )}{2 f^2}-\frac {\left (3 d^2\right ) \int (c+d x) \text {Li}_2\left (-e^{2 (e+f x)}\right ) \, dx}{f^2}\\ &=-\frac {(c+d x)^4}{4 d}+\frac {(c+d x)^3 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {3 d (c+d x)^2 \text {Li}_2\left (-e^{2 (e+f x)}\right )}{2 f^2}-\frac {3 d^2 (c+d x) \text {Li}_3\left (-e^{2 (e+f x)}\right )}{2 f^3}+\frac {\left (3 d^3\right ) \int \text {Li}_3\left (-e^{2 (e+f x)}\right ) \, dx}{2 f^3}\\ &=-\frac {(c+d x)^4}{4 d}+\frac {(c+d x)^3 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {3 d (c+d x)^2 \text {Li}_2\left (-e^{2 (e+f x)}\right )}{2 f^2}-\frac {3 d^2 (c+d x) \text {Li}_3\left (-e^{2 (e+f x)}\right )}{2 f^3}+\frac {\left (3 d^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{4 f^4}\\ &=-\frac {(c+d x)^4}{4 d}+\frac {(c+d x)^3 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {3 d (c+d x)^2 \text {Li}_2\left (-e^{2 (e+f x)}\right )}{2 f^2}-\frac {3 d^2 (c+d x) \text {Li}_3\left (-e^{2 (e+f x)}\right )}{2 f^3}+\frac {3 d^3 \text {Li}_4\left (-e^{2 (e+f x)}\right )}{4 f^4}\\ \end {align*}
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Mathematica [A]
time = 1.28, size = 199, normalized size = 1.70 \begin {gather*} \frac {1}{4} \left (-4 c^3 x-6 c^2 d x^2-4 c d^2 x^3-d^3 x^4+\frac {4 c^3 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {12 c^2 d x \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {12 c d^2 x^2 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {4 d^3 x^3 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {6 d (c+d x)^2 \text {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{f^2}-\frac {6 d^2 (c+d x) \text {PolyLog}\left (3,-e^{2 (e+f x)}\right )}{f^3}+\frac {3 d^3 \text {PolyLog}\left (4,-e^{2 (e+f x)}\right )}{f^4}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(401\) vs.
\(2(109)=218\).
time = 1.70, size = 402, normalized size = 3.44
method | result | size |
risch | \(\frac {3 d^{2} c \polylog \left (2, -{\mathrm e}^{2 f x +2 e}\right ) x}{f^{2}}+\frac {3 d^{3} \polylog \left (4, -{\mathrm e}^{2 f x +2 e}\right )}{4 f^{4}}-\frac {6 c^{2} d e x}{f}-\frac {3 c^{2} d \,e^{2}}{f^{2}}+\frac {4 c \,e^{3} d^{2}}{f^{3}}-\frac {3 d^{3} e^{4}}{2 f^{4}}-\frac {2 c^{3} \ln \left ({\mathrm e}^{f x +e}\right )}{f}+\frac {c^{3} \ln \left (1+{\mathrm e}^{2 f x +2 e}\right )}{f}+\frac {d^{3} \ln \left (1+{\mathrm e}^{2 f x +2 e}\right ) x^{3}}{f}+\frac {2 d^{3} e^{3} \ln \left ({\mathrm e}^{f x +e}\right )}{f^{4}}+c^{3} x +\frac {6 c \,e^{2} d^{2} x}{f^{2}}+\frac {3 d \,c^{2} \ln \left (1+{\mathrm e}^{2 f x +2 e}\right ) x}{f}+\frac {6 d e \,c^{2} \ln \left ({\mathrm e}^{f x +e}\right )}{f^{2}}-\frac {6 d^{2} e^{2} c \ln \left ({\mathrm e}^{f x +e}\right )}{f^{3}}+\frac {3 d^{2} c \ln \left (1+{\mathrm e}^{2 f x +2 e}\right ) x^{2}}{f}-\frac {2 d^{3} e^{3} x}{f^{3}}+\frac {3 d^{3} \polylog \left (2, -{\mathrm e}^{2 f x +2 e}\right ) x^{2}}{2 f^{2}}-\frac {3 d^{3} \polylog \left (3, -{\mathrm e}^{2 f x +2 e}\right ) x}{2 f^{3}}+\frac {3 d \,c^{2} \polylog \left (2, -{\mathrm e}^{2 f x +2 e}\right )}{2 f^{2}}-\frac {3 d^{2} c \polylog \left (3, -{\mathrm e}^{2 f x +2 e}\right )}{2 f^{3}}-d^{2} c \,x^{3}-\frac {3 d \,c^{2} x^{2}}{2}-\frac {d^{3} x^{4}}{4}+\frac {c^{4}}{4 d}\) | \(402\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 297 vs.
\(2 (112) = 224\).
time = 0.53, size = 297, normalized size = 2.54 \begin {gather*} \frac {1}{4} \, d^{3} x^{4} + c d^{2} x^{3} + \frac {3}{2} \, c^{2} d x^{2} + \frac {c^{3} \log \left (e^{\left (2 \, f x + 2 \, e\right )} + 1\right )}{2 \, f} + \frac {c^{3} \log \left (e^{\left (-2 \, f x - 2 \, e\right )} + 1\right )}{2 \, f} + \frac {3 \, {\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 )} c^{2} d}{2 \, f^{2}} + \frac {3 \, {\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 )} c d^{2}}{2 \, f^{3}} + \frac {{\left (4 \, f^{3} x^{3} \log \left (e^{\left (2 \, f x + 2 \, e\right )} + 1\right ) + 6 \, f^{2} x^{2} {\rm Li}_2\left (-e^{\left (2 \, f x + 2 \, e\right )}\right ) - 6 \, f x {\rm Li}_{3}(-e^{\left (2 \, f x + 2 \, e\right )}) + 3 \, {\rm Li}_{4}(-e^{\left (2 \, f x + 2 \, e\right )})\right )} d^{3}}{3 \, f^{4}} - \frac {d^{3} f^{4} x^{4} + 4 \, c d^{2} f^{4} x^{3} + 6 \, c^{2} d f^{4} x^{2}}{2 \, f^{4}} \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 = 0.48, size = 837, normalized size = 7.15 \begin {gather*} -\frac {d^{3} f^{4} x^{4} + 4 \, c d^{2} f^{4} x^{3} + 6 \, c^{2} d f^{4} x^{2} + 4 \, c^{3} f^{4} x - 24 \, d^{3} {\rm polylog}\left (4, i \, \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + i \, \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )\right ) - 24 \, d^{3} {\rm polylog}\left (4, -i \, \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) - i \, \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )\right ) - 12 \, {\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x + c^{2} d f^{2}\right )} {\rm Li}_2\left (i \, \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + i \, \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )\right ) - 12 \, {\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x + c^{2} d f^{2}\right )} {\rm Li}_2\left (-i \, \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) - i \, \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )\right ) - 4 \, {\left (c^{3} f^{3} - 3 \, c^{2} d f^{2} \cosh \left (1\right ) + 3 \, c d^{2} f \cosh \left (1\right )^{2} - d^{3} \cosh \left (1\right )^{3} - d^{3} \sinh \left (1\right )^{3} + 3 \, {\left (c d^{2} f - d^{3} \cosh \left (1\right )\right )} \sinh \left (1\right )^{2} - 3 \, {\left (c^{2} d f^{2} - 2 \, c d^{2} f \cosh \left (1\right ) + d^{3} \cosh \left (1\right )^{2}\right )} \sinh \left (1\right )\right )} \log \left (\cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + i\right ) - 4 \, {\left (c^{3} f^{3} - 3 \, c^{2} d f^{2} \cosh \left (1\right ) + 3 \, c d^{2} f \cosh \left (1\right )^{2} - d^{3} \cosh \left (1\right )^{3} - d^{3} \sinh \left (1\right )^{3} + 3 \, {\left (c d^{2} f - d^{3} \cosh \left (1\right )\right )} \sinh \left (1\right )^{2} - 3 \, {\left (c^{2} d f^{2} - 2 \, c d^{2} f \cosh \left (1\right ) + d^{3} \cosh \left (1\right )^{2}\right )} \sinh \left (1\right )\right )} \log \left (\cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) - i\right ) - 4 \, {\left (d^{3} f^{3} x^{3} + 3 \, c d^{2} f^{3} x^{2} + 3 \, c^{2} d f^{3} x + 3 \, c^{2} d f^{2} \cosh \left (1\right ) - 3 \, c d^{2} f \cosh \left (1\right )^{2} + d^{3} \cosh \left (1\right )^{3} + d^{3} \sinh \left (1\right )^{3} - 3 \, {\left (c d^{2} f - d^{3} \cosh \left (1\right )\right )} \sinh \left (1\right )^{2} + 3 \, {\left (c^{2} d f^{2} - 2 \, c d^{2} f \cosh \left (1\right ) + d^{3} \cosh \left (1\right )^{2}\right )} \sinh \left (1\right )\right )} \log \left (i \, \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + i \, \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + 1\right ) - 4 \, {\left (d^{3} f^{3} x^{3} + 3 \, c d^{2} f^{3} x^{2} + 3 \, c^{2} d f^{3} x + 3 \, c^{2} d f^{2} \cosh \left (1\right ) - 3 \, c d^{2} f \cosh \left (1\right )^{2} + d^{3} \cosh \left (1\right )^{3} + d^{3} \sinh \left (1\right )^{3} - 3 \, {\left (c d^{2} f - d^{3} \cosh \left (1\right )\right )} \sinh \left (1\right )^{2} + 3 \, {\left (c^{2} d f^{2} - 2 \, c d^{2} f \cosh \left (1\right ) + d^{3} \cosh \left (1\right )^{2}\right )} \sinh \left (1\right )\right )} \log \left (-i \, \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) - i \, \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + 1\right ) + 24 \, {\left (d^{3} f x + c d^{2} f\right )} {\rm polylog}\left (3, i \, \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + i \, \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )\right ) + 24 \, {\left (d^{3} f x + c d^{2} f\right )} {\rm polylog}\left (3, -i \, \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) - i \, \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )\right )}{4 \, f^{4}} \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 (c + d x\right )^{3} \tanh {\left (e + f x \right )}\, 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 = -0.01 \begin {gather*} \int \mathrm {tanh}\left (e+f\,x\right )\,{\left (c+d\,x\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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