Optimal. Leaf size=133 \[ \frac {a (c+d x)^4}{4 d}-\frac {b (c+d x)^4}{4 d}+\frac {b (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {3 b d (c+d x)^2 \text {PolyLog}\left (2,e^{2 (e+f x)}\right )}{2 f^2}-\frac {3 b d^2 (c+d x) \text {PolyLog}\left (3,e^{2 (e+f x)}\right )}{2 f^3}+\frac {3 b d^3 \text {PolyLog}\left (4,e^{2 (e+f x)}\right )}{4 f^4} \]
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Rubi [A]
time = 0.18, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {3803, 3797,
2221, 2611, 6744, 2320, 6724} \begin {gather*} \frac {a (c+d x)^4}{4 d}-\frac {3 b d^2 (c+d x) \text {Li}_3\left (e^{2 (e+f x)}\right )}{2 f^3}+\frac {3 b d (c+d x)^2 \text {Li}_2\left (e^{2 (e+f x)}\right )}{2 f^2}+\frac {b (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}-\frac {b (c+d x)^4}{4 d}+\frac {3 b 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 3797
Rule 3803
Rule 6724
Rule 6744
Rubi steps
\begin {align*} \int (c+d x)^3 (a+b \coth (e+f x)) \, dx &=\int \left (a (c+d x)^3+b (c+d x)^3 \coth (e+f x)\right ) \, dx\\ &=\frac {a (c+d x)^4}{4 d}+b \int (c+d x)^3 \coth (e+f x) \, dx\\ &=\frac {a (c+d x)^4}{4 d}-\frac {b (c+d x)^4}{4 d}-(2 b) \int \frac {e^{2 (e+f x)} (c+d x)^3}{1-e^{2 (e+f x)}} \, dx\\ &=\frac {a (c+d x)^4}{4 d}-\frac {b (c+d x)^4}{4 d}+\frac {b (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}-\frac {(3 b d) \int (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right ) \, dx}{f}\\ &=\frac {a (c+d x)^4}{4 d}-\frac {b (c+d x)^4}{4 d}+\frac {b (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {3 b d (c+d x)^2 \text {Li}_2\left (e^{2 (e+f x)}\right )}{2 f^2}-\frac {\left (3 b d^2\right ) \int (c+d x) \text {Li}_2\left (e^{2 (e+f x)}\right ) \, dx}{f^2}\\ &=\frac {a (c+d x)^4}{4 d}-\frac {b (c+d x)^4}{4 d}+\frac {b (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {3 b d (c+d x)^2 \text {Li}_2\left (e^{2 (e+f x)}\right )}{2 f^2}-\frac {3 b d^2 (c+d x) \text {Li}_3\left (e^{2 (e+f x)}\right )}{2 f^3}+\frac {\left (3 b d^3\right ) \int \text {Li}_3\left (e^{2 (e+f x)}\right ) \, dx}{2 f^3}\\ &=\frac {a (c+d x)^4}{4 d}-\frac {b (c+d x)^4}{4 d}+\frac {b (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {3 b d (c+d x)^2 \text {Li}_2\left (e^{2 (e+f x)}\right )}{2 f^2}-\frac {3 b d^2 (c+d x) \text {Li}_3\left (e^{2 (e+f x)}\right )}{2 f^3}+\frac {\left (3 b 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 {a (c+d x)^4}{4 d}-\frac {b (c+d x)^4}{4 d}+\frac {b (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {3 b d (c+d x)^2 \text {Li}_2\left (e^{2 (e+f x)}\right )}{2 f^2}-\frac {3 b d^2 (c+d x) \text {Li}_3\left (e^{2 (e+f x)}\right )}{2 f^3}+\frac {3 b d^3 \text {Li}_4\left (e^{2 (e+f x)}\right )}{4 f^4}\\ \end {align*}
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Mathematica [A]
time = 1.44, size = 247, normalized size = 1.86 \begin {gather*} \frac {1}{4} \left (4 a c^3 x-4 b c^3 x+6 a c^2 d x^2-6 b c^2 d x^2+4 a c d^2 x^3-4 b c d^2 x^3+a d^3 x^4-b d^3 x^4+\frac {4 b c^3 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {12 b c^2 d x \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {12 b c d^2 x^2 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {4 b d^3 x^3 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {6 b d (c+d x)^2 \text {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^2}-\frac {6 b d^2 (c+d x) \text {PolyLog}\left (3,e^{2 (e+f x)}\right )}{f^3}+\frac {3 b 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. \(765\) vs.
\(2(123)=246\).
time = 2.66, size = 766, normalized size = 5.76
method | result | size |
risch | \(\frac {3 b \,d^{2} e^{2} c \ln \left ({\mathrm e}^{f x +e}-1\right )}{f^{3}}-\frac {6 b c \,d^{2} \polylog \left (3, {\mathrm e}^{f x +e}\right )}{f^{3}}-\frac {6 b c \,d^{2} \polylog \left (3, -{\mathrm e}^{f x +e}\right )}{f^{3}}+\frac {3 b \,d^{3} \polylog \left (2, {\mathrm e}^{f x +e}\right ) x^{2}}{f^{2}}+a c \,d^{2} x^{3}+\frac {3 a \,c^{2} d \,x^{2}}{2}+c^{3} a x +\frac {a \,d^{3} x^{4}}{4}+\frac {3 b c \,d^{2} \ln \left ({\mathrm e}^{f x +e}+1\right ) x^{2}}{f}-\frac {3 b c \,d^{2} \ln \left (1-{\mathrm e}^{f x +e}\right ) e^{2}}{f^{3}}-\frac {3 b d e \,c^{2} \ln \left ({\mathrm e}^{f x +e}-1\right )}{f^{2}}+\frac {3 b \,c^{2} d \ln \left ({\mathrm e}^{f x +e}+1\right ) x}{f}+\frac {3 b \,c^{2} d \ln \left (1-{\mathrm e}^{f x +e}\right ) e}{f^{2}}+\frac {6 b c \,d^{2} \polylog \left (2, {\mathrm e}^{f x +e}\right ) x}{f^{2}}+\frac {6 b c \,d^{2} \polylog \left (2, -{\mathrm e}^{f x +e}\right ) x}{f^{2}}+\frac {6 b d e \,c^{2} \ln \left ({\mathrm e}^{f x +e}\right )}{f^{2}}-\frac {6 b \,d^{2} e^{2} c \ln \left ({\mathrm e}^{f x +e}\right )}{f^{3}}-\frac {d^{3} b \,x^{4}}{4}+\frac {a \,c^{4}}{4 d}+\frac {b \,c^{4}}{4 d}-d^{2} b c \,x^{3}+\frac {b \,c^{3} \ln \left ({\mathrm e}^{f x +e}-1\right )}{f}-\frac {2 b \,c^{3} \ln \left ({\mathrm e}^{f x +e}\right )}{f}+\frac {6 b \,d^{3} \polylog \left (4, {\mathrm e}^{f x +e}\right )}{f^{4}}+\frac {6 b \,d^{3} \polylog \left (4, -{\mathrm e}^{f x +e}\right )}{f^{4}}-\frac {2 b \,d^{3} e^{3} x}{f^{3}}+\frac {3 b c \,d^{2} \ln \left (1-{\mathrm e}^{f x +e}\right ) x^{2}}{f}+\frac {3 b \,c^{2} d \ln \left (1-{\mathrm e}^{f x +e}\right ) x}{f}-\frac {6 b \,c^{2} d e x}{f}+\frac {6 b c \,e^{2} d^{2} x}{f^{2}}+\frac {b \,d^{3} \ln \left ({\mathrm e}^{f x +e}+1\right ) x^{3}}{f}+\frac {2 b \,d^{3} e^{3} \ln \left ({\mathrm e}^{f x +e}\right )}{f^{4}}+\frac {b \,d^{3} \ln \left (1-{\mathrm e}^{f x +e}\right ) x^{3}}{f}+\frac {b \,d^{3} e^{3} \ln \left (1-{\mathrm e}^{f x +e}\right )}{f^{4}}+\frac {3 b \,d^{3} \polylog \left (2, -{\mathrm e}^{f x +e}\right ) x^{2}}{f^{2}}-\frac {6 b \,d^{3} \polylog \left (3, -{\mathrm e}^{f x +e}\right ) x}{f^{3}}-\frac {6 b \,d^{3} \polylog \left (3, {\mathrm e}^{f x +e}\right ) x}{f^{3}}+\frac {3 b \,c^{2} d \polylog \left (2, -{\mathrm e}^{f x +e}\right )}{f^{2}}+\frac {3 b \,c^{2} d \polylog \left (2, {\mathrm e}^{f x +e}\right )}{f^{2}}-\frac {3 b \,c^{2} d \,e^{2}}{f^{2}}+\frac {4 b c \,e^{3} d^{2}}{f^{3}}-\frac {b \,d^{3} e^{3} \ln \left ({\mathrm e}^{f x +e}-1\right )}{f^{4}}-\frac {3 b \,d^{3} e^{4}}{2 f^{4}}+\frac {b \,c^{3} \ln \left ({\mathrm e}^{f x +e}+1\right )}{f}-\frac {3 d b \,c^{2} x^{2}}{2}+b \,c^{3} x\) | \(766\) |
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. 438 vs.
\(2 (126) = 252\).
time = 0.36, size = 438, normalized size = 3.29 \begin {gather*} \frac {1}{4} \, a d^{3} x^{4} + \frac {1}{4} \, b d^{3} x^{4} + a c d^{2} x^{3} + b c d^{2} x^{3} + \frac {3}{2} \, a c^{2} d x^{2} + \frac {3}{2} \, b c^{2} d x^{2} + a c^{3} x + \frac {b c^{3} \log \left (\sinh \left (f x + e\right )\right )}{f} + \frac {3 \, {\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 )} b c^{2} d}{f^{2}} + \frac {3 \, {\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 )} b c^{2} d}{f^{2}} + \frac {3 \, {\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 )} b c d^{2}}{f^{3}} + \frac {3 \, {\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 )} b c d^{2}}{f^{3}} + \frac {{\left (f^{3} x^{3} \log \left (e^{\left (f x + e\right )} + 1\right ) + 3 \, f^{2} x^{2} {\rm Li}_2\left (-e^{\left (f x + e\right )}\right ) - 6 \, f x {\rm Li}_{3}(-e^{\left (f x + e\right )}) + 6 \, {\rm Li}_{4}(-e^{\left (f x + e\right )})\right )} b d^{3}}{f^{4}} + \frac {{\left (f^{3} x^{3} \log \left (-e^{\left (f x + e\right )} + 1\right ) + 3 \, f^{2} x^{2} {\rm Li}_2\left (e^{\left (f x + e\right )}\right ) - 6 \, f x {\rm Li}_{3}(e^{\left (f x + e\right )}) + 6 \, {\rm Li}_{4}(e^{\left (f x + e\right )})\right )} b d^{3}}{f^{4}} - \frac {b d^{3} f^{4} x^{4} + 4 \, b c d^{2} f^{4} x^{3} + 6 \, b 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 [B] Leaf count of result is larger than twice the leaf count of optimal. 677 vs.
\(2 (126) = 252\).
time = 0.37, size = 677, normalized size = 5.09 \begin {gather*} \frac {{\left (a - b\right )} d^{3} f^{4} x^{4} + 4 \, {\left (a - b\right )} c d^{2} f^{4} x^{3} + 6 \, {\left (a - b\right )} c^{2} d f^{4} x^{2} + 4 \, {\left (a - b\right )} c^{3} f^{4} x + 24 \, b d^{3} {\rm polylog}\left (4, \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 )\right ) + 24 \, b d^{3} {\rm polylog}\left (4, -\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 )\right ) + 12 \, {\left (b d^{3} f^{2} x^{2} + 2 \, b c d^{2} f^{2} x + b c^{2} d f^{2}\right )} {\rm Li}_2\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 )\right ) + 12 \, {\left (b d^{3} f^{2} x^{2} + 2 \, b c d^{2} f^{2} x + b c^{2} d f^{2}\right )} {\rm Li}_2\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 )\right ) + 4 \, {\left (b d^{3} f^{3} x^{3} + 3 \, b c d^{2} f^{3} x^{2} + 3 \, b c^{2} d f^{3} x + b c^{3} f^{3}\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 ) + 1\right ) + 4 \, {\left (b c^{3} f^{3} - 3 \, b c^{2} d f^{2} \cosh \left (1\right ) + 3 \, b c d^{2} f \cosh \left (1\right )^{2} - b d^{3} \cosh \left (1\right )^{3} - b d^{3} \sinh \left (1\right )^{3} + 3 \, {\left (b c d^{2} f - b d^{3} \cosh \left (1\right )\right )} \sinh \left (1\right )^{2} - 3 \, {\left (b c^{2} d f^{2} - 2 \, b c d^{2} f \cosh \left (1\right ) + b 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 ) - 1\right ) + 4 \, {\left (b d^{3} f^{3} x^{3} + 3 \, b c d^{2} f^{3} x^{2} + 3 \, b c^{2} d f^{3} x + 3 \, b c^{2} d f^{2} \cosh \left (1\right ) - 3 \, b c d^{2} f \cosh \left (1\right )^{2} + b d^{3} \cosh \left (1\right )^{3} + b d^{3} \sinh \left (1\right )^{3} - 3 \, {\left (b c d^{2} f - b d^{3} \cosh \left (1\right )\right )} \sinh \left (1\right )^{2} + 3 \, {\left (b c^{2} d f^{2} - 2 \, b c d^{2} f \cosh \left (1\right ) + b 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 ) + 1\right ) - 24 \, {\left (b d^{3} f x + b c d^{2} f\right )} {\rm polylog}\left (3, \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 )\right ) - 24 \, {\left (b d^{3} f x + b c d^{2} f\right )} {\rm polylog}\left (3, -\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 )\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 (a + b \coth {\left (e + f x \right )}\right ) \left (c + d x\right )^{3}\, 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 \left (a+b\,\mathrm {coth}\left (e+f\,x\right )\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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