Optimal. Leaf size=256 \[ -\frac {6 d^2 (c+d x) \tanh ^{-1}\left (e^{a+b x}\right )}{b^3}+\frac {(c+d x)^3 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {3 d (c+d x)^2 \text {csch}(a+b x)}{2 b^2}-\frac {(c+d x)^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}-\frac {3 d^3 \text {PolyLog}\left (2,-e^{a+b x}\right )}{b^4}+\frac {3 d (c+d x)^2 \text {PolyLog}\left (2,-e^{a+b x}\right )}{2 b^2}+\frac {3 d^3 \text {PolyLog}\left (2,e^{a+b x}\right )}{b^4}-\frac {3 d (c+d x)^2 \text {PolyLog}\left (2,e^{a+b x}\right )}{2 b^2}-\frac {3 d^2 (c+d x) \text {PolyLog}\left (3,-e^{a+b x}\right )}{b^3}+\frac {3 d^2 (c+d x) \text {PolyLog}\left (3,e^{a+b x}\right )}{b^3}+\frac {3 d^3 \text {PolyLog}\left (4,-e^{a+b x}\right )}{b^4}-\frac {3 d^3 \text {PolyLog}\left (4,e^{a+b x}\right )}{b^4} \]
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Rubi [A]
time = 0.20, antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4271, 4267,
2317, 2438, 2611, 6744, 2320, 6724} \begin {gather*} -\frac {3 d^3 \text {Li}_2\left (-e^{a+b x}\right )}{b^4}+\frac {3 d^3 \text {Li}_2\left (e^{a+b x}\right )}{b^4}+\frac {3 d^3 \text {Li}_4\left (-e^{a+b x}\right )}{b^4}-\frac {3 d^3 \text {Li}_4\left (e^{a+b x}\right )}{b^4}-\frac {3 d^2 (c+d x) \text {Li}_3\left (-e^{a+b x}\right )}{b^3}+\frac {3 d^2 (c+d x) \text {Li}_3\left (e^{a+b x}\right )}{b^3}-\frac {6 d^2 (c+d x) \tanh ^{-1}\left (e^{a+b x}\right )}{b^3}+\frac {3 d (c+d x)^2 \text {Li}_2\left (-e^{a+b x}\right )}{2 b^2}-\frac {3 d (c+d x)^2 \text {Li}_2\left (e^{a+b x}\right )}{2 b^2}-\frac {3 d (c+d x)^2 \text {csch}(a+b x)}{2 b^2}+\frac {(c+d x)^3 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {(c+d x)^3 \coth (a+b x) \text {csch}(a+b x)}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 4267
Rule 4271
Rule 6724
Rule 6744
Rubi steps
\begin {align*} \int (c+d x)^3 \text {csch}^3(a+b x) \, dx &=-\frac {3 d (c+d x)^2 \text {csch}(a+b x)}{2 b^2}-\frac {(c+d x)^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}-\frac {1}{2} \int (c+d x)^3 \text {csch}(a+b x) \, dx+\frac {\left (3 d^2\right ) \int (c+d x) \text {csch}(a+b x) \, dx}{b^2}\\ &=-\frac {6 d^2 (c+d x) \tanh ^{-1}\left (e^{a+b x}\right )}{b^3}+\frac {(c+d x)^3 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {3 d (c+d x)^2 \text {csch}(a+b x)}{2 b^2}-\frac {(c+d x)^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}+\frac {(3 d) \int (c+d x)^2 \log \left (1-e^{a+b x}\right ) \, dx}{2 b}-\frac {(3 d) \int (c+d x)^2 \log \left (1+e^{a+b x}\right ) \, dx}{2 b}-\frac {\left (3 d^3\right ) \int \log \left (1-e^{a+b x}\right ) \, dx}{b^3}+\frac {\left (3 d^3\right ) \int \log \left (1+e^{a+b x}\right ) \, dx}{b^3}\\ &=-\frac {6 d^2 (c+d x) \tanh ^{-1}\left (e^{a+b x}\right )}{b^3}+\frac {(c+d x)^3 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {3 d (c+d x)^2 \text {csch}(a+b x)}{2 b^2}-\frac {(c+d x)^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}+\frac {3 d (c+d x)^2 \text {Li}_2\left (-e^{a+b x}\right )}{2 b^2}-\frac {3 d (c+d x)^2 \text {Li}_2\left (e^{a+b x}\right )}{2 b^2}-\frac {\left (3 d^2\right ) \int (c+d x) \text {Li}_2\left (-e^{a+b x}\right ) \, dx}{b^2}+\frac {\left (3 d^2\right ) \int (c+d x) \text {Li}_2\left (e^{a+b x}\right ) \, dx}{b^2}-\frac {\left (3 d^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{a+b x}\right )}{b^4}+\frac {\left (3 d^3\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{a+b x}\right )}{b^4}\\ &=-\frac {6 d^2 (c+d x) \tanh ^{-1}\left (e^{a+b x}\right )}{b^3}+\frac {(c+d x)^3 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {3 d (c+d x)^2 \text {csch}(a+b x)}{2 b^2}-\frac {(c+d x)^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}-\frac {3 d^3 \text {Li}_2\left (-e^{a+b x}\right )}{b^4}+\frac {3 d (c+d x)^2 \text {Li}_2\left (-e^{a+b x}\right )}{2 b^2}+\frac {3 d^3 \text {Li}_2\left (e^{a+b x}\right )}{b^4}-\frac {3 d (c+d x)^2 \text {Li}_2\left (e^{a+b x}\right )}{2 b^2}-\frac {3 d^2 (c+d x) \text {Li}_3\left (-e^{a+b x}\right )}{b^3}+\frac {3 d^2 (c+d x) \text {Li}_3\left (e^{a+b x}\right )}{b^3}+\frac {\left (3 d^3\right ) \int \text {Li}_3\left (-e^{a+b x}\right ) \, dx}{b^3}-\frac {\left (3 d^3\right ) \int \text {Li}_3\left (e^{a+b x}\right ) \, dx}{b^3}\\ &=-\frac {6 d^2 (c+d x) \tanh ^{-1}\left (e^{a+b x}\right )}{b^3}+\frac {(c+d x)^3 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {3 d (c+d x)^2 \text {csch}(a+b x)}{2 b^2}-\frac {(c+d x)^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}-\frac {3 d^3 \text {Li}_2\left (-e^{a+b x}\right )}{b^4}+\frac {3 d (c+d x)^2 \text {Li}_2\left (-e^{a+b x}\right )}{2 b^2}+\frac {3 d^3 \text {Li}_2\left (e^{a+b x}\right )}{b^4}-\frac {3 d (c+d x)^2 \text {Li}_2\left (e^{a+b x}\right )}{2 b^2}-\frac {3 d^2 (c+d x) \text {Li}_3\left (-e^{a+b x}\right )}{b^3}+\frac {3 d^2 (c+d x) \text {Li}_3\left (e^{a+b x}\right )}{b^3}+\frac {\left (3 d^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{a+b x}\right )}{b^4}-\frac {\left (3 d^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{a+b x}\right )}{b^4}\\ &=-\frac {6 d^2 (c+d x) \tanh ^{-1}\left (e^{a+b x}\right )}{b^3}+\frac {(c+d x)^3 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {3 d (c+d x)^2 \text {csch}(a+b x)}{2 b^2}-\frac {(c+d x)^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}-\frac {3 d^3 \text {Li}_2\left (-e^{a+b x}\right )}{b^4}+\frac {3 d (c+d x)^2 \text {Li}_2\left (-e^{a+b x}\right )}{2 b^2}+\frac {3 d^3 \text {Li}_2\left (e^{a+b x}\right )}{b^4}-\frac {3 d (c+d x)^2 \text {Li}_2\left (e^{a+b x}\right )}{2 b^2}-\frac {3 d^2 (c+d x) \text {Li}_3\left (-e^{a+b x}\right )}{b^3}+\frac {3 d^2 (c+d x) \text {Li}_3\left (e^{a+b x}\right )}{b^3}+\frac {3 d^3 \text {Li}_4\left (-e^{a+b x}\right )}{b^4}-\frac {3 d^3 \text {Li}_4\left (e^{a+b x}\right )}{b^4}\\ \end {align*}
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Mathematica [A]
time = 5.98, size = 398, normalized size = 1.55 \begin {gather*} -\frac {-2 b^3 c^3 \tanh ^{-1}\left (e^{a+b x}\right )+12 b c d^2 \tanh ^{-1}\left (e^{a+b x}\right )+b^2 (c+d x)^2 (3 d+b (c+d x) \coth (a+b x)) \text {csch}(a+b x)+3 b^3 c^2 d x \log \left (1-e^{a+b x}\right )-6 b d^3 x \log \left (1-e^{a+b x}\right )+3 b^3 c d^2 x^2 \log \left (1-e^{a+b x}\right )+b^3 d^3 x^3 \log \left (1-e^{a+b x}\right )-3 b^3 c^2 d x \log \left (1+e^{a+b x}\right )+6 b d^3 x \log \left (1+e^{a+b x}\right )-3 b^3 c d^2 x^2 \log \left (1+e^{a+b x}\right )-b^3 d^3 x^3 \log \left (1+e^{a+b x}\right )-3 d \left (-2 d^2+b^2 (c+d x)^2\right ) \text {PolyLog}\left (2,-e^{a+b x}\right )+3 d \left (-2 d^2+b^2 (c+d x)^2\right ) \text {PolyLog}\left (2,e^{a+b x}\right )+6 b c d^2 \text {PolyLog}\left (3,-e^{a+b x}\right )+6 b d^3 x \text {PolyLog}\left (3,-e^{a+b x}\right )-6 b c d^2 \text {PolyLog}\left (3,e^{a+b x}\right )-6 b d^3 x \text {PolyLog}\left (3,e^{a+b x}\right )-6 d^3 \text {PolyLog}\left (4,-e^{a+b x}\right )+6 d^3 \text {PolyLog}\left (4,e^{a+b x}\right )}{2 b^4} \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. \(875\) vs.
\(2(238)=476\).
time = 1.07, size = 876, normalized size = 3.42
method | result | size |
risch | \(\frac {3 d^{2} c \polylog \left (2, -{\mathrm e}^{b x +a}\right ) x}{b^{2}}-\frac {d^{3} \ln \left (1-{\mathrm e}^{b x +a}\right ) x^{3}}{2 b}-\frac {d^{3} \ln \left (1-{\mathrm e}^{b x +a}\right ) a^{3}}{2 b^{4}}-\frac {3 d^{2} c \polylog \left (2, {\mathrm e}^{b x +a}\right ) x}{b^{2}}-\frac {3 d a \,c^{2} \arctanh \left ({\mathrm e}^{b x +a}\right )}{b^{2}}-\frac {{\mathrm e}^{b x +a} \left (b \,d^{3} x^{3} {\mathrm e}^{2 b x +2 a}+3 b c \,d^{2} x^{2} {\mathrm e}^{2 b x +2 a}+3 b \,c^{2} d x \,{\mathrm e}^{2 b x +2 a}+b \,d^{3} x^{3}+3 d^{3} x^{2} {\mathrm e}^{2 b x +2 a}+b \,c^{3} {\mathrm e}^{2 b x +2 a}+3 b c \,d^{2} x^{2}+6 c \,d^{2} x \,{\mathrm e}^{2 b x +2 a}+3 b \,c^{2} d x +3 c^{2} d \,{\mathrm e}^{2 b x +2 a}-3 d^{3} x^{2}+b \,c^{3}-6 c \,d^{2} x -3 c^{2} d \right )}{b^{2} \left ({\mathrm e}^{2 b x +2 a}-1\right )^{2}}-\frac {3 d^{3} \polylog \left (3, -{\mathrm e}^{b x +a}\right ) x}{b^{3}}-\frac {3 d^{3} \polylog \left (2, {\mathrm e}^{b x +a}\right ) x^{2}}{2 b^{2}}+\frac {3 d^{3} \polylog \left (3, {\mathrm e}^{b x +a}\right ) x}{b^{3}}+\frac {3 d \,c^{2} \polylog \left (2, -{\mathrm e}^{b x +a}\right )}{2 b^{2}}+\frac {3 d^{2} c \polylog \left (3, {\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {3 d^{2} c \polylog \left (3, -{\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {3 d^{3} \polylog \left (2, -{\mathrm e}^{b x +a}\right ) x^{2}}{2 b^{2}}+\frac {3 d^{3} \polylog \left (4, -{\mathrm e}^{b x +a}\right )}{b^{4}}-\frac {3 d^{3} \polylog \left (4, {\mathrm e}^{b x +a}\right )}{b^{4}}-\frac {3 d^{3} \polylog \left (2, -{\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {c^{3} \arctanh \left ({\mathrm e}^{b x +a}\right )}{b}+\frac {3 d^{3} \polylog \left (2, {\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {d^{3} \ln \left ({\mathrm e}^{b x +a}+1\right ) x^{3}}{2 b}+\frac {d^{3} \ln \left ({\mathrm e}^{b x +a}+1\right ) a^{3}}{2 b^{4}}+\frac {3 d \,c^{2} \ln \left ({\mathrm e}^{b x +a}+1\right ) x}{2 b}+\frac {3 d \,c^{2} \ln \left ({\mathrm e}^{b x +a}+1\right ) a}{2 b^{2}}-\frac {3 d^{2} a^{2} c \ln \left ({\mathrm e}^{b x +a}+1\right )}{2 b^{3}}-\frac {3 d \,c^{2} \ln \left (1-{\mathrm e}^{b x +a}\right ) x}{2 b}-\frac {3 d \,c^{2} \ln \left (1-{\mathrm e}^{b x +a}\right ) a}{2 b^{2}}+\frac {3 d^{2} a^{2} c \ln \left (1-{\mathrm e}^{b x +a}\right )}{2 b^{3}}-\frac {3 d^{2} c \ln \left (1-{\mathrm e}^{b x +a}\right ) x^{2}}{2 b}+\frac {3 d^{2} c \ln \left ({\mathrm e}^{b x +a}+1\right ) x^{2}}{2 b}+\frac {3 d^{2} a^{2} c \arctanh \left ({\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {3 d \,c^{2} \polylog \left (2, {\mathrm e}^{b x +a}\right )}{2 b^{2}}-\frac {d^{3} a^{3} \arctanh \left ({\mathrm e}^{b x +a}\right )}{b^{4}}-\frac {6 c \,d^{2} \arctanh \left ({\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {6 d^{3} a \arctanh \left ({\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {3 d^{3} \ln \left (1-{\mathrm e}^{b x +a}\right ) x}{b^{3}}-\frac {3 d^{3} \ln \left ({\mathrm e}^{b x +a}+1\right ) x}{b^{3}}+\frac {3 d^{3} a \ln \left (1-{\mathrm e}^{b x +a}\right )}{b^{4}}-\frac {3 d^{3} a \ln \left ({\mathrm e}^{b x +a}+1\right )}{b^{4}}\) | \(876\) |
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. 605 vs.
\(2 (234) = 468\).
time = 0.42, size = 605, normalized size = 2.36 \begin {gather*} \frac {1}{2} \, c^{3} {\left (\frac {\log \left (e^{\left (-b x - a\right )} + 1\right )}{b} - \frac {\log \left (e^{\left (-b x - a\right )} - 1\right )}{b} + \frac {2 \, {\left (e^{\left (-b x - a\right )} + e^{\left (-3 \, b x - 3 \, a\right )}\right )}}{b {\left (2 \, e^{\left (-2 \, b x - 2 \, a\right )} - e^{\left (-4 \, b x - 4 \, a\right )} - 1\right )}}\right )} + \frac {3 \, {\left (b^{2} x^{2} \log \left (e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (b x + a\right )})\right )} c d^{2}}{2 \, b^{3}} - \frac {3 \, {\left (b^{2} x^{2} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (b x + a\right )})\right )} c d^{2}}{2 \, b^{3}} - \frac {3 \, c d^{2} \log \left (e^{\left (b x + a\right )} + 1\right )}{b^{3}} + \frac {3 \, c d^{2} \log \left (e^{\left (b x + a\right )} - 1\right )}{b^{3}} - \frac {{\left (b d^{3} x^{3} e^{\left (3 \, a\right )} + 3 \, c^{2} d e^{\left (3 \, a\right )} + 3 \, {\left (b c d^{2} + d^{3}\right )} x^{2} e^{\left (3 \, a\right )} + 3 \, {\left (b c^{2} d + 2 \, c d^{2}\right )} x e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )} + {\left (b d^{3} x^{3} e^{a} - 3 \, c^{2} d e^{a} + 3 \, {\left (b c d^{2} - d^{3}\right )} x^{2} e^{a} + 3 \, {\left (b c^{2} d - 2 \, c d^{2}\right )} x e^{a}\right )} e^{\left (b x\right )}}{b^{2} e^{\left (4 \, b x + 4 \, a\right )} - 2 \, b^{2} e^{\left (2 \, b x + 2 \, a\right )} + b^{2}} + \frac {{\left (b^{3} x^{3} \log \left (e^{\left (b x + a\right )} + 1\right ) + 3 \, b^{2} x^{2} {\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 6 \, b x {\rm Li}_{3}(-e^{\left (b x + a\right )}) + 6 \, {\rm Li}_{4}(-e^{\left (b x + a\right )})\right )} d^{3}}{2 \, b^{4}} - \frac {{\left (b^{3} x^{3} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 3 \, b^{2} x^{2} {\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 6 \, b x {\rm Li}_{3}(e^{\left (b x + a\right )}) + 6 \, {\rm Li}_{4}(e^{\left (b x + a\right )})\right )} d^{3}}{2 \, b^{4}} + \frac {3 \, {\left (b^{2} c^{2} d - 2 \, d^{3}\right )} {\left (b x \log \left (e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (b x + a\right )}\right )\right )}}{2 \, b^{4}} - \frac {3 \, {\left (b^{2} c^{2} d - 2 \, d^{3}\right )} {\left (b x \log \left (-e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (b x + a\right )}\right )\right )}}{2 \, b^{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. 4008 vs.
\(2 (234) = 468\).
time = 0.41, size = 4008, normalized size = 15.66 \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 (c + d x\right )^{3} \operatorname {csch}^{3}{\left (a + b 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 = -0.00 \begin {gather*} \int \frac {{\left (c+d\,x\right )}^3}{{\mathrm {sinh}\left (a+b\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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