Optimal. Leaf size=149 \[ -\frac {6 d^3 \text {Li}_4\left (-e^{a+b x}\right )}{b^4}+\frac {6 d^3 \text {Li}_4\left (e^{a+b x}\right )}{b^4}+\frac {6 d^2 (c+d x) \text {Li}_3\left (-e^{a+b x}\right )}{b^3}-\frac {6 d^2 (c+d x) \text {Li}_3\left (e^{a+b x}\right )}{b^3}-\frac {3 d (c+d x)^2 \text {Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac {3 d (c+d x)^2 \text {Li}_2\left (e^{a+b x}\right )}{b^2}-\frac {2 (c+d x)^3 \tanh ^{-1}\left (e^{a+b x}\right )}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.14, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {4182, 2531, 6609, 2282, 6589} \[ \frac {6 d^2 (c+d x) \text {PolyLog}\left (3,-e^{a+b x}\right )}{b^3}-\frac {6 d^2 (c+d x) \text {PolyLog}\left (3,e^{a+b x}\right )}{b^3}-\frac {3 d (c+d x)^2 \text {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}+\frac {3 d (c+d x)^2 \text {PolyLog}\left (2,e^{a+b x}\right )}{b^2}-\frac {6 d^3 \text {PolyLog}\left (4,-e^{a+b x}\right )}{b^4}+\frac {6 d^3 \text {PolyLog}\left (4,e^{a+b x}\right )}{b^4}-\frac {2 (c+d x)^3 \tanh ^{-1}\left (e^{a+b x}\right )}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2282
Rule 2531
Rule 4182
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int (c+d x)^3 \text {csch}(a+b x) \, dx &=-\frac {2 (c+d x)^3 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {(3 d) \int (c+d x)^2 \log \left (1-e^{a+b x}\right ) \, dx}{b}+\frac {(3 d) \int (c+d x)^2 \log \left (1+e^{a+b x}\right ) \, dx}{b}\\ &=-\frac {2 (c+d x)^3 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {3 d (c+d x)^2 \text {Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac {3 d (c+d x)^2 \text {Li}_2\left (e^{a+b x}\right )}{b^2}+\frac {\left (6 d^2\right ) \int (c+d x) \text {Li}_2\left (-e^{a+b x}\right ) \, dx}{b^2}-\frac {\left (6 d^2\right ) \int (c+d x) \text {Li}_2\left (e^{a+b x}\right ) \, dx}{b^2}\\ &=-\frac {2 (c+d x)^3 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {3 d (c+d x)^2 \text {Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac {3 d (c+d x)^2 \text {Li}_2\left (e^{a+b x}\right )}{b^2}+\frac {6 d^2 (c+d x) \text {Li}_3\left (-e^{a+b x}\right )}{b^3}-\frac {6 d^2 (c+d x) \text {Li}_3\left (e^{a+b x}\right )}{b^3}-\frac {\left (6 d^3\right ) \int \text {Li}_3\left (-e^{a+b x}\right ) \, dx}{b^3}+\frac {\left (6 d^3\right ) \int \text {Li}_3\left (e^{a+b x}\right ) \, dx}{b^3}\\ &=-\frac {2 (c+d x)^3 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {3 d (c+d x)^2 \text {Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac {3 d (c+d x)^2 \text {Li}_2\left (e^{a+b x}\right )}{b^2}+\frac {6 d^2 (c+d x) \text {Li}_3\left (-e^{a+b x}\right )}{b^3}-\frac {6 d^2 (c+d x) \text {Li}_3\left (e^{a+b x}\right )}{b^3}-\frac {\left (6 d^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{a+b x}\right )}{b^4}+\frac {\left (6 d^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{a+b x}\right )}{b^4}\\ &=-\frac {2 (c+d x)^3 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {3 d (c+d x)^2 \text {Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac {3 d (c+d x)^2 \text {Li}_2\left (e^{a+b x}\right )}{b^2}+\frac {6 d^2 (c+d x) \text {Li}_3\left (-e^{a+b x}\right )}{b^3}-\frac {6 d^2 (c+d x) \text {Li}_3\left (e^{a+b x}\right )}{b^3}-\frac {6 d^3 \text {Li}_4\left (-e^{a+b x}\right )}{b^4}+\frac {6 d^3 \text {Li}_4\left (e^{a+b x}\right )}{b^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 2.65, size = 191, normalized size = 1.28 \[ \frac {-2 b^3 (c+d x)^3 \tanh ^{-1}(\sinh (a+b x)+\cosh (a+b x))-3 d \left (b^2 (c+d x)^2 \text {Li}_2(-\cosh (a+b x)-\sinh (a+b x))-2 b d (c+d x) \text {Li}_3(-\cosh (a+b x)-\sinh (a+b x))+2 d^2 \text {Li}_4(-\cosh (a+b x)-\sinh (a+b x))\right )+3 d \left (b^2 (c+d x)^2 \text {Li}_2(\cosh (a+b x)+\sinh (a+b x))-2 b d (c+d x) \text {Li}_3(\cosh (a+b x)+\sinh (a+b x))+2 d^2 \text {Li}_4(\cosh (a+b x)+\sinh (a+b x))\right )}{b^4} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [C] time = 0.47, size = 396, normalized size = 2.66 \[ \frac {6 \, d^{3} {\rm polylog}\left (4, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - 6 \, d^{3} {\rm polylog}\left (4, -\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) + 3 \, {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} c^{2} d\right )} {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - 3 \, {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} c^{2} d\right )} {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) - {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right ) - 6 \, {\left (b d^{3} x + b c d^{2}\right )} {\rm polylog}\left (3, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + 6 \, {\left (b d^{3} x + b c d^{2}\right )} {\rm polylog}\left (3, -\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )}^{3} \operatorname {csch}\left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.11, size = 541, normalized size = 3.63 \[ \frac {3 d^{3} \polylog \left (2, {\mathrm e}^{b x +a}\right ) x^{2}}{b^{2}}-\frac {6 d^{3} \polylog \left (3, {\mathrm e}^{b x +a}\right ) x}{b^{3}}+\frac {2 d^{3} a^{3} \arctanh \left ({\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {6 c \,d^{2} \polylog \left (3, -{\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {6 c \,d^{2} \polylog \left (3, {\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {3 c^{2} d \polylog \left (2, -{\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {3 c^{2} d \polylog \left (2, {\mathrm e}^{b x +a}\right )}{b^{2}}-\frac {3 d^{3} \polylog \left (2, -{\mathrm e}^{b x +a}\right ) x^{2}}{b^{2}}+\frac {6 d^{3} \polylog \left (3, -{\mathrm e}^{b x +a}\right ) x}{b^{3}}-\frac {d^{3} \ln \left (1+{\mathrm e}^{b x +a}\right ) x^{3}}{b}-\frac {d^{3} \ln \left (1+{\mathrm e}^{b x +a}\right ) a^{3}}{b^{4}}+\frac {d^{3} \ln \left (1-{\mathrm e}^{b x +a}\right ) x^{3}}{b}+\frac {d^{3} \ln \left (1-{\mathrm e}^{b x +a}\right ) a^{3}}{b^{4}}-\frac {6 c \,d^{2} a^{2} \arctanh \left ({\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {3 c \,d^{2} a^{2} \ln \left (1+{\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {3 c \,d^{2} a^{2} \ln \left (1-{\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {3 c \,d^{2} \ln \left (1-{\mathrm e}^{b x +a}\right ) x^{2}}{b}-\frac {3 c^{2} d \ln \left (1+{\mathrm e}^{b x +a}\right ) x}{b}-\frac {3 c^{2} d \ln \left (1+{\mathrm e}^{b x +a}\right ) a}{b^{2}}+\frac {3 c^{2} d \ln \left (1-{\mathrm e}^{b x +a}\right ) x}{b}+\frac {3 c^{2} d \ln \left (1-{\mathrm e}^{b x +a}\right ) a}{b^{2}}-\frac {3 c \,d^{2} \ln \left (1+{\mathrm e}^{b x +a}\right ) x^{2}}{b}-\frac {6 c \,d^{2} \polylog \left (2, -{\mathrm e}^{b x +a}\right ) x}{b^{2}}+\frac {6 c \,d^{2} \polylog \left (2, {\mathrm e}^{b x +a}\right ) x}{b^{2}}+\frac {6 c^{2} d a \arctanh \left ({\mathrm e}^{b x +a}\right )}{b^{2}}-\frac {6 d^{3} \polylog \left (4, -{\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {6 d^{3} \polylog \left (4, {\mathrm e}^{b x +a}\right )}{b^{4}}-\frac {2 c^{3} \arctanh \left ({\mathrm e}^{b x +a}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.64, size = 333, normalized size = 2.23 \[ -c^{3} {\left (\frac {\log \left (e^{\left (-b x - a\right )} + 1\right )}{b} - \frac {\log \left (e^{\left (-b x - a\right )} - 1\right )}{b}\right )} - \frac {3 \, {\left (b x \log \left (e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (b x + a\right )}\right )\right )} c^{2} d}{b^{2}} + \frac {3 \, {\left (b x \log \left (-e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (b x + a\right )}\right )\right )} c^{2} d}{b^{2}} - \frac {3 \, {\left (b^{2} x^{2} \log \left (e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (b x + a\right )})\right )} c d^{2}}{b^{3}} + \frac {3 \, {\left (b^{2} x^{2} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (b x + a\right )})\right )} c d^{2}}{b^{3}} - \frac {{\left (b^{3} x^{3} \log \left (e^{\left (b x + a\right )} + 1\right ) + 3 \, b^{2} x^{2} {\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 6 \, b x {\rm Li}_{3}(-e^{\left (b x + a\right )}) + 6 \, {\rm Li}_{4}(-e^{\left (b x + a\right )})\right )} d^{3}}{b^{4}} + \frac {{\left (b^{3} x^{3} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 3 \, b^{2} x^{2} {\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 6 \, b x {\rm Li}_{3}(e^{\left (b x + a\right )}) + 6 \, {\rm Li}_{4}(e^{\left (b x + a\right )})\right )} d^{3}}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c+d\,x\right )}^3}{\mathrm {sinh}\left (a+b\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c + d x\right )^{3} \operatorname {csch}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________