Integrand size = 14, antiderivative size = 149 \[ \int (c+d x)^3 \text {csch}(a+b x) \, dx=-\frac {2 (c+d x)^3 \text {arctanh}\left (e^{a+b x}\right )}{b}-\frac {3 d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}+\frac {3 d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^2}+\frac {6 d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{a+b x}\right )}{b^3}-\frac {6 d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{a+b x}\right )}{b^3}-\frac {6 d^3 \operatorname {PolyLog}\left (4,-e^{a+b x}\right )}{b^4}+\frac {6 d^3 \operatorname {PolyLog}\left (4,e^{a+b x}\right )}{b^4} \]
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Time = 0.10 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {4267, 2611, 6744, 2320, 6724} \[ \int (c+d x)^3 \text {csch}(a+b x) \, dx=-\frac {2 (c+d x)^3 \text {arctanh}\left (e^{a+b x}\right )}{b}-\frac {6 d^3 \operatorname {PolyLog}\left (4,-e^{a+b x}\right )}{b^4}+\frac {6 d^3 \operatorname {PolyLog}\left (4,e^{a+b x}\right )}{b^4}+\frac {6 d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{a+b x}\right )}{b^3}-\frac {6 d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{a+b x}\right )}{b^3}-\frac {3 d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}+\frac {3 d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^2} \]
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Rule 2320
Rule 2611
Rule 4267
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (c+d x)^3 \text {arctanh}\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 \text {arctanh}\left (e^{a+b x}\right )}{b}-\frac {3 d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}+\frac {3 d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^2}+\frac {\left (6 d^2\right ) \int (c+d x) \operatorname {PolyLog}\left (2,-e^{a+b x}\right ) \, dx}{b^2}-\frac {\left (6 d^2\right ) \int (c+d x) \operatorname {PolyLog}\left (2,e^{a+b x}\right ) \, dx}{b^2} \\ & = -\frac {2 (c+d x)^3 \text {arctanh}\left (e^{a+b x}\right )}{b}-\frac {3 d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}+\frac {3 d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^2}+\frac {6 d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{a+b x}\right )}{b^3}-\frac {6 d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{a+b x}\right )}{b^3}-\frac {\left (6 d^3\right ) \int \operatorname {PolyLog}\left (3,-e^{a+b x}\right ) \, dx}{b^3}+\frac {\left (6 d^3\right ) \int \operatorname {PolyLog}\left (3,e^{a+b x}\right ) \, dx}{b^3} \\ & = -\frac {2 (c+d x)^3 \text {arctanh}\left (e^{a+b x}\right )}{b}-\frac {3 d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}+\frac {3 d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^2}+\frac {6 d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{a+b x}\right )}{b^3}-\frac {6 d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{a+b x}\right )}{b^3}-\frac {\left (6 d^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{a+b x}\right )}{b^4}+\frac {\left (6 d^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{a+b x}\right )}{b^4} \\ & = -\frac {2 (c+d x)^3 \text {arctanh}\left (e^{a+b x}\right )}{b}-\frac {3 d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}+\frac {3 d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^2}+\frac {6 d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{a+b x}\right )}{b^3}-\frac {6 d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{a+b x}\right )}{b^3}-\frac {6 d^3 \operatorname {PolyLog}\left (4,-e^{a+b x}\right )}{b^4}+\frac {6 d^3 \operatorname {PolyLog}\left (4,e^{a+b x}\right )}{b^4} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.13 \[ \int (c+d x)^3 \text {csch}(a+b x) \, dx=\frac {(c+d x)^3 \log \left (1-e^{a+b x}\right )-(c+d x)^3 \log \left (1+e^{a+b x}\right )-\frac {3 d \left (b^2 (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )-2 b d (c+d x) \operatorname {PolyLog}\left (3,-e^{a+b x}\right )+2 d^2 \operatorname {PolyLog}\left (4,-e^{a+b x}\right )\right )}{b^3}+\frac {3 d \left (b^2 (c+d x)^2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )-2 b d (c+d x) \operatorname {PolyLog}\left (3,e^{a+b x}\right )+2 d^2 \operatorname {PolyLog}\left (4,e^{a+b x}\right )\right )}{b^3}}{b} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(540\) vs. \(2(142)=284\).
Time = 1.86 (sec) , antiderivative size = 541, normalized size of antiderivative = 3.63
method | result | size |
risch | \(\frac {3 d^{3} \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right ) x^{2}}{b^{2}}-\frac {6 d^{3} \operatorname {polylog}\left (3, {\mathrm e}^{b x +a}\right ) x}{b^{3}}-\frac {d^{3} \ln \left ({\mathrm e}^{b x +a}+1\right ) x^{3}}{b}-\frac {d^{3} \ln \left ({\mathrm e}^{b x +a}+1\right ) a^{3}}{b^{4}}-\frac {3 d^{3} \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right ) x^{2}}{b^{2}}+\frac {6 d^{3} \operatorname {polylog}\left (3, -{\mathrm e}^{b x +a}\right ) x}{b^{3}}-\frac {6 c \,d^{2} \operatorname {polylog}\left (3, {\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {6 c \,d^{2} \operatorname {polylog}\left (3, -{\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {3 c^{2} d \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )}{b^{2}}-\frac {3 c^{2} d \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {2 d^{3} a^{3} \operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{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 {3 c^{2} d \ln \left ({\mathrm e}^{b x +a}+1\right ) a}{b^{2}}-\frac {6 d^{2} a^{2} c \,\operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {6 d a \,c^{2} \operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{b^{2}}-\frac {6 c \,d^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right ) x}{b^{2}}+\frac {3 c \,d^{2} \ln \left (1-{\mathrm e}^{b x +a}\right ) x^{2}}{b}-\frac {3 c \,d^{2} \ln \left (1-{\mathrm e}^{b x +a}\right ) a^{2}}{b^{3}}+\frac {6 c \,d^{2} \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right ) x}{b^{2}}-\frac {3 c \,d^{2} \ln \left ({\mathrm e}^{b x +a}+1\right ) x^{2}}{b}+\frac {3 c \,d^{2} \ln \left ({\mathrm e}^{b x +a}+1\right ) a^{2}}{b^{3}}+\frac {3 c^{2} d \ln \left (1-{\mathrm e}^{b x +a}\right ) x}{b}-\frac {2 c^{3} \operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{b}+\frac {6 d^{3} \operatorname {polylog}\left (4, {\mathrm e}^{b x +a}\right )}{b^{4}}-\frac {6 d^{3} \operatorname {polylog}\left (4, -{\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {3 c^{2} d \ln \left (1-{\mathrm e}^{b x +a}\right ) a}{b^{2}}-\frac {3 c^{2} d \ln \left ({\mathrm e}^{b x +a}+1\right ) x}{b}\) | \(541\) |
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Leaf count of result is larger than twice the leaf count of optimal. 396 vs. \(2 (140) = 280\).
Time = 0.25 (sec) , antiderivative size = 396, normalized size of antiderivative = 2.66 \[ \int (c+d x)^3 \text {csch}(a+b x) \, dx=\frac {6 \, d^{3} {\rm polylog}\left (4, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - 6 \, d^{3} {\rm polylog}\left (4, -\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) + 3 \, {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} c^{2} d\right )} {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - 3 \, {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} c^{2} d\right )} {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) - {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right ) - 6 \, {\left (b d^{3} x + b c d^{2}\right )} {\rm polylog}\left (3, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + 6 \, {\left (b d^{3} x + b c d^{2}\right )} {\rm polylog}\left (3, -\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right )}{b^{4}} \]
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\[ \int (c+d x)^3 \text {csch}(a+b x) \, dx=\int \left (c + d x\right )^{3} \operatorname {csch}{\left (a + b x \right )}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (140) = 280\).
Time = 0.26 (sec) , antiderivative size = 333, normalized size of antiderivative = 2.23 \[ \int (c+d x)^3 \text {csch}(a+b x) \, dx=-c^{3} {\left (\frac {\log \left (e^{\left (-b x - a\right )} + 1\right )}{b} - \frac {\log \left (e^{\left (-b x - a\right )} - 1\right )}{b}\right )} - \frac {3 \, {\left (b x \log \left (e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (b x + a\right )}\right )\right )} c^{2} d}{b^{2}} + \frac {3 \, {\left (b x \log \left (-e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (b x + a\right )}\right )\right )} c^{2} d}{b^{2}} - \frac {3 \, {\left (b^{2} x^{2} \log \left (e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (b x + a\right )})\right )} c d^{2}}{b^{3}} + \frac {3 \, {\left (b^{2} x^{2} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (b x + a\right )})\right )} c d^{2}}{b^{3}} - \frac {{\left (b^{3} x^{3} \log \left (e^{\left (b x + a\right )} + 1\right ) + 3 \, b^{2} x^{2} {\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 6 \, b x {\rm Li}_{3}(-e^{\left (b x + a\right )}) + 6 \, {\rm Li}_{4}(-e^{\left (b x + a\right )})\right )} d^{3}}{b^{4}} + \frac {{\left (b^{3} x^{3} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 3 \, b^{2} x^{2} {\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 6 \, b x {\rm Li}_{3}(e^{\left (b x + a\right )}) + 6 \, {\rm Li}_{4}(e^{\left (b x + a\right )})\right )} d^{3}}{b^{4}} \]
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\[ \int (c+d x)^3 \text {csch}(a+b x) \, dx=\int { {\left (d x + c\right )}^{3} \operatorname {csch}\left (b x + a\right ) \,d x } \]
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Timed out. \[ \int (c+d x)^3 \text {csch}(a+b x) \, dx=\int \frac {{\left (c+d\,x\right )}^3}{\mathrm {sinh}\left (a+b\,x\right )} \,d x \]
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