Integrand size = 14, antiderivative size = 99 \[ \int (c+d x)^2 \text {csch}(a+b x) \, dx=-\frac {2 (c+d x)^2 \text {arctanh}\left (e^{a+b x}\right )}{b}-\frac {2 d (c+d x) \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}+\frac {2 d (c+d x) \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^2}+\frac {2 d^2 \operatorname {PolyLog}\left (3,-e^{a+b x}\right )}{b^3}-\frac {2 d^2 \operatorname {PolyLog}\left (3,e^{a+b x}\right )}{b^3} \]
[Out]
Time = 0.07 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4267, 2611, 2320, 6724} \[ \int (c+d x)^2 \text {csch}(a+b x) \, dx=-\frac {2 (c+d x)^2 \text {arctanh}\left (e^{a+b x}\right )}{b}+\frac {2 d^2 \operatorname {PolyLog}\left (3,-e^{a+b x}\right )}{b^3}-\frac {2 d^2 \operatorname {PolyLog}\left (3,e^{a+b x}\right )}{b^3}-\frac {2 d (c+d x) \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}+\frac {2 d (c+d x) \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^2} \]
[In]
[Out]
Rule 2320
Rule 2611
Rule 4267
Rule 6724
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (c+d x)^2 \text {arctanh}\left (e^{a+b x}\right )}{b}-\frac {(2 d) \int (c+d x) \log \left (1-e^{a+b x}\right ) \, dx}{b}+\frac {(2 d) \int (c+d x) \log \left (1+e^{a+b x}\right ) \, dx}{b} \\ & = -\frac {2 (c+d x)^2 \text {arctanh}\left (e^{a+b x}\right )}{b}-\frac {2 d (c+d x) \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}+\frac {2 d (c+d x) \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^2}+\frac {\left (2 d^2\right ) \int \operatorname {PolyLog}\left (2,-e^{a+b x}\right ) \, dx}{b^2}-\frac {\left (2 d^2\right ) \int \operatorname {PolyLog}\left (2,e^{a+b x}\right ) \, dx}{b^2} \\ & = -\frac {2 (c+d x)^2 \text {arctanh}\left (e^{a+b x}\right )}{b}-\frac {2 d (c+d x) \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}+\frac {2 d (c+d x) \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^2}+\frac {\left (2 d^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}-\frac {\left (2 d^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{a+b x}\right )}{b^3} \\ & = -\frac {2 (c+d x)^2 \text {arctanh}\left (e^{a+b x}\right )}{b}-\frac {2 d (c+d x) \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}+\frac {2 d (c+d x) \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^2}+\frac {2 d^2 \operatorname {PolyLog}\left (3,-e^{a+b x}\right )}{b^3}-\frac {2 d^2 \operatorname {PolyLog}\left (3,e^{a+b x}\right )}{b^3} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.19 \[ \int (c+d x)^2 \text {csch}(a+b x) \, dx=\frac {(c+d x)^2 \log \left (1-e^{a+b x}\right )-(c+d x)^2 \log \left (1+e^{a+b x}\right )-\frac {2 d \left (b (c+d x) \operatorname {PolyLog}\left (2,-e^{a+b x}\right )-d \operatorname {PolyLog}\left (3,-e^{a+b x}\right )\right )}{b^2}+\frac {2 d \left (b (c+d x) \operatorname {PolyLog}\left (2,e^{a+b x}\right )-d \operatorname {PolyLog}\left (3,e^{a+b x}\right )\right )}{b^2}}{b} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(305\) vs. \(2(94)=188\).
Time = 0.94 (sec) , antiderivative size = 306, normalized size of antiderivative = 3.09
method | result | size |
risch | \(\frac {2 c d \ln \left (1-{\mathrm e}^{b x +a}\right ) x}{b}+\frac {2 c d \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )}{b^{2}}-\frac {2 c d \ln \left ({\mathrm e}^{b x +a}+1\right ) x}{b}-\frac {2 c d \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {2 d^{2} \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right ) x}{b^{2}}+\frac {d^{2} \ln \left ({\mathrm e}^{b x +a}+1\right ) a^{2}}{b^{3}}-\frac {2 d^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right ) x}{b^{2}}-\frac {d^{2} \ln \left (1-{\mathrm e}^{b x +a}\right ) a^{2}}{b^{3}}-\frac {2 d^{2} a^{2} \operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {d^{2} \ln \left (1-{\mathrm e}^{b x +a}\right ) x^{2}}{b}-\frac {2 d^{2} \operatorname {polylog}\left (3, {\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {d^{2} \ln \left ({\mathrm e}^{b x +a}+1\right ) x^{2}}{b}+\frac {2 d^{2} \operatorname {polylog}\left (3, -{\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {2 c d \ln \left (1-{\mathrm e}^{b x +a}\right ) a}{b^{2}}-\frac {2 c d \ln \left ({\mathrm e}^{b x +a}+1\right ) a}{b^{2}}+\frac {4 d a c \,\operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{b^{2}}-\frac {2 c^{2} \operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{b}\) | \(306\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 242 vs. \(2 (92) = 184\).
Time = 0.27 (sec) , antiderivative size = 242, normalized size of antiderivative = 2.44 \[ \int (c+d x)^2 \text {csch}(a+b x) \, dx=-\frac {2 \, d^{2} {\rm polylog}\left (3, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - 2 \, d^{2} {\rm polylog}\left (3, -\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) - 2 \, {\left (b d^{2} x + b c d\right )} {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + 2 \, {\left (b d^{2} x + b c d\right )} {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) - {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) - {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + 2 \, a b c d - a^{2} d^{2}\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right )}{b^{3}} \]
[In]
[Out]
\[ \int (c+d x)^2 \text {csch}(a+b x) \, dx=\int \left (c + d x\right )^{2} \operatorname {csch}{\left (a + b x \right )}\, dx \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 195 vs. \(2 (92) = 184\).
Time = 0.28 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.97 \[ \int (c+d x)^2 \text {csch}(a+b x) \, dx=-c^{2} {\left (\frac {\log \left (e^{\left (-b x - a\right )} + 1\right )}{b} - \frac {\log \left (e^{\left (-b x - a\right )} - 1\right )}{b}\right )} - \frac {2 \, {\left (b x \log \left (e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (b x + a\right )}\right )\right )} c d}{b^{2}} + \frac {2 \, {\left (b x \log \left (-e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (b x + a\right )}\right )\right )} c d}{b^{2}} - \frac {{\left (b^{2} x^{2} \log \left (e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (b x + a\right )})\right )} d^{2}}{b^{3}} + \frac {{\left (b^{2} x^{2} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (b x + a\right )})\right )} d^{2}}{b^{3}} \]
[In]
[Out]
\[ \int (c+d x)^2 \text {csch}(a+b x) \, dx=\int { {\left (d x + c\right )}^{2} \operatorname {csch}\left (b x + a\right ) \,d x } \]
[In]
[Out]
Timed out. \[ \int (c+d x)^2 \text {csch}(a+b x) \, dx=\int \frac {{\left (c+d\,x\right )}^2}{\mathrm {sinh}\left (a+b\,x\right )} \,d x \]
[In]
[Out]