Integrand size = 23, antiderivative size = 141 \[ \int \frac {\text {csch}^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\frac {(a+4 b) \text {arctanh}(\cosh (c+d x))}{2 a^3 d}-\frac {\sqrt {b} (3 a+4 b) \text {arctanh}\left (\frac {\sqrt {b} \text {sech}(c+d x)}{\sqrt {a+b}}\right )}{2 a^3 \sqrt {a+b} d}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d \left (a+b-b \text {sech}^2(c+d x)\right )}-\frac {b \text {sech}(c+d x)}{a^2 d \left (a+b-b \text {sech}^2(c+d x)\right )} \]
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Time = 0.16 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3745, 482, 541, 536, 213, 214} \[ \int \frac {\text {csch}^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\frac {(a+4 b) \text {arctanh}(\cosh (c+d x))}{2 a^3 d}-\frac {\sqrt {b} (3 a+4 b) \text {arctanh}\left (\frac {\sqrt {b} \text {sech}(c+d x)}{\sqrt {a+b}}\right )}{2 a^3 d \sqrt {a+b}}-\frac {b \text {sech}(c+d x)}{a^2 d \left (a-b \text {sech}^2(c+d x)+b\right )}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d \left (a-b \text {sech}^2(c+d x)+b\right )} \]
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Rule 213
Rule 214
Rule 482
Rule 536
Rule 541
Rule 3745
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {x^2}{\left (-1+x^2\right )^2 \left (a+b-b x^2\right )^2} \, dx,x,\text {sech}(c+d x)\right )}{d} \\ & = -\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d \left (a+b-b \text {sech}^2(c+d x)\right )}-\frac {\text {Subst}\left (\int \frac {a+b+3 b x^2}{\left (-1+x^2\right ) \left (a+b-b x^2\right )^2} \, dx,x,\text {sech}(c+d x)\right )}{2 a d} \\ & = -\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d \left (a+b-b \text {sech}^2(c+d x)\right )}-\frac {b \text {sech}(c+d x)}{a^2 d \left (a+b-b \text {sech}^2(c+d x)\right )}-\frac {\text {Subst}\left (\int \frac {2 (a+b) (a+2 b)+4 b (a+b) x^2}{\left (-1+x^2\right ) \left (a+b-b x^2\right )} \, dx,x,\text {sech}(c+d x)\right )}{4 a^2 (a+b) d} \\ & = -\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d \left (a+b-b \text {sech}^2(c+d x)\right )}-\frac {b \text {sech}(c+d x)}{a^2 d \left (a+b-b \text {sech}^2(c+d x)\right )}-\frac {(a+4 b) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\text {sech}(c+d x)\right )}{2 a^3 d}-\frac {(b (3 a+4 b)) \text {Subst}\left (\int \frac {1}{a+b-b x^2} \, dx,x,\text {sech}(c+d x)\right )}{2 a^3 d} \\ & = \frac {(a+4 b) \text {arctanh}(\cosh (c+d x))}{2 a^3 d}-\frac {\sqrt {b} (3 a+4 b) \text {arctanh}\left (\frac {\sqrt {b} \text {sech}(c+d x)}{\sqrt {a+b}}\right )}{2 a^3 \sqrt {a+b} d}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d \left (a+b-b \text {sech}^2(c+d x)\right )}-\frac {b \text {sech}(c+d x)}{a^2 d \left (a+b-b \text {sech}^2(c+d x)\right )} \\ \end{align*}
Result contains complex when optimal does not.
Time = 5.85 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.57 \[ \int \frac {\text {csch}^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=-\frac {\frac {4 i \sqrt {b} (3 a+4 b) \arctan \left (\frac {-i \sqrt {a+b}-\sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b}}\right )}{\sqrt {a+b}}+\frac {4 i \sqrt {b} (3 a+4 b) \arctan \left (\frac {-i \sqrt {a+b}+\sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b}}\right )}{\sqrt {a+b}}+\frac {8 a b \cosh (c+d x)}{a-b+(a+b) \cosh (2 (c+d x))}+a \text {csch}^2\left (\frac {1}{2} (c+d x)\right )-4 (a+4 b) \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )+4 (a+4 b) \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )+a \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{8 a^3 d} \]
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Time = 2.45 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.33
method | result | size |
derivativedivides | \(\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 a^{2}}-\frac {1}{8 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-2 a -8 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{3}}+\frac {2 b \left (\frac {\left (-\frac {a}{2}-b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-\frac {a}{2}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a}-\frac {\left (3 a +4 b \right ) \operatorname {arctanh}\left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 a +4 b}{4 \sqrt {a b +b^{2}}}\right )}{4 \sqrt {a b +b^{2}}}\right )}{a^{3}}}{d}\) | \(187\) |
default | \(\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 a^{2}}-\frac {1}{8 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-2 a -8 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{3}}+\frac {2 b \left (\frac {\left (-\frac {a}{2}-b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-\frac {a}{2}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a}-\frac {\left (3 a +4 b \right ) \operatorname {arctanh}\left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 a +4 b}{4 \sqrt {a b +b^{2}}}\right )}{4 \sqrt {a b +b^{2}}}\right )}{a^{3}}}{d}\) | \(187\) |
risch | \(-\frac {{\mathrm e}^{d x +c} \left (a \,{\mathrm e}^{6 d x +6 c}+2 b \,{\mathrm e}^{6 d x +6 c}+3 a \,{\mathrm e}^{4 d x +4 c}-2 b \,{\mathrm e}^{4 d x +4 c}+3 \,{\mathrm e}^{2 d x +2 c} a -2 b \,{\mathrm e}^{2 d x +2 c}+a +2 b \right )}{d \,a^{2} \left (a \,{\mathrm e}^{4 d x +4 c}+b \,{\mathrm e}^{4 d x +4 c}+2 \,{\mathrm e}^{2 d x +2 c} a -2 b \,{\mathrm e}^{2 d x +2 c}+a +b \right ) \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}+\frac {\ln \left ({\mathrm e}^{d x +c}+1\right )}{2 a^{2} d}+\frac {2 \ln \left ({\mathrm e}^{d x +c}+1\right ) b}{d \,a^{3}}-\frac {\ln \left ({\mathrm e}^{d x +c}-1\right )}{2 a^{2} d}-\frac {2 \ln \left ({\mathrm e}^{d x +c}-1\right ) b}{d \,a^{3}}+\frac {3 \sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {\left (a +b \right ) b}\, {\mathrm e}^{d x +c}}{a +b}+1\right )}{4 \left (a +b \right ) d \,a^{2}}+\frac {\sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {\left (a +b \right ) b}\, {\mathrm e}^{d x +c}}{a +b}+1\right ) b}{\left (a +b \right ) d \,a^{3}}-\frac {3 \sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {\left (a +b \right ) b}\, {\mathrm e}^{d x +c}}{a +b}+1\right )}{4 \left (a +b \right ) d \,a^{2}}-\frac {\sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {\left (a +b \right ) b}\, {\mathrm e}^{d x +c}}{a +b}+1\right ) b}{\left (a +b \right ) d \,a^{3}}\) | \(435\) |
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Leaf count of result is larger than twice the leaf count of optimal. 3368 vs. \(2 (132) = 264\).
Time = 0.35 (sec) , antiderivative size = 6335, normalized size of antiderivative = 44.93 \[ \int \frac {\text {csch}^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\text {Too large to display} \]
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\[ \int \frac {\text {csch}^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\int \frac {\operatorname {csch}^{3}{\left (c + d x \right )}}{\left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{2}}\, dx \]
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\[ \int \frac {\text {csch}^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )^{3}}{{\left (b \tanh \left (d x + c\right )^{2} + a\right )}^{2}} \,d x } \]
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\[ \int \frac {\text {csch}^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )^{3}}{{\left (b \tanh \left (d x + c\right )^{2} + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\text {csch}^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\int \frac {1}{{\mathrm {sinh}\left (c+d\,x\right )}^3\,{\left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )}^2} \,d x \]
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