Integrand size = 23, antiderivative size = 75 \[ \int \frac {\sinh ^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {a \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \text {sech}(c+d x)}{\sqrt {a+b}}\right )}{(a+b)^{5/2} d}-\frac {a \cosh (c+d x)}{(a+b)^2 d}+\frac {\cosh ^3(c+d x)}{3 (a+b) d} \]
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Time = 0.10 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3745, 464, 331, 214} \[ \int \frac {\sinh ^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {a \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \text {sech}(c+d x)}{\sqrt {a+b}}\right )}{d (a+b)^{5/2}}+\frac {\cosh ^3(c+d x)}{3 d (a+b)}-\frac {a \cosh (c+d x)}{d (a+b)^2} \]
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Rule 214
Rule 331
Rule 464
Rule 3745
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {-1+x^2}{x^4 \left (a+b-b x^2\right )} \, dx,x,\text {sech}(c+d x)\right )}{d} \\ & = \frac {\cosh ^3(c+d x)}{3 (a+b) d}+\frac {a \text {Subst}\left (\int \frac {1}{x^2 \left (a+b-b x^2\right )} \, dx,x,\text {sech}(c+d x)\right )}{(a+b) d} \\ & = -\frac {a \cosh (c+d x)}{(a+b)^2 d}+\frac {\cosh ^3(c+d x)}{3 (a+b) d}+\frac {(a b) \text {Subst}\left (\int \frac {1}{a+b-b x^2} \, dx,x,\text {sech}(c+d x)\right )}{(a+b)^2 d} \\ & = \frac {a \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \text {sech}(c+d x)}{\sqrt {a+b}}\right )}{(a+b)^{5/2} d}-\frac {a \cosh (c+d x)}{(a+b)^2 d}+\frac {\cosh ^3(c+d x)}{3 (a+b) d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.23 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.80 \[ \int \frac {\sinh ^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {12 i a \sqrt {b} \left (\arctan \left (\frac {-i \sqrt {a+b}-\sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b}}\right )+\arctan \left (\frac {-i \sqrt {a+b}+\sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b}}\right )\right )-3 (3 a-b) \sqrt {a+b} \cosh (c+d x)+(a+b)^{3/2} \cosh (3 (c+d x))}{12 (a+b)^{5/2} d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(201\) vs. \(2(65)=130\).
Time = 3.90 (sec) , antiderivative size = 202, normalized size of antiderivative = 2.69
method | result | size |
derivativedivides | \(\frac {-\frac {8}{\left (16 a +16 b \right ) \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {16}{3 \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3} \left (16 a +16 b \right )}-\frac {a -b}{2 \left (a +b \right )^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {a b \,\operatorname {arctanh}\left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 a +4 b}{4 \sqrt {a b +b^{2}}}\right )}{\left (a +b \right )^{2} \sqrt {a b +b^{2}}}-\frac {16}{3 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (16 a +16 b \right )}-\frac {8}{\left (16 a +16 b \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-a +b}{2 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\) | \(202\) |
default | \(\frac {-\frac {8}{\left (16 a +16 b \right ) \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {16}{3 \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3} \left (16 a +16 b \right )}-\frac {a -b}{2 \left (a +b \right )^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {a b \,\operatorname {arctanh}\left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 a +4 b}{4 \sqrt {a b +b^{2}}}\right )}{\left (a +b \right )^{2} \sqrt {a b +b^{2}}}-\frac {16}{3 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (16 a +16 b \right )}-\frac {8}{\left (16 a +16 b \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-a +b}{2 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\) | \(202\) |
risch | \(\frac {{\mathrm e}^{3 d x +3 c}}{24 d \left (a +b \right )}-\frac {3 \,{\mathrm e}^{d x +c} a}{8 \left (a +b \right )^{2} d}+\frac {{\mathrm e}^{d x +c} b}{8 \left (a +b \right )^{2} d}-\frac {3 \,{\mathrm e}^{-d x -c} a}{8 \left (a +b \right )^{2} d}+\frac {{\mathrm e}^{-d x -c} b}{8 \left (a +b \right )^{2} d}+\frac {{\mathrm e}^{-3 d x -3 c}}{24 d \left (a +b \right )}+\frac {\sqrt {\left (a +b \right ) b}\, a \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {\left (a +b \right ) b}\, {\mathrm e}^{d x +c}}{a +b}+1\right )}{2 \left (a +b \right )^{3} d}-\frac {\sqrt {\left (a +b \right ) b}\, a \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {\left (a +b \right ) b}\, {\mathrm e}^{d x +c}}{a +b}+1\right )}{2 \left (a +b \right )^{3} d}\) | \(214\) |
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Leaf count of result is larger than twice the leaf count of optimal. 616 vs. \(2 (65) = 130\).
Time = 0.28 (sec) , antiderivative size = 1367, normalized size of antiderivative = 18.23 \[ \int \frac {\sinh ^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\text {Too large to display} \]
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\[ \int \frac {\sinh ^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\int \frac {\sinh ^{3}{\left (c + d x \right )}}{a + b \tanh ^{2}{\left (c + d x \right )}}\, dx \]
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\[ \int \frac {\sinh ^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\int { \frac {\sinh \left (d x + c\right )^{3}}{b \tanh \left (d x + c\right )^{2} + a} \,d x } \]
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\[ \int \frac {\sinh ^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\int { \frac {\sinh \left (d x + c\right )^{3}}{b \tanh \left (d x + c\right )^{2} + a} \,d x } \]
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Time = 3.34 (sec) , antiderivative size = 955, normalized size of antiderivative = 12.73 \[ \int \frac {\sinh ^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {{\mathrm {e}}^{-3\,c-3\,d\,x}}{24\,d\,\left (a+b\right )}+\frac {{\mathrm {e}}^{3\,c+3\,d\,x}}{24\,d\,\left (a+b\right )}-\frac {\sqrt {a^2\,b}\,\left (2\,\mathrm {atan}\left (\frac {\left ({\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (\frac {4\,\left (2\,a^2\,b^3\,d\,\sqrt {a^2\,b}+4\,a^3\,b^2\,d\,\sqrt {a^2\,b}+2\,a^4\,b\,d\,\sqrt {a^2\,b}\right )}{a\,\left (a+b\right )\,\sqrt {-d^2\,{\left (a+b\right )}^5}\,\left (a^2+2\,a\,b+b^2\right )\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )\,\sqrt {-a^5\,d^2-5\,a^4\,b\,d^2-10\,a^3\,b^2\,d^2-10\,a^2\,b^3\,d^2-5\,a\,b^4\,d^2-b^5\,d^2}}+\frac {2\,a^3\,b}{d\,{\left (a+b\right )}^3\,\sqrt {a^2\,b}\,\left (a^2+2\,a\,b+b^2\right )\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )}\right )+\frac {2\,a^3\,b\,{\mathrm {e}}^{3\,c}\,{\mathrm {e}}^{3\,d\,x}}{d\,{\left (a+b\right )}^3\,\sqrt {a^2\,b}\,\left (a^2+2\,a\,b+b^2\right )\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )}\right )\,\left (a^6\,\sqrt {-a^5\,d^2-5\,a^4\,b\,d^2-10\,a^3\,b^2\,d^2-10\,a^2\,b^3\,d^2-5\,a\,b^4\,d^2-b^5\,d^2}+b^6\,\sqrt {-a^5\,d^2-5\,a^4\,b\,d^2-10\,a^3\,b^2\,d^2-10\,a^2\,b^3\,d^2-5\,a\,b^4\,d^2-b^5\,d^2}+15\,a^2\,b^4\,\sqrt {-a^5\,d^2-5\,a^4\,b\,d^2-10\,a^3\,b^2\,d^2-10\,a^2\,b^3\,d^2-5\,a\,b^4\,d^2-b^5\,d^2}+20\,a^3\,b^3\,\sqrt {-a^5\,d^2-5\,a^4\,b\,d^2-10\,a^3\,b^2\,d^2-10\,a^2\,b^3\,d^2-5\,a\,b^4\,d^2-b^5\,d^2}+15\,a^4\,b^2\,\sqrt {-a^5\,d^2-5\,a^4\,b\,d^2-10\,a^3\,b^2\,d^2-10\,a^2\,b^3\,d^2-5\,a\,b^4\,d^2-b^5\,d^2}+6\,a\,b^5\,\sqrt {-a^5\,d^2-5\,a^4\,b\,d^2-10\,a^3\,b^2\,d^2-10\,a^2\,b^3\,d^2-5\,a\,b^4\,d^2-b^5\,d^2}+6\,a^5\,b\,\sqrt {-a^5\,d^2-5\,a^4\,b\,d^2-10\,a^3\,b^2\,d^2-10\,a^2\,b^3\,d^2-5\,a\,b^4\,d^2-b^5\,d^2}\right )}{4\,a^2\,b}\right )-2\,\mathrm {atan}\left (\frac {a\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-d^2\,{\left (a+b\right )}^5}}{2\,d\,{\left (a+b\right )}^2\,\sqrt {a^2\,b}}\right )\right )}{2\,\sqrt {-a^5\,d^2-5\,a^4\,b\,d^2-10\,a^3\,b^2\,d^2-10\,a^2\,b^3\,d^2-5\,a\,b^4\,d^2-b^5\,d^2}}-\frac {{\mathrm {e}}^{-c-d\,x}\,\left (3\,a-b\right )}{8\,d\,{\left (a+b\right )}^2}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (3\,a-b\right )}{8\,d\,{\left (a+b\right )}^2} \]
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