Integrand size = 17, antiderivative size = 131 \[ \int \frac {\tanh ^2(x)}{\left (a+b \coth ^2(x)\right )^{5/2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{(a+b)^{5/2}}+\frac {b \tanh (x)}{3 a (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}+\frac {b (7 a+4 b) \tanh (x)}{3 a^2 (a+b)^2 \sqrt {a+b \coth ^2(x)}}-\frac {(3 a+2 b) (a+4 b) \sqrt {a+b \coth ^2(x)} \tanh (x)}{3 a^3 (a+b)^2} \]
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Time = 0.17 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {3751, 483, 593, 597, 12, 385, 212} \[ \int \frac {\tanh ^2(x)}{\left (a+b \coth ^2(x)\right )^{5/2}} \, dx=-\frac {(3 a+2 b) (a+4 b) \tanh (x) \sqrt {a+b \coth ^2(x)}}{3 a^3 (a+b)^2}+\frac {b (7 a+4 b) \tanh (x)}{3 a^2 (a+b)^2 \sqrt {a+b \coth ^2(x)}}+\frac {\text {arctanh}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{(a+b)^{5/2}}+\frac {b \tanh (x)}{3 a (a+b) \left (a+b \coth ^2(x)\right )^{3/2}} \]
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Rule 12
Rule 212
Rule 385
Rule 483
Rule 593
Rule 597
Rule 3751
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{x^2 \left (1-x^2\right ) \left (a+b x^2\right )^{5/2}} \, dx,x,\coth (x)\right ) \\ & = \frac {b \tanh (x)}{3 a (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}-\frac {\text {Subst}\left (\int \frac {-3 a-4 b+4 b x^2}{x^2 \left (1-x^2\right ) \left (a+b x^2\right )^{3/2}} \, dx,x,\coth (x)\right )}{3 a (a+b)} \\ & = \frac {b \tanh (x)}{3 a (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}+\frac {b (7 a+4 b) \tanh (x)}{3 a^2 (a+b)^2 \sqrt {a+b \coth ^2(x)}}+\frac {\text {Subst}\left (\int \frac {(3 a+2 b) (a+4 b)-2 b (7 a+4 b) x^2}{x^2 \left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\coth (x)\right )}{3 a^2 (a+b)^2} \\ & = \frac {b \tanh (x)}{3 a (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}+\frac {b (7 a+4 b) \tanh (x)}{3 a^2 (a+b)^2 \sqrt {a+b \coth ^2(x)}}-\frac {(3 a+2 b) (a+4 b) \sqrt {a+b \coth ^2(x)} \tanh (x)}{3 a^3 (a+b)^2}-\frac {\text {Subst}\left (\int -\frac {3 a^3}{\left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\coth (x)\right )}{3 a^3 (a+b)^2} \\ & = \frac {b \tanh (x)}{3 a (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}+\frac {b (7 a+4 b) \tanh (x)}{3 a^2 (a+b)^2 \sqrt {a+b \coth ^2(x)}}-\frac {(3 a+2 b) (a+4 b) \sqrt {a+b \coth ^2(x)} \tanh (x)}{3 a^3 (a+b)^2}+\frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\coth (x)\right )}{(a+b)^2} \\ & = \frac {b \tanh (x)}{3 a (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}+\frac {b (7 a+4 b) \tanh (x)}{3 a^2 (a+b)^2 \sqrt {a+b \coth ^2(x)}}-\frac {(3 a+2 b) (a+4 b) \sqrt {a+b \coth ^2(x)} \tanh (x)}{3 a^3 (a+b)^2}+\frac {\text {Subst}\left (\int \frac {1}{1-(a+b) x^2} \, dx,x,\frac {\coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{(a+b)^2} \\ & = \frac {\text {arctanh}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{(a+b)^{5/2}}+\frac {b \tanh (x)}{3 a (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}+\frac {b (7 a+4 b) \tanh (x)}{3 a^2 (a+b)^2 \sqrt {a+b \coth ^2(x)}}-\frac {(3 a+2 b) (a+4 b) \sqrt {a+b \coth ^2(x)} \tanh (x)}{3 a^3 (a+b)^2} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 8.54 (sec) , antiderivative size = 1350, normalized size of antiderivative = 10.31 \[ \int \frac {\tanh ^2(x)}{\left (a+b \coth ^2(x)\right )^{5/2}} \, dx=\frac {\sinh ^2(x) \left (\frac {16 b^3 \left (-i \coth (x)+i \coth ^3(x)\right )^2}{a (a+b)^2}+\frac {40 b \text {csch}^2(x)}{a+b}+\frac {160 b^2 \coth ^2(x) \text {csch}^2(x)}{3 a (a+b)}+\frac {64 b^3 \coth ^4(x) \text {csch}^2(x)}{3 a^2 (a+b)}-\frac {40 b^2 \text {csch}^4(x)}{(a+b)^2}+\frac {92 (a+b) \cosh ^2(x) \operatorname {Hypergeometric2F1}\left (2,2,\frac {9}{2},\frac {(a+b) \cosh ^2(x)}{a}\right )}{105 a}+\frac {124 b (a+b) \cosh ^2(x) \coth ^2(x) \operatorname {Hypergeometric2F1}\left (2,2,\frac {9}{2},\frac {(a+b) \cosh ^2(x)}{a}\right )}{35 a^2}+\frac {152 b^2 (a+b) \cosh ^2(x) \coth ^4(x) \operatorname {Hypergeometric2F1}\left (2,2,\frac {9}{2},\frac {(a+b) \cosh ^2(x)}{a}\right )}{35 a^3}+\frac {176 b^3 (a+b) \cosh ^2(x) \coth ^6(x) \operatorname {Hypergeometric2F1}\left (2,2,\frac {9}{2},\frac {(a+b) \cosh ^2(x)}{a}\right )}{105 a^4}+\frac {24 (a+b) \cosh ^2(x) \, _3F_2\left (2,2,2;1,\frac {9}{2};\frac {(a+b) \cosh ^2(x)}{a}\right )}{35 a}+\frac {16 b (a+b) \cosh ^2(x) \coth ^2(x) \, _3F_2\left (2,2,2;1,\frac {9}{2};\frac {(a+b) \cosh ^2(x)}{a}\right )}{7 a^2}+\frac {88 b^2 (a+b) \cosh ^2(x) \coth ^4(x) \, _3F_2\left (2,2,2;1,\frac {9}{2};\frac {(a+b) \cosh ^2(x)}{a}\right )}{35 a^3}+\frac {32 b^3 (a+b) \cosh ^2(x) \coth ^6(x) \, _3F_2\left (2,2,2;1,\frac {9}{2};\frac {(a+b) \cosh ^2(x)}{a}\right )}{35 a^4}+\frac {16 (a+b) \cosh ^2(x) \, _4F_3\left (2,2,2,2;1,1,\frac {9}{2};\frac {(a+b) \cosh ^2(x)}{a}\right )}{105 a}+\frac {16 b (a+b) \cosh ^2(x) \coth ^2(x) \, _4F_3\left (2,2,2,2;1,1,\frac {9}{2};\frac {(a+b) \cosh ^2(x)}{a}\right )}{35 a^2}+\frac {16 b^2 (a+b) \cosh ^2(x) \coth ^4(x) \, _4F_3\left (2,2,2,2;1,1,\frac {9}{2};\frac {(a+b) \cosh ^2(x)}{a}\right )}{35 a^3}+\frac {16 b^3 (a+b) \cosh ^2(x) \coth ^6(x) \, _4F_3\left (2,2,2,2;1,1,\frac {9}{2};\frac {(a+b) \cosh ^2(x)}{a}\right )}{105 a^4}+\frac {20 a \text {sech}^2(x)}{3 (a+b)}-\frac {30 a b \text {csch}^2(x) \text {sech}^2(x)}{(a+b)^2}-\frac {5 a^2 \text {sech}^4(x)}{(a+b)^2}+\frac {5 \arcsin \left (\sqrt {\frac {(a+b) \cosh ^2(x)}{a}}\right )}{\left (\frac {(a+b) \cosh ^2(x)}{a}\right )^{5/2} \sqrt {-\frac {\left (a+b \coth ^2(x)\right ) \sinh ^2(x)}{a}}}+\frac {30 b \arcsin \left (\sqrt {\frac {(a+b) \cosh ^2(x)}{a}}\right ) \coth ^2(x)}{a \left (\frac {(a+b) \cosh ^2(x)}{a}\right )^{5/2} \sqrt {-\frac {\left (a+b \coth ^2(x)\right ) \sinh ^2(x)}{a}}}+\frac {40 b^2 \arcsin \left (\sqrt {\frac {(a+b) \cosh ^2(x)}{a}}\right ) \coth ^4(x)}{a^2 \left (\frac {(a+b) \cosh ^2(x)}{a}\right )^{5/2} \sqrt {-\frac {\left (a+b \coth ^2(x)\right ) \sinh ^2(x)}{a}}}+\frac {16 b^3 \arcsin \left (\sqrt {\frac {(a+b) \cosh ^2(x)}{a}}\right ) \coth ^6(x)}{a^3 \left (\frac {(a+b) \cosh ^2(x)}{a}\right )^{5/2} \sqrt {-\frac {\left (a+b \coth ^2(x)\right ) \sinh ^2(x)}{a}}}+\frac {5 \arcsin \left (\sqrt {\frac {(a+b) \cosh ^2(x)}{a}}\right )}{\sqrt {-\frac {(a+b) \cosh ^2(x) \left (a+b \coth ^2(x)\right ) \sinh ^2(x)}{a^2}}}+\frac {30 b \arcsin \left (\sqrt {\frac {(a+b) \cosh ^2(x)}{a}}\right ) \coth ^2(x)}{a \sqrt {-\frac {(a+b) \cosh ^2(x) \left (a+b \coth ^2(x)\right ) \sinh ^2(x)}{a^2}}}+\frac {40 b^2 \arcsin \left (\sqrt {\frac {(a+b) \cosh ^2(x)}{a}}\right ) \coth ^4(x)}{a^2 \sqrt {-\frac {(a+b) \cosh ^2(x) \left (a+b \coth ^2(x)\right ) \sinh ^2(x)}{a^2}}}+\frac {16 b^3 \arcsin \left (\sqrt {\frac {(a+b) \cosh ^2(x)}{a}}\right ) \coth ^6(x)}{a^3 \sqrt {-\frac {(a+b) \cosh ^2(x) \left (a+b \coth ^2(x)\right ) \sinh ^2(x)}{a^2}}}-\frac {60 b \arcsin \left (\sqrt {\frac {(a+b) \cosh ^2(x)}{a}}\right ) \text {csch}^2(x)}{(a+b) \sqrt {-\frac {(a+b) \cosh ^2(x) \left (a+b \coth ^2(x)\right ) \sinh ^2(x)}{a^2}}}-\frac {80 b^2 \arcsin \left (\sqrt {\frac {(a+b) \cosh ^2(x)}{a}}\right ) \coth ^2(x) \text {csch}^2(x)}{a (a+b) \sqrt {-\frac {(a+b) \cosh ^2(x) \left (a+b \coth ^2(x)\right ) \sinh ^2(x)}{a^2}}}-\frac {32 b^3 \arcsin \left (\sqrt {\frac {(a+b) \cosh ^2(x)}{a}}\right ) \coth ^4(x) \text {csch}^2(x)}{a^2 (a+b) \sqrt {-\frac {(a+b) \cosh ^2(x) \left (a+b \coth ^2(x)\right ) \sinh ^2(x)}{a^2}}}-\frac {10 a \arcsin \left (\sqrt {\frac {(a+b) \cosh ^2(x)}{a}}\right ) \text {sech}^2(x)}{(a+b) \sqrt {-\frac {(a+b) \cosh ^2(x) \left (a+b \coth ^2(x)\right ) \sinh ^2(x)}{a^2}}}\right ) \tanh (x)}{a^2 \sqrt {a+b \coth ^2(x)} \left (1+\frac {b \coth ^2(x)}{a}\right )} \]
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\[\int \frac {\tanh \left (x \right )^{2}}{\left (a +b \coth \left (x \right )^{2}\right )^{\frac {5}{2}}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 5085 vs. \(2 (113) = 226\).
Time = 1.30 (sec) , antiderivative size = 10729, normalized size of antiderivative = 81.90 \[ \int \frac {\tanh ^2(x)}{\left (a+b \coth ^2(x)\right )^{5/2}} \, dx=\text {Too large to display} \]
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\[ \int \frac {\tanh ^2(x)}{\left (a+b \coth ^2(x)\right )^{5/2}} \, dx=\int \frac {\tanh ^{2}{\left (x \right )}}{\left (a + b \coth ^{2}{\left (x \right )}\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {\tanh ^2(x)}{\left (a+b \coth ^2(x)\right )^{5/2}} \, dx=\int { \frac {\tanh \left (x\right )^{2}}{{\left (b \coth \left (x\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 1133 vs. \(2 (113) = 226\).
Time = 0.77 (sec) , antiderivative size = 1133, normalized size of antiderivative = 8.65 \[ \int \frac {\tanh ^2(x)}{\left (a+b \coth ^2(x)\right )^{5/2}} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\tanh ^2(x)}{\left (a+b \coth ^2(x)\right )^{5/2}} \, dx=\int \frac {{\mathrm {tanh}\left (x\right )}^2}{{\left (b\,{\mathrm {coth}\left (x\right )}^2+a\right )}^{5/2}} \,d x \]
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