Integrand size = 16, antiderivative size = 193 \[ \int \frac {1}{\left (a+b \text {csch}^2(c+d x)\right )^{7/2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a-b+b \coth ^2(c+d x)}}\right )}{a^{7/2} d}+\frac {b \coth (c+d x)}{5 a (a-b) d \left (a-b+b \coth ^2(c+d x)\right )^{5/2}}+\frac {(9 a-5 b) b \coth (c+d x)}{15 a^2 (a-b)^2 d \left (a-b+b \coth ^2(c+d x)\right )^{3/2}}+\frac {b \left (33 a^2-40 a b+15 b^2\right ) \coth (c+d x)}{15 a^3 (a-b)^3 d \sqrt {a-b+b \coth ^2(c+d x)}} \]
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Time = 0.14 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4213, 425, 541, 12, 385, 212} \[ \int \frac {1}{\left (a+b \text {csch}^2(c+d x)\right )^{7/2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a+b \coth ^2(c+d x)-b}}\right )}{a^{7/2} d}+\frac {b (9 a-5 b) \coth (c+d x)}{15 a^2 d (a-b)^2 \left (a+b \coth ^2(c+d x)-b\right )^{3/2}}+\frac {b \left (33 a^2-40 a b+15 b^2\right ) \coth (c+d x)}{15 a^3 d (a-b)^3 \sqrt {a+b \coth ^2(c+d x)-b}}+\frac {b \coth (c+d x)}{5 a d (a-b) \left (a+b \coth ^2(c+d x)-b\right )^{5/2}} \]
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Rule 12
Rule 212
Rule 385
Rule 425
Rule 541
Rule 4213
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \left (a-b+b x^2\right )^{7/2}} \, dx,x,\coth (c+d x)\right )}{d} \\ & = \frac {b \coth (c+d x)}{5 a (a-b) d \left (a-b+b \coth ^2(c+d x)\right )^{5/2}}-\frac {\text {Subst}\left (\int \frac {-5 a+b+4 b x^2}{\left (1-x^2\right ) \left (a-b+b x^2\right )^{5/2}} \, dx,x,\coth (c+d x)\right )}{5 a (a-b) d} \\ & = \frac {b \coth (c+d x)}{5 a (a-b) d \left (a-b+b \coth ^2(c+d x)\right )^{5/2}}+\frac {(9 a-5 b) b \coth (c+d x)}{15 a^2 (a-b)^2 d \left (a-b+b \coth ^2(c+d x)\right )^{3/2}}+\frac {\text {Subst}\left (\int \frac {15 a^2-12 a b+5 b^2-2 (9 a-5 b) b x^2}{\left (1-x^2\right ) \left (a-b+b x^2\right )^{3/2}} \, dx,x,\coth (c+d x)\right )}{15 a^2 (a-b)^2 d} \\ & = \frac {b \coth (c+d x)}{5 a (a-b) d \left (a-b+b \coth ^2(c+d x)\right )^{5/2}}+\frac {(9 a-5 b) b \coth (c+d x)}{15 a^2 (a-b)^2 d \left (a-b+b \coth ^2(c+d x)\right )^{3/2}}+\frac {b \left (33 a^2-40 a b+15 b^2\right ) \coth (c+d x)}{15 a^3 (a-b)^3 d \sqrt {a-b+b \coth ^2(c+d x)}}-\frac {\text {Subst}\left (\int -\frac {15 (a-b)^3}{\left (1-x^2\right ) \sqrt {a-b+b x^2}} \, dx,x,\coth (c+d x)\right )}{15 a^3 (a-b)^3 d} \\ & = \frac {b \coth (c+d x)}{5 a (a-b) d \left (a-b+b \coth ^2(c+d x)\right )^{5/2}}+\frac {(9 a-5 b) b \coth (c+d x)}{15 a^2 (a-b)^2 d \left (a-b+b \coth ^2(c+d x)\right )^{3/2}}+\frac {b \left (33 a^2-40 a b+15 b^2\right ) \coth (c+d x)}{15 a^3 (a-b)^3 d \sqrt {a-b+b \coth ^2(c+d x)}}+\frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {a-b+b x^2}} \, dx,x,\coth (c+d x)\right )}{a^3 d} \\ & = \frac {b \coth (c+d x)}{5 a (a-b) d \left (a-b+b \coth ^2(c+d x)\right )^{5/2}}+\frac {(9 a-5 b) b \coth (c+d x)}{15 a^2 (a-b)^2 d \left (a-b+b \coth ^2(c+d x)\right )^{3/2}}+\frac {b \left (33 a^2-40 a b+15 b^2\right ) \coth (c+d x)}{15 a^3 (a-b)^3 d \sqrt {a-b+b \coth ^2(c+d x)}}+\frac {\text {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\coth (c+d x)}{\sqrt {a-b+b \coth ^2(c+d x)}}\right )}{a^3 d} \\ & = \frac {\text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a-b+b \coth ^2(c+d x)}}\right )}{a^{7/2} d}+\frac {b \coth (c+d x)}{5 a (a-b) d \left (a-b+b \coth ^2(c+d x)\right )^{5/2}}+\frac {(9 a-5 b) b \coth (c+d x)}{15 a^2 (a-b)^2 d \left (a-b+b \coth ^2(c+d x)\right )^{3/2}}+\frac {b \left (33 a^2-40 a b+15 b^2\right ) \coth (c+d x)}{15 a^3 (a-b)^3 d \sqrt {a-b+b \coth ^2(c+d x)}} \\ \end{align*}
Time = 1.09 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.21 \[ \int \frac {1}{\left (a+b \text {csch}^2(c+d x)\right )^{7/2}} \, dx=\frac {\text {csch}^7(c+d x) \left (\frac {b \cosh (c+d x) (-a+2 b+a \cosh (2 (c+d x))) \left (135 a^4-480 a^3 b+709 a^2 b^2-460 a b^3+120 b^4-4 a \left (45 a^3-135 a^2 b+117 a b^2-35 b^3\right ) \cosh (2 (c+d x))+a^2 \left (45 a^2-60 a b+23 b^2\right ) \cosh (4 (c+d x))\right )}{15 a^3 (a-b)^3}+\frac {\sqrt {2} (-a+2 b+a \cosh (2 (c+d x)))^{7/2} \log \left (\sqrt {2} \sqrt {a} \cosh (c+d x)+\sqrt {-a+2 b+a \cosh (2 (c+d x))}\right )}{a^{7/2}}\right )}{16 d \left (a+b \text {csch}^2(c+d x)\right )^{7/2}} \]
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\[\int \frac {1}{\left (a +b \operatorname {csch}\left (d x +c \right )^{2}\right )^{\frac {7}{2}}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 9002 vs. \(2 (175) = 350\).
Time = 1.80 (sec) , antiderivative size = 18681, normalized size of antiderivative = 96.79 \[ \int \frac {1}{\left (a+b \text {csch}^2(c+d x)\right )^{7/2}} \, dx=\text {Too large to display} \]
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\[ \int \frac {1}{\left (a+b \text {csch}^2(c+d x)\right )^{7/2}} \, dx=\int \frac {1}{\left (a + b \operatorname {csch}^{2}{\left (c + d x \right )}\right )^{\frac {7}{2}}}\, dx \]
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\[ \int \frac {1}{\left (a+b \text {csch}^2(c+d x)\right )^{7/2}} \, dx=\int { \frac {1}{{\left (b \operatorname {csch}\left (d x + c\right )^{2} + a\right )}^{\frac {7}{2}}} \,d x } \]
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\[ \int \frac {1}{\left (a+b \text {csch}^2(c+d x)\right )^{7/2}} \, dx=\int { \frac {1}{{\left (b \operatorname {csch}\left (d x + c\right )^{2} + a\right )}^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (a+b \text {csch}^2(c+d x)\right )^{7/2}} \, dx=\int \frac {1}{{\left (a+\frac {b}{{\mathrm {sinh}\left (c+d\,x\right )}^2}\right )}^{7/2}} \,d x \]
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