Integrand size = 22, antiderivative size = 118 \[ \int \frac {1}{\sqrt {x} \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\frac {2 \sqrt {x}}{a^2}+\frac {4 b \left (2 a^2+b^2\right ) \text {arctanh}\left (\frac {a-b \tanh \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right )}{\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {2 b^2 \coth \left (c+d \sqrt {x}\right )}{a \left (a^2+b^2\right ) d \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )} \]
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Time = 0.15 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {5545, 3870, 4004, 3916, 2739, 632, 210} \[ \int \frac {1}{\sqrt {x} \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\frac {4 b \left (2 a^2+b^2\right ) \text {arctanh}\left (\frac {a-b \tanh \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right )}{\sqrt {a^2+b^2}}\right )}{a^2 d \left (a^2+b^2\right )^{3/2}}-\frac {2 b^2 \coth \left (c+d \sqrt {x}\right )}{a d \left (a^2+b^2\right ) \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )}+\frac {2 \sqrt {x}}{a^2} \]
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Rule 210
Rule 632
Rule 2739
Rule 3870
Rule 3916
Rule 4004
Rule 5545
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {1}{(a+b \text {csch}(c+d x))^2} \, dx,x,\sqrt {x}\right ) \\ & = -\frac {2 b^2 \coth \left (c+d \sqrt {x}\right )}{a \left (a^2+b^2\right ) d \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )}-\frac {2 \text {Subst}\left (\int \frac {-a^2-b^2+a b \text {csch}(c+d x)}{a+b \text {csch}(c+d x)} \, dx,x,\sqrt {x}\right )}{a \left (a^2+b^2\right )} \\ & = \frac {2 \sqrt {x}}{a^2}-\frac {2 b^2 \coth \left (c+d \sqrt {x}\right )}{a \left (a^2+b^2\right ) d \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )}-\frac {\left (2 b \left (2 a^2+b^2\right )\right ) \text {Subst}\left (\int \frac {\text {csch}(c+d x)}{a+b \text {csch}(c+d x)} \, dx,x,\sqrt {x}\right )}{a^2 \left (a^2+b^2\right )} \\ & = \frac {2 \sqrt {x}}{a^2}-\frac {2 b^2 \coth \left (c+d \sqrt {x}\right )}{a \left (a^2+b^2\right ) d \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )}-\frac {\left (2 \left (2 a^2+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{1+\frac {a \sinh (c+d x)}{b}} \, dx,x,\sqrt {x}\right )}{a^2 \left (a^2+b^2\right )} \\ & = \frac {2 \sqrt {x}}{a^2}-\frac {2 b^2 \coth \left (c+d \sqrt {x}\right )}{a \left (a^2+b^2\right ) d \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )}+\frac {\left (4 i \left (2 a^2+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-\frac {2 i a x}{b}+x^2} \, dx,x,i \tanh \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right )\right )}{a^2 \left (a^2+b^2\right ) d} \\ & = \frac {2 \sqrt {x}}{a^2}-\frac {2 b^2 \coth \left (c+d \sqrt {x}\right )}{a \left (a^2+b^2\right ) d \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )}-\frac {\left (8 i \left (2 a^2+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{-4 \left (1+\frac {a^2}{b^2}\right )-x^2} \, dx,x,-\frac {2 i a}{b}+2 i \tanh \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right )\right )}{a^2 \left (a^2+b^2\right ) d} \\ & = \frac {2 \sqrt {x}}{a^2}+\frac {4 b \left (2 a^2+b^2\right ) \text {arctanh}\left (\frac {b \left (\frac {a}{b}-\tanh \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right )\right )}{\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {2 b^2 \coth \left (c+d \sqrt {x}\right )}{a \left (a^2+b^2\right ) d \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )} \\ \end{align*}
Time = 0.36 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.48 \[ \int \frac {1}{\sqrt {x} \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\frac {2 \text {csch}\left (c+d \sqrt {x}\right ) \left (-\frac {a b^2 \coth \left (c+d \sqrt {x}\right )}{a^2+b^2}+\left (c+d \sqrt {x}\right ) \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )+\frac {2 b \left (2 a^2+b^2\right ) \arctan \left (\frac {a-b \tanh \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right )}{\sqrt {-a^2-b^2}}\right ) \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )}{\left (-a^2-b^2\right )^{3/2}}\right ) \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )}{a^2 d \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \]
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Time = 0.35 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.60
method | result | size |
derivativedivides | \(\frac {\frac {2 \ln \left (\tanh \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )+1\right )}{a^{2}}-\frac {4 b \left (\frac {\frac {a^{2} \tanh \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )}{2 a^{2}+2 b^{2}}+\frac {a b}{2 a^{2}+2 b^{2}}}{-\frac {\tanh \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )^{2} b}{2}+a \tanh \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )+\frac {b}{2}}-\frac {2 \left (2 a^{2}+b^{2}\right ) \operatorname {arctanh}\left (\frac {-2 b \tanh \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (2 a^{2}+2 b^{2}\right ) \sqrt {a^{2}+b^{2}}}\right )}{a^{2}}-\frac {2 \ln \left (\tanh \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )-1\right )}{a^{2}}}{d}\) | \(189\) |
default | \(\frac {\frac {2 \ln \left (\tanh \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )+1\right )}{a^{2}}-\frac {4 b \left (\frac {\frac {a^{2} \tanh \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )}{2 a^{2}+2 b^{2}}+\frac {a b}{2 a^{2}+2 b^{2}}}{-\frac {\tanh \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )^{2} b}{2}+a \tanh \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )+\frac {b}{2}}-\frac {2 \left (2 a^{2}+b^{2}\right ) \operatorname {arctanh}\left (\frac {-2 b \tanh \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (2 a^{2}+2 b^{2}\right ) \sqrt {a^{2}+b^{2}}}\right )}{a^{2}}-\frac {2 \ln \left (\tanh \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )-1\right )}{a^{2}}}{d}\) | \(189\) |
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Leaf count of result is larger than twice the leaf count of optimal. 670 vs. \(2 (107) = 214\).
Time = 0.32 (sec) , antiderivative size = 670, normalized size of antiderivative = 5.68 \[ \int \frac {1}{\sqrt {x} \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx=-\frac {2 \, {\left (2 \, a^{3} b^{2} + 2 \, a b^{4} - {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d \sqrt {x} \cosh \left (d \sqrt {x} + c\right )^{2} - {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d \sqrt {x} \sinh \left (d \sqrt {x} + c\right )^{2} + {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d \sqrt {x} - 2 \, {\left (a^{2} b^{3} + b^{5} + {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} d \sqrt {x}\right )} \cosh \left (d \sqrt {x} + c\right ) - {\left ({\left (2 \, a^{3} b + a b^{3}\right )} \sqrt {a^{2} + b^{2}} \cosh \left (d \sqrt {x} + c\right )^{2} + {\left (2 \, a^{3} b + a b^{3}\right )} \sqrt {a^{2} + b^{2}} \sinh \left (d \sqrt {x} + c\right )^{2} + 2 \, {\left (2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} + b^{2}} \cosh \left (d \sqrt {x} + c\right ) + 2 \, {\left ({\left (2 \, a^{3} b + a b^{3}\right )} \sqrt {a^{2} + b^{2}} \cosh \left (d \sqrt {x} + c\right ) + {\left (2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} + b^{2}}\right )} \sinh \left (d \sqrt {x} + c\right ) - {\left (2 \, a^{3} b + a b^{3}\right )} \sqrt {a^{2} + b^{2}}\right )} \log \left (\frac {a b + {\left (a^{2} + b^{2} + \sqrt {a^{2} + b^{2}} b\right )} \cosh \left (d \sqrt {x} + c\right ) - {\left (b^{2} + \sqrt {a^{2} + b^{2}} b\right )} \sinh \left (d \sqrt {x} + c\right ) + \sqrt {a^{2} + b^{2}} a}{a \sinh \left (d \sqrt {x} + c\right ) + b}\right ) - 2 \, {\left (a^{2} b^{3} + b^{5} + {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d \sqrt {x} \cosh \left (d \sqrt {x} + c\right ) + {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} d \sqrt {x}\right )} \sinh \left (d \sqrt {x} + c\right )\right )}}{{\left (a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d \cosh \left (d \sqrt {x} + c\right )^{2} + {\left (a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d \sinh \left (d \sqrt {x} + c\right )^{2} + 2 \, {\left (a^{6} b + 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d \cosh \left (d \sqrt {x} + c\right ) - {\left (a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d + 2 \, {\left ({\left (a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d \cosh \left (d \sqrt {x} + c\right ) + {\left (a^{6} b + 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d\right )} \sinh \left (d \sqrt {x} + c\right )} \]
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\[ \int \frac {1}{\sqrt {x} \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\int \frac {1}{\sqrt {x} \left (a + b \operatorname {csch}{\left (c + d \sqrt {x} \right )}\right )^{2}}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.69 \[ \int \frac {1}{\sqrt {x} \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx=-\frac {2 \, {\left (2 \, a^{2} b + b^{3}\right )} \log \left (\frac {a e^{\left (-d \sqrt {x} - c\right )} - b - \sqrt {a^{2} + b^{2}}}{a e^{\left (-d \sqrt {x} - c\right )} - b + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{4} + a^{2} b^{2}\right )} \sqrt {a^{2} + b^{2}} d} - \frac {4 \, {\left (b^{3} e^{\left (-d \sqrt {x} - c\right )} + a b^{2}\right )}}{{\left (a^{5} + a^{3} b^{2} + 2 \, {\left (a^{4} b + a^{2} b^{3}\right )} e^{\left (-d \sqrt {x} - c\right )} - {\left (a^{5} + a^{3} b^{2}\right )} e^{\left (-2 \, d \sqrt {x} - 2 \, c\right )}\right )} d} + \frac {2 \, {\left (d \sqrt {x} + c\right )}}{a^{2} d} \]
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Time = 0.35 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.51 \[ \int \frac {1}{\sqrt {x} \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx=-\frac {2 \, {\left (2 \, a^{2} b + b^{3}\right )} \log \left (\frac {{\left | 2 \, a e^{\left (d \sqrt {x} + c\right )} + 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a e^{\left (d \sqrt {x} + c\right )} + 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{4} d + a^{2} b^{2} d\right )} \sqrt {a^{2} + b^{2}}} + \frac {4 \, {\left (b^{3} e^{\left (d \sqrt {x} + c\right )} - a b^{2}\right )}}{{\left (a^{4} d + a^{2} b^{2} d\right )} {\left (a e^{\left (2 \, d \sqrt {x} + 2 \, c\right )} + 2 \, b e^{\left (d \sqrt {x} + c\right )} - a\right )}} + \frac {2 \, {\left (d \sqrt {x} + c\right )}}{a^{2} d} \]
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Time = 2.82 (sec) , antiderivative size = 319, normalized size of antiderivative = 2.70 \[ \int \frac {1}{\sqrt {x} \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\frac {2\,\sqrt {x}}{a^2}-\frac {\frac {4\,b^2\,\sqrt {x}}{d\,\left (a^3\,\sqrt {x}+a\,b^2\,\sqrt {x}\right )}-\frac {4\,b^3\,\sqrt {x}\,{\mathrm {e}}^{c+d\,\sqrt {x}}}{a\,d\,\left (a^3\,\sqrt {x}+a\,b^2\,\sqrt {x}\right )}}{2\,b\,{\mathrm {e}}^{c+d\,\sqrt {x}}-a+a\,{\mathrm {e}}^{2\,c+2\,d\,\sqrt {x}}}-\frac {2\,b\,\ln \left (\frac {2\,{\mathrm {e}}^{c+d\,\sqrt {x}}\,\left (2\,a^2\,b+b^3\right )}{a^3\,\sqrt {x}\,\left (a^2+b^2\right )}-\frac {2\,b\,\left (2\,a^2+b^2\right )\,\left (a-b\,{\mathrm {e}}^{c+d\,\sqrt {x}}\right )}{a^3\,\sqrt {x}\,{\left (a^2+b^2\right )}^{3/2}}\right )\,\left (2\,a^2+b^2\right )}{a^2\,d\,{\left (a^2+b^2\right )}^{3/2}}+\frac {2\,b\,\ln \left (\frac {2\,{\mathrm {e}}^{c+d\,\sqrt {x}}\,\left (2\,a^2\,b+b^3\right )}{a^3\,\sqrt {x}\,\left (a^2+b^2\right )}+\frac {2\,b\,\left (2\,a^2+b^2\right )\,\left (a-b\,{\mathrm {e}}^{c+d\,\sqrt {x}}\right )}{a^3\,\sqrt {x}\,{\left (a^2+b^2\right )}^{3/2}}\right )\,\left (2\,a^2+b^2\right )}{a^2\,d\,{\left (a^2+b^2\right )}^{3/2}} \]
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