Integrand size = 22, antiderivative size = 127 \[ \int \frac {1}{\sqrt {x} \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx=\frac {2 \sqrt {x}}{a^2}-\frac {4 b \left (2 a^2-b^2\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right )}{\sqrt {a+b}}\right )}{a^2 (a-b)^{3/2} (a+b)^{3/2} d}+\frac {2 b^2 \tan \left (c+d \sqrt {x}\right )}{a \left (a^2-b^2\right ) d \left (a+b \sec \left (c+d \sqrt {x}\right )\right )} \]
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Time = 0.27 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4289, 3870, 4004, 3916, 2738, 214} \[ \int \frac {1}{\sqrt {x} \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx=-\frac {4 b \left (2 a^2-b^2\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right )}{\sqrt {a+b}}\right )}{a^2 d (a-b)^{3/2} (a+b)^{3/2}}+\frac {2 b^2 \tan \left (c+d \sqrt {x}\right )}{a d \left (a^2-b^2\right ) \left (a+b \sec \left (c+d \sqrt {x}\right )\right )}+\frac {2 \sqrt {x}}{a^2} \]
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Rule 214
Rule 2738
Rule 3870
Rule 3916
Rule 4004
Rule 4289
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {1}{(a+b \sec (c+d x))^2} \, dx,x,\sqrt {x}\right ) \\ & = \frac {2 b^2 \tan \left (c+d \sqrt {x}\right )}{a \left (a^2-b^2\right ) d \left (a+b \sec \left (c+d \sqrt {x}\right )\right )}-\frac {2 \text {Subst}\left (\int \frac {-a^2+b^2+a b \sec (c+d x)}{a+b \sec (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 \tan \left (c+d \sqrt {x}\right )}{a \left (a^2-b^2\right ) d \left (a+b \sec \left (c+d \sqrt {x}\right )\right )}-\frac {\left (2 b \left (2 a^2-b^2\right )\right ) \text {Subst}\left (\int \frac {\sec (c+d x)}{a+b \sec (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 \tan \left (c+d \sqrt {x}\right )}{a \left (a^2-b^2\right ) d \left (a+b \sec \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 \cos (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 \tan \left (c+d \sqrt {x}\right )}{a \left (a^2-b^2\right ) d \left (a+b \sec \left (c+d \sqrt {x}\right )\right )}-\frac {\left (4 \left (2 a^2-b^2\right )\right ) \text {Subst}\left (\int \frac {1}{1+\frac {a}{b}+\left (1-\frac {a}{b}\right ) x^2} \, dx,x,\tan \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 {\sqrt {a-b} \tan \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right )}{\sqrt {a+b}}\right )}{a^2 (a-b)^{3/2} (a+b)^{3/2} d}+\frac {2 b^2 \tan \left (c+d \sqrt {x}\right )}{a \left (a^2-b^2\right ) d \left (a+b \sec \left (c+d \sqrt {x}\right )\right )} \\ \end{align*}
Time = 1.00 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.28 \[ \int \frac {1}{\sqrt {x} \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx=\frac {2 \left (-\frac {2 b \left (-2 a^2+b^2\right ) \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+\frac {a \left (a^2-b^2\right ) \left (c+d \sqrt {x}\right ) \cos \left (c+d \sqrt {x}\right )+b \left (\left (a^2-b^2\right ) \left (c+d \sqrt {x}\right )+a b \sin \left (c+d \sqrt {x}\right )\right )}{b+a \cos \left (c+d \sqrt {x}\right )}\right )}{a^2 (a-b) (a+b) d} \]
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Time = 0.34 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.28
method | result | size |
derivativedivides | \(\frac {\frac {4 \arctan \left (\tan \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )\right )}{a^{2}}+\frac {4 b \left (-\frac {b a \tan \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (\tan \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )^{2} a -\tan \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )^{2} b -a -b \right )}-\frac {\left (2 a^{2}-b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a -b \right ) \left (a +b \right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{2}}}{d}\) | \(162\) |
default | \(\frac {\frac {4 \arctan \left (\tan \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )\right )}{a^{2}}+\frac {4 b \left (-\frac {b a \tan \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (\tan \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )^{2} a -\tan \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )^{2} b -a -b \right )}-\frac {\left (2 a^{2}-b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a -b \right ) \left (a +b \right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{2}}}{d}\) | \(162\) |
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Leaf count of result is larger than twice the leaf count of optimal. 254 vs. \(2 (110) = 220\).
Time = 0.33 (sec) , antiderivative size = 574, normalized size of antiderivative = 4.52 \[ \int \frac {1}{\sqrt {x} \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx=\left [\frac {2 \, {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d \sqrt {x} \cos \left (d \sqrt {x} + c\right ) + 2 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d \sqrt {x} + {\left ({\left (2 \, a^{3} b - a b^{3}\right )} \sqrt {a^{2} - b^{2}} \cos \left (d \sqrt {x} + c\right ) + {\left (2 \, a^{2} b^{2} - b^{4}\right )} \sqrt {a^{2} - b^{2}}\right )} \log \left (\frac {2 \, a b \cos \left (d \sqrt {x} + c\right ) - {\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d \sqrt {x} + c\right )^{2} + 2 \, a^{2} - b^{2} - 2 \, {\left (\sqrt {a^{2} - b^{2}} b \cos \left (d \sqrt {x} + c\right ) + \sqrt {a^{2} - b^{2}} a\right )} \sin \left (d \sqrt {x} + c\right )}{a^{2} \cos \left (d \sqrt {x} + c\right )^{2} + 2 \, a b \cos \left (d \sqrt {x} + c\right ) + b^{2}}\right ) + 2 \, {\left (a^{3} b^{2} - a b^{4}\right )} \sin \left (d \sqrt {x} + c\right )}{{\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d \cos \left (d \sqrt {x} + c\right ) + {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d}, \frac {2 \, {\left ({\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d \sqrt {x} \cos \left (d \sqrt {x} + c\right ) + {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d \sqrt {x} - {\left ({\left (2 \, a^{3} b - a b^{3}\right )} \sqrt {-a^{2} + b^{2}} \cos \left (d \sqrt {x} + c\right ) + {\left (2 \, a^{2} b^{2} - b^{4}\right )} \sqrt {-a^{2} + b^{2}}\right )} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} b \cos \left (d \sqrt {x} + c\right ) + \sqrt {-a^{2} + b^{2}} a}{{\left (a^{2} - b^{2}\right )} \sin \left (d \sqrt {x} + c\right )}\right ) + {\left (a^{3} b^{2} - a b^{4}\right )} \sin \left (d \sqrt {x} + c\right )\right )}}{{\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d \cos \left (d \sqrt {x} + c\right ) + {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d}\right ] \]
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\[ \int \frac {1}{\sqrt {x} \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx=\int \frac {1}{\sqrt {x} \left (a + b \sec {\left (c + d \sqrt {x} \right )}\right )^{2}}\, dx \]
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Exception generated. \[ \int \frac {1}{\sqrt {x} \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.30 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.54 \[ \int \frac {1}{\sqrt {x} \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx=-\frac {4 \, b^{2} \tan \left (\frac {1}{2} \, d \sqrt {x} + \frac {1}{2} \, c\right )}{{\left (a^{3} d - a b^{2} d\right )} {\left (a \tan \left (\frac {1}{2} \, d \sqrt {x} + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d \sqrt {x} + \frac {1}{2} \, c\right )^{2} - a - b\right )}} + \frac {4 \, {\left (2 \, a^{2} b - b^{3}\right )} {\left (\pi \left \lfloor \frac {d \sqrt {x} + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a - 2 \, b\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d \sqrt {x} + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d \sqrt {x} + \frac {1}{2} \, c\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{{\left (a^{4} d - a^{2} b^{2} d\right )} \sqrt {-a^{2} + b^{2}}} + \frac {2 \, {\left (d \sqrt {x} + c\right )}}{a^{2} d} \]
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Time = 18.69 (sec) , antiderivative size = 330, normalized size of antiderivative = 2.60 \[ \int \frac {1}{\sqrt {x} \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx=\frac {\frac {b^2\,4{}\mathrm {i}}{a\,d\,\left (a^2-b^2\right )}+\frac {b^3\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,\sqrt {x}\,1{}\mathrm {i}}\,4{}\mathrm {i}}{a^2\,d\,\left (a^2-b^2\right )}}{a+a\,{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,\sqrt {x}\,2{}\mathrm {i}}+2\,b\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,\sqrt {x}\,1{}\mathrm {i}}}+\frac {2\,\sqrt {x}}{a^2}+\frac {\ln \left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,\sqrt {x}\,1{}\mathrm {i}}\,\left (4\,a^2\,b-2\,b^3\right )-\frac {\left (4\,a^2\,b-2\,b^3\right )\,\left (a^2-b^2\right )\,\left (a+b\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,\sqrt {x}\,1{}\mathrm {i}}\right )\,1{}\mathrm {i}}{{\left (a+b\right )}^{3/2}\,{\left (a-b\right )}^{3/2}}\right )\,\left (4\,a^2\,b-2\,b^3\right )}{a^2\,d\,{\left (a+b\right )}^{3/2}\,{\left (a-b\right )}^{3/2}}-\frac {2\,b\,\ln \left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,\sqrt {x}\,1{}\mathrm {i}}\,\left (4\,a^2\,b-2\,b^3\right )+\frac {b\,\left (a^2-b^2\right )\,\left (2\,a^2-b^2\right )\,\left (a+b\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,\sqrt {x}\,1{}\mathrm {i}}\right )\,2{}\mathrm {i}}{{\left (a+b\right )}^{3/2}\,{\left (a-b\right )}^{3/2}}\right )\,\left (2\,a^2-b^2\right )}{a^2\,d\,{\left (a+b\right )}^{3/2}\,{\left (a-b\right )}^{3/2}} \]
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