Integrand size = 23, antiderivative size = 129 \[ \int \frac {\csc (e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a+b-b \cos ^2(e+f x)}}\right )}{a^{5/2} f}+\frac {b \cos (e+f x)}{3 a (a+b) f \left (a+b-b \cos ^2(e+f x)\right )^{3/2}}+\frac {b (5 a+3 b) \cos (e+f x)}{3 a^2 (a+b)^2 f \sqrt {a+b-b \cos ^2(e+f x)}} \]
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Time = 0.10 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3265, 425, 541, 12, 385, 212} \[ \int \frac {\csc (e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a-b \cos ^2(e+f x)+b}}\right )}{a^{5/2} f}+\frac {b (5 a+3 b) \cos (e+f x)}{3 a^2 f (a+b)^2 \sqrt {a-b \cos ^2(e+f x)+b}}+\frac {b \cos (e+f x)}{3 a f (a+b) \left (a-b \cos ^2(e+f x)+b\right )^{3/2}} \]
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Rule 12
Rule 212
Rule 385
Rule 425
Rule 541
Rule 3265
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 )^{5/2}} \, dx,x,\cos (e+f x)\right )}{f} \\ & = \frac {b \cos (e+f x)}{3 a (a+b) f \left (a+b-b \cos ^2(e+f x)\right )^{3/2}}+\frac {\text {Subst}\left (\int \frac {-3 a-b-2 b x^2}{\left (1-x^2\right ) \left (a+b-b x^2\right )^{3/2}} \, dx,x,\cos (e+f x)\right )}{3 a (a+b) f} \\ & = \frac {b \cos (e+f x)}{3 a (a+b) f \left (a+b-b \cos ^2(e+f x)\right )^{3/2}}+\frac {b (5 a+3 b) \cos (e+f x)}{3 a^2 (a+b)^2 f \sqrt {a+b-b \cos ^2(e+f x)}}-\frac {\text {Subst}\left (\int \frac {3 (a+b)^2}{\left (1-x^2\right ) \sqrt {a+b-b x^2}} \, dx,x,\cos (e+f x)\right )}{3 a^2 (a+b)^2 f} \\ & = \frac {b \cos (e+f x)}{3 a (a+b) f \left (a+b-b \cos ^2(e+f x)\right )^{3/2}}+\frac {b (5 a+3 b) \cos (e+f x)}{3 a^2 (a+b)^2 f \sqrt {a+b-b \cos ^2(e+f x)}}-\frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {a+b-b x^2}} \, dx,x,\cos (e+f x)\right )}{a^2 f} \\ & = \frac {b \cos (e+f x)}{3 a (a+b) f \left (a+b-b \cos ^2(e+f x)\right )^{3/2}}+\frac {b (5 a+3 b) \cos (e+f x)}{3 a^2 (a+b)^2 f \sqrt {a+b-b \cos ^2(e+f x)}}-\frac {\text {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\cos (e+f x)}{\sqrt {a+b-b \cos ^2(e+f x)}}\right )}{a^2 f} \\ & = -\frac {\text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a+b-b \cos ^2(e+f x)}}\right )}{a^{5/2} f}+\frac {b \cos (e+f x)}{3 a (a+b) f \left (a+b-b \cos ^2(e+f x)\right )^{3/2}}+\frac {b (5 a+3 b) \cos (e+f x)}{3 a^2 (a+b)^2 f \sqrt {a+b-b \cos ^2(e+f x)}} \\ \end{align*}
Time = 0.54 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.98 \[ \int \frac {\csc (e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\frac {-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \cos (e+f x)}{\sqrt {2 a+b-b \cos (2 (e+f x))}}\right )}{a^{5/2}}+\frac {\sqrt {2} b \cos (e+f x) \left (12 a^2+13 a b+3 b^2-b (5 a+3 b) \cos (2 (e+f x))\right )}{3 a^2 (a+b)^2 (2 a+b-b \cos (2 (e+f x)))^{3/2}}}{f} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(248\) vs. \(2(115)=230\).
Time = 1.36 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.93
method | result | size |
default | \(\frac {\sqrt {-\left (-b \left (\sin ^{2}\left (f x +e \right )\right )-a \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, \left (-\frac {\ln \left (\frac {2 a +\left (-a +b \right ) \left (\sin ^{2}\left (f x +e \right )\right )+2 \sqrt {a}\, \sqrt {-\left (-b \left (\sin ^{2}\left (f x +e \right )\right )-a \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}{\sin \left (f x +e \right )^{2}}\right )}{2 a^{\frac {5}{2}}}+\frac {b \left (2 b \left (\sin ^{2}\left (f x +e \right )\right )+3 a +b \right ) \left (\cos ^{2}\left (f x +e \right )\right )}{3 a \sqrt {-\left (-b \left (\sin ^{2}\left (f x +e \right )\right )-a \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, \left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a^{2} \left (a +b \right ) \sqrt {-\left (-b \left (\sin ^{2}\left (f x +e \right )\right )-a \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}\right )}{\cos \left (f x +e \right ) \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}\, f}\) | \(249\) |
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Leaf count of result is larger than twice the leaf count of optimal. 348 vs. \(2 (115) = 230\).
Time = 0.56 (sec) , antiderivative size = 752, normalized size of antiderivative = 5.83 \[ \int \frac {\csc (e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\left [\frac {3 \, {\left ({\left (a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right )} \cos \left (f x + e\right )^{4} + a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4} - 2 \, {\left (a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {a} \log \left (\frac {2 \, {\left ({\left (a^{2} - 6 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{4} + 2 \, {\left (3 \, a^{2} + 2 \, a b - b^{2}\right )} \cos \left (f x + e\right )^{2} - 4 \, {\left ({\left (a - b\right )} \cos \left (f x + e\right )^{3} + {\left (a + b\right )} \cos \left (f x + e\right )\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {a} + a^{2} + 2 \, a b + b^{2}\right )}}{\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1}\right ) - 4 \, {\left ({\left (5 \, a^{2} b^{2} + 3 \, a b^{3}\right )} \cos \left (f x + e\right )^{3} - 3 \, {\left (2 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right )} \cos \left (f x + e\right )\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b}}{12 \, {\left ({\left (a^{5} b^{2} + 2 \, a^{4} b^{3} + a^{3} b^{4}\right )} f \cos \left (f x + e\right )^{4} - 2 \, {\left (a^{6} b + 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} + a^{3} b^{4}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right )} f\right )}}, \frac {3 \, {\left ({\left (a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right )} \cos \left (f x + e\right )^{4} + a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4} - 2 \, {\left (a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-a} \arctan \left (-\frac {{\left ({\left (a - b\right )} \cos \left (f x + e\right )^{2} + a + b\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {-a}}{2 \, {\left (a b \cos \left (f x + e\right )^{3} - {\left (a^{2} + a b\right )} \cos \left (f x + e\right )\right )}}\right ) - 2 \, {\left ({\left (5 \, a^{2} b^{2} + 3 \, a b^{3}\right )} \cos \left (f x + e\right )^{3} - 3 \, {\left (2 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right )} \cos \left (f x + e\right )\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b}}{6 \, {\left ({\left (a^{5} b^{2} + 2 \, a^{4} b^{3} + a^{3} b^{4}\right )} f \cos \left (f x + e\right )^{4} - 2 \, {\left (a^{6} b + 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} + a^{3} b^{4}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right )} f\right )}}\right ] \]
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\[ \int \frac {\csc (e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {\csc {\left (e + f x \right )}}{\left (a + b \sin ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 306 vs. \(2 (115) = 230\).
Time = 0.41 (sec) , antiderivative size = 306, normalized size of antiderivative = 2.37 \[ \int \frac {\csc (e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\frac {\frac {4 \, b^{3} \cos \left (f x + e\right )}{\sqrt {-b \cos \left (f x + e\right )^{2} + a + b} a^{3} b^{2} + 2 \, \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} a^{2} b^{3} + \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} a b^{4}} + \frac {2 \, b^{2} \cos \left (f x + e\right )}{{\left (-b \cos \left (f x + e\right )^{2} + a + b\right )}^{\frac {3}{2}} a^{2} b + {\left (-b \cos \left (f x + e\right )^{2} + a + b\right )}^{\frac {3}{2}} a b^{2}} + \frac {6 \, b^{2} \cos \left (f x + e\right )}{\sqrt {-b \cos \left (f x + e\right )^{2} + a + b} a^{3} b + \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} a^{2} b^{2}} - \frac {3 \, \log \left (b - \frac {\sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {a}}{\cos \left (f x + e\right ) - 1} - \frac {a}{\cos \left (f x + e\right ) - 1}\right )}{a^{\frac {5}{2}}} + \frac {3 \, \log \left (-b + \frac {\sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {a}}{\cos \left (f x + e\right ) + 1} + \frac {a}{\cos \left (f x + e\right ) + 1}\right )}{a^{\frac {5}{2}}}}{6 \, f} \]
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\[ \int \frac {\csc (e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\csc \left (f x + e\right )}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\csc (e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {1}{\sin \left (e+f\,x\right )\,{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{5/2}} \,d x \]
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