Integrand size = 25, antiderivative size = 197 \[ \int \csc ^7(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=-\frac {(5 a-b) (a+b)^2 \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a+b-b \cos ^2(e+f x)}}\right )}{16 a^{3/2} f}-\frac {(5 a-b) (a+b) \sqrt {a+b-b \cos ^2(e+f x)} \cot (e+f x) \csc (e+f x)}{16 a f}-\frac {(5 a-b) \left (a+b-b \cos ^2(e+f x)\right )^{3/2} \cot (e+f x) \csc ^3(e+f x)}{24 a f}-\frac {\left (a+b-b \cos ^2(e+f x)\right )^{5/2} \cot (e+f x) \csc ^5(e+f x)}{6 a f} \]
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Time = 0.11 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3265, 390, 386, 385, 212} \[ \int \csc ^7(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=-\frac {(5 a-b) (a+b)^2 \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a-b \cos ^2(e+f x)+b}}\right )}{16 a^{3/2} f}-\frac {\cot (e+f x) \csc ^5(e+f x) \left (a-b \cos ^2(e+f x)+b\right )^{5/2}}{6 a f}-\frac {(5 a-b) \cot (e+f x) \csc ^3(e+f x) \left (a-b \cos ^2(e+f x)+b\right )^{3/2}}{24 a f}-\frac {(5 a-b) (a+b) \cot (e+f x) \csc (e+f x) \sqrt {a-b \cos ^2(e+f x)+b}}{16 a f} \]
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Rule 212
Rule 385
Rule 386
Rule 390
Rule 3265
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\left (a+b-b x^2\right )^{3/2}}{\left (1-x^2\right )^4} \, dx,x,\cos (e+f x)\right )}{f} \\ & = -\frac {\left (a+b-b \cos ^2(e+f x)\right )^{5/2} \cot (e+f x) \csc ^5(e+f x)}{6 a f}-\frac {(5 a-b) \text {Subst}\left (\int \frac {\left (a+b-b x^2\right )^{3/2}}{\left (1-x^2\right )^3} \, dx,x,\cos (e+f x)\right )}{6 a f} \\ & = -\frac {(5 a-b) \left (a+b-b \cos ^2(e+f x)\right )^{3/2} \cot (e+f x) \csc ^3(e+f x)}{24 a f}-\frac {\left (a+b-b \cos ^2(e+f x)\right )^{5/2} \cot (e+f x) \csc ^5(e+f x)}{6 a f}-\frac {((5 a-b) (a+b)) \text {Subst}\left (\int \frac {\sqrt {a+b-b x^2}}{\left (1-x^2\right )^2} \, dx,x,\cos (e+f x)\right )}{8 a f} \\ & = -\frac {(5 a-b) (a+b) \sqrt {a+b-b \cos ^2(e+f x)} \cot (e+f x) \csc (e+f x)}{16 a f}-\frac {(5 a-b) \left (a+b-b \cos ^2(e+f x)\right )^{3/2} \cot (e+f x) \csc ^3(e+f x)}{24 a f}-\frac {\left (a+b-b \cos ^2(e+f x)\right )^{5/2} \cot (e+f x) \csc ^5(e+f x)}{6 a f}-\frac {\left ((5 a-b) (a+b)^2\right ) \text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {a+b-b x^2}} \, dx,x,\cos (e+f x)\right )}{16 a f} \\ & = -\frac {(5 a-b) (a+b) \sqrt {a+b-b \cos ^2(e+f x)} \cot (e+f x) \csc (e+f x)}{16 a f}-\frac {(5 a-b) \left (a+b-b \cos ^2(e+f x)\right )^{3/2} \cot (e+f x) \csc ^3(e+f x)}{24 a f}-\frac {\left (a+b-b \cos ^2(e+f x)\right )^{5/2} \cot (e+f x) \csc ^5(e+f x)}{6 a f}-\frac {\left ((5 a-b) (a+b)^2\right ) \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 )}{16 a f} \\ & = -\frac {(5 a-b) (a+b)^2 \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a+b-b \cos ^2(e+f x)}}\right )}{16 a^{3/2} f}-\frac {(5 a-b) (a+b) \sqrt {a+b-b \cos ^2(e+f x)} \cot (e+f x) \csc (e+f x)}{16 a f}-\frac {(5 a-b) \left (a+b-b \cos ^2(e+f x)\right )^{3/2} \cot (e+f x) \csc ^3(e+f x)}{24 a f}-\frac {\left (a+b-b \cos ^2(e+f x)\right )^{5/2} \cot (e+f x) \csc ^5(e+f x)}{6 a f} \\ \end{align*}
Time = 1.60 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.82 \[ \int \csc ^7(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\frac {-6 (5 a-b) (a+b)^2 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \cos (e+f x)}{\sqrt {2 a+b-b \cos (2 (e+f x))}}\right )-\sqrt {2} \sqrt {a} \sqrt {2 a+b-b \cos (2 (e+f x))} \csc ^2(e+f x) \left (\left (15 a^2+22 a b+3 b^2\right ) \cos (e+f x)+2 a \cot (e+f x) \csc (e+f x) \left (5 a+7 b+4 a \csc ^2(e+f x)\right )\right )}{96 a^{3/2} f} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(564\) vs. \(2(177)=354\).
Time = 1.39 (sec) , antiderivative size = 565, normalized size of antiderivative = 2.87
method | result | size |
default | \(-\frac {\sqrt {\left (\cos ^{2}\left (f x +e \right )\right ) \left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}\, \left (15 a^{4} \ln \left (\frac {\left (a -b \right ) \left (\cos ^{2}\left (f x +e \right )\right )+2 \sqrt {a}\, \sqrt {-b \left (\cos ^{4}\left (f x +e \right )\right )+\left (a +b \right ) \left (\cos ^{2}\left (f x +e \right )\right )}+a +b}{\sin \left (f x +e \right )^{2}}\right ) \left (\sin ^{6}\left (f x +e \right )\right )+27 a^{3} b \ln \left (\frac {\left (a -b \right ) \left (\cos ^{2}\left (f x +e \right )\right )+2 \sqrt {a}\, \sqrt {-b \left (\cos ^{4}\left (f x +e \right )\right )+\left (a +b \right ) \left (\cos ^{2}\left (f x +e \right )\right )}+a +b}{\sin \left (f x +e \right )^{2}}\right ) \left (\sin ^{6}\left (f x +e \right )\right )+9 b^{2} \ln \left (\frac {\left (a -b \right ) \left (\cos ^{2}\left (f x +e \right )\right )+2 \sqrt {a}\, \sqrt {-b \left (\cos ^{4}\left (f x +e \right )\right )+\left (a +b \right ) \left (\cos ^{2}\left (f x +e \right )\right )}+a +b}{\sin \left (f x +e \right )^{2}}\right ) \left (\sin ^{6}\left (f x +e \right )\right ) a^{2}-3 b^{3} \ln \left (\frac {\left (a -b \right ) \left (\cos ^{2}\left (f x +e \right )\right )+2 \sqrt {a}\, \sqrt {-b \left (\cos ^{4}\left (f x +e \right )\right )+\left (a +b \right ) \left (\cos ^{2}\left (f x +e \right )\right )}+a +b}{\sin \left (f x +e \right )^{2}}\right ) \left (\sin ^{6}\left (f x +e \right )\right ) a +30 \left (\sin ^{4}\left (f x +e \right )\right ) \sqrt {\left (\cos ^{2}\left (f x +e \right )\right ) \left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}\, a^{\frac {7}{2}}+44 \left (\sin ^{4}\left (f x +e \right )\right ) \sqrt {\left (\cos ^{2}\left (f x +e \right )\right ) \left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}\, a^{\frac {5}{2}} b +6 \left (\sin ^{4}\left (f x +e \right )\right ) \sqrt {\left (\cos ^{2}\left (f x +e \right )\right ) \left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}\, a^{\frac {3}{2}} b^{2}+20 \left (\sin ^{2}\left (f x +e \right )\right ) \sqrt {\left (\cos ^{2}\left (f x +e \right )\right ) \left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}\, a^{\frac {7}{2}}+28 \left (\sin ^{2}\left (f x +e \right )\right ) \sqrt {\left (\cos ^{2}\left (f x +e \right )\right ) \left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}\, a^{\frac {5}{2}} b +16 a^{\frac {7}{2}} \sqrt {\left (\cos ^{2}\left (f x +e \right )\right ) \left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}\right )}{96 \sin \left (f x +e \right )^{6} a^{\frac {5}{2}} \cos \left (f x +e \right ) \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}\, f}\) | \(565\) |
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Time = 1.13 (sec) , antiderivative size = 752, normalized size of antiderivative = 3.82 \[ \int \csc ^7(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\left [-\frac {3 \, {\left ({\left (5 \, a^{3} + 9 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \cos \left (f x + e\right )^{6} - 3 \, {\left (5 \, a^{3} + 9 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \cos \left (f x + e\right )^{4} - 5 \, a^{3} - 9 \, a^{2} b - 3 \, a b^{2} + b^{3} + 3 \, {\left (5 \, a^{3} + 9 \, a^{2} b + 3 \, a b^{2} - b^{3}\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 (15 \, a^{3} + 22 \, a^{2} b + 3 \, a b^{2}\right )} \cos \left (f x + e\right )^{5} - 2 \, {\left (20 \, a^{3} + 29 \, a^{2} b + 3 \, a b^{2}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (11 \, a^{3} + 12 \, a^{2} b + a b^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b}}{192 \, {\left (a^{2} f \cos \left (f x + e\right )^{6} - 3 \, a^{2} f \cos \left (f x + e\right )^{4} + 3 \, a^{2} f \cos \left (f x + e\right )^{2} - a^{2} f\right )}}, \frac {3 \, {\left ({\left (5 \, a^{3} + 9 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \cos \left (f x + e\right )^{6} - 3 \, {\left (5 \, a^{3} + 9 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \cos \left (f x + e\right )^{4} - 5 \, a^{3} - 9 \, a^{2} b - 3 \, a b^{2} + b^{3} + 3 \, {\left (5 \, a^{3} + 9 \, a^{2} b + 3 \, a b^{2} - b^{3}\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 (15 \, a^{3} + 22 \, a^{2} b + 3 \, a b^{2}\right )} \cos \left (f x + e\right )^{5} - 2 \, {\left (20 \, a^{3} + 29 \, a^{2} b + 3 \, a b^{2}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (11 \, a^{3} + 12 \, a^{2} b + a b^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b}}{96 \, {\left (a^{2} f \cos \left (f x + e\right )^{6} - 3 \, a^{2} f \cos \left (f x + e\right )^{4} + 3 \, a^{2} f \cos \left (f x + e\right )^{2} - a^{2} f\right )}}\right ] \]
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Timed out. \[ \int \csc ^7(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\text {Timed out} \]
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\[ \int \csc ^7(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \csc \left (f x + e\right )^{7} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 1623 vs. \(2 (177) = 354\).
Time = 0.94 (sec) , antiderivative size = 1623, normalized size of antiderivative = 8.24 \[ \int \csc ^7(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \csc ^7(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\int \frac {{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{3/2}}{{\sin \left (e+f\,x\right )}^7} \,d x \]
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