Integrand size = 25, antiderivative size = 511 \[ \int \frac {1}{(a+b \sec (c+d x)) (e \sin (c+d x))^{7/2}} \, dx=\frac {a^{5/2} b \arctan \left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{\left (a^2-b^2\right )^{9/4} d e^{7/2}}-\frac {a^{5/2} b \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{\left (a^2-b^2\right )^{9/4} d e^{7/2}}+\frac {2 (b-a \cos (c+d x))}{5 \left (a^2-b^2\right ) d e (e \sin (c+d x))^{5/2}}+\frac {2 \left (5 a^2 b-a \left (3 a^2+2 b^2\right ) \cos (c+d x)\right )}{5 \left (a^2-b^2\right )^2 d e^3 \sqrt {e \sin (c+d x)}}-\frac {a^2 b^2 \operatorname {EllipticPi}\left (\frac {2 a}{a-\sqrt {a^2-b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{\left (a^2-b^2\right )^2 \left (a-\sqrt {a^2-b^2}\right ) d e^3 \sqrt {e \sin (c+d x)}}-\frac {a^2 b^2 \operatorname {EllipticPi}\left (\frac {2 a}{a+\sqrt {a^2-b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{\left (a^2-b^2\right )^2 \left (a+\sqrt {a^2-b^2}\right ) d e^3 \sqrt {e \sin (c+d x)}}-\frac {2 a \left (3 a^2+2 b^2\right ) E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 \left (a^2-b^2\right )^2 d e^4 \sqrt {\sin (c+d x)}} \]
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Time = 1.66 (sec) , antiderivative size = 511, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {3957, 2945, 2946, 2721, 2719, 2780, 2886, 2884, 335, 304, 211, 214} \[ \int \frac {1}{(a+b \sec (c+d x)) (e \sin (c+d x))^{7/2}} \, dx=-\frac {2 a \left (3 a^2+2 b^2\right ) E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 d e^4 \left (a^2-b^2\right )^2 \sqrt {\sin (c+d x)}}+\frac {2 \left (5 a^2 b-a \left (3 a^2+2 b^2\right ) \cos (c+d x)\right )}{5 d e^3 \left (a^2-b^2\right )^2 \sqrt {e \sin (c+d x)}}-\frac {a^2 b^2 \sqrt {\sin (c+d x)} \operatorname {EllipticPi}\left (\frac {2 a}{a-\sqrt {a^2-b^2}},\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{d e^3 \left (a^2-b^2\right )^2 \left (a-\sqrt {a^2-b^2}\right ) \sqrt {e \sin (c+d x)}}-\frac {a^2 b^2 \sqrt {\sin (c+d x)} \operatorname {EllipticPi}\left (\frac {2 a}{a+\sqrt {a^2-b^2}},\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{d e^3 \left (a^2-b^2\right )^2 \left (\sqrt {a^2-b^2}+a\right ) \sqrt {e \sin (c+d x)}}+\frac {2 (b-a \cos (c+d x))}{5 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2}}+\frac {a^{5/2} b \arctan \left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt {e} \sqrt [4]{a^2-b^2}}\right )}{d e^{7/2} \left (a^2-b^2\right )^{9/4}}-\frac {a^{5/2} b \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt {e} \sqrt [4]{a^2-b^2}}\right )}{d e^{7/2} \left (a^2-b^2\right )^{9/4}} \]
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Rule 211
Rule 214
Rule 304
Rule 335
Rule 2719
Rule 2721
Rule 2780
Rule 2884
Rule 2886
Rule 2945
Rule 2946
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int \frac {\cos (c+d x)}{(-b-a \cos (c+d x)) (e \sin (c+d x))^{7/2}} \, dx \\ & = \frac {2 (b-a \cos (c+d x))}{5 \left (a^2-b^2\right ) d e (e \sin (c+d x))^{5/2}}+\frac {2 \int \frac {a b-\frac {3}{2} a^2 \cos (c+d x)}{(-b-a \cos (c+d x)) (e \sin (c+d x))^{3/2}} \, dx}{5 \left (a^2-b^2\right ) e^2} \\ & = \frac {2 (b-a \cos (c+d x))}{5 \left (a^2-b^2\right ) d e (e \sin (c+d x))^{5/2}}+\frac {2 \left (5 a^2 b-a \left (3 a^2+2 b^2\right ) \cos (c+d x)\right )}{5 \left (a^2-b^2\right )^2 d e^3 \sqrt {e \sin (c+d x)}}+\frac {4 \int \frac {\left (\frac {1}{2} a b \left (4 a^2+b^2\right )+\frac {1}{4} a^2 \left (3 a^2+2 b^2\right ) \cos (c+d x)\right ) \sqrt {e \sin (c+d x)}}{-b-a \cos (c+d x)} \, dx}{5 \left (a^2-b^2\right )^2 e^4} \\ & = \frac {2 (b-a \cos (c+d x))}{5 \left (a^2-b^2\right ) d e (e \sin (c+d x))^{5/2}}+\frac {2 \left (5 a^2 b-a \left (3 a^2+2 b^2\right ) \cos (c+d x)\right )}{5 \left (a^2-b^2\right )^2 d e^3 \sqrt {e \sin (c+d x)}}+\frac {\left (a^3 b\right ) \int \frac {\sqrt {e \sin (c+d x)}}{-b-a \cos (c+d x)} \, dx}{\left (a^2-b^2\right )^2 e^4}-\frac {\left (a \left (3 a^2+2 b^2\right )\right ) \int \sqrt {e \sin (c+d x)} \, dx}{5 \left (a^2-b^2\right )^2 e^4} \\ & = \frac {2 (b-a \cos (c+d x))}{5 \left (a^2-b^2\right ) d e (e \sin (c+d x))^{5/2}}+\frac {2 \left (5 a^2 b-a \left (3 a^2+2 b^2\right ) \cos (c+d x)\right )}{5 \left (a^2-b^2\right )^2 d e^3 \sqrt {e \sin (c+d x)}}+\frac {\left (a^2 b^2\right ) \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {a^2-b^2}-a \sin (c+d x)\right )} \, dx}{2 \left (a^2-b^2\right )^2 e^3}-\frac {\left (a^2 b^2\right ) \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {a^2-b^2}+a \sin (c+d x)\right )} \, dx}{2 \left (a^2-b^2\right )^2 e^3}+\frac {\left (a^4 b\right ) \text {Subst}\left (\int \frac {\sqrt {x}}{\left (-a^2+b^2\right ) e^2+a^2 x^2} \, dx,x,e \sin (c+d x)\right )}{\left (a^2-b^2\right )^2 d e^3}-\frac {\left (a \left (3 a^2+2 b^2\right ) \sqrt {e \sin (c+d x)}\right ) \int \sqrt {\sin (c+d x)} \, dx}{5 \left (a^2-b^2\right )^2 e^4 \sqrt {\sin (c+d x)}} \\ & = \frac {2 (b-a \cos (c+d x))}{5 \left (a^2-b^2\right ) d e (e \sin (c+d x))^{5/2}}+\frac {2 \left (5 a^2 b-a \left (3 a^2+2 b^2\right ) \cos (c+d x)\right )}{5 \left (a^2-b^2\right )^2 d e^3 \sqrt {e \sin (c+d x)}}-\frac {2 a \left (3 a^2+2 b^2\right ) E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 \left (a^2-b^2\right )^2 d e^4 \sqrt {\sin (c+d x)}}+\frac {\left (2 a^4 b\right ) \text {Subst}\left (\int \frac {x^2}{\left (-a^2+b^2\right ) e^2+a^2 x^4} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{\left (a^2-b^2\right )^2 d e^3}+\frac {\left (a^2 b^2 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)} \left (\sqrt {a^2-b^2}-a \sin (c+d x)\right )} \, dx}{2 \left (a^2-b^2\right )^2 e^3 \sqrt {e \sin (c+d x)}}-\frac {\left (a^2 b^2 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)} \left (\sqrt {a^2-b^2}+a \sin (c+d x)\right )} \, dx}{2 \left (a^2-b^2\right )^2 e^3 \sqrt {e \sin (c+d x)}} \\ & = \frac {2 (b-a \cos (c+d x))}{5 \left (a^2-b^2\right ) d e (e \sin (c+d x))^{5/2}}+\frac {2 \left (5 a^2 b-a \left (3 a^2+2 b^2\right ) \cos (c+d x)\right )}{5 \left (a^2-b^2\right )^2 d e^3 \sqrt {e \sin (c+d x)}}-\frac {a^2 b^2 \operatorname {EllipticPi}\left (\frac {2 a}{a-\sqrt {a^2-b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{\left (a^2-b^2\right )^2 \left (a-\sqrt {a^2-b^2}\right ) d e^3 \sqrt {e \sin (c+d x)}}-\frac {a^2 b^2 \operatorname {EllipticPi}\left (\frac {2 a}{a+\sqrt {a^2-b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{\left (a^2-b^2\right )^2 \left (a+\sqrt {a^2-b^2}\right ) d e^3 \sqrt {e \sin (c+d x)}}-\frac {2 a \left (3 a^2+2 b^2\right ) E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 \left (a^2-b^2\right )^2 d e^4 \sqrt {\sin (c+d x)}}-\frac {\left (a^3 b\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a^2-b^2} e-a x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{\left (a^2-b^2\right )^2 d e^3}+\frac {\left (a^3 b\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a^2-b^2} e+a x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{\left (a^2-b^2\right )^2 d e^3} \\ & = \frac {a^{5/2} b \arctan \left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{\left (a^2-b^2\right )^{9/4} d e^{7/2}}-\frac {a^{5/2} b \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{\left (a^2-b^2\right )^{9/4} d e^{7/2}}+\frac {2 (b-a \cos (c+d x))}{5 \left (a^2-b^2\right ) d e (e \sin (c+d x))^{5/2}}+\frac {2 \left (5 a^2 b-a \left (3 a^2+2 b^2\right ) \cos (c+d x)\right )}{5 \left (a^2-b^2\right )^2 d e^3 \sqrt {e \sin (c+d x)}}-\frac {a^2 b^2 \operatorname {EllipticPi}\left (\frac {2 a}{a-\sqrt {a^2-b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{\left (a^2-b^2\right )^2 \left (a-\sqrt {a^2-b^2}\right ) d e^3 \sqrt {e \sin (c+d x)}}-\frac {a^2 b^2 \operatorname {EllipticPi}\left (\frac {2 a}{a+\sqrt {a^2-b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{\left (a^2-b^2\right )^2 \left (a+\sqrt {a^2-b^2}\right ) d e^3 \sqrt {e \sin (c+d x)}}-\frac {2 a \left (3 a^2+2 b^2\right ) E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 \left (a^2-b^2\right )^2 d e^4 \sqrt {\sin (c+d x)}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 8.40 (sec) , antiderivative size = 797, normalized size of antiderivative = 1.56 \[ \int \frac {1}{(a+b \sec (c+d x)) (e \sin (c+d x))^{7/2}} \, dx=\frac {(b+a \cos (c+d x)) \sin ^3(c+d x) \left (-\frac {2 \left (\left (a^2-b^2\right ) (-b+a \cos (c+d x)) \csc ^2(c+d x) \sec (c+d x)+a \left (3 a^2+2 b^2-5 a b \sec (c+d x)\right )\right )}{\left (a^2-b^2\right )^2}-\frac {\left (b+a \sqrt {\cos ^2(c+d x)}\right ) \sec ^2(c+d x) \sqrt {\sin (c+d x)} \left (\left (3 a^3+2 a b^2\right ) \cos (c+d x) \left (3 \sqrt {2} b \left (-a^2+b^2\right )^{3/4} \left (2 \arctan \left (1-\frac {\sqrt {2} \sqrt {a} \sqrt {\sin (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )-2 \arctan \left (1+\frac {\sqrt {2} \sqrt {a} \sqrt {\sin (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )-\log \left (\sqrt {-a^2+b^2}-\sqrt {2} \sqrt {a} \sqrt [4]{-a^2+b^2} \sqrt {\sin (c+d x)}+a \sin (c+d x)\right )+\log \left (\sqrt {-a^2+b^2}+\sqrt {2} \sqrt {a} \sqrt [4]{-a^2+b^2} \sqrt {\sin (c+d x)}+a \sin (c+d x)\right )\right )+8 a^{5/2} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{2},1,\frac {7}{4},\sin ^2(c+d x),\frac {a^2 \sin ^2(c+d x)}{a^2-b^2}\right ) \sin ^{\frac {3}{2}}(c+d x)\right )+(2+2 i) a b \left (4 a^2+b^2\right ) \sqrt {\cos ^2(c+d x)} \left (3 \left (a^2-b^2\right )^{3/4} \left (2 \arctan \left (1-\frac {(1+i) \sqrt {a} \sqrt {\sin (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-2 \arctan \left (1+\frac {(1+i) \sqrt {a} \sqrt {\sin (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-\log \left (\sqrt {a^2-b^2}-(1+i) \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\sin (c+d x)}+i a \sin (c+d x)\right )+\log \left (\sqrt {a^2-b^2}+(1+i) \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\sin (c+d x)}+i a \sin (c+d x)\right )\right )-(4-4 i) \sqrt {a} b \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\sin ^2(c+d x),\frac {a^2 \sin ^2(c+d x)}{a^2-b^2}\right ) \sin ^{\frac {3}{2}}(c+d x)\right )\right )}{12 \sqrt {a} (a-b)^2 (a+b)^2 \left (a^2-b^2\right ) (b+a \cos (c+d x))}\right )}{5 d (a+b \sec (c+d x)) (e \sin (c+d x))^{7/2}} \]
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Time = 11.09 (sec) , antiderivative size = 892, normalized size of antiderivative = 1.75
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Timed out. \[ \int \frac {1}{(a+b \sec (c+d x)) (e \sin (c+d x))^{7/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {1}{(a+b \sec (c+d x)) (e \sin (c+d x))^{7/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {1}{(a+b \sec (c+d x)) (e \sin (c+d x))^{7/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{(a+b \sec (c+d x)) (e \sin (c+d x))^{7/2}} \, dx=\int { \frac {1}{{\left (b \sec \left (d x + c\right ) + a\right )} \left (e \sin \left (d x + c\right )\right )^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(a+b \sec (c+d x)) (e \sin (c+d x))^{7/2}} \, dx=\int \frac {\cos \left (c+d\,x\right )}{{\left (e\,\sin \left (c+d\,x\right )\right )}^{7/2}\,\left (b+a\,\cos \left (c+d\,x\right )\right )} \,d x \]
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