Integrand size = 25, antiderivative size = 611 \[ \int \frac {1}{(a+b \cos (c+d x))^3 (e \sin (c+d x))^{3/2}} \, dx=-\frac {5 b^{3/2} \left (7 a^2+2 b^2\right ) \arctan \left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 \left (-a^2+b^2\right )^{13/4} d e^{3/2}}+\frac {5 b^{3/2} \left (7 a^2+2 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 \left (-a^2+b^2\right )^{13/4} d e^{3/2}}-\frac {b}{2 \left (a^2-b^2\right ) d e (a+b \cos (c+d x))^2 \sqrt {e \sin (c+d x)}}-\frac {9 a b}{4 \left (a^2-b^2\right )^2 d e (a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}}+\frac {5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \cos (c+d x)}{4 \left (a^2-b^2\right )^3 d e \sqrt {e \sin (c+d x)}}-\frac {5 a b \left (7 a^2+2 b^2\right ) \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{8 \left (a^2-b^2\right )^3 \left (b-\sqrt {-a^2+b^2}\right ) d e \sqrt {e \sin (c+d x)}}-\frac {5 a b \left (7 a^2+2 b^2\right ) \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{8 \left (a^2-b^2\right )^3 \left (b+\sqrt {-a^2+b^2}\right ) d e \sqrt {e \sin (c+d x)}}-\frac {a \left (8 a^2+37 b^2\right ) E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{4 \left (a^2-b^2\right )^3 d e^2 \sqrt {\sin (c+d x)}} \]
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Time = 1.93 (sec) , antiderivative size = 611, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {2773, 2943, 2945, 2946, 2721, 2719, 2780, 2886, 2884, 335, 304, 211, 214} \[ \int \frac {1}{(a+b \cos (c+d x))^3 (e \sin (c+d x))^{3/2}} \, dx=-\frac {5 b^{3/2} \left (7 a^2+2 b^2\right ) \arctan \left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{8 d e^{3/2} \left (b^2-a^2\right )^{13/4}}+\frac {5 b^{3/2} \left (7 a^2+2 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{8 d e^{3/2} \left (b^2-a^2\right )^{13/4}}-\frac {a \left (8 a^2+37 b^2\right ) E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{4 d e^2 \left (a^2-b^2\right )^3 \sqrt {\sin (c+d x)}}-\frac {9 a b}{4 d e \left (a^2-b^2\right )^2 \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))}-\frac {b}{2 d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))^2}+\frac {5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \cos (c+d x)}{4 d e \left (a^2-b^2\right )^3 \sqrt {e \sin (c+d x)}}-\frac {5 a b \left (7 a^2+2 b^2\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {b^2-a^2}},\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{8 d e \left (a^2-b^2\right )^3 \left (b-\sqrt {b^2-a^2}\right ) \sqrt {e \sin (c+d x)}}-\frac {5 a b \left (7 a^2+2 b^2\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {b^2-a^2}},\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{8 d e \left (a^2-b^2\right )^3 \left (\sqrt {b^2-a^2}+b\right ) \sqrt {e \sin (c+d x)}} \]
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Rule 211
Rule 214
Rule 304
Rule 335
Rule 2719
Rule 2721
Rule 2773
Rule 2780
Rule 2884
Rule 2886
Rule 2943
Rule 2945
Rule 2946
Rubi steps \begin{align*} \text {integral}& = -\frac {b}{2 \left (a^2-b^2\right ) d e (a+b \cos (c+d x))^2 \sqrt {e \sin (c+d x)}}-\frac {\int \frac {-2 a+\frac {5}{2} b \cos (c+d x)}{(a+b \cos (c+d x))^2 (e \sin (c+d x))^{3/2}} \, dx}{2 \left (a^2-b^2\right )} \\ & = -\frac {b}{2 \left (a^2-b^2\right ) d e (a+b \cos (c+d x))^2 \sqrt {e \sin (c+d x)}}-\frac {9 a b}{4 \left (a^2-b^2\right )^2 d e (a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}}+\frac {\int \frac {\frac {1}{2} \left (4 a^2+5 b^2\right )-\frac {27}{4} a b \cos (c+d x)}{(a+b \cos (c+d x)) (e \sin (c+d x))^{3/2}} \, dx}{2 \left (a^2-b^2\right )^2} \\ & = -\frac {b}{2 \left (a^2-b^2\right ) d e (a+b \cos (c+d x))^2 \sqrt {e \sin (c+d x)}}-\frac {9 a b}{4 \left (a^2-b^2\right )^2 d e (a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}}+\frac {5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \cos (c+d x)}{4 \left (a^2-b^2\right )^3 d e \sqrt {e \sin (c+d x)}}-\frac {\int \frac {\left (\frac {1}{4} \left (4 a^4+36 a^2 b^2+5 b^4\right )+\frac {1}{8} a b \left (8 a^2+37 b^2\right ) \cos (c+d x)\right ) \sqrt {e \sin (c+d x)}}{a+b \cos (c+d x)} \, dx}{\left (a^2-b^2\right )^3 e^2} \\ & = -\frac {b}{2 \left (a^2-b^2\right ) d e (a+b \cos (c+d x))^2 \sqrt {e \sin (c+d x)}}-\frac {9 a b}{4 \left (a^2-b^2\right )^2 d e (a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}}+\frac {5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \cos (c+d x)}{4 \left (a^2-b^2\right )^3 d e \sqrt {e \sin (c+d x)}}-\frac {\left (5 b^2 \left (7 a^2+2 b^2\right )\right ) \int \frac {\sqrt {e \sin (c+d x)}}{a+b \cos (c+d x)} \, dx}{8 \left (a^2-b^2\right )^3 e^2}-\frac {\left (a \left (8 a^2+37 b^2\right )\right ) \int \sqrt {e \sin (c+d x)} \, dx}{8 \left (a^2-b^2\right )^3 e^2} \\ & = -\frac {b}{2 \left (a^2-b^2\right ) d e (a+b \cos (c+d x))^2 \sqrt {e \sin (c+d x)}}-\frac {9 a b}{4 \left (a^2-b^2\right )^2 d e (a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}}+\frac {5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \cos (c+d x)}{4 \left (a^2-b^2\right )^3 d e \sqrt {e \sin (c+d x)}}+\frac {\left (5 a b \left (7 a^2+2 b^2\right )\right ) \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {-a^2+b^2}-b \sin (c+d x)\right )} \, dx}{16 \left (a^2-b^2\right )^3 e}-\frac {\left (5 a b \left (7 a^2+2 b^2\right )\right ) \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {-a^2+b^2}+b \sin (c+d x)\right )} \, dx}{16 \left (a^2-b^2\right )^3 e}+\frac {\left (5 b^3 \left (7 a^2+2 b^2\right )\right ) \text {Subst}\left (\int \frac {\sqrt {x}}{\left (a^2-b^2\right ) e^2+b^2 x^2} \, dx,x,e \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^3 d e}-\frac {\left (a \left (8 a^2+37 b^2\right ) \sqrt {e \sin (c+d x)}\right ) \int \sqrt {\sin (c+d x)} \, dx}{8 \left (a^2-b^2\right )^3 e^2 \sqrt {\sin (c+d x)}} \\ & = -\frac {b}{2 \left (a^2-b^2\right ) d e (a+b \cos (c+d x))^2 \sqrt {e \sin (c+d x)}}-\frac {9 a b}{4 \left (a^2-b^2\right )^2 d e (a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}}+\frac {5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \cos (c+d x)}{4 \left (a^2-b^2\right )^3 d e \sqrt {e \sin (c+d x)}}-\frac {a \left (8 a^2+37 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)}}{4 \left (a^2-b^2\right )^3 d e^2 \sqrt {\sin (c+d x)}}+\frac {\left (5 b^3 \left (7 a^2+2 b^2\right )\right ) \text {Subst}\left (\int \frac {x^2}{\left (a^2-b^2\right ) e^2+b^2 x^4} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{4 \left (a^2-b^2\right )^3 d e}+\frac {\left (5 a b \left (7 a^2+2 b^2\right ) \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)} \left (\sqrt {-a^2+b^2}-b \sin (c+d x)\right )} \, dx}{16 \left (a^2-b^2\right )^3 e \sqrt {e \sin (c+d x)}}-\frac {\left (5 a b \left (7 a^2+2 b^2\right ) \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)} \left (\sqrt {-a^2+b^2}+b \sin (c+d x)\right )} \, dx}{16 \left (a^2-b^2\right )^3 e \sqrt {e \sin (c+d x)}} \\ & = -\frac {b}{2 \left (a^2-b^2\right ) d e (a+b \cos (c+d x))^2 \sqrt {e \sin (c+d x)}}-\frac {9 a b}{4 \left (a^2-b^2\right )^2 d e (a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}}+\frac {5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \cos (c+d x)}{4 \left (a^2-b^2\right )^3 d e \sqrt {e \sin (c+d x)}}-\frac {5 a b \left (7 a^2+2 b^2\right ) \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{8 \left (a^2-b^2\right )^3 \left (b-\sqrt {-a^2+b^2}\right ) d e \sqrt {e \sin (c+d x)}}-\frac {5 a b \left (7 a^2+2 b^2\right ) \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{8 \left (a^2-b^2\right )^3 \left (b+\sqrt {-a^2+b^2}\right ) d e \sqrt {e \sin (c+d x)}}-\frac {a \left (8 a^2+37 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)}}{4 \left (a^2-b^2\right )^3 d e^2 \sqrt {\sin (c+d x)}}-\frac {\left (5 b^2 \left (7 a^2+2 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} e-b x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{8 \left (a^2-b^2\right )^3 d e}+\frac {\left (5 b^2 \left (7 a^2+2 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} e+b x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{8 \left (a^2-b^2\right )^3 d e} \\ & = -\frac {5 b^{3/2} \left (7 a^2+2 b^2\right ) \arctan \left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 \left (-a^2+b^2\right )^{13/4} d e^{3/2}}+\frac {5 b^{3/2} \left (7 a^2+2 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 \left (-a^2+b^2\right )^{13/4} d e^{3/2}}-\frac {b}{2 \left (a^2-b^2\right ) d e (a+b \cos (c+d x))^2 \sqrt {e \sin (c+d x)}}-\frac {9 a b}{4 \left (a^2-b^2\right )^2 d e (a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}}+\frac {5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \cos (c+d x)}{4 \left (a^2-b^2\right )^3 d e \sqrt {e \sin (c+d x)}}-\frac {5 a b \left (7 a^2+2 b^2\right ) \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{8 \left (a^2-b^2\right )^3 \left (b-\sqrt {-a^2+b^2}\right ) d e \sqrt {e \sin (c+d x)}}-\frac {5 a b \left (7 a^2+2 b^2\right ) \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{8 \left (a^2-b^2\right )^3 \left (b+\sqrt {-a^2+b^2}\right ) d e \sqrt {e \sin (c+d x)}}-\frac {a \left (8 a^2+37 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)}}{4 \left (a^2-b^2\right )^3 d e^2 \sqrt {\sin (c+d x)}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 6.89 (sec) , antiderivative size = 922, normalized size of antiderivative = 1.51 \[ \int \frac {1}{(a+b \cos (c+d x))^3 (e \sin (c+d x))^{3/2}} \, dx=\frac {\sin ^2(c+d x) \left (-\frac {2 \left (-3 a^2 b-b^3+a^3 \cos (c+d x)+3 a b^2 \cos (c+d x)\right ) \csc (c+d x)}{\left (a^2-b^2\right )^3}+\frac {b^3 \sin (c+d x)}{2 \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}+\frac {13 a b^3 \sin (c+d x)}{4 \left (a^2-b^2\right )^3 (a+b \cos (c+d x))}\right )}{d (e \sin (c+d x))^{3/2}}-\frac {\sin ^{\frac {3}{2}}(c+d x) \left (\frac {\left (8 a^3 b+37 a b^3\right ) \cos ^2(c+d x) \left (3 \sqrt {2} a \left (a^2-b^2\right )^{3/4} \left (2 \arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {\sin (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-2 \arctan \left (1+\frac {\sqrt {2} \sqrt {b} \sqrt {\sin (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-\log \left (\sqrt {a^2-b^2}-\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\sin (c+d x)}+b \sin (c+d x)\right )+\log \left (\sqrt {a^2-b^2}+\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\sin (c+d x)}+b \sin (c+d x)\right )\right )+8 b^{5/2} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{2},1,\frac {7}{4},\sin ^2(c+d x),\frac {b^2 \sin ^2(c+d x)}{-a^2+b^2}\right ) \sin ^{\frac {3}{2}}(c+d x)\right ) \left (a+b \sqrt {1-\sin ^2(c+d x)}\right )}{12 b^{3/2} \left (-a^2+b^2\right ) (a+b \cos (c+d x)) \left (1-\sin ^2(c+d x)\right )}+\frac {2 \left (8 a^4+72 a^2 b^2+10 b^4\right ) \cos (c+d x) \left (\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) \left (2 \arctan \left (1-\frac {(1+i) \sqrt {b} \sqrt {\sin (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )-2 \arctan \left (1+\frac {(1+i) \sqrt {b} \sqrt {\sin (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )-\log \left (\sqrt {-a^2+b^2}-(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\sin (c+d x)}+i b \sin (c+d x)\right )+\log \left (\sqrt {-a^2+b^2}+(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\sin (c+d x)}+i b \sin (c+d x)\right )\right )}{\sqrt {b} \sqrt [4]{-a^2+b^2}}+\frac {a \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\sin ^2(c+d x),\frac {b^2 \sin ^2(c+d x)}{-a^2+b^2}\right ) \sin ^{\frac {3}{2}}(c+d x)}{3 \left (a^2-b^2\right )}\right ) \left (a+b \sqrt {1-\sin ^2(c+d x)}\right )}{(a+b \cos (c+d x)) \sqrt {1-\sin ^2(c+d x)}}\right )}{8 (a-b)^3 (a+b)^3 d (e \sin (c+d x))^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(2836\) vs. \(2(631)=1262\).
Time = 10.14 (sec) , antiderivative size = 2837, normalized size of antiderivative = 4.64
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Timed out. \[ \int \frac {1}{(a+b \cos (c+d x))^3 (e \sin (c+d x))^{3/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {1}{(a+b \cos (c+d x))^3 (e \sin (c+d x))^{3/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {1}{(a+b \cos (c+d x))^3 (e \sin (c+d x))^{3/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{(a+b \cos (c+d x))^3 (e \sin (c+d x))^{3/2}} \, dx=\int { \frac {1}{{\left (b \cos \left (d x + c\right ) + a\right )}^{3} \left (e \sin \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(a+b \cos (c+d x))^3 (e \sin (c+d x))^{3/2}} \, dx=\int \frac {1}{{\left (e\,\sin \left (c+d\,x\right )\right )}^{3/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3} \,d x \]
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