Integrand size = 25, antiderivative size = 529 \[ \int \frac {\sqrt {e \sin (c+d x)}}{(a+b \cos (c+d x))^3} \, dx=-\frac {\left (3 a^2+2 b^2\right ) \sqrt {e} \arctan \left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 \sqrt {b} \left (-a^2+b^2\right )^{9/4} d}+\frac {\left (3 a^2+2 b^2\right ) \sqrt {e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 \sqrt {b} \left (-a^2+b^2\right )^{9/4} d}+\frac {a \left (3 a^2+2 b^2\right ) e \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{8 b \left (a^2-b^2\right )^2 \left (b-\sqrt {-a^2+b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {a \left (3 a^2+2 b^2\right ) e \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{8 b \left (a^2-b^2\right )^2 \left (b+\sqrt {-a^2+b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {5 a E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{4 \left (a^2-b^2\right )^2 d \sqrt {\sin (c+d x)}}-\frac {b (e \sin (c+d x))^{3/2}}{2 \left (a^2-b^2\right ) d e (a+b \cos (c+d x))^2}-\frac {5 a b (e \sin (c+d x))^{3/2}}{4 \left (a^2-b^2\right )^2 d e (a+b \cos (c+d x))} \]
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Time = 1.59 (sec) , antiderivative size = 529, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {2773, 2943, 2946, 2721, 2719, 2780, 2886, 2884, 335, 304, 211, 214} \[ \int \frac {\sqrt {e \sin (c+d x)}}{(a+b \cos (c+d x))^3} \, dx=-\frac {\sqrt {e} \left (3 a^2+2 b^2\right ) \arctan \left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{8 \sqrt {b} d \left (b^2-a^2\right )^{9/4}}+\frac {\sqrt {e} \left (3 a^2+2 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{8 \sqrt {b} d \left (b^2-a^2\right )^{9/4}}-\frac {5 a b (e \sin (c+d x))^{3/2}}{4 d e \left (a^2-b^2\right )^2 (a+b \cos (c+d x))}-\frac {b (e \sin (c+d x))^{3/2}}{2 d e \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}+\frac {5 a E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{4 d \left (a^2-b^2\right )^2 \sqrt {\sin (c+d x)}}+\frac {a e \left (3 a^2+2 b^2\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {b^2-a^2}},\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{8 b d \left (a^2-b^2\right )^2 \left (b-\sqrt {b^2-a^2}\right ) \sqrt {e \sin (c+d x)}}+\frac {a e \left (3 a^2+2 b^2\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {b^2-a^2}},\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{8 b d \left (a^2-b^2\right )^2 \left (\sqrt {b^2-a^2}+b\right ) \sqrt {e \sin (c+d x)}} \]
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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 2946
Rubi steps \begin{align*} \text {integral}& = -\frac {b (e \sin (c+d x))^{3/2}}{2 \left (a^2-b^2\right ) d e (a+b \cos (c+d x))^2}-\frac {\int \frac {\left (-2 a+\frac {1}{2} b \cos (c+d x)\right ) \sqrt {e \sin (c+d x)}}{(a+b \cos (c+d x))^2} \, dx}{2 \left (a^2-b^2\right )} \\ & = -\frac {b (e \sin (c+d x))^{3/2}}{2 \left (a^2-b^2\right ) d e (a+b \cos (c+d x))^2}-\frac {5 a b (e \sin (c+d x))^{3/2}}{4 \left (a^2-b^2\right )^2 d e (a+b \cos (c+d x))}+\frac {\int \frac {\left (\frac {1}{2} \left (4 a^2+b^2\right )+\frac {5}{4} a b \cos (c+d x)\right ) \sqrt {e \sin (c+d x)}}{a+b \cos (c+d x)} \, dx}{2 \left (a^2-b^2\right )^2} \\ & = -\frac {b (e \sin (c+d x))^{3/2}}{2 \left (a^2-b^2\right ) d e (a+b \cos (c+d x))^2}-\frac {5 a b (e \sin (c+d x))^{3/2}}{4 \left (a^2-b^2\right )^2 d e (a+b \cos (c+d x))}+\frac {(5 a) \int \sqrt {e \sin (c+d x)} \, dx}{8 \left (a^2-b^2\right )^2}+\frac {\left (3 a^2+2 b^2\right ) \int \frac {\sqrt {e \sin (c+d x)}}{a+b \cos (c+d x)} \, dx}{8 \left (a^2-b^2\right )^2} \\ & = -\frac {b (e \sin (c+d x))^{3/2}}{2 \left (a^2-b^2\right ) d e (a+b \cos (c+d x))^2}-\frac {5 a b (e \sin (c+d x))^{3/2}}{4 \left (a^2-b^2\right )^2 d e (a+b \cos (c+d x))}-\frac {\left (a \left (3 a^2+2 b^2\right ) e\right ) \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {-a^2+b^2}-b \sin (c+d x)\right )} \, dx}{16 b \left (a^2-b^2\right )^2}+\frac {\left (a \left (3 a^2+2 b^2\right ) e\right ) \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {-a^2+b^2}+b \sin (c+d x)\right )} \, dx}{16 b \left (a^2-b^2\right )^2}-\frac {\left (b \left (3 a^2+2 b^2\right ) e\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 )^2 d}+\frac {\left (5 a \sqrt {e \sin (c+d x)}\right ) \int \sqrt {\sin (c+d x)} \, dx}{8 \left (a^2-b^2\right )^2 \sqrt {\sin (c+d x)}} \\ & = \frac {5 a 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 )^2 d \sqrt {\sin (c+d x)}}-\frac {b (e \sin (c+d x))^{3/2}}{2 \left (a^2-b^2\right ) d e (a+b \cos (c+d x))^2}-\frac {5 a b (e \sin (c+d x))^{3/2}}{4 \left (a^2-b^2\right )^2 d e (a+b \cos (c+d x))}-\frac {\left (b \left (3 a^2+2 b^2\right ) e\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 )^2 d}-\frac {\left (a \left (3 a^2+2 b^2\right ) e \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 b \left (a^2-b^2\right )^2 \sqrt {e \sin (c+d x)}}+\frac {\left (a \left (3 a^2+2 b^2\right ) e \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 b \left (a^2-b^2\right )^2 \sqrt {e \sin (c+d x)}} \\ & = \frac {a \left (3 a^2+2 b^2\right ) e \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 b \left (a^2-b^2\right )^2 \left (b-\sqrt {-a^2+b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {a \left (3 a^2+2 b^2\right ) e \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 b \left (a^2-b^2\right )^2 \left (b+\sqrt {-a^2+b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {5 a 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 )^2 d \sqrt {\sin (c+d x)}}-\frac {b (e \sin (c+d x))^{3/2}}{2 \left (a^2-b^2\right ) d e (a+b \cos (c+d x))^2}-\frac {5 a b (e \sin (c+d x))^{3/2}}{4 \left (a^2-b^2\right )^2 d e (a+b \cos (c+d x))}+\frac {\left (\left (3 a^2+2 b^2\right ) e\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 )^2 d}-\frac {\left (\left (3 a^2+2 b^2\right ) e\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 )^2 d} \\ & = -\frac {\left (3 a^2+2 b^2\right ) \sqrt {e} \arctan \left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 \sqrt {b} \left (-a^2+b^2\right )^{9/4} d}+\frac {\left (3 a^2+2 b^2\right ) \sqrt {e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 \sqrt {b} \left (-a^2+b^2\right )^{9/4} d}+\frac {a \left (3 a^2+2 b^2\right ) e \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 b \left (a^2-b^2\right )^2 \left (b-\sqrt {-a^2+b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {a \left (3 a^2+2 b^2\right ) e \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 b \left (a^2-b^2\right )^2 \left (b+\sqrt {-a^2+b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {5 a 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 )^2 d \sqrt {\sin (c+d x)}}-\frac {b (e \sin (c+d x))^{3/2}}{2 \left (a^2-b^2\right ) d e (a+b \cos (c+d x))^2}-\frac {5 a b (e \sin (c+d x))^{3/2}}{4 \left (a^2-b^2\right )^2 d e (a+b \cos (c+d x))} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 12.24 (sec) , antiderivative size = 748, normalized size of antiderivative = 1.41 \[ \int \frac {\sqrt {e \sin (c+d x)}}{(a+b \cos (c+d x))^3} \, dx=\frac {\sqrt {e \sin (c+d x)} \left (-\frac {2 b \left (7 a^2-2 b^2+5 a b \cos (c+d x)\right ) \sin (c+d x)}{\left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}+\frac {\cos (c+d x) \left (a+b \sqrt {\cos ^2(c+d x)}\right ) \left (\frac {5 a \sec (c+d x) \left (3 \sqrt {2} a \left (a^2-b^2\right )^{3/4} \left (2 \arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {\sin (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-2 \arctan \left (1+\frac {\sqrt {2} \sqrt {b} \sqrt {\sin (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-\log \left (\sqrt {a^2-b^2}-\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\sin (c+d x)}+b \sin (c+d x)\right )+\log \left (\sqrt {a^2-b^2}+\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\sin (c+d x)}+b \sin (c+d x)\right )\right )+8 b^{5/2} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{2},1,\frac {7}{4},\sin ^2(c+d x),\frac {b^2 \sin ^2(c+d x)}{-a^2+b^2}\right ) \sin ^{\frac {3}{2}}(c+d x)\right )}{\sqrt {b} \left (-a^2+b^2\right )}+\frac {48 \left (4 a^2+b^2\right ) \left (\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) \left (2 \arctan \left (1-\frac {(1+i) \sqrt {b} \sqrt {\sin (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )-2 \arctan \left (1+\frac {(1+i) \sqrt {b} \sqrt {\sin (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )-\log \left (\sqrt {-a^2+b^2}-(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\sin (c+d x)}+i b \sin (c+d x)\right )+\log \left (\sqrt {-a^2+b^2}+(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\sin (c+d x)}+i b \sin (c+d x)\right )\right )}{\sqrt {b} \sqrt [4]{-a^2+b^2}}+\frac {a \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\sin ^2(c+d x),\frac {b^2 \sin ^2(c+d x)}{-a^2+b^2}\right ) \sin ^{\frac {3}{2}}(c+d x)}{3 \left (a^2-b^2\right )}\right )}{\sqrt {\cos ^2(c+d x)}}\right )}{12 (a-b)^2 (a+b)^2 (a+b \cos (c+d x)) \sqrt {\sin (c+d x)}}\right )}{8 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(2364\) vs. \(2(553)=1106\).
Time = 7.73 (sec) , antiderivative size = 2365, normalized size of antiderivative = 4.47
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Timed out. \[ \int \frac {\sqrt {e \sin (c+d x)}}{(a+b \cos (c+d x))^3} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\sqrt {e \sin (c+d x)}}{(a+b \cos (c+d x))^3} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt {e \sin (c+d x)}}{(a+b \cos (c+d x))^3} \, dx=\int { \frac {\sqrt {e \sin \left (d x + c\right )}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{3}} \,d x } \]
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\[ \int \frac {\sqrt {e \sin (c+d x)}}{(a+b \cos (c+d x))^3} \, dx=\int { \frac {\sqrt {e \sin \left (d x + c\right )}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {e \sin (c+d x)}}{(a+b \cos (c+d x))^3} \, dx=\int \frac {\sqrt {e\,\sin \left (c+d\,x\right )}}{{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3} \,d x \]
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