Integrand size = 25, antiderivative size = 507 \[ \int \frac {1}{(a+b \cos (c+d x))^2 (e \sin (c+d x))^{3/2}} \, dx=\frac {5 a b^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{2 \left (-a^2+b^2\right )^{9/4} d e^{3/2}}-\frac {5 a b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{2 \left (-a^2+b^2\right )^{9/4} d e^{3/2}}-\frac {b}{\left (a^2-b^2\right ) d e (a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}}+\frac {5 a b-\left (2 a^2+3 b^2\right ) \cos (c+d x)}{\left (a^2-b^2\right )^2 d e \sqrt {e \sin (c+d x)}}-\frac {5 a^2 b \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{2 \left (a^2-b^2\right )^2 \left (b-\sqrt {-a^2+b^2}\right ) d e \sqrt {e \sin (c+d x)}}-\frac {5 a^2 b \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{2 \left (a^2-b^2\right )^2 \left (b+\sqrt {-a^2+b^2}\right ) d e \sqrt {e \sin (c+d x)}}-\frac {\left (2 a^2+3 b^2\right ) E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{\left (a^2-b^2\right )^2 d e^2 \sqrt {\sin (c+d x)}} \]
[Out]
Time = 1.68 (sec) , antiderivative size = 507, 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, 2945, 2946, 2721, 2719, 2780, 2886, 2884, 335, 304, 211, 214} \[ \int \frac {1}{(a+b \cos (c+d x))^2 (e \sin (c+d x))^{3/2}} \, dx=\frac {5 a b^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{2 d e^{3/2} \left (b^2-a^2\right )^{9/4}}-\frac {5 a b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{2 d e^{3/2} \left (b^2-a^2\right )^{9/4}}-\frac {\left (2 a^2+3 b^2\right ) E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d e^2 \left (a^2-b^2\right )^2 \sqrt {\sin (c+d x)}}+\frac {5 a b-\left (2 a^2+3 b^2\right ) \cos (c+d x)}{d e \left (a^2-b^2\right )^2 \sqrt {e \sin (c+d x)}}-\frac {b}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))}-\frac {5 a^2 b \sqrt {\sin (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {b^2-a^2}},\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{2 d e \left (a^2-b^2\right )^2 \left (b-\sqrt {b^2-a^2}\right ) \sqrt {e \sin (c+d x)}}-\frac {5 a^2 b \sqrt {\sin (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {b^2-a^2}},\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{2 d e \left (a^2-b^2\right )^2 \left (\sqrt {b^2-a^2}+b\right ) \sqrt {e \sin (c+d x)}} \]
[In]
[Out]
Rule 211
Rule 214
Rule 304
Rule 335
Rule 2719
Rule 2721
Rule 2773
Rule 2780
Rule 2884
Rule 2886
Rule 2945
Rule 2946
Rubi steps \begin{align*} \text {integral}& = -\frac {b}{\left (a^2-b^2\right ) d e (a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}}+\frac {\int \frac {-a+\frac {3}{2} b \cos (c+d x)}{(a+b \cos (c+d x)) (e \sin (c+d x))^{3/2}} \, dx}{-a^2+b^2} \\ & = -\frac {b}{\left (a^2-b^2\right ) d e (a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}}+\frac {5 a b-\left (2 a^2+3 b^2\right ) \cos (c+d x)}{\left (a^2-b^2\right )^2 d e \sqrt {e \sin (c+d x)}}+\frac {2 \int \frac {\left (-\frac {1}{2} a \left (a^2+4 b^2\right )-\frac {1}{4} b \left (2 a^2+3 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 )^2 e^2} \\ & = -\frac {b}{\left (a^2-b^2\right ) d e (a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}}+\frac {5 a b-\left (2 a^2+3 b^2\right ) \cos (c+d x)}{\left (a^2-b^2\right )^2 d e \sqrt {e \sin (c+d x)}}-\frac {\left (5 a b^2\right ) \int \frac {\sqrt {e \sin (c+d x)}}{a+b \cos (c+d x)} \, dx}{2 \left (a^2-b^2\right )^2 e^2}-\frac {\left (2 a^2+3 b^2\right ) \int \sqrt {e \sin (c+d x)} \, dx}{2 \left (a^2-b^2\right )^2 e^2} \\ & = -\frac {b}{\left (a^2-b^2\right ) d e (a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}}+\frac {5 a b-\left (2 a^2+3 b^2\right ) \cos (c+d x)}{\left (a^2-b^2\right )^2 d e \sqrt {e \sin (c+d x)}}+\frac {\left (5 a^2 b\right ) \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {-a^2+b^2}-b \sin (c+d x)\right )} \, dx}{4 \left (a^2-b^2\right )^2 e}-\frac {\left (5 a^2 b\right ) \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {-a^2+b^2}+b \sin (c+d x)\right )} \, dx}{4 \left (a^2-b^2\right )^2 e}+\frac {\left (5 a b^3\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 )}{2 \left (a^2-b^2\right )^2 d e}-\frac {\left (\left (2 a^2+3 b^2\right ) \sqrt {e \sin (c+d x)}\right ) \int \sqrt {\sin (c+d x)} \, dx}{2 \left (a^2-b^2\right )^2 e^2 \sqrt {\sin (c+d x)}} \\ & = -\frac {b}{\left (a^2-b^2\right ) d e (a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}}+\frac {5 a b-\left (2 a^2+3 b^2\right ) \cos (c+d x)}{\left (a^2-b^2\right )^2 d e \sqrt {e \sin (c+d x)}}-\frac {\left (2 a^2+3 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)}}{\left (a^2-b^2\right )^2 d e^2 \sqrt {\sin (c+d x)}}+\frac {\left (5 a b^3\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 )}{\left (a^2-b^2\right )^2 d e}+\frac {\left (5 a^2 b \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}{4 \left (a^2-b^2\right )^2 e \sqrt {e \sin (c+d x)}}-\frac {\left (5 a^2 b \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}{4 \left (a^2-b^2\right )^2 e \sqrt {e \sin (c+d x)}} \\ & = -\frac {b}{\left (a^2-b^2\right ) d e (a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}}+\frac {5 a b-\left (2 a^2+3 b^2\right ) \cos (c+d x)}{\left (a^2-b^2\right )^2 d e \sqrt {e \sin (c+d x)}}-\frac {5 a^2 b \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)}}{2 \left (a^2-b^2\right )^2 \left (b-\sqrt {-a^2+b^2}\right ) d e \sqrt {e \sin (c+d x)}}-\frac {5 a^2 b \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)}}{2 \left (a^2-b^2\right )^2 \left (b+\sqrt {-a^2+b^2}\right ) d e \sqrt {e \sin (c+d x)}}-\frac {\left (2 a^2+3 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)}}{\left (a^2-b^2\right )^2 d e^2 \sqrt {\sin (c+d x)}}-\frac {\left (5 a b^2\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 )}{2 \left (a^2-b^2\right )^2 d e}+\frac {\left (5 a b^2\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 )}{2 \left (a^2-b^2\right )^2 d e} \\ & = \frac {5 a b^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{2 \left (-a^2+b^2\right )^{9/4} d e^{3/2}}-\frac {5 a b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{2 \left (-a^2+b^2\right )^{9/4} d e^{3/2}}-\frac {b}{\left (a^2-b^2\right ) d e (a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}}+\frac {5 a b-\left (2 a^2+3 b^2\right ) \cos (c+d x)}{\left (a^2-b^2\right )^2 d e \sqrt {e \sin (c+d x)}}-\frac {5 a^2 b \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)}}{2 \left (a^2-b^2\right )^2 \left (b-\sqrt {-a^2+b^2}\right ) d e \sqrt {e \sin (c+d x)}}-\frac {5 a^2 b \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)}}{2 \left (a^2-b^2\right )^2 \left (b+\sqrt {-a^2+b^2}\right ) d e \sqrt {e \sin (c+d x)}}-\frac {\left (2 a^2+3 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)}}{\left (a^2-b^2\right )^2 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 = 6.54 (sec) , antiderivative size = 774, normalized size of antiderivative = 1.53 \[ \int \frac {1}{(a+b \cos (c+d x))^2 (e \sin (c+d x))^{3/2}} \, dx=-\frac {\sin ^{\frac {3}{2}}(c+d x) \left (\frac {12 \left (-6 a^2 b+b^3+4 a \left (a^2-b^2\right ) \cos (c+d x)+b \left (2 a^2+3 b^2\right ) \cos (2 (c+d x))\right )}{\left (a^2-b^2\right )^2 \sqrt {\sin (c+d x)}}+\frac {\cos (c+d x) \left (a+b \sqrt {\cos ^2(c+d x)}\right ) \left (\frac {\left (2 a^2+3 b^2\right ) \sec (c+d x) \left (3 \sqrt {2} a \left (a^2-b^2\right )^{3/4} \left (2 \arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {\sin (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-2 \arctan \left (1+\frac {\sqrt {2} \sqrt {b} \sqrt {\sin (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-\log \left (\sqrt {a^2-b^2}-\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\sin (c+d x)}+b \sin (c+d x)\right )+\log \left (\sqrt {a^2-b^2}+\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\sin (c+d x)}+b \sin (c+d x)\right )\right )+8 b^{5/2} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{2},1,\frac {7}{4},\sin ^2(c+d x),\frac {b^2 \sin ^2(c+d x)}{-a^2+b^2}\right ) \sin ^{\frac {3}{2}}(c+d x)\right )}{\sqrt {b} \left (-a^2+b^2\right )}+\frac {48 a \left (a^2+4 b^2\right ) \left (\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) \left (2 \arctan \left (1-\frac {(1+i) \sqrt {b} \sqrt {\sin (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )-2 \arctan \left (1+\frac {(1+i) \sqrt {b} \sqrt {\sin (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )-\log \left (\sqrt {-a^2+b^2}-(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\sin (c+d x)}+i b \sin (c+d x)\right )+\log \left (\sqrt {-a^2+b^2}+(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\sin (c+d x)}+i b \sin (c+d x)\right )\right )}{\sqrt {b} \sqrt [4]{-a^2+b^2}}+\frac {a \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\sin ^2(c+d x),\frac {b^2 \sin ^2(c+d x)}{-a^2+b^2}\right ) \sin ^{\frac {3}{2}}(c+d x)}{3 \left (a^2-b^2\right )}\right )}{\sqrt {\cos ^2(c+d x)}}\right )}{(a-b)^2 (a+b)^2}\right )}{24 d (a+b \cos (c+d x)) (e \sin (c+d x))^{3/2}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(2001\) vs. \(2(536)=1072\).
Time = 4.67 (sec) , antiderivative size = 2002, normalized size of antiderivative = 3.95
[In]
[Out]
Timed out. \[ \int \frac {1}{(a+b \cos (c+d x))^2 (e \sin (c+d x))^{3/2}} \, dx=\text {Timed out} \]
[In]
[Out]
Timed out. \[ \int \frac {1}{(a+b \cos (c+d x))^2 (e \sin (c+d x))^{3/2}} \, dx=\text {Timed out} \]
[In]
[Out]
Timed out. \[ \int \frac {1}{(a+b \cos (c+d x))^2 (e \sin (c+d x))^{3/2}} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {1}{(a+b \cos (c+d x))^2 (e \sin (c+d x))^{3/2}} \, dx=\int { \frac {1}{{\left (b \cos \left (d x + c\right ) + a\right )}^{2} \left (e \sin \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {1}{(a+b \cos (c+d x))^2 (e \sin (c+d x))^{3/2}} \, dx=\int \frac {1}{{\left (e\,\sin \left (c+d\,x\right )\right )}^{3/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^2} \,d x \]
[In]
[Out]