Integrand size = 27, antiderivative size = 217 \[ \int \frac {1}{(3+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}} \, dx=-\frac {2 d^2 \cos (e+f x)}{(b c-3 d) \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}-\frac {2 d E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{(b c-3 d) \left (c^2-d^2\right ) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {2 b \operatorname {EllipticPi}\left (\frac {2 b}{3+b},\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{(3+b) (b c-3 d) f \sqrt {c+d \sin (e+f x)}} \]
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Time = 0.41 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2881, 3138, 2734, 2732, 12, 2886, 2884} \[ \int \frac {1}{(3+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}} \, dx=-\frac {2 d^2 \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}-\frac {2 d \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {2 b \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{f (a+b) (b c-a d) \sqrt {c+d \sin (e+f x)}} \]
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Rule 12
Rule 2732
Rule 2734
Rule 2881
Rule 2884
Rule 2886
Rule 3138
Rubi steps \begin{align*} \text {integral}& = -\frac {2 d^2 \cos (e+f x)}{(b c-a d) \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}+\frac {2 \int \frac {\frac {1}{2} \left (-a c d+b \left (c^2-d^2\right )\right )-\frac {1}{2} d (b c+a d) \sin (e+f x)-\frac {1}{2} b d^2 \sin ^2(e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx}{(b c-a d) \left (c^2-d^2\right )} \\ & = -\frac {2 d^2 \cos (e+f x)}{(b c-a d) \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}-\frac {2 \int -\frac {b^2 d \left (c^2-d^2\right )}{2 (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx}{b d (b c-a d) \left (c^2-d^2\right )}-\frac {d \int \sqrt {c+d \sin (e+f x)} \, dx}{(b c-a d) \left (c^2-d^2\right )} \\ & = -\frac {2 d^2 \cos (e+f x)}{(b c-a d) \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}+\frac {b \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx}{b c-a d}-\frac {\left (d \sqrt {c+d \sin (e+f x)}\right ) \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}} \, dx}{(b c-a d) \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}} \\ & = -\frac {2 d^2 \cos (e+f x)}{(b c-a d) \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}-\frac {2 d E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{(b c-a d) \left (c^2-d^2\right ) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left (b \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right ) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}} \, dx}{(b c-a d) \sqrt {c+d \sin (e+f x)}} \\ & = -\frac {2 d^2 \cos (e+f x)}{(b c-a d) \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}-\frac {2 d E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{(b c-a d) \left (c^2-d^2\right ) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {2 b \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{(a+b) (b c-a d) f \sqrt {c+d \sin (e+f x)}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 4.72 (sec) , antiderivative size = 603, normalized size of antiderivative = 2.78 \[ \int \frac {1}{(3+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}} \, dx=-\frac {\frac {4 d^2 \cos (e+f x)}{\left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}+\frac {-\frac {4 i (b c+3 d) \left ((b c-3 d) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )+3 d \operatorname {EllipticPi}\left (\frac {b (c+d)}{b c-3 d},i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )\right ) \sec (e+f x) \sqrt {-\frac {d (-1+\sin (e+f x))}{c+d}} \sqrt {\frac {d (1+\sin (e+f x))}{-c+d}}}{b (b c-3 d) \sqrt {-\frac {1}{c+d}}}-\frac {2 i \left (-2 b (b c-3 d) (c-d) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right )|\frac {c+d}{c-d}\right )+d \left (-2 (3+b) (b c-3 d) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )+\left (-18+b^2\right ) d \operatorname {EllipticPi}\left (\frac {b (c+d)}{b c-3 d},i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )\right )\right ) \sec (e+f x) \sqrt {-\frac {d (-1+\sin (e+f x))}{c+d}} \sqrt {\frac {d (1+\sin (e+f x))}{-c+d}}}{b (b c-3 d) \sqrt {-\frac {1}{c+d}}}+\frac {2 \left (2 b c^2-6 c d-3 b d^2\right ) \operatorname {EllipticPi}\left (\frac {2 b}{3+b},\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{(3+b) \sqrt {c+d \sin (e+f x)}}}{(c-d) (c+d)}}{2 (b c-3 d) f} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(609\) vs. \(2(276)=552\).
Time = 3.29 (sec) , antiderivative size = 610, normalized size of antiderivative = 2.81
method | result | size |
default | \(\frac {\sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, \left (\frac {d \left (\frac {2 d \left (\cos ^{2}\left (f x +e \right )\right )}{\left (c^{2}-d^{2}\right ) \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}+\frac {2 c \left (\frac {c}{d}-1\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {\frac {d \left (1-\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {\frac {\left (-\sin \left (f x +e \right )-1\right ) d}{c -d}}\, F\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )}{\left (c^{2}-d^{2}\right ) \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}+\frac {2 d \left (\frac {c}{d}-1\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {\frac {d \left (1-\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {\frac {\left (-\sin \left (f x +e \right )-1\right ) d}{c -d}}\, \left (\left (-\frac {c}{d}-1\right ) E\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )+F\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )\right )}{\left (c^{2}-d^{2}\right ) \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}\right )}{d a -c b}-\frac {2 \left (\frac {c}{d}-1\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {\frac {d \left (1-\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {\frac {\left (-\sin \left (f x +e \right )-1\right ) d}{c -d}}\, \Pi \left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \frac {-\frac {c}{d}+1}{-\frac {c}{d}+\frac {a}{b}}, \sqrt {\frac {c -d}{c +d}}\right )}{\left (d a -c b \right ) \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, \left (-\frac {c}{d}+\frac {a}{b}\right )}\right )}{\cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}\) | \(610\) |
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Timed out. \[ \int \frac {1}{(3+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{(3+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}} \, dx=\int \frac {1}{\left (a + b \sin {\left (e + f x \right )}\right ) \left (c + d \sin {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {1}{(3+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}} \, dx=\int { \frac {1}{{\left (b \sin \left (f x + e\right ) + a\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {1}{(3+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}} \, dx=\int { \frac {1}{{\left (b \sin \left (f x + e\right ) + a\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(3+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}} \, dx=\int \frac {1}{\left (a+b\,\sin \left (e+f\,x\right )\right )\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]
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