Integrand size = 25, antiderivative size = 193 \[ \int \frac {3+b \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx=-\frac {2 (b c-3 d) \cos (e+f x)}{\left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}-\frac {2 (b c-3 d) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{d \left (c^2-d^2\right ) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {2 b \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{d f \sqrt {c+d \sin (e+f x)}} \]
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Time = 0.15 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2833, 2831, 2742, 2740, 2734, 2732} \[ \int \frac {3+b \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx=-\frac {2 (b c-a d) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}-\frac {2 (b c-a d) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {2 b \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{d f \sqrt {c+d \sin (e+f x)}} \]
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2831
Rule 2833
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (b c-a d) \cos (e+f x)}{\left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}-\frac {2 \int \frac {\frac {1}{2} (-a c+b d)+\frac {1}{2} (b c-a d) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx}{c^2-d^2} \\ & = -\frac {2 (b c-a d) \cos (e+f x)}{\left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}+\frac {b \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{d}-\frac {(b c-a d) \int \sqrt {c+d \sin (e+f x)} \, dx}{d \left (c^2-d^2\right )} \\ & = -\frac {2 (b c-a d) \cos (e+f x)}{\left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}-\frac {\left ((b c-a d) \sqrt {c+d \sin (e+f x)}\right ) \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}} \, dx}{d \left (c^2-d^2\right ) \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}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}} \, dx}{d \sqrt {c+d \sin (e+f x)}} \\ & = -\frac {2 (b c-a d) \cos (e+f x)}{\left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}-\frac {2 (b c-a 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)}}{d \left (c^2-d^2\right ) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {2 b \operatorname {EllipticF}\left (\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}}}{d f \sqrt {c+d \sin (e+f x)}} \\ \end{align*}
Time = 0.54 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.82 \[ \int \frac {3+b \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx=\frac {2 \left (d (-b c+3 d) \cos (e+f x)+(b c-3 d) (c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}-b \left (c^2-d^2\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right )}{(c-d) d (c+d) f \sqrt {c+d \sin (e+f x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(566\) vs. \(2(251)=502\).
Time = 7.34 (sec) , antiderivative size = 567, normalized size of antiderivative = 2.94
method | result | size |
default | \(\frac {\sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, \left (\frac {2 b \left (\frac {c}{d}-1\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {\frac {d \left (1-\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {\frac {\left (-\sin \left (f x +e \right )-1\right ) d}{c -d}}\, F\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )}{d \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}+\frac {\left (d a -c b \right ) \left (\frac {2 d \left (\cos ^{2}\left (f x +e \right )\right )}{\left (c^{2}-d^{2}\right ) \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}+\frac {2 c \left (\frac {c}{d}-1\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {\frac {d \left (1-\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {\frac {\left (-\sin \left (f x +e \right )-1\right ) d}{c -d}}\, F\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )}{\left (c^{2}-d^{2}\right ) \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}+\frac {2 d \left (\frac {c}{d}-1\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {\frac {d \left (1-\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {\frac {\left (-\sin \left (f x +e \right )-1\right ) d}{c -d}}\, \left (\left (-\frac {c}{d}-1\right ) E\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )+F\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )\right )}{\left (c^{2}-d^{2}\right ) \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}\right )}{d}\right )}{\cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}\) | \(567\) |
parts | \(\frac {2 a \left (c^{2} \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (\sin \left (f x +e \right )+1\right )}{c -d}}\, F\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )-\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (\sin \left (f x +e \right )+1\right )}{c -d}}\, F\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) d^{2}-\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (\sin \left (f x +e \right )+1\right )}{c -d}}\, E\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c^{2}+\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (\sin \left (f x +e \right )+1\right )}{c -d}}\, E\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) d^{2}-\left (\sin ^{2}\left (f x +e \right )\right ) d^{2}+d^{2}\right )}{d \left (c^{2}-d^{2}\right ) \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}-\frac {2 b \left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (\sin \left (f x +e \right )+1\right )}{c -d}}\, F\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c^{2} d -\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (\sin \left (f x +e \right )+1\right )}{c -d}}\, F\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) d^{3}-\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (\sin \left (f x +e \right )+1\right )}{c -d}}\, E\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c^{3}+\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (\sin \left (f x +e \right )+1\right )}{c -d}}\, E\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c \,d^{2}-\left (\sin ^{2}\left (f x +e \right )\right ) c \,d^{2}+c \,d^{2}\right )}{d^{2} \left (c^{2}-d^{2}\right ) \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}\) | \(893\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 614, normalized size of antiderivative = 3.18 \[ \int \frac {3+b \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx=-\frac {6 \, {\left (b c d^{2} - a d^{3}\right )} \sqrt {d \sin \left (f x + e\right ) + c} \cos \left (f x + e\right ) - {\left (\sqrt {2} {\left (2 \, b c^{2} d + a c d^{2} - 3 \, b d^{3}\right )} \sin \left (f x + e\right ) + \sqrt {2} {\left (2 \, b c^{3} + a c^{2} d - 3 \, b c d^{2}\right )}\right )} \sqrt {i \, d} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) - 3 i \, d \sin \left (f x + e\right ) - 2 i \, c}{3 \, d}\right ) - {\left (\sqrt {2} {\left (2 \, b c^{2} d + a c d^{2} - 3 \, b d^{3}\right )} \sin \left (f x + e\right ) + \sqrt {2} {\left (2 \, b c^{3} + a c^{2} d - 3 \, b c d^{2}\right )}\right )} \sqrt {-i \, d} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) + 3 i \, d \sin \left (f x + e\right ) + 2 i \, c}{3 \, d}\right ) - 3 \, {\left (\sqrt {2} {\left (i \, b c d^{2} - i \, a d^{3}\right )} \sin \left (f x + e\right ) + \sqrt {2} {\left (i \, b c^{2} d - i \, a c d^{2}\right )}\right )} \sqrt {i \, d} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) - 3 i \, d \sin \left (f x + e\right ) - 2 i \, c}{3 \, d}\right )\right ) - 3 \, {\left (\sqrt {2} {\left (-i \, b c d^{2} + i \, a d^{3}\right )} \sin \left (f x + e\right ) + \sqrt {2} {\left (-i \, b c^{2} d + i \, a c d^{2}\right )}\right )} \sqrt {-i \, d} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) + 3 i \, d \sin \left (f x + e\right ) + 2 i \, c}{3 \, d}\right )\right )}{3 \, {\left ({\left (c^{2} d^{3} - d^{5}\right )} f \sin \left (f x + e\right ) + {\left (c^{3} d^{2} - c d^{4}\right )} f\right )}} \]
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Timed out. \[ \int \frac {3+b \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {3+b \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx=\int { \frac {b \sin \left (f x + e\right ) + a}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {3+b \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx=\int { \frac {b \sin \left (f x + e\right ) + a}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {3+b \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx=\int \frac {a+b\,\sin \left (e+f\,x\right )}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]
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