Integrand size = 27, antiderivative size = 180 \[ \int \frac {(3+3 \sin (e+f x))^2}{(c+d \sin (e+f x))^{3/2}} \, dx=\frac {18 (c-d) \cos (e+f x)}{d (c+d) f \sqrt {c+d \sin (e+f x)}}+\frac {36 c E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{d^2 (c+d) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {36 (c-d) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{d^2 f \sqrt {c+d \sin (e+f x)}} \]
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Time = 0.17 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2841, 2831, 2742, 2740, 2734, 2732} \[ \int \frac {(3+3 \sin (e+f x))^2}{(c+d \sin (e+f x))^{3/2}} \, dx=-\frac {4 a^2 (c-d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{d^2 f \sqrt {c+d \sin (e+f x)}}+\frac {4 a^2 c \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d^2 f (c+d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {2 a^2 (c-d) \cos (e+f x)}{d f (c+d) \sqrt {c+d \sin (e+f x)}} \]
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2831
Rule 2841
Rubi steps \begin{align*} \text {integral}& = \frac {2 a^2 (c-d) \cos (e+f x)}{d (c+d) f \sqrt {c+d \sin (e+f x)}}-\frac {(2 a) \int \frac {-a d-a c \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx}{d (c+d)} \\ & = \frac {2 a^2 (c-d) \cos (e+f x)}{d (c+d) f \sqrt {c+d \sin (e+f x)}}-\frac {\left (2 a^2 (c-d)\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{d^2}+\frac {\left (2 a^2 c\right ) \int \sqrt {c+d \sin (e+f x)} \, dx}{d^2 (c+d)} \\ & = \frac {2 a^2 (c-d) \cos (e+f x)}{d (c+d) f \sqrt {c+d \sin (e+f x)}}+\frac {\left (2 a^2 c \sqrt {c+d \sin (e+f x)}\right ) \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}} \, dx}{d^2 (c+d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (2 a^2 (c-d) \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^2 \sqrt {c+d \sin (e+f x)}} \\ & = \frac {2 a^2 (c-d) \cos (e+f x)}{d (c+d) f \sqrt {c+d \sin (e+f x)}}+\frac {4 a^2 c 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^2 (c+d) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {4 a^2 (c-d) \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^2 f \sqrt {c+d \sin (e+f x)}} \\ \end{align*}
Time = 0.53 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.77 \[ \int \frac {(3+3 \sin (e+f x))^2}{(c+d \sin (e+f x))^{3/2}} \, dx=\frac {9 \left (-4 c (c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}+2 (c-d) \left (d \cos (e+f x)+2 (c+d) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right )\right )}{d^2 (c+d) f \sqrt {c+d \sin (e+f x)}} \]
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Time = 3.80 (sec) , antiderivative size = 464, normalized size of antiderivative = 2.58
method | result | size |
default | \(\frac {2 \left (2 \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (\sin \left (f x +e \right )+1\right )}{c -d}}\, F\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c^{2} d -2 \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (\sin \left (f x +e \right )+1\right )}{c -d}}\, F\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) d^{3}-2 \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (\sin \left (f x +e \right )+1\right )}{c -d}}\, E\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c^{3}+2 \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (\sin \left (f x +e \right )+1\right )}{c -d}}\, E\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c \,d^{2}-\left (\sin ^{2}\left (f x +e \right )\right ) c \,d^{2}+d^{3} \left (\sin ^{2}\left (f x +e \right )\right )+c \,d^{2}-d^{3}\right ) a^{2}}{d^{3} \left (c +d \right ) \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}\) | \(464\) |
parts | \(\text {Expression too large to display}\) | \(1649\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 579, normalized size of antiderivative = 3.22 \[ \int \frac {(3+3 \sin (e+f x))^2}{(c+d \sin (e+f x))^{3/2}} \, dx=\frac {2 \, {\left (3 \, {\left (a^{2} c d^{2} - a^{2} d^{3}\right )} \sqrt {d \sin \left (f x + e\right ) + c} \cos \left (f x + e\right ) - {\left (\sqrt {2} {\left (2 \, a^{2} c^{2} d - 3 \, a^{2} d^{3}\right )} \sin \left (f x + e\right ) + \sqrt {2} {\left (2 \, a^{2} c^{3} - 3 \, a^{2} 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 \, a^{2} c^{2} d - 3 \, a^{2} d^{3}\right )} \sin \left (f x + e\right ) + \sqrt {2} {\left (2 \, a^{2} c^{3} - 3 \, a^{2} 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 (i \, \sqrt {2} a^{2} c d^{2} \sin \left (f x + e\right ) + i \, \sqrt {2} a^{2} c^{2} d\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 (-i \, \sqrt {2} a^{2} c d^{2} \sin \left (f x + e\right ) - i \, \sqrt {2} a^{2} c^{2} d\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 )\right )}}{3 \, {\left ({\left (c d^{4} + d^{5}\right )} f \sin \left (f x + e\right ) + {\left (c^{2} d^{3} + c d^{4}\right )} f\right )}} \]
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Timed out. \[ \int \frac {(3+3 \sin (e+f x))^2}{(c+d \sin (e+f x))^{3/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(3+3 \sin (e+f x))^2}{(c+d \sin (e+f x))^{3/2}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{2}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(3+3 \sin (e+f x))^2}{(c+d \sin (e+f x))^{3/2}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{2}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(3+3 \sin (e+f x))^2}{(c+d \sin (e+f x))^{3/2}} \, dx=\int \frac {{\left (a+a\,\sin \left (e+f\,x\right )\right )}^2}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]
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