Integrand size = 27, antiderivative size = 227 \[ \int (3+3 \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)} \, dx=\frac {12 (c-5 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 d f}-\frac {18 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}-\frac {12 \left (c^2-5 c d-12 d^2\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{5 d^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {12 (c-5 d) \left (c^2-d^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{5 d^2 f \sqrt {c+d \sin (e+f x)}} \]
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Time = 0.27 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.05, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2842, 2832, 2831, 2742, 2740, 2734, 2732} \[ \int (3+3 \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)} \, dx=\frac {4 a^2 (c-5 d) \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{15 d^2 f \sqrt {c+d \sin (e+f x)}}-\frac {4 a^2 \left (c^2-5 c d-12 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{15 d^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}+\frac {4 a^2 (c-5 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{15 d f} \]
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2831
Rule 2832
Rule 2842
Rubi steps \begin{align*} \text {integral}& = -\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}+\frac {2 \int \left (4 a^2 d-a^2 (c-5 d) \sin (e+f x)\right ) \sqrt {c+d \sin (e+f x)} \, dx}{5 d} \\ & = \frac {4 a^2 (c-5 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{15 d f}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}+\frac {4 \int \frac {\frac {1}{2} a^2 d (11 c+5 d)-\frac {1}{2} a^2 \left (c^2-5 c d-12 d^2\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx}{15 d} \\ & = \frac {4 a^2 (c-5 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{15 d f}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}-\frac {\left (2 a^2 \left (c^2-5 c d-12 d^2\right )\right ) \int \sqrt {c+d \sin (e+f x)} \, dx}{15 d^2}+\frac {\left (2 a^2 (c-5 d) \left (c^2-d^2\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{15 d^2} \\ & = \frac {4 a^2 (c-5 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{15 d f}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}-\frac {\left (2 a^2 \left (c^2-5 c d-12 d^2\right ) \sqrt {c+d \sin (e+f x)}\right ) \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}} \, dx}{15 d^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left (2 a^2 (c-5 d) \left (c^2-d^2\right ) \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}{15 d^2 \sqrt {c+d \sin (e+f x)}} \\ & = \frac {4 a^2 (c-5 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{15 d f}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}-\frac {4 a^2 \left (c^2-5 c d-12 d^2\right ) 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)}}{15 d^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {4 a^2 (c-5 d) \left (c^2-d^2\right ) \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}}}{15 d^2 f \sqrt {c+d \sin (e+f x)}} \\ \end{align*}
Time = 0.74 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.91 \[ \int (3+3 \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)} \, dx=\frac {3 \left (4 \left (c^3-4 c^2 d-17 c d^2-12 d^3\right ) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}-4 \left (c^3-5 c^2 d-c d^2+5 d^3\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}+d \cos (e+f x) \left (-2 c^2-20 c d-3 d^2+3 d^2 \cos (2 (e+f x))-4 d (2 c+5 d) \sin (e+f x)\right )\right )}{5 d^2 f \sqrt {c+d \sin (e+f x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1034\) vs. \(2(285)=570\).
Time = 4.90 (sec) , antiderivative size = 1035, normalized size of antiderivative = 4.56
method | result | size |
default | \(\text {Expression too large to display}\) | \(1035\) |
parts | \(\text {Expression too large to display}\) | \(1499\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 542, normalized size of antiderivative = 2.39 \[ \int (3+3 \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)} \, dx=\frac {2 \, {\left (\sqrt {2} {\left (2 \, a^{2} c^{3} - 10 \, a^{2} c^{2} d + 9 \, a^{2} c d^{2} + 15 \, a^{2} d^{3}\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 ) + \sqrt {2} {\left (2 \, a^{2} c^{3} - 10 \, a^{2} c^{2} d + 9 \, a^{2} c d^{2} + 15 \, a^{2} d^{3}\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 \, \sqrt {2} {\left (-i \, a^{2} c^{2} d + 5 i \, a^{2} c d^{2} + 12 i \, a^{2} d^{3}\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 \, \sqrt {2} {\left (i \, a^{2} c^{2} d - 5 i \, a^{2} c d^{2} - 12 i \, a^{2} d^{3}\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 (3 \, a^{2} d^{3} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + {\left (a^{2} c d^{2} + 10 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )\right )} \sqrt {d \sin \left (f x + e\right ) + c}\right )}}{45 \, d^{3} f} \]
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\[ \int (3+3 \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)} \, dx=a^{2} \left (\int 2 \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )}\, dx + \int \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )}\, dx + \int \sqrt {c + d \sin {\left (e + f x \right )}}\, dx\right ) \]
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\[ \int (3+3 \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{2} \sqrt {d \sin \left (f x + e\right ) + c} \,d x } \]
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\[ \int (3+3 \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{2} \sqrt {d \sin \left (f x + e\right ) + c} \,d x } \]
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Timed out. \[ \int (3+3 \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)} \, dx=\int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^2\,\sqrt {c+d\,\sin \left (e+f\,x\right )} \,d x \]
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