Integrand size = 27, antiderivative size = 333 \[ \int (3+b \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2} \, dx=-\frac {2 \left (-6 b c (b c-21 d)+5 \left (63+5 b^2\right ) d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{105 d f}+\frac {4 b (b c-21 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d f}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}+\frac {4 \left (630 c d^2+63 b d \left (c^2+3 d^2\right )-b^2 \left (3 c^3-41 c d^2\right )\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{105 d^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 \left (c^2-d^2\right ) \left (126 b c d+315 d^2-b^2 \left (6 c^2-25 d^2\right )\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{105 d^2 f \sqrt {c+d \sin (e+f x)}} \]
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Time = 0.43 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2870, 2832, 2831, 2742, 2740, 2734, 2732} \[ \int (3+b \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2} \, dx=-\frac {2 \left (c^2-d^2\right ) \left (35 a^2 d^2+42 a b c d-\left (b^2 \left (6 c^2-25 d^2\right )\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{105 d^2 f \sqrt {c+d \sin (e+f x)}}+\frac {4 \left (70 a^2 c d^2+21 a b d \left (c^2+3 d^2\right )-\left (b^2 \left (3 c^3-41 c d^2\right )\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{105 d^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 \left (5 d^2 \left (7 a^2+5 b^2\right )-6 b c (b c-7 a d)\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{105 d f}+\frac {4 b (b c-7 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d f}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f} \]
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2831
Rule 2832
Rule 2870
Rubi steps \begin{align*} \text {integral}& = -\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}+\frac {2 \int (c+d \sin (e+f x))^{3/2} \left (\frac {1}{2} \left (7 a^2+5 b^2\right ) d-b (b c-7 a d) \sin (e+f x)\right ) \, dx}{7 d} \\ & = \frac {4 b (b c-7 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d f}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}+\frac {4 \int \sqrt {c+d \sin (e+f x)} \left (\frac {1}{4} d \left (35 a^2 c+19 b^2 c+42 a b d\right )+\frac {1}{4} \left (5 \left (7 a^2+5 b^2\right ) d^2-6 b c (b c-7 a d)\right ) \sin (e+f x)\right ) \, dx}{35 d} \\ & = -\frac {2 \left (5 \left (7 a^2+5 b^2\right ) d^2-6 b c (b c-7 a d)\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{105 d f}+\frac {4 b (b c-7 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d f}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}+\frac {8 \int \frac {\frac {1}{8} d \left (168 a b c d+35 a^2 \left (3 c^2+d^2\right )+b^2 \left (51 c^2+25 d^2\right )\right )+\frac {1}{4} \left (70 a^2 c d^2+21 a b d \left (c^2+3 d^2\right )-b^2 \left (3 c^3-41 c d^2\right )\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx}{105 d} \\ & = -\frac {2 \left (5 \left (7 a^2+5 b^2\right ) d^2-6 b c (b c-7 a d)\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{105 d f}+\frac {4 b (b c-7 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d f}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}-\frac {\left (\left (c^2-d^2\right ) \left (42 a b c d+35 a^2 d^2-b^2 \left (6 c^2-25 d^2\right )\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{105 d^2}+\frac {\left (2 \left (70 a^2 c d^2+21 a b d \left (c^2+3 d^2\right )-b^2 \left (3 c^3-41 c d^2\right )\right )\right ) \int \sqrt {c+d \sin (e+f x)} \, dx}{105 d^2} \\ & = -\frac {2 \left (5 \left (7 a^2+5 b^2\right ) d^2-6 b c (b c-7 a d)\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{105 d f}+\frac {4 b (b c-7 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d f}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}+\frac {\left (2 \left (70 a^2 c d^2+21 a b d \left (c^2+3 d^2\right )-b^2 \left (3 c^3-41 c d^2\right )\right ) \sqrt {c+d \sin (e+f x)}\right ) \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}} \, dx}{105 d^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (\left (c^2-d^2\right ) \left (42 a b c d+35 a^2 d^2-b^2 \left (6 c^2-25 d^2\right )\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}{105 d^2 \sqrt {c+d \sin (e+f x)}} \\ & = -\frac {2 \left (5 \left (7 a^2+5 b^2\right ) d^2-6 b c (b c-7 a d)\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{105 d f}+\frac {4 b (b c-7 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d f}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}+\frac {4 \left (70 a^2 c d^2+21 a b d \left (c^2+3 d^2\right )-b^2 \left (3 c^3-41 c d^2\right )\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)}}{105 d^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 \left (c^2-d^2\right ) \left (42 a b c d+35 a^2 d^2-b^2 \left (6 c^2-25 d^2\right )\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}}}{105 d^2 f \sqrt {c+d \sin (e+f x)}} \\ \end{align*}
Time = 2.36 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.83 \[ \int (3+b \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2} \, dx=\frac {4 \left (-d^2 \left (\left (945+51 b^2\right ) c^2+504 b c d+5 \left (63+5 b^2\right ) d^2\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )+2 \left (-630 c d^2-63 b d \left (c^2+3 d^2\right )+b^2 \left (3 c^3-41 c d^2\right )\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}-d (c+d \sin (e+f x)) \left (\left (1008 b c d+1260 d^2+b^2 \left (12 c^2+115 d^2\right )\right ) \cos (e+f x)+3 b d (-5 b d \cos (3 (e+f x))+4 (4 b c+21 d) \sin (2 (e+f x)))\right )}{210 d^2 f \sqrt {c+d \sin (e+f x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1574\) vs. \(2(389)=778\).
Time = 13.22 (sec) , antiderivative size = 1575, normalized size of antiderivative = 4.73
method | result | size |
default | \(\text {Expression too large to display}\) | \(1575\) |
parts | \(\text {Expression too large to display}\) | \(2397\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 671, normalized size of antiderivative = 2.02 \[ \int (3+b \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2} \, dx=\frac {\sqrt {2} {\left (12 \, b^{2} c^{4} - 84 \, a b c^{3} d + 252 \, a b c d^{3} + {\left (35 \, a^{2} - 11 \, b^{2}\right )} c^{2} d^{2} + 15 \, {\left (7 \, a^{2} + 5 \, b^{2}\right )} d^{4}\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 (12 \, b^{2} c^{4} - 84 \, a b c^{3} d + 252 \, a b c d^{3} + {\left (35 \, a^{2} - 11 \, b^{2}\right )} c^{2} d^{2} + 15 \, {\left (7 \, a^{2} + 5 \, b^{2}\right )} d^{4}\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 ) - 6 \, \sqrt {2} {\left (-3 i \, b^{2} c^{3} d + 21 i \, a b c^{2} d^{2} + 63 i \, a b d^{4} + i \, {\left (70 \, a^{2} + 41 \, b^{2}\right )} c 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 ) - 6 \, \sqrt {2} {\left (3 i \, b^{2} c^{3} d - 21 i \, a b c^{2} d^{2} - 63 i \, a b d^{4} - i \, {\left (70 \, a^{2} + 41 \, b^{2}\right )} c 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 ) + 6 \, {\left (15 \, b^{2} d^{4} \cos \left (f x + e\right )^{3} - 6 \, {\left (4 \, b^{2} c d^{3} + 7 \, a b d^{4}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - {\left (3 \, b^{2} c^{2} d^{2} + 84 \, a b c d^{3} + 5 \, {\left (7 \, a^{2} + 8 \, b^{2}\right )} d^{4}\right )} \cos \left (f x + e\right )\right )} \sqrt {d \sin \left (f x + e\right ) + c}}{315 \, d^{3} f} \]
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\[ \int (3+b \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2} \, dx=\int \left (a + b \sin {\left (e + f x \right )}\right )^{2} \left (c + d \sin {\left (e + f x \right )}\right )^{\frac {3}{2}}\, dx \]
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\[ \int (3+b \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2} \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{2} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}} \,d x } \]
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\[ \int (3+b \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2} \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{2} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int (3+b \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2} \, dx=\int {\left (a+b\,\sin \left (e+f\,x\right )\right )}^2\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2} \,d x \]
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