Integrand size = 27, antiderivative size = 358 \[ \int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2} \, dx=\frac {4 \left (5 c^3-45 c^2 d-141 c d^2-75 d^3\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{35 d f}+\frac {4 \left (5 c (c-9 d)-56 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d f}+\frac {4 (c-9 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}-\frac {2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{d f}-\frac {4 \left (5 c^4-45 c^3 d-381 c^2 d^2-435 c d^3-168 d^4\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{35 d^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {4 \left (c^2-d^2\right ) \left (5 c^3-45 c^2 d-141 c d^2-75 d^3\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{35 d^2 f \sqrt {c+d \sin (e+f x)}} \]
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Time = 0.46 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2842, 2832, 2831, 2742, 2740, 2734, 2732} \[ \int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2} \, dx=\frac {4 a^2 \left (5 c^3-45 c^2 d-141 c d^2-75 d^3\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{315 d f}+\frac {4 a^2 \left (c^2-d^2\right ) \left (5 c^3-45 c^2 d-141 c d^2-75 d^3\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{315 d^2 f \sqrt {c+d \sin (e+f x)}}-\frac {4 a^2 \left (5 c^4-45 c^3 d-381 c^2 d^2-435 c d^3-168 d^4\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{315 d^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {4 a^2 \left (5 c (c-9 d)-56 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d f}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}+\frac {4 a^2 (c-9 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d f} \]
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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))^{7/2}}{9 d f}+\frac {2 \int \left (8 a^2 d-a^2 (c-9 d) \sin (e+f x)\right ) (c+d \sin (e+f x))^{5/2} \, dx}{9 d} \\ & = \frac {4 a^2 (c-9 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d f}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}+\frac {4 \int (c+d \sin (e+f x))^{3/2} \left (\frac {3}{2} a^2 d (17 c+15 d)-\frac {1}{2} a^2 \left (5 c (c-9 d)-56 d^2\right ) \sin (e+f x)\right ) \, dx}{63 d} \\ & = \frac {4 a^2 \left (5 c (c-9 d)-56 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d f}+\frac {4 a^2 (c-9 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d f}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}+\frac {8 \int \sqrt {c+d \sin (e+f x)} \left (6 a^2 d \left (10 c^2+15 c d+7 d^2\right )-\frac {3}{4} a^2 \left (5 c^3-45 c^2 d-141 c d^2-75 d^3\right ) \sin (e+f x)\right ) \, dx}{315 d} \\ & = \frac {4 a^2 \left (5 c^3-45 c^2 d-141 c d^2-75 d^3\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{315 d f}+\frac {4 a^2 \left (5 c (c-9 d)-56 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d f}+\frac {4 a^2 (c-9 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d f}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}+\frac {16 \int \frac {\frac {3}{8} a^2 d \left (235 c^3+405 c^2 d+309 c d^2+75 d^3\right )-\frac {3}{8} a^2 \left (5 c^4-45 c^3 d-381 c^2 d^2-435 c d^3-168 d^4\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx}{945 d} \\ & = \frac {4 a^2 \left (5 c^3-45 c^2 d-141 c d^2-75 d^3\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{315 d f}+\frac {4 a^2 \left (5 c (c-9 d)-56 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d f}+\frac {4 a^2 (c-9 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d f}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}+\frac {\left (2 a^2 \left (c^2-d^2\right ) \left (5 c^3-45 c^2 d-141 c d^2-75 d^3\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{315 d^2}-\frac {\left (2 a^2 \left (5 c^4-45 c^3 d-381 c^2 d^2-435 c d^3-168 d^4\right )\right ) \int \sqrt {c+d \sin (e+f x)} \, dx}{315 d^2} \\ & = \frac {4 a^2 \left (5 c^3-45 c^2 d-141 c d^2-75 d^3\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{315 d f}+\frac {4 a^2 \left (5 c (c-9 d)-56 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d f}+\frac {4 a^2 (c-9 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d f}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}-\frac {\left (2 a^2 \left (5 c^4-45 c^3 d-381 c^2 d^2-435 c d^3-168 d^4\right ) \sqrt {c+d \sin (e+f x)}\right ) \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}} \, dx}{315 d^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left (2 a^2 \left (c^2-d^2\right ) \left (5 c^3-45 c^2 d-141 c d^2-75 d^3\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}{315 d^2 \sqrt {c+d \sin (e+f x)}} \\ & = \frac {4 a^2 \left (5 c^3-45 c^2 d-141 c d^2-75 d^3\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{315 d f}+\frac {4 a^2 \left (5 c (c-9 d)-56 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d f}+\frac {4 a^2 (c-9 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d f}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}-\frac {4 a^2 \left (5 c^4-45 c^3 d-381 c^2 d^2-435 c d^3-168 d^4\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)}}{315 d^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {4 a^2 \left (c^2-d^2\right ) \left (5 c^3-45 c^2 d-141 c d^2-75 d^3\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}}}{315 d^2 f \sqrt {c+d \sin (e+f x)}} \\ \end{align*}
Time = 1.00 (sec) , antiderivative size = 286, normalized size of antiderivative = 0.80 \[ \int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2} \, dx=\frac {16 \left (-d^2 \left (235 c^3+405 c^2 d+309 c d^2+75 d^3\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )+\left (5 c^4-45 c^3 d-381 c^2 d^2-435 c d^3-168 d^4\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}-d (c+d \sin (e+f x)) \left (2 \left (20 c^3+1080 c^2 d+1671 c d^2+690 d^3\right ) \cos (e+f x)+2 d \left (-5 d (19 c+18 d) \cos (3 (e+f x))+\left (150 c^2+540 c d+259 d^2-35 d^2 \cos (2 (e+f x))\right ) \sin (2 (e+f x))\right )\right )}{140 d^2 f \sqrt {c+d \sin (e+f x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1613\) vs. \(2(416)=832\).
Time = 12.37 (sec) , antiderivative size = 1614, normalized size of antiderivative = 4.51
method | result | size |
default | \(\text {Expression too large to display}\) | \(1614\) |
parts | \(\text {Expression too large to display}\) | \(3031\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.15 (sec) , antiderivative size = 723, normalized size of antiderivative = 2.02 \[ \int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2} \, dx=\frac {2 \, {\left (\sqrt {2} {\left (10 \, a^{2} c^{5} - 90 \, a^{2} c^{4} d - 57 \, a^{2} c^{3} d^{2} + 345 \, a^{2} c^{2} d^{3} + 591 \, a^{2} c d^{4} + 225 \, a^{2} d^{5}\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 (10 \, a^{2} c^{5} - 90 \, a^{2} c^{4} d - 57 \, a^{2} c^{3} d^{2} + 345 \, a^{2} c^{2} d^{3} + 591 \, a^{2} c d^{4} + 225 \, a^{2} d^{5}\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 (-5 i \, a^{2} c^{4} d + 45 i \, a^{2} c^{3} d^{2} + 381 i \, a^{2} c^{2} d^{3} + 435 i \, a^{2} c d^{4} + 168 i \, a^{2} d^{5}\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 (5 i \, a^{2} c^{4} d - 45 i \, a^{2} c^{3} d^{2} - 381 i \, a^{2} c^{2} d^{3} - 435 i \, a^{2} c d^{4} - 168 i \, a^{2} d^{5}\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 (5 \, {\left (19 \, a^{2} c d^{4} + 18 \, a^{2} d^{5}\right )} \cos \left (f x + e\right )^{3} - {\left (5 \, a^{2} c^{3} d^{2} + 270 \, a^{2} c^{2} d^{3} + 489 \, a^{2} c d^{4} + 240 \, a^{2} d^{5}\right )} \cos \left (f x + e\right ) + {\left (35 \, a^{2} d^{5} \cos \left (f x + e\right )^{3} - 3 \, {\left (25 \, a^{2} c^{2} d^{3} + 90 \, a^{2} c d^{4} + 49 \, a^{2} d^{5}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {d \sin \left (f x + e\right ) + c}\right )}}{945 \, d^{3} f} \]
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\[ \int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2} \, dx=a^{2} \left (\int c^{2} \sqrt {c + d \sin {\left (e + f x \right )}}\, dx + \int 2 c^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )}\, dx + \int c^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )}\, dx + \int d^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )}\, dx + \int 2 d^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{3}{\left (e + f x \right )}\, dx + \int d^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{4}{\left (e + f x \right )}\, dx + \int 2 c d \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )}\, dx + \int 4 c d \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )}\, dx + \int 2 c d \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{3}{\left (e + f x \right )}\, dx\right ) \]
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\[ \int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{2} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \]
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\[ \int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{2} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \]
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Timed out. \[ \int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2} \, dx=\int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^2\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2} \,d x \]
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