Integrand size = 29, antiderivative size = 394 \[ \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^{3/2} \, dx=-\frac {6 a \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{143 d}-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}+\frac {8 \left (32 a^6-137 a^4 b^2+258 a^2 b^4+231 b^6\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{15015 b^5 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {8 a \left (32 a^6-145 a^4 b^2+290 a^2 b^4-177 b^6\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{15015 b^5 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (2 a^2-5 b^2\right )-7 b \left (a^2+11 b^2\right ) \sin (c+d x)\right )}{3003 b^2 d}+\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^4-113 a^2 b^2+177 b^4\right )-3 b \left (8 a^4-27 a^2 b^2-77 b^4\right ) \sin (c+d x)\right )}{15015 b^4 d} \]
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Time = 0.70 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2941, 2944, 2831, 2742, 2740, 2734, 2732} \[ \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^{3/2} \, dx=-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (2 a^2-5 b^2\right )-7 b \left (a^2+11 b^2\right ) \sin (c+d x)\right )}{3003 b^2 d}+\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^4-113 a^2 b^2+177 b^4\right )-3 b \left (8 a^4-27 a^2 b^2-77 b^4\right ) \sin (c+d x)\right )}{15015 b^4 d}-\frac {8 a \left (32 a^6-145 a^4 b^2+290 a^2 b^4-177 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{15015 b^5 d \sqrt {a+b \sin (c+d x)}}+\frac {8 \left (32 a^6-137 a^4 b^2+258 a^2 b^4+231 b^6\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{15015 b^5 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}-\frac {6 a \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{143 d} \]
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2831
Rule 2941
Rule 2944
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}+\frac {2}{13} \int \cos ^4(c+d x) \left (\frac {3 b}{2}+\frac {3}{2} a \sin (c+d x)\right ) \sqrt {a+b \sin (c+d x)} \, dx \\ & = -\frac {6 a \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{143 d}-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}+\frac {4}{143} \int \frac {\cos ^4(c+d x) \left (9 a b+\frac {3}{4} \left (a^2+11 b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx \\ & = -\frac {6 a \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{143 d}-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (2 a^2-5 b^2\right )-7 b \left (a^2+11 b^2\right ) \sin (c+d x)\right )}{3003 b^2 d}+\frac {16 \int \frac {\cos ^2(c+d x) \left (-\frac {3}{8} a b \left (a^2-97 b^2\right )-\frac {3}{8} \left (8 a^4-27 a^2 b^2-77 b^4\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{3003 b^2} \\ & = -\frac {6 a \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{143 d}-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (2 a^2-5 b^2\right )-7 b \left (a^2+11 b^2\right ) \sin (c+d x)\right )}{3003 b^2 d}+\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^4-113 a^2 b^2+177 b^4\right )-3 b \left (8 a^4-27 a^2 b^2-77 b^4\right ) \sin (c+d x)\right )}{15015 b^4 d}+\frac {64 \int \frac {\frac {3}{2} a b \left (a^4-4 a^2 b^2+51 b^4\right )+\frac {3}{16} \left (32 a^6-137 a^4 b^2+258 a^2 b^4+231 b^6\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{45045 b^4} \\ & = -\frac {6 a \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{143 d}-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (2 a^2-5 b^2\right )-7 b \left (a^2+11 b^2\right ) \sin (c+d x)\right )}{3003 b^2 d}+\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^4-113 a^2 b^2+177 b^4\right )-3 b \left (8 a^4-27 a^2 b^2-77 b^4\right ) \sin (c+d x)\right )}{15015 b^4 d}-\frac {\left (4 a \left (32 a^6-145 a^4 b^2+290 a^2 b^4-177 b^6\right )\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{15015 b^5}+\frac {\left (4 \left (32 a^6-137 a^4 b^2+258 a^2 b^4+231 b^6\right )\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{15015 b^5} \\ & = -\frac {6 a \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{143 d}-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (2 a^2-5 b^2\right )-7 b \left (a^2+11 b^2\right ) \sin (c+d x)\right )}{3003 b^2 d}+\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^4-113 a^2 b^2+177 b^4\right )-3 b \left (8 a^4-27 a^2 b^2-77 b^4\right ) \sin (c+d x)\right )}{15015 b^4 d}+\frac {\left (4 \left (32 a^6-137 a^4 b^2+258 a^2 b^4+231 b^6\right ) \sqrt {a+b \sin (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}} \, dx}{15015 b^5 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (4 a \left (32 a^6-145 a^4 b^2+290 a^2 b^4-177 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}} \, dx}{15015 b^5 \sqrt {a+b \sin (c+d x)}} \\ & = -\frac {6 a \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{143 d}-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}+\frac {8 \left (32 a^6-137 a^4 b^2+258 a^2 b^4+231 b^6\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{15015 b^5 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {8 a \left (32 a^6-145 a^4 b^2+290 a^2 b^4-177 b^6\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{15015 b^5 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (2 a^2-5 b^2\right )-7 b \left (a^2+11 b^2\right ) \sin (c+d x)\right )}{3003 b^2 d}+\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^4-113 a^2 b^2+177 b^4\right )-3 b \left (8 a^4-27 a^2 b^2-77 b^4\right ) \sin (c+d x)\right )}{15015 b^4 d} \\ \end{align*}
Time = 8.60 (sec) , antiderivative size = 382, normalized size of antiderivative = 0.97 \[ \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\frac {-384 \left (32 a^7+32 a^6 b-137 a^5 b^2-137 a^4 b^3+258 a^3 b^4+258 a^2 b^5+231 a b^6+231 b^7\right ) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}+384 a \left (32 a^6-145 a^4 b^2+290 a^2 b^4-177 b^6\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}-3 b \cos (c+d x) \left (-2048 a^6+8640 a^4 b^2+1980 a^2 b^4-6622 b^6+\left (-128 a^4 b^2+24512 a^2 b^4+8547 b^6\right ) \cos (2 (c+d x))+70 \left (86 a^2 b^4-11 b^6\right ) \cos (4 (c+d x))-1155 b^6 \cos (6 (c+d x))-512 a^5 b \sin (c+d x)+2088 a^3 b^3 \sin (c+d x)-19492 a b^5 \sin (c+d x)+40 a^3 b^3 \sin (3 (c+d x))+11870 a b^5 \sin (3 (c+d x))+5250 a b^5 \sin (5 (c+d x))\right )}{720720 b^5 d \sqrt {a+b \sin (c+d x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1618\) vs. \(2(432)=864\).
Time = 2.51 (sec) , antiderivative size = 1619, normalized size of antiderivative = 4.11
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.17 (sec) , antiderivative size = 632, normalized size of antiderivative = 1.60 \[ \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^{3/2} \, dx=-\frac {2 \, {\left (4 \, \sqrt {2} {\left (32 \, a^{7} - 149 \, a^{5} b^{2} + 306 \, a^{3} b^{4} - 381 \, a b^{6}\right )} \sqrt {i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) - 2 i \, a}{3 \, b}\right ) + 4 \, \sqrt {2} {\left (32 \, a^{7} - 149 \, a^{5} b^{2} + 306 \, a^{3} b^{4} - 381 \, a b^{6}\right )} \sqrt {-i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 i \, a}{3 \, b}\right ) + 6 \, \sqrt {2} {\left (32 i \, a^{6} b - 137 i \, a^{4} b^{3} + 258 i \, a^{2} b^{5} + 231 i \, b^{7}\right )} \sqrt {i \, b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) - 2 i \, a}{3 \, b}\right )\right ) + 6 \, \sqrt {2} {\left (-32 i \, a^{6} b + 137 i \, a^{4} b^{3} - 258 i \, a^{2} b^{5} - 231 i \, b^{7}\right )} \sqrt {-i \, b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 i \, a}{3 \, b}\right )\right ) + 3 \, {\left (1470 \, a b^{6} \cos \left (d x + c\right )^{5} + 20 \, {\left (2 \, a^{3} b^{4} - 5 \, a b^{6}\right )} \cos \left (d x + c\right )^{3} - 2 \, {\left (32 \, a^{5} b^{2} - 113 \, a^{3} b^{4} + 177 \, a b^{6}\right )} \cos \left (d x + c\right ) + {\left (1155 \, b^{7} \cos \left (d x + c\right )^{5} - 35 \, {\left (a^{2} b^{5} + 11 \, b^{7}\right )} \cos \left (d x + c\right )^{3} + 6 \, {\left (8 \, a^{4} b^{3} - 27 \, a^{2} b^{5} - 77 \, b^{7}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )} \sqrt {b \sin \left (d x + c\right ) + a}\right )}}{45045 \, b^{6} d} \]
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Timed out. \[ \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\text {Timed out} \]
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\[ \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) \,d x } \]
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\[ \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) \,d x } \]
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Timed out. \[ \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\int {\cos \left (c+d\,x\right )}^4\,\sin \left (c+d\,x\right )\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{3/2} \,d x \]
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