Integrand size = 29, antiderivative size = 261 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=-\frac {8 a \left (32 a^2-29 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{35 b^5 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {8 \left (32 a^4-37 a^2 b^2+5 b^4\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{35 b^5 d \sqrt {a+b \sin (c+d x)}}+\frac {2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-5 b^2-24 a b \sin (c+d x)\right )}{35 b^4 d} \]
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Time = 0.30 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2942, 2944, 2831, 2742, 2740, 2734, 2732} \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=-\frac {8 a \left (32 a^2-29 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{35 b^5 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-24 a b \sin (c+d x)-5 b^2\right )}{35 b^4 d}+\frac {8 \left (32 a^4-37 a^2 b^2+5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{35 b^5 d \sqrt {a+b \sin (c+d x)}}+\frac {2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt {a+b \sin (c+d x)}} \]
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2831
Rule 2942
Rule 2944
Rubi steps \begin{align*} \text {integral}& = \frac {2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {12 \int \frac {\cos ^2(c+d x) \left (-\frac {b}{2}-4 a \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{7 b^2} \\ & = \frac {2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-5 b^2-24 a b \sin (c+d x)\right )}{35 b^4 d}-\frac {16 \int \frac {\frac {1}{4} b \left (8 a^2-5 b^2\right )+\frac {1}{4} a \left (32 a^2-29 b^2\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{35 b^4} \\ & = \frac {2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-5 b^2-24 a b \sin (c+d x)\right )}{35 b^4 d}-\frac {\left (4 a \left (32 a^2-29 b^2\right )\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{35 b^5}+\frac {\left (4 \left (32 a^4-37 a^2 b^2+5 b^4\right )\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{35 b^5} \\ & = \frac {2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-5 b^2-24 a b \sin (c+d x)\right )}{35 b^4 d}-\frac {\left (4 a \left (32 a^2-29 b^2\right ) \sqrt {a+b \sin (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}} \, dx}{35 b^5 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (4 \left (32 a^4-37 a^2 b^2+5 b^4\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}{35 b^5 \sqrt {a+b \sin (c+d x)}} \\ & = -\frac {8 a \left (32 a^2-29 b^2\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)}}{35 b^5 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {8 \left (32 a^4-37 a^2 b^2+5 b^4\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}}}{35 b^5 d \sqrt {a+b \sin (c+d x)}}+\frac {2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-5 b^2-24 a b \sin (c+d x)\right )}{35 b^4 d} \\ \end{align*}
Time = 2.71 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.85 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\frac {16 a \left (32 a^3+32 a^2 b-29 a b^2-29 b^3\right ) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}-16 \left (32 a^4-37 a^2 b^2+5 b^4\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}+b \cos (c+d x) \left (-256 a^3+216 a b^2-16 a b^2 \cos (2 (c+d x))+\left (-64 a^2 b+45 b^3\right ) \sin (c+d x)+5 b^3 \sin (3 (c+d x))\right )}{70 b^5 d \sqrt {a+b \sin (c+d x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(942\) vs. \(2(307)=614\).
Time = 1.76 (sec) , antiderivative size = 943, normalized size of antiderivative = 3.61
method | result | size |
default | \(-\frac {2 \left (-5 b^{5} \left (\sin ^{5}\left (d x +c \right )\right )+128 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (d x +c \right )\right ) b}{a -b}}\, F\left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{4} b -96 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (d x +c \right )\right ) b}{a -b}}\, F\left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{3} b^{2}-148 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (d x +c \right )\right ) b}{a -b}}\, F\left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{2} b^{3}+96 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (d x +c \right )\right ) b}{a -b}}\, F\left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a \,b^{4}+20 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (d x +c \right )\right ) b}{a -b}}\, F\left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) b^{5}-128 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (d x +c \right )\right ) b}{a -b}}\, E\left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{5}+244 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (d x +c \right )\right ) b}{a -b}}\, E\left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{3} b^{2}-116 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (d x +c \right )\right ) b}{a -b}}\, E\left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a \,b^{4}+8 a \,b^{4} \left (\sin ^{4}\left (d x +c \right )\right )-16 a^{2} b^{3} \left (\sin ^{3}\left (d x +c \right )\right )+20 b^{5} \left (\sin ^{3}\left (d x +c \right )\right )-64 a^{3} b^{2} \left (\sin ^{2}\left (d x +c \right )\right )+42 a \,b^{4} \left (\sin ^{2}\left (d x +c \right )\right )+16 a^{2} b^{3} \sin \left (d x +c \right )-15 b^{5} \sin \left (d x +c \right )+64 a^{3} b^{2}-50 a \,b^{4}\right )}{35 b^{6} \cos \left (d x +c \right ) \sqrt {a +b \sin \left (d x +c \right )}\, d}\) | \(943\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.19 (sec) , antiderivative size = 659, normalized size of antiderivative = 2.52 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\frac {2 \, {\left (2 \, {\left (\sqrt {2} {\left (64 \, a^{4} b - 82 \, a^{2} b^{3} + 15 \, b^{5}\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (64 \, a^{5} - 82 \, a^{3} b^{2} + 15 \, a b^{4}\right )}\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 ) + 2 \, {\left (\sqrt {2} {\left (64 \, a^{4} b - 82 \, a^{2} b^{3} + 15 \, b^{5}\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (64 \, a^{5} - 82 \, a^{3} b^{2} + 15 \, a b^{4}\right )}\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 \, {\left (\sqrt {2} {\left (-32 i \, a^{3} b^{2} + 29 i \, a b^{4}\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (-32 i \, a^{4} b + 29 i \, a^{2} b^{3}\right )}\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 \, {\left (\sqrt {2} {\left (32 i \, a^{3} b^{2} - 29 i \, a b^{4}\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (32 i \, a^{4} b - 29 i \, a^{2} b^{3}\right )}\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 (8 \, a b^{4} \cos \left (d x + c\right )^{3} + 2 \, {\left (32 \, a^{3} b^{2} - 29 \, a b^{4}\right )} \cos \left (d x + c\right ) - {\left (5 \, b^{5} \cos \left (d x + c\right )^{3} - 2 \, {\left (8 \, a^{2} b^{3} - 5 \, b^{5}\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 )}}{105 \, {\left (b^{7} d \sin \left (d x + c\right ) + a b^{6} d\right )}} \]
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Timed out. \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\int { \frac {\cos \left (d x + c\right )^{4} \sin \left (d x + c\right )}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^4\,\sin \left (c+d\,x\right )}{{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
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