Integrand size = 23, antiderivative size = 313 \[ \int \frac {\cos ^6(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=-\frac {2 \cos ^5(c+d x)}{b d \sqrt {a+b \sin (c+d x)}}+\frac {20 \cos ^3(c+d x) (8 a-7 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{63 b^3 d}-\frac {16 \left (32 a^4-57 a^2 b^2+21 b^4\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{63 b^6 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {16 a \left (32 a^4-65 a^2 b^2+33 b^4\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{63 b^6 d \sqrt {a+b \sin (c+d x)}}-\frac {8 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^2-33 b^2\right )-3 b \left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{63 b^5 d} \]
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Time = 0.38 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {2772, 2944, 2831, 2742, 2740, 2734, 2732} \[ \int \frac {\cos ^6(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=-\frac {8 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^2-33 b^2\right )-3 b \left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{63 b^5 d}+\frac {16 a \left (32 a^4-65 a^2 b^2+33 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{63 b^6 d \sqrt {a+b \sin (c+d x)}}-\frac {16 \left (32 a^4-57 a^2 b^2+21 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{63 b^6 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {20 \cos ^3(c+d x) (8 a-7 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{63 b^3 d}-\frac {2 \cos ^5(c+d x)}{b d \sqrt {a+b \sin (c+d x)}} \]
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2772
Rule 2831
Rule 2944
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \cos ^5(c+d x)}{b d \sqrt {a+b \sin (c+d x)}}-\frac {10 \int \frac {\cos ^4(c+d x) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{b} \\ & = -\frac {2 \cos ^5(c+d x)}{b d \sqrt {a+b \sin (c+d x)}}+\frac {20 \cos ^3(c+d x) (8 a-7 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{63 b^3 d}-\frac {40 \int \frac {\cos ^2(c+d x) \left (-\frac {a b}{2}-\frac {1}{2} \left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{21 b^3} \\ & = -\frac {2 \cos ^5(c+d x)}{b d \sqrt {a+b \sin (c+d x)}}+\frac {20 \cos ^3(c+d x) (8 a-7 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{63 b^3 d}-\frac {8 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^2-33 b^2\right )-3 b \left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{63 b^5 d}-\frac {32 \int \frac {a b \left (2 a^2-3 b^2\right )+\frac {1}{4} \left (32 a^4-57 a^2 b^2+21 b^4\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{63 b^5} \\ & = -\frac {2 \cos ^5(c+d x)}{b d \sqrt {a+b \sin (c+d x)}}+\frac {20 \cos ^3(c+d x) (8 a-7 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{63 b^3 d}-\frac {8 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^2-33 b^2\right )-3 b \left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{63 b^5 d}-\frac {\left (8 \left (32 a^4-57 a^2 b^2+21 b^4\right )\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{63 b^6}+\frac {\left (8 a \left (32 a^4-65 a^2 b^2+33 b^4\right )\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{63 b^6} \\ & = -\frac {2 \cos ^5(c+d x)}{b d \sqrt {a+b \sin (c+d x)}}+\frac {20 \cos ^3(c+d x) (8 a-7 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{63 b^3 d}-\frac {8 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^2-33 b^2\right )-3 b \left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{63 b^5 d}-\frac {\left (8 \left (32 a^4-57 a^2 b^2+21 b^4\right ) \sqrt {a+b \sin (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}} \, dx}{63 b^6 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (8 a \left (32 a^4-65 a^2 b^2+33 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}{63 b^6 \sqrt {a+b \sin (c+d x)}} \\ & = -\frac {2 \cos ^5(c+d x)}{b d \sqrt {a+b \sin (c+d x)}}+\frac {20 \cos ^3(c+d x) (8 a-7 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{63 b^3 d}-\frac {16 \left (32 a^4-57 a^2 b^2+21 b^4\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)}}{63 b^6 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {16 a \left (32 a^4-65 a^2 b^2+33 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}}}{63 b^6 d \sqrt {a+b \sin (c+d x)}}-\frac {8 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^2-33 b^2\right )-3 b \left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{63 b^5 d} \\ \end{align*}
Time = 1.04 (sec) , antiderivative size = 273, normalized size of antiderivative = 0.87 \[ \int \frac {\cos ^6(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\frac {64 \left (32 a^5+32 a^4 b-57 a^3 b^2-57 a^2 b^3+21 a b^4+21 b^5\right ) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}-64 a \left (32 a^4-65 a^2 b^2+33 b^4\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}+b \cos (c+d x) \left (-1024 a^4+1760 a^2 b^2-595 b^4+\left (-64 a^2 b^2+84 b^4\right ) \cos (2 (c+d x))+7 b^4 \cos (4 (c+d x))-256 a^3 b \sin (c+d x)+404 a b^3 \sin (c+d x)+20 a b^3 \sin (3 (c+d x))\right )}{252 b^6 d \sqrt {a+b \sin (c+d x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1194\) vs. \(2(357)=714\).
Time = 3.50 (sec) , antiderivative size = 1195, normalized size of antiderivative = 3.82
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.17 (sec) , antiderivative size = 723, normalized size of antiderivative = 2.31 \[ \int \frac {\cos ^6(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\frac {2 \, {\left (8 \, {\left (\sqrt {2} {\left (32 \, a^{5} b - 69 \, a^{3} b^{3} + 39 \, a b^{5}\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (32 \, a^{6} - 69 \, a^{4} b^{2} + 39 \, a^{2} 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 ) + 8 \, {\left (\sqrt {2} {\left (32 \, a^{5} b - 69 \, a^{3} b^{3} + 39 \, a b^{5}\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (32 \, a^{6} - 69 \, a^{4} b^{2} + 39 \, a^{2} 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 ) - 12 \, {\left (\sqrt {2} {\left (-32 i \, a^{4} b^{2} + 57 i \, a^{2} b^{4} - 21 i \, b^{6}\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (-32 i \, a^{5} b + 57 i \, a^{3} b^{3} - 21 i \, a b^{5}\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 ) - 12 \, {\left (\sqrt {2} {\left (32 i \, a^{4} b^{2} - 57 i \, a^{2} b^{4} + 21 i \, b^{6}\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (32 i \, a^{5} b - 57 i \, a^{3} b^{3} + 21 i \, a b^{5}\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 (7 \, b^{6} \cos \left (d x + c\right )^{5} - 2 \, {\left (8 \, a^{2} b^{4} - 7 \, b^{6}\right )} \cos \left (d x + c\right )^{3} - 4 \, {\left (32 \, a^{4} b^{2} - 57 \, a^{2} b^{4} + 21 \, b^{6}\right )} \cos \left (d x + c\right ) + 2 \, {\left (5 \, a b^{5} \cos \left (d x + c\right )^{3} - 8 \, {\left (2 \, a^{3} b^{3} - 3 \, a 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 )}}{189 \, {\left (b^{8} d \sin \left (d x + c\right ) + a b^{7} d\right )}} \]
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Timed out. \[ \int \frac {\cos ^6(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {\cos ^6(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\int { \frac {\cos \left (d x + c\right )^{6}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\cos ^6(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\cos ^6(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^6}{{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
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