Integrand size = 23, antiderivative size = 229 \[ \int \frac {\cos ^4(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=-\frac {2 \cos ^3(c+d x)}{b d \sqrt {a+b \sin (c+d x)}}+\frac {4 \cos (c+d x) (4 a-3 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{5 b^3 d}+\frac {8 \left (4 a^2-3 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{5 b^4 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {32 a \left (a^2-b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{5 b^4 d \sqrt {a+b \sin (c+d x)}} \]
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Time = 0.23 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {2772, 2944, 2831, 2742, 2740, 2734, 2732} \[ \int \frac {\cos ^4(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=-\frac {32 a \left (a^2-b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{5 b^4 d \sqrt {a+b \sin (c+d x)}}+\frac {8 \left (4 a^2-3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{5 b^4 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {4 \cos (c+d x) (4 a-3 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{5 b^3 d}-\frac {2 \cos ^3(c+d x)}{b d \sqrt {a+b \sin (c+d x)}} \]
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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 ^3(c+d x)}{b d \sqrt {a+b \sin (c+d x)}}-\frac {6 \int \frac {\cos ^2(c+d x) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{b} \\ & = -\frac {2 \cos ^3(c+d x)}{b d \sqrt {a+b \sin (c+d x)}}+\frac {4 \cos (c+d x) (4 a-3 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{5 b^3 d}-\frac {8 \int \frac {-\frac {a b}{2}-\frac {1}{2} \left (4 a^2-3 b^2\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{5 b^3} \\ & = -\frac {2 \cos ^3(c+d x)}{b d \sqrt {a+b \sin (c+d x)}}+\frac {4 \cos (c+d x) (4 a-3 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{5 b^3 d}+\frac {\left (4 \left (4 a^2-3 b^2\right )\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{5 b^4}-\frac {\left (16 a \left (a^2-b^2\right )\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{5 b^4} \\ & = -\frac {2 \cos ^3(c+d x)}{b d \sqrt {a+b \sin (c+d x)}}+\frac {4 \cos (c+d x) (4 a-3 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{5 b^3 d}+\frac {\left (4 \left (4 a^2-3 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}{5 b^4 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (16 a \left (a^2-b^2\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}{5 b^4 \sqrt {a+b \sin (c+d x)}} \\ & = -\frac {2 \cos ^3(c+d x)}{b d \sqrt {a+b \sin (c+d x)}}+\frac {4 \cos (c+d x) (4 a-3 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{5 b^3 d}+\frac {8 \left (4 a^2-3 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)}}{5 b^4 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {32 a \left (a^2-b^2\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}}}{5 b^4 d \sqrt {a+b \sin (c+d x)}} \\ \end{align*}
Time = 0.79 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.82 \[ \int \frac {\cos ^4(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\frac {-8 \left (4 a^3+4 a^2 b-3 a b^2-3 b^3\right ) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}+32 a \left (a^2-b^2\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}+b \cos (c+d x) \left (16 a^2-11 b^2+b^2 \cos (2 (c+d x))+4 a b \sin (c+d x)\right )}{5 b^4 d \sqrt {a+b \sin (c+d x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(796\) vs. \(2(277)=554\).
Time = 1.76 (sec) , antiderivative size = 797, normalized size of antiderivative = 3.48
method | result | size |
default | \(\frac {\frac {32 \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}{5}-\frac {24 a^{2} \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^{2}}{5}-\frac {32 a \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^{3}}{5}+\frac {24 \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^{4}}{5}-\frac {32 \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^{4}}{5}+\frac {56 \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^{2} b^{2}}{5}-\frac {24 \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 ) b^{4}}{5}+\frac {2 b^{4} \left (\sin ^{4}\left (d x +c \right )\right )}{5}-\frac {4 a \,b^{3} \left (\sin ^{3}\left (d x +c \right )\right )}{5}-\frac {16 a^{2} b^{2} \left (\sin ^{2}\left (d x +c \right )\right )}{5}+\frac {8 b^{4} \left (\sin ^{2}\left (d x +c \right )\right )}{5}+\frac {4 a \,b^{3} \sin \left (d x +c \right )}{5}+\frac {16 a^{2} b^{2}}{5}-2 b^{4}}{b^{5} \cos \left (d x +c \right ) \sqrt {a +b \sin \left (d x +c \right )}\, d}\) | \(797\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.15 (sec) , antiderivative size = 598, normalized size of antiderivative = 2.61 \[ \int \frac {\cos ^4(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=-\frac {2 \, {\left (2 \, {\left (\sqrt {2} {\left (8 \, a^{3} b - 9 \, a b^{3}\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (8 \, a^{4} - 9 \, a^{2} b^{2}\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 (8 \, a^{3} b - 9 \, a b^{3}\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (8 \, a^{4} - 9 \, a^{2} b^{2}\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 (4 i \, a^{2} b^{2} - 3 i \, b^{4}\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (4 i \, a^{3} b - 3 i \, a 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 (-4 i \, a^{2} b^{2} + 3 i \, b^{4}\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (-4 i \, a^{3} b + 3 i \, a 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 (b^{4} \cos \left (d x + c\right )^{3} + 2 \, a b^{3} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 2 \, {\left (4 \, a^{2} b^{2} - 3 \, b^{4}\right )} \cos \left (d x + c\right )\right )} \sqrt {b \sin \left (d x + c\right ) + a}\right )}}{15 \, {\left (b^{6} d \sin \left (d x + c\right ) + a b^{5} d\right )}} \]
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\[ \int \frac {\cos ^4(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\int \frac {\cos ^{4}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {\cos ^4(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\int { \frac {\cos \left (d x + c\right )^{4}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\cos ^4(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\cos ^4(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^4}{{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
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