Integrand size = 23, antiderivative size = 215 \[ \int \cos ^2(c+d x) \sqrt {a+b \sin (c+d x)} \, dx=-\frac {4 a \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{15 b d}+\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 b d}+\frac {4 \left (a^2+3 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{15 b^2 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {4 a \left (a^2-b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{15 b^2 d \sqrt {a+b \sin (c+d x)}} \]
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Time = 0.17 (sec) , antiderivative size = 215, 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 = {2774, 2832, 2831, 2742, 2740, 2734, 2732} \[ \int \cos ^2(c+d x) \sqrt {a+b \sin (c+d x)} \, dx=-\frac {4 a \left (a^2-b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{15 b^2 d \sqrt {a+b \sin (c+d x)}}+\frac {4 \left (a^2+3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{15 b^2 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 b d}-\frac {4 a \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{15 b d} \]
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2774
Rule 2831
Rule 2832
Rubi steps \begin{align*} \text {integral}& = \frac {2 \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 b d}+\frac {2 \int (b+a \sin (c+d x)) \sqrt {a+b \sin (c+d x)} \, dx}{5 b} \\ & = -\frac {4 a \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{15 b d}+\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 b d}+\frac {4 \int \frac {2 a b+\frac {1}{2} \left (a^2+3 b^2\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{15 b} \\ & = -\frac {4 a \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{15 b d}+\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 b d}-\frac {\left (2 a \left (a^2-b^2\right )\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{15 b^2}+\frac {\left (2 \left (a^2+3 b^2\right )\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{15 b^2} \\ & = -\frac {4 a \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{15 b d}+\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 b d}+\frac {\left (2 \left (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}{15 b^2 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (2 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}{15 b^2 \sqrt {a+b \sin (c+d x)}} \\ & = -\frac {4 a \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{15 b d}+\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 b d}+\frac {4 \left (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)}}{15 b^2 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {4 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}}}{15 b^2 d \sqrt {a+b \sin (c+d x)}} \\ \end{align*}
Time = 0.59 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.86 \[ \int \cos ^2(c+d x) \sqrt {a+b \sin (c+d x)} \, dx=\frac {-4 \left (a^3+a^2 b+3 a b^2+3 b^3\right ) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}+4 a \left (a^2-b^2\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}+b \cos (c+d x) \left (2 a^2+3 b^2-3 b^2 \cos (2 (c+d x))+8 a b \sin (c+d x)\right )}{15 b^2 d \sqrt {a+b \sin (c+d x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(791\) vs. \(2(261)=522\).
Time = 2.11 (sec) , antiderivative size = 792, normalized size of antiderivative = 3.68
method | result | size |
default | \(\frac {\frac {4 \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}{15}+\frac {4 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 {4 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}}{15}-\frac {4 \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 {4 \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}}{15}-\frac {8 \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}}{15}+\frac {4 \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 {8 a \,b^{3} \left (\sin ^{3}\left (d x +c \right )\right )}{15}-\frac {2 a^{2} b^{2} \left (\sin ^{2}\left (d x +c \right )\right )}{15}+\frac {2 b^{4} \left (\sin ^{2}\left (d x +c \right )\right )}{5}+\frac {8 a \,b^{3} \sin \left (d x +c \right )}{15}+\frac {2 a^{2} b^{2}}{15}}{b^{3} \cos \left (d x +c \right ) \sqrt {a +b \sin \left (d x +c \right )}\, d}\) | \(792\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.13 (sec) , antiderivative size = 448, normalized size of antiderivative = 2.08 \[ \int \cos ^2(c+d x) \sqrt {a+b \sin (c+d x)} \, dx=-\frac {2 \, {\left (2 \, \sqrt {2} {\left (a^{3} - 3 \, a b^{2}\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 \, \sqrt {2} {\left (a^{3} - 3 \, a b^{2}\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 ) + 3 \, \sqrt {2} {\left (i \, a^{2} b + 3 i \, b^{3}\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 \, \sqrt {2} {\left (-i \, a^{2} b - 3 i \, b^{3}\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 (3 \, b^{3} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a b^{2} \cos \left (d x + c\right )\right )} \sqrt {b \sin \left (d x + c\right ) + a}\right )}}{45 \, b^{3} d} \]
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\[ \int \cos ^2(c+d x) \sqrt {a+b \sin (c+d x)} \, dx=\int \sqrt {a + b \sin {\left (c + d x \right )}} \cos ^{2}{\left (c + d x \right )}\, dx \]
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\[ \int \cos ^2(c+d x) \sqrt {a+b \sin (c+d x)} \, dx=\int { \sqrt {b \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{2} \,d x } \]
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\[ \int \cos ^2(c+d x) \sqrt {a+b \sin (c+d x)} \, dx=\int { \sqrt {b \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{2} \,d x } \]
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Timed out. \[ \int \cos ^2(c+d x) \sqrt {a+b \sin (c+d x)} \, dx=\int {\cos \left (c+d\,x\right )}^2\,\sqrt {a+b\,\sin \left (c+d\,x\right )} \,d x \]
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