Integrand size = 25, antiderivative size = 169 \[ \int \frac {(a+b \cos (c+d x))^3}{(e \sin (c+d x))^{5/2}} \, dx=-\frac {2 (b+a \cos (c+d x)) (a+b \cos (c+d x))^2}{3 d e (e \sin (c+d x))^{3/2}}+\frac {2 a \left (a^2-6 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{3 d e^2 \sqrt {e \sin (c+d x)}}-\frac {2 b \left (a^2+4 b^2\right ) \sqrt {e \sin (c+d x)}}{3 d e^3}-\frac {2 a b (a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}}{3 d e^3} \]
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Time = 0.34 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2770, 2941, 2748, 2721, 2720} \[ \int \frac {(a+b \cos (c+d x))^3}{(e \sin (c+d x))^{5/2}} \, dx=-\frac {2 b \left (a^2+4 b^2\right ) \sqrt {e \sin (c+d x)}}{3 d e^3}+\frac {2 a \left (a^2-6 b^2\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{3 d e^2 \sqrt {e \sin (c+d x)}}-\frac {2 a b \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))}{3 d e^3}-\frac {2 (a \cos (c+d x)+b) (a+b \cos (c+d x))^2}{3 d e (e \sin (c+d x))^{3/2}} \]
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Rule 2720
Rule 2721
Rule 2748
Rule 2770
Rule 2941
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (b+a \cos (c+d x)) (a+b \cos (c+d x))^2}{3 d e (e \sin (c+d x))^{3/2}}-\frac {2 \int \frac {(a+b \cos (c+d x)) \left (-\frac {a^2}{2}+2 b^2+\frac {3}{2} a b \cos (c+d x)\right )}{\sqrt {e \sin (c+d x)}} \, dx}{3 e^2} \\ & = -\frac {2 (b+a \cos (c+d x)) (a+b \cos (c+d x))^2}{3 d e (e \sin (c+d x))^{3/2}}-\frac {2 a b (a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}}{3 d e^3}-\frac {4 \int \frac {-\frac {3}{4} a \left (a^2-6 b^2\right )+\frac {3}{4} b \left (a^2+4 b^2\right ) \cos (c+d x)}{\sqrt {e \sin (c+d x)}} \, dx}{9 e^2} \\ & = -\frac {2 (b+a \cos (c+d x)) (a+b \cos (c+d x))^2}{3 d e (e \sin (c+d x))^{3/2}}-\frac {2 b \left (a^2+4 b^2\right ) \sqrt {e \sin (c+d x)}}{3 d e^3}-\frac {2 a b (a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}}{3 d e^3}+\frac {\left (a \left (a^2-6 b^2\right )\right ) \int \frac {1}{\sqrt {e \sin (c+d x)}} \, dx}{3 e^2} \\ & = -\frac {2 (b+a \cos (c+d x)) (a+b \cos (c+d x))^2}{3 d e (e \sin (c+d x))^{3/2}}-\frac {2 b \left (a^2+4 b^2\right ) \sqrt {e \sin (c+d x)}}{3 d e^3}-\frac {2 a b (a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}}{3 d e^3}+\frac {\left (a \left (a^2-6 b^2\right ) \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{3 e^2 \sqrt {e \sin (c+d x)}} \\ & = -\frac {2 (b+a \cos (c+d x)) (a+b \cos (c+d x))^2}{3 d e (e \sin (c+d x))^{3/2}}+\frac {2 a \left (a^2-6 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{3 d e^2 \sqrt {e \sin (c+d x)}}-\frac {2 b \left (a^2+4 b^2\right ) \sqrt {e \sin (c+d x)}}{3 d e^3}-\frac {2 a b (a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}}{3 d e^3} \\ \end{align*}
Time = 1.83 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.60 \[ \int \frac {(a+b \cos (c+d x))^3}{(e \sin (c+d x))^{5/2}} \, dx=-\frac {6 a^2 b+5 b^3+2 a \left (a^2+3 b^2\right ) \cos (c+d x)-3 b^3 \cos (2 (c+d x))+2 a \left (a^2-6 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),2\right ) \sin ^{\frac {3}{2}}(c+d x)}{3 d e (e \sin (c+d x))^{3/2}} \]
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Time = 3.77 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.34
method | result | size |
default | \(\frac {-\frac {2 b \left (-3 b^{2} \left (\cos ^{2}\left (d x +c \right )\right )+3 a^{2}+4 b^{2}\right )}{3 e \left (e \sin \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {a \left (\sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sin ^{\frac {5}{2}}\left (d x +c \right )\right ) F\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) a^{2}-6 b^{2} \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sin ^{\frac {5}{2}}\left (d x +c \right )\right ) F\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )+2 a^{2} \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+6 b^{2} \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )\right )}{3 e^{2} \sin \left (d x +c \right )^{2} \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}}}{d}\) | \(226\) |
parts | \(-\frac {a^{3} \left (\sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sin ^{\frac {5}{2}}\left (d x +c \right )\right ) F\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-2 \left (\sin ^{3}\left (d x +c \right )\right )+2 \sin \left (d x +c \right )\right )}{3 e^{2} \sin \left (d x +c \right )^{2} \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d}-\frac {2 b^{3} \left (\sqrt {e \sin \left (d x +c \right )}+\frac {e^{2}}{3 \left (e \sin \left (d x +c \right )\right )^{\frac {3}{2}}}\right )}{d \,e^{3}}+\frac {2 a \,b^{2} \left (\sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sin ^{\frac {5}{2}}\left (d x +c \right )\right ) F\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )+\sin ^{3}\left (d x +c \right )-\sin \left (d x +c \right )\right )}{e^{2} \sin \left (d x +c \right )^{2} \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d}-\frac {2 a^{2} b}{\left (e \sin \left (d x +c \right )\right )^{\frac {3}{2}} e d}\) | \(274\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.21 \[ \int \frac {(a+b \cos (c+d x))^3}{(e \sin (c+d x))^{5/2}} \, dx=\frac {{\left (\sqrt {2} {\left (a^{3} - 6 \, a b^{2}\right )} \cos \left (d x + c\right )^{2} - \sqrt {2} {\left (a^{3} - 6 \, a b^{2}\right )}\right )} \sqrt {-i \, e} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + {\left (\sqrt {2} {\left (a^{3} - 6 \, a b^{2}\right )} \cos \left (d x + c\right )^{2} - \sqrt {2} {\left (a^{3} - 6 \, a b^{2}\right )}\right )} \sqrt {i \, e} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 2 \, {\left (3 \, b^{3} \cos \left (d x + c\right )^{2} - 3 \, a^{2} b - 4 \, b^{3} - {\left (a^{3} + 3 \, a b^{2}\right )} \cos \left (d x + c\right )\right )} \sqrt {e \sin \left (d x + c\right )}}{3 \, {\left (d e^{3} \cos \left (d x + c\right )^{2} - d e^{3}\right )}} \]
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\[ \int \frac {(a+b \cos (c+d x))^3}{(e \sin (c+d x))^{5/2}} \, dx=\int \frac {\left (a + b \cos {\left (c + d x \right )}\right )^{3}}{\left (e \sin {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {(a+b \cos (c+d x))^3}{(e \sin (c+d x))^{5/2}} \, dx=\int { \frac {{\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\left (e \sin \left (d x + c\right )\right )^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {(a+b \cos (c+d x))^3}{(e \sin (c+d x))^{5/2}} \, dx=\int { \frac {{\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\left (e \sin \left (d x + c\right )\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \cos (c+d x))^3}{(e \sin (c+d x))^{5/2}} \, dx=\int \frac {{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3}{{\left (e\,\sin \left (c+d\,x\right )\right )}^{5/2}} \,d x \]
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