Integrand size = 12, antiderivative size = 103 \[ \int (c \sin (a+b x))^{7/2} \, dx=\frac {10 c^4 \operatorname {EllipticF}\left (\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right ),2\right ) \sqrt {\sin (a+b x)}}{21 b \sqrt {c \sin (a+b x)}}-\frac {10 c^3 \cos (a+b x) \sqrt {c \sin (a+b x)}}{21 b}-\frac {2 c \cos (a+b x) (c \sin (a+b x))^{5/2}}{7 b} \]
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Time = 0.04 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2715, 2721, 2720} \[ \int (c \sin (a+b x))^{7/2} \, dx=\frac {10 c^4 \sqrt {\sin (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right ),2\right )}{21 b \sqrt {c \sin (a+b x)}}-\frac {10 c^3 \cos (a+b x) \sqrt {c \sin (a+b x)}}{21 b}-\frac {2 c \cos (a+b x) (c \sin (a+b x))^{5/2}}{7 b} \]
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Rule 2715
Rule 2720
Rule 2721
Rubi steps \begin{align*} \text {integral}& = -\frac {2 c \cos (a+b x) (c \sin (a+b x))^{5/2}}{7 b}+\frac {1}{7} \left (5 c^2\right ) \int (c \sin (a+b x))^{3/2} \, dx \\ & = -\frac {10 c^3 \cos (a+b x) \sqrt {c \sin (a+b x)}}{21 b}-\frac {2 c \cos (a+b x) (c \sin (a+b x))^{5/2}}{7 b}+\frac {1}{21} \left (5 c^4\right ) \int \frac {1}{\sqrt {c \sin (a+b x)}} \, dx \\ & = -\frac {10 c^3 \cos (a+b x) \sqrt {c \sin (a+b x)}}{21 b}-\frac {2 c \cos (a+b x) (c \sin (a+b x))^{5/2}}{7 b}+\frac {\left (5 c^4 \sqrt {\sin (a+b x)}\right ) \int \frac {1}{\sqrt {\sin (a+b x)}} \, dx}{21 \sqrt {c \sin (a+b x)}} \\ & = \frac {10 c^4 \operatorname {EllipticF}\left (\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right ),2\right ) \sqrt {\sin (a+b x)}}{21 b \sqrt {c \sin (a+b x)}}-\frac {10 c^3 \cos (a+b x) \sqrt {c \sin (a+b x)}}{21 b}-\frac {2 c \cos (a+b x) (c \sin (a+b x))^{5/2}}{7 b} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.78 \[ \int (c \sin (a+b x))^{7/2} \, dx=\frac {c^3 \left (-20 \operatorname {EllipticF}\left (\frac {1}{4} (-2 a+\pi -2 b x),2\right )+(-23 \cos (a+b x)+3 \cos (3 (a+b x))) \sqrt {\sin (a+b x)}\right ) \sqrt {c \sin (a+b x)}}{42 b \sqrt {\sin (a+b x)}} \]
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Time = 0.24 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.05
method | result | size |
default | \(-\frac {c^{4} \left (-6 \left (\sin ^{5}\left (b x +a \right )\right )+5 \sqrt {-\sin \left (b x +a \right )+1}\, \sqrt {2 \sin \left (b x +a \right )+2}\, \left (\sqrt {\sin }\left (b x +a \right )\right ) F\left (\sqrt {-\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )-4 \left (\sin ^{3}\left (b x +a \right )\right )+10 \sin \left (b x +a \right )\right )}{21 \cos \left (b x +a \right ) \sqrt {c \sin \left (b x +a \right )}\, b}\) | \(108\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.02 \[ \int (c \sin (a+b x))^{7/2} \, dx=\frac {5 \, \sqrt {2} \sqrt {-i \, c} c^{3} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + 5 \, \sqrt {2} \sqrt {i \, c} c^{3} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) + 2 \, {\left (3 \, c^{3} \cos \left (b x + a\right )^{3} - 8 \, c^{3} \cos \left (b x + a\right )\right )} \sqrt {c \sin \left (b x + a\right )}}{21 \, b} \]
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Timed out. \[ \int (c \sin (a+b x))^{7/2} \, dx=\text {Timed out} \]
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\[ \int (c \sin (a+b x))^{7/2} \, dx=\int { \left (c \sin \left (b x + a\right )\right )^{\frac {7}{2}} \,d x } \]
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\[ \int (c \sin (a+b x))^{7/2} \, dx=\int { \left (c \sin \left (b x + a\right )\right )^{\frac {7}{2}} \,d x } \]
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Timed out. \[ \int (c \sin (a+b x))^{7/2} \, dx=\int {\left (c\,\sin \left (a+b\,x\right )\right )}^{7/2} \,d x \]
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