Integrand size = 25, antiderivative size = 93 \[ \int \frac {(c \sin (a+b x))^{3/2}}{\sqrt {d \cos (a+b x)}} \, dx=-\frac {c \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}}{b d}+\frac {c^2 \operatorname {EllipticF}\left (a-\frac {\pi }{4}+b x,2\right ) \sqrt {\sin (2 a+2 b x)}}{2 b \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}} \]
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Time = 0.07 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2648, 2653, 2720} \[ \int \frac {(c \sin (a+b x))^{3/2}}{\sqrt {d \cos (a+b x)}} \, dx=\frac {c^2 \sqrt {\sin (2 a+2 b x)} \operatorname {EllipticF}\left (a+b x-\frac {\pi }{4},2\right )}{2 b \sqrt {c \sin (a+b x)} \sqrt {d \cos (a+b x)}}-\frac {c \sqrt {c \sin (a+b x)} \sqrt {d \cos (a+b x)}}{b d} \]
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Rule 2648
Rule 2653
Rule 2720
Rubi steps \begin{align*} \text {integral}& = -\frac {c \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}}{b d}+\frac {1}{2} c^2 \int \frac {1}{\sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}} \, dx \\ & = -\frac {c \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}}{b d}+\frac {\left (c^2 \sqrt {\sin (2 a+2 b x)}\right ) \int \frac {1}{\sqrt {\sin (2 a+2 b x)}} \, dx}{2 \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}} \\ & = -\frac {c \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}}{b d}+\frac {c^2 \operatorname {EllipticF}\left (a-\frac {\pi }{4}+b x,2\right ) \sqrt {\sin (2 a+2 b x)}}{2 b \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.06 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.72 \[ \int \frac {(c \sin (a+b x))^{3/2}}{\sqrt {d \cos (a+b x)}} \, dx=\frac {2 \cos ^2(a+b x)^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {5}{4},\frac {9}{4},\sin ^2(a+b x)\right ) (c \sin (a+b x))^{3/2} \tan (a+b x)}{5 b \sqrt {d \cos (a+b x)}} \]
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Time = 1.34 (sec) , antiderivative size = 211, normalized size of antiderivative = 2.27
method | result | size |
default | \(-\frac {\sqrt {2}\, \sqrt {c \sin \left (b x +a \right )}\, c \left (-\sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, F\left (\sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right ) \cot \left (b x +a \right )-\sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, F\left (\sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right ) \csc \left (b x +a \right )+\sqrt {2}\, \cos \left (b x +a \right )\right )}{2 b \sqrt {d \cos \left (b x +a \right )}}\) | \(211\) |
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\[ \int \frac {(c \sin (a+b x))^{3/2}}{\sqrt {d \cos (a+b x)}} \, dx=\int { \frac {\left (c \sin \left (b x + a\right )\right )^{\frac {3}{2}}}{\sqrt {d \cos \left (b x + a\right )}} \,d x } \]
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\[ \int \frac {(c \sin (a+b x))^{3/2}}{\sqrt {d \cos (a+b x)}} \, dx=\int \frac {\left (c \sin {\left (a + b x \right )}\right )^{\frac {3}{2}}}{\sqrt {d \cos {\left (a + b x \right )}}}\, dx \]
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\[ \int \frac {(c \sin (a+b x))^{3/2}}{\sqrt {d \cos (a+b x)}} \, dx=\int { \frac {\left (c \sin \left (b x + a\right )\right )^{\frac {3}{2}}}{\sqrt {d \cos \left (b x + a\right )}} \,d x } \]
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\[ \int \frac {(c \sin (a+b x))^{3/2}}{\sqrt {d \cos (a+b x)}} \, dx=\int { \frac {\left (c \sin \left (b x + a\right )\right )^{\frac {3}{2}}}{\sqrt {d \cos \left (b x + a\right )}} \,d x } \]
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Timed out. \[ \int \frac {(c \sin (a+b x))^{3/2}}{\sqrt {d \cos (a+b x)}} \, dx=\int \frac {{\left (c\,\sin \left (a+b\,x\right )\right )}^{3/2}}{\sqrt {d\,\cos \left (a+b\,x\right )}} \,d x \]
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