Integrand size = 10, antiderivative size = 47 \[ \int \sin ^{\frac {3}{2}}(a+b x) \, dx=\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right ),2\right )}{3 b}-\frac {2 \cos (a+b x) \sqrt {\sin (a+b x)}}{3 b} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2715, 2720} \[ \int \sin ^{\frac {3}{2}}(a+b x) \, dx=\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right ),2\right )}{3 b}-\frac {2 \sqrt {\sin (a+b x)} \cos (a+b x)}{3 b} \]
[In]
[Out]
Rule 2715
Rule 2720
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \cos (a+b x) \sqrt {\sin (a+b x)}}{3 b}+\frac {1}{3} \int \frac {1}{\sqrt {\sin (a+b x)}} \, dx \\ & = \frac {2 \operatorname {EllipticF}\left (\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right ),2\right )}{3 b}-\frac {2 \cos (a+b x) \sqrt {\sin (a+b x)}}{3 b} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.85 \[ \int \sin ^{\frac {3}{2}}(a+b x) \, dx=-\frac {2 \left (\operatorname {EllipticF}\left (\frac {1}{4} (-2 a+\pi -2 b x),2\right )+\cos (a+b x) \sqrt {\sin (a+b x)}\right )}{3 b} \]
[In]
[Out]
Time = 0.05 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.87
method | result | size |
default | \(\frac {\frac {\sqrt {\sin \left (b x +a \right )+1}\, \sqrt {-2 \sin \left (b x +a \right )+2}\, \sqrt {-\sin \left (b x +a \right )}\, F\left (\sqrt {\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )}{3}-\frac {2 \left (\cos ^{2}\left (b x +a \right )\right ) \sin \left (b x +a \right )}{3}}{\cos \left (b x +a \right ) \sqrt {\sin \left (b x +a \right )}\, b}\) | \(88\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.53 \[ \int \sin ^{\frac {3}{2}}(a+b x) \, dx=\frac {\sqrt {2} \sqrt {-i} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + \sqrt {2} \sqrt {i} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) - 2 \, \cos \left (b x + a\right ) \sqrt {\sin \left (b x + a\right )}}{3 \, b} \]
[In]
[Out]
\[ \int \sin ^{\frac {3}{2}}(a+b x) \, dx=\int \sin ^{\frac {3}{2}}{\left (a + b x \right )}\, dx \]
[In]
[Out]
\[ \int \sin ^{\frac {3}{2}}(a+b x) \, dx=\int { \sin \left (b x + a\right )^{\frac {3}{2}} \,d x } \]
[In]
[Out]
\[ \int \sin ^{\frac {3}{2}}(a+b x) \, dx=\int { \sin \left (b x + a\right )^{\frac {3}{2}} \,d x } \]
[In]
[Out]
Time = 0.10 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.89 \[ \int \sin ^{\frac {3}{2}}(a+b x) \, dx=-\frac {\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^{5/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{2};\ {\cos \left (a+b\,x\right )}^2\right )}{b\,{\left ({\sin \left (a+b\,x\right )}^2\right )}^{5/4}} \]
[In]
[Out]