Optimal. Leaf size=67 \[ \frac {2 \cos (a+b x)}{3 b \sqrt {\csc (a+b x)}}+\frac {4 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} F\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{3 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2628, 3771, 2641} \[ \frac {2 \cos (a+b x)}{3 b \sqrt {\csc (a+b x)}}+\frac {4 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} F\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{3 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2628
Rule 2641
Rule 3771
Rubi steps
\begin {align*} \int \cos ^2(a+b x) \sqrt {\csc (a+b x)} \, dx &=\frac {2 \cos (a+b x)}{3 b \sqrt {\csc (a+b x)}}+\frac {2}{3} \int \sqrt {\csc (a+b x)} \, dx\\ &=\frac {2 \cos (a+b x)}{3 b \sqrt {\csc (a+b x)}}+\frac {1}{3} \left (2 \sqrt {\csc (a+b x)} \sqrt {\sin (a+b x)}\right ) \int \frac {1}{\sqrt {\sin (a+b x)}} \, dx\\ &=\frac {2 \cos (a+b x)}{3 b \sqrt {\csc (a+b x)}}+\frac {4 \sqrt {\csc (a+b x)} F\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right ) \sqrt {\sin (a+b x)}}{3 b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.10, size = 53, normalized size = 0.79 \[ \frac {\sqrt {\csc (a+b x)} \left (\sin (2 (a+b x))-4 \sqrt {\sin (a+b x)} F\left (\left .\frac {1}{4} (-2 a-2 b x+\pi )\right |2\right )\right )}{3 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\cos \left (b x + a\right )^{2} \sqrt {\csc \left (b x + a\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos \left (b x + a\right )^{2} \sqrt {\csc \left (b x + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.16, size = 88, normalized size = 1.31 \[ \frac {\frac {2 \sqrt {\sin \left (b x +a \right )+1}\, \sqrt {-2 \sin \left (b x +a \right )+2}\, \sqrt {-\sin \left (b x +a \right )}\, \EllipticF \left (\sqrt {\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )}{3}+\frac {2 \left (\cos ^{2}\left (b x +a \right )\right ) \sin \left (b x +a \right )}{3}}{\cos \left (b x +a \right ) \sqrt {\sin \left (b x +a \right )}\, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos \left (b x + a\right )^{2} \sqrt {\csc \left (b x + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\cos \left (a+b\,x\right )}^2\,\sqrt {\frac {1}{\sin \left (a+b\,x\right )}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos ^{2}{\left (a + b x \right )} \sqrt {\csc {\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________