Optimal. Leaf size=104 \[ -\frac {4 \sin (a+b x) \sqrt {\sin (2 a+2 b x)}}{b}+\frac {2 \sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{b}-\frac {\sin ^{\frac {5}{2}}(2 a+2 b x) \csc ^3(a+b x)}{b}+\frac {2 \log \left (\sin (a+b x)+\sqrt {\sin (2 a+2 b x)}+\cos (a+b x)\right )}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4300, 4308, 4301, 4306} \[ -\frac {4 \sin (a+b x) \sqrt {\sin (2 a+2 b x)}}{b}+\frac {2 \sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{b}-\frac {\sin ^{\frac {5}{2}}(2 a+2 b x) \csc ^3(a+b x)}{b}+\frac {2 \log \left (\sin (a+b x)+\sqrt {\sin (2 a+2 b x)}+\cos (a+b x)\right )}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4300
Rule 4301
Rule 4306
Rule 4308
Rubi steps
\begin {align*} \int \csc ^3(a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x) \, dx &=-\frac {\csc ^3(a+b x) \sin ^{\frac {5}{2}}(2 a+2 b x)}{b}-4 \int \csc (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x) \, dx\\ &=-\frac {\csc ^3(a+b x) \sin ^{\frac {5}{2}}(2 a+2 b x)}{b}-8 \int \cos (a+b x) \sqrt {\sin (2 a+2 b x)} \, dx\\ &=-\frac {4 \sin (a+b x) \sqrt {\sin (2 a+2 b x)}}{b}-\frac {\csc ^3(a+b x) \sin ^{\frac {5}{2}}(2 a+2 b x)}{b}-4 \int \frac {\sin (a+b x)}{\sqrt {\sin (2 a+2 b x)}} \, dx\\ &=\frac {2 \sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{b}+\frac {2 \log \left (\cos (a+b x)+\sin (a+b x)+\sqrt {\sin (2 a+2 b x)}\right )}{b}-\frac {4 \sin (a+b x) \sqrt {\sin (2 a+2 b x)}}{b}-\frac {\csc ^3(a+b x) \sin ^{\frac {5}{2}}(2 a+2 b x)}{b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.10, size = 68, normalized size = 0.65 \[ \frac {2 \left (\sin ^{-1}(\cos (a+b x)-\sin (a+b x))-2 \sqrt {\sin (2 (a+b x))} \csc (a+b x)+\log \left (\sin (a+b x)+\sqrt {\sin (2 (a+b x))}+\cos (a+b x)\right )\right )}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.54, size = 295, normalized size = 2.84 \[ -\frac {2 \, \arctan \left (-\frac {\sqrt {2} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} {\left (\cos \left (b x + a\right ) - \sin \left (b x + a\right )\right )} + \cos \left (b x + a\right ) \sin \left (b x + a\right )}{\cos \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) - 1}\right ) \sin \left (b x + a\right ) - 2 \, \arctan \left (-\frac {2 \, \sqrt {2} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} - \cos \left (b x + a\right ) - \sin \left (b x + a\right )}{\cos \left (b x + a\right ) - \sin \left (b x + a\right )}\right ) \sin \left (b x + a\right ) + \log \left (-32 \, \cos \left (b x + a\right )^{4} + 4 \, \sqrt {2} {\left (4 \, \cos \left (b x + a\right )^{3} - {\left (4 \, \cos \left (b x + a\right )^{2} + 1\right )} \sin \left (b x + a\right ) - 5 \, \cos \left (b x + a\right )\right )} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} + 32 \, \cos \left (b x + a\right )^{2} + 16 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) + 8 \, \sqrt {2} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} + 8 \, \sin \left (b x + a\right )}{2 \, b \sin \left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \csc \left (b x + a\right )^{3} \sin \left (2 \, b x + 2 \, a\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 86.50, size = 542, normalized size = 5.21 \[ \frac {4 \sqrt {-\frac {\tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )-1}}\, \left (4 \sqrt {\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1}\, \sqrt {-2 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )+2}\, \sqrt {-\tan \left (\frac {b x}{2}+\frac {a}{2}\right )}\, \sqrt {\tan \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right ) \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )}\, \sqrt {\tan \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )}\, \EllipticE \left (\sqrt {\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1}, \frac {\sqrt {2}}{2}\right )-2 \sqrt {\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1}\, \sqrt {-2 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )+2}\, \sqrt {-\tan \left (\frac {b x}{2}+\frac {a}{2}\right )}\, \sqrt {\tan \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right ) \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )}\, \sqrt {\tan \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )}\, \EllipticF \left (\sqrt {\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1}, \frac {\sqrt {2}}{2}\right )+2 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \sqrt {\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )-\tan \left (\frac {b x}{2}+\frac {a}{2}\right )}\, \sqrt {\tan \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )}+\sqrt {\tan \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right ) \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )}\, \sqrt {\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )-\tan \left (\frac {b x}{2}+\frac {a}{2}\right )}\, \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-\sqrt {\tan \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right ) \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )}\, \sqrt {\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )-\tan \left (\frac {b x}{2}+\frac {a}{2}\right )}\right )}{\tan \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {\tan \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right ) \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )}\, \sqrt {\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )-\tan \left (\frac {b x}{2}+\frac {a}{2}\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 \[ \int \csc \left (b x + a\right )^{3} \sin \left (2 \, b x + 2 \, a\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\sin \left (2\,a+2\,b\,x\right )}^{3/2}}{{\sin \left (a+b\,x\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________