Optimal. Leaf size=63 \[ -\frac {2 \cos (a+b x) \sqrt {\csc (a+b x)}}{b}-\frac {2 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3768, 3771, 2639} \[ -\frac {2 \cos (a+b x) \sqrt {\csc (a+b x)}}{b}-\frac {2 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2639
Rule 3768
Rule 3771
Rubi steps
\begin {align*} \int \csc ^{\frac {3}{2}}(a+b x) \, dx &=-\frac {2 \cos (a+b x) \sqrt {\csc (a+b x)}}{b}-\int \frac {1}{\sqrt {\csc (a+b x)}} \, dx\\ &=-\frac {2 \cos (a+b x) \sqrt {\csc (a+b x)}}{b}-\left (\sqrt {\csc (a+b x)} \sqrt {\sin (a+b x)}\right ) \int \sqrt {\sin (a+b x)} \, dx\\ &=-\frac {2 \cos (a+b x) \sqrt {\csc (a+b x)}}{b}-\frac {2 \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right ) \sqrt {\sin (a+b x)}}{b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 49, normalized size = 0.78 \[ -\frac {2 \sqrt {\csc (a+b x)} \left (\cos (a+b x)-\sqrt {\sin (a+b x)} E\left (\left .\frac {1}{4} (-2 a-2 b x+\pi )\right |2\right )\right )}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\csc \left (b x + a\right )^{\frac {3}{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \csc \left (b x + a\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 3.18, size = 132, normalized size = 2.10 \[ \frac {2 \sqrt {\sin \left (b x +a \right )+1}\, \sqrt {-2 \sin \left (b x +a \right )+2}\, \sqrt {-\sin \left (b x +a \right )}\, \EllipticE \left (\sqrt {\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )-\sqrt {\sin \left (b x +a \right )+1}\, \sqrt {-2 \sin \left (b x +a \right )+2}\, \sqrt {-\sin \left (b x +a \right )}\, \EllipticF \left (\sqrt {\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )-2 \left (\cos ^{2}\left (b x +a \right )\right )}{\cos \left (b x +a \right ) \sqrt {\sin \left (b x +a \right )}\, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \csc \left (b x + a\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (\frac {1}{\sin \left (a+b\,x\right )}\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \csc ^{\frac {3}{2}}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________