Optimal. Leaf size=77 \[ \frac{10 \text{EllipticF}\left (a+b x-\frac{\pi }{4},2\right )}{21 b}-\frac{10 \cos (2 a+2 b x)}{21 b \sin ^{\frac{3}{2}}(2 a+2 b x)}-\frac{\csc ^2(a+b x)}{7 b \sin ^{\frac{3}{2}}(2 a+2 b x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0478643, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {4300, 2636, 2641} \[ \frac{10 F\left (\left .a+b x-\frac{\pi }{4}\right |2\right )}{21 b}-\frac{10 \cos (2 a+2 b x)}{21 b \sin ^{\frac{3}{2}}(2 a+2 b x)}-\frac{\csc ^2(a+b x)}{7 b \sin ^{\frac{3}{2}}(2 a+2 b x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4300
Rule 2636
Rule 2641
Rubi steps
\begin{align*} \int \frac{\csc ^2(a+b x)}{\sin ^{\frac{5}{2}}(2 a+2 b x)} \, dx &=-\frac{\csc ^2(a+b x)}{7 b \sin ^{\frac{3}{2}}(2 a+2 b x)}+\frac{10}{7} \int \frac{1}{\sin ^{\frac{5}{2}}(2 a+2 b x)} \, dx\\ &=-\frac{10 \cos (2 a+2 b x)}{21 b \sin ^{\frac{3}{2}}(2 a+2 b x)}-\frac{\csc ^2(a+b x)}{7 b \sin ^{\frac{3}{2}}(2 a+2 b x)}+\frac{10}{21} \int \frac{1}{\sqrt{\sin (2 a+2 b x)}} \, dx\\ &=\frac{10 F\left (\left .a-\frac{\pi }{4}+b x\right |2\right )}{21 b}-\frac{10 \cos (2 a+2 b x)}{21 b \sin ^{\frac{3}{2}}(2 a+2 b x)}-\frac{\csc ^2(a+b x)}{7 b \sin ^{\frac{3}{2}}(2 a+2 b x)}\\ \end{align*}
Mathematica [A] time = 0.46149, size = 66, normalized size = 0.86 \[ \frac{40 \text{EllipticF}\left (a+b x-\frac{\pi }{4},2\right )+\sqrt{\sin (2 (a+b x))} \left (-3 \csc ^4(a+b x)-13 \csc ^2(a+b x)+7 \sec ^2(a+b x)\right )}{84 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 9.525, size = 154, normalized size = 2. \begin{align*}{\frac{\sqrt{2}}{16\,b} \left ( -{\frac{16\,\sqrt{2}}{7} \left ( \sin \left ( 2\,bx+2\,a \right ) \right ) ^{-{\frac{7}{2}}}}+{\frac{8\,\sqrt{2}}{21\,\cos \left ( 2\,bx+2\,a \right ) } \left ( 5\,\sqrt{\sin \left ( 2\,bx+2\,a \right ) +1}\sqrt{-2\,\sin \left ( 2\,bx+2\,a \right ) +2}\sqrt{-\sin \left ( 2\,bx+2\,a \right ) }{\it EllipticF} \left ( \sqrt{\sin \left ( 2\,bx+2\,a \right ) +1},1/2\,\sqrt{2} \right ) \left ( \sin \left ( 2\,bx+2\,a \right ) \right ) ^{3}+10\, \left ( \sin \left ( 2\,bx+2\,a \right ) \right ) ^{4}-4\, \left ( \sin \left ( 2\,bx+2\,a \right ) \right ) ^{2}-6 \right ) \left ( \sin \left ( 2\,bx+2\,a \right ) \right ) ^{-{\frac{7}{2}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc \left (b x + a\right )^{2}}{\sin \left (2 \, b x + 2 \, a\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\csc \left (b x + a\right )^{2}}{{\left (\cos \left (2 \, b x + 2 \, a\right )^{2} - 1\right )} \sqrt{\sin \left (2 \, b x + 2 \, a\right )}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc \left (b x + a\right )^{2}}{\sin \left (2 \, b x + 2 \, a\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]