Optimal. Leaf size=51 \[ -\frac{3 \cos (a+b x) \sqrt [3]{\csc (a+b x)} \text{Hypergeometric2F1}\left (-\frac{1}{6},\frac{1}{2},\frac{5}{6},\sin ^2(a+b x)\right )}{b \sqrt{\cos ^2(a+b x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0223791, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3772, 2643} \[ -\frac{3 \cos (a+b x) \sqrt [3]{\csc (a+b x)} \, _2F_1\left (-\frac{1}{6},\frac{1}{2};\frac{5}{6};\sin ^2(a+b x)\right )}{b \sqrt{\cos ^2(a+b x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3772
Rule 2643
Rubi steps
\begin{align*} \int \csc ^{\frac{4}{3}}(a+b x) \, dx &=\sqrt [3]{\csc (a+b x)} \sqrt [3]{\sin (a+b x)} \int \frac{1}{\sin ^{\frac{4}{3}}(a+b x)} \, dx\\ &=-\frac{3 \cos (a+b x) \sqrt [3]{\csc (a+b x)} \, _2F_1\left (-\frac{1}{6},\frac{1}{2};\frac{5}{6};\sin ^2(a+b x)\right )}{b \sqrt{\cos ^2(a+b x)}}\\ \end{align*}
Mathematica [A] time = 0.0775629, size = 54, normalized size = 1.06 \[ \frac{\cos (a+b x) \sqrt [3]{\csc (a+b x)} \left (2 \sqrt [6]{\sin ^2(a+b x)} \text{Hypergeometric2F1}\left (\frac{1}{6},\frac{1}{2},\frac{3}{2},\cos ^2(a+b x)\right )-3\right )}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.104, size = 0, normalized size = 0. \begin{align*} \int \left ( \csc \left ( bx+a \right ) \right ) ^{{\frac{4}{3}}}\, dx \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 \csc \left (b x + a\right )^{\frac{4}{3}}\,{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 (\csc \left (b x + a\right )^{\frac{4}{3}}, 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 \csc \left (b x + a\right )^{\frac{4}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]