Optimal. Leaf size=53 \[ \frac{3 \cos (a+b x) \text{Hypergeometric2F1}\left (\frac{1}{3},\frac{1}{2},\frac{4}{3},\sin ^2(a+b x)\right )}{2 b \sqrt{\cos ^2(a+b x)} \csc ^{\frac{2}{3}}(a+b x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0216941, antiderivative size = 53, 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} \[ \frac{3 \cos (a+b x) \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};\sin ^2(a+b x)\right )}{2 b \sqrt{\cos ^2(a+b x)} \csc ^{\frac{2}{3}}(a+b x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3772
Rule 2643
Rubi steps
\begin{align*} \int \sqrt [3]{\csc (a+b x)} \, dx &=\sqrt [3]{\csc (a+b x)} \sqrt [3]{\sin (a+b x)} \int \frac{1}{\sqrt [3]{\sin (a+b x)}} \, dx\\ &=\frac{3 \cos (a+b x) \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};\sin ^2(a+b x)\right )}{2 b \sqrt{\cos ^2(a+b x)} \csc ^{\frac{2}{3}}(a+b x)}\\ \end{align*}
Mathematica [A] time = 0.053962, size = 51, normalized size = 0.96 \[ -\frac{\cos (a+b x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{2}{3},\frac{3}{2},\cos ^2(a+b x)\right )}{b \sqrt [3]{\sin ^2(a+b x)} \csc ^{\frac{2}{3}}(a+b x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.166, size = 0, normalized size = 0. \begin{align*} \int \sqrt [3]{\csc \left ( bx+a \right ) }\, 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{1}{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{1}{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt [3]{\csc{\left (a + b x \right )}}\, dx \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{1}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]