Optimal. Leaf size=54 \[ \frac {3 c \cos (a+b x) \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {7}{6};\sin ^2(a+b x)\right )}{b \sqrt {\cos ^2(a+b x)} \sqrt [3]{c \csc (a+b x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3772, 2643} \[ \frac {3 c \cos (a+b x) \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {7}{6};\sin ^2(a+b x)\right )}{b \sqrt {\cos ^2(a+b x)} \sqrt [3]{c \csc (a+b x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2643
Rule 3772
Rubi steps
\begin {align*} \int (c \csc (a+b x))^{2/3} \, dx &=(c \csc (a+b x))^{2/3} \left (\frac {\sin (a+b x)}{c}\right )^{2/3} \int \frac {1}{\left (\frac {\sin (a+b x)}{c}\right )^{2/3}} \, dx\\ &=\frac {3 \cos (a+b x) (c \csc (a+b x))^{2/3} \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {7}{6};\sin ^2(a+b x)\right ) \sin (a+b x)}{b \sqrt {\cos ^2(a+b x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 59, normalized size = 1.09 \[ -\frac {\sin (a+b x) \cos (a+b x) (c \csc (a+b x))^{2/3} \, _2F_1\left (\frac {1}{2},\frac {5}{6};\frac {3}{2};\cos ^2(a+b x)\right )}{b \sqrt [6]{\sin ^2(a+b x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (c \csc \left (b x + a\right )\right )^{\frac {2}{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c \csc \left (b x + a\right )\right )^{\frac {2}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.65, size = 0, normalized size = 0.00 \[ \int \left (c \csc \left (b x +a \right )\right )^{\frac {2}{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c \csc \left (b x + a\right )\right )^{\frac {2}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (\frac {c}{\sin \left (a+b\,x\right )}\right )}^{2/3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c \csc {\left (a + b x \right )}\right )^{\frac {2}{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________