Optimal. Leaf size=53 \[ -\frac {3 \cos ^{\frac {4}{3}}(a+b x) \text {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},\cos ^2(a+b x)\right ) \sin (a+b x)}{4 b \sqrt {\sin ^2(a+b x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2722}
\begin {gather*} -\frac {3 \sin (a+b x) \cos ^{\frac {4}{3}}(a+b x) \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};\cos ^2(a+b x)\right )}{4 b \sqrt {\sin ^2(a+b x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2722
Rubi steps
\begin {align*} \int \sqrt [3]{\cos (a+b x)} \, dx &=-\frac {3 \cos ^{\frac {4}{3}}(a+b x) \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};\cos ^2(a+b x)\right ) \sin (a+b x)}{4 b \sqrt {\sin ^2(a+b x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 53, normalized size = 1.00 \begin {gather*} -\frac {3 \cos ^{\frac {4}{3}}(a+b x) \csc (a+b x) \text {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},\cos ^2(a+b x)\right ) \sqrt {\sin ^2(a+b x)}}{4 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \cos ^{\frac {1}{3}}\left (b x +a \right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt [3]{\cos {\left (a + b x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.18, size = 42, normalized size = 0.79 \begin {gather*} -\frac {3\,{\cos \left (a+b\,x\right )}^{4/3}\,\sin \left (a+b\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {2}{3};\ \frac {5}{3};\ {\cos \left (a+b\,x\right )}^2\right )}{4\,b\,\sqrt {{\sin \left (a+b\,x\right )}^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________