Optimal. Leaf size=51 \[ \frac {3 \, _2F_1\left (-\frac {1}{6},\frac {1}{2};\frac {5}{6};\cos ^2(a+b x)\right ) \sqrt [3]{\sec (a+b x)} \sin (a+b x)}{b \sqrt {\sin ^2(a+b x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3857, 2722}
\begin {gather*} \frac {3 \sin (a+b x) \sqrt [3]{\sec (a+b x)} \, _2F_1\left (-\frac {1}{6},\frac {1}{2};\frac {5}{6};\cos ^2(a+b x)\right )}{b \sqrt {\sin ^2(a+b x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2722
Rule 3857
Rubi steps
\begin {align*} \int \sec ^{\frac {4}{3}}(a+b x) \, dx &=\sqrt [3]{\cos (a+b x)} \sqrt [3]{\sec (a+b x)} \int \frac {1}{\cos ^{\frac {4}{3}}(a+b x)} \, dx\\ &=\frac {3 \, _2F_1\left (-\frac {1}{6},\frac {1}{2};\frac {5}{6};\cos ^2(a+b x)\right ) \sqrt [3]{\sec (a+b x)} \sin (a+b x)}{b \sqrt {\sin ^2(a+b x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.06, size = 55, normalized size = 1.08 \begin {gather*} \frac {3 \csc (a+b x) \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};\sec ^2(a+b x)\right ) \sqrt [3]{\sec (a+b x)} \sqrt {-\tan ^2(a+b x)}}{4 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.22, size = 0, normalized size = 0.00 \[\int \sec ^{\frac {4}{3}}\left (b x +a \right )\, dx\]
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 \sec ^{\frac {4}{3}}{\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 [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\left (\frac {1}{\cos \left (a+b\,x\right )}\right )}^{4/3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________