Optimal. Leaf size=58 \[ -\frac {3 \sin (c+d x) (b \cos (c+d x))^{4/3} \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};\cos ^2(c+d x)\right )}{4 b^2 d \sqrt {\sin ^2(c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {16, 2643} \[ -\frac {3 \sin (c+d x) (b \cos (c+d x))^{4/3} \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};\cos ^2(c+d x)\right )}{4 b^2 d \sqrt {\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 16
Rule 2643
Rubi steps
\begin {align*} \int \frac {\cos (c+d x)}{(b \cos (c+d x))^{2/3}} \, dx &=\frac {\int \sqrt [3]{b \cos (c+d x)} \, dx}{b}\\ &=-\frac {3 (b \cos (c+d x))^{4/3} \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};\cos ^2(c+d x)\right ) \sin (c+d x)}{4 b^2 d \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 58, normalized size = 1.00 \[ -\frac {3 \sqrt {\sin ^2(c+d x)} \cot (c+d x) \sqrt [3]{b \cos (c+d x)} \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};\cos ^2(c+d x)\right )}{4 b d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (b \cos \left (d x + c\right )\right )^{\frac {1}{3}}}{b}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (d x + c\right )}{\left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (d x +c \right )}{\left (b \cos \left (d x +c \right )\right )^{\frac {2}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (d x + c\right )}{\left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\cos \left (c+d\,x\right )}{{\left (b\,\cos \left (c+d\,x\right )\right )}^{2/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos {\left (c + d x \right )}}{\left (b \cos {\left (c + d x \right )}\right )^{\frac {2}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________