Optimal. Leaf size=95 \[ \frac{3 C \sin (c+d x) (b \cos (c+d x))^{4/3}}{7 b^2 d}-\frac{3 (7 A+4 C) \sin (c+d x) (b \cos (c+d x))^{4/3} \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right )}{28 b^2 d \sqrt{\sin ^2(c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0640115, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {16, 3014, 2643} \[ \frac{3 C \sin (c+d x) (b \cos (c+d x))^{4/3}}{7 b^2 d}-\frac{3 (7 A+4 C) \sin (c+d x) (b \cos (c+d x))^{4/3} \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right )}{28 b^2 d \sqrt{\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 16
Rule 3014
Rule 2643
Rubi steps
\begin{align*} \int \frac{\cos (c+d x) \left (A+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{2/3}} \, dx &=\frac{\int \sqrt [3]{b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx}{b}\\ &=\frac{3 C (b \cos (c+d x))^{4/3} \sin (c+d x)}{7 b^2 d}+\frac{(7 A+4 C) \int \sqrt [3]{b \cos (c+d x)} \, dx}{7 b}\\ &=\frac{3 C (b \cos (c+d x))^{4/3} \sin (c+d x)}{7 b^2 d}-\frac{3 (7 A+4 C) (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)}{28 b^2 d \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.112143, size = 91, normalized size = 0.96 \[ -\frac{3 \sqrt{\sin ^2(c+d x)} \cot (c+d x) \sqrt [3]{b \cos (c+d x)} \left (5 A \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right )+2 C \cos ^2(c+d x) \, _2F_1\left (\frac{1}{2},\frac{5}{3};\frac{8}{3};\cos ^2(c+d x)\right )\right )}{20 b d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.343, size = 0, normalized size = 0. \begin{align*} \int{\cos \left ( dx+c \right ) \left ( A+C \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \left ( b\cos \left ( dx+c \right ) \right ) ^{-{\frac{2}{3}}}}\, 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 \frac{{\left (C \cos \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )}{\left (b \cos \left (d x + c\right )\right )^{\frac{2}{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 (\frac{{\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac{1}{3}}}{b}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 \frac{{\left (C \cos \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )}{\left (b \cos \left (d x + c\right )\right )^{\frac{2}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]