Optimal. Leaf size=144 \[ \frac{3 C \sin (c+d x) \cos ^{m+1}(c+d x)}{d (3 m+4) (b \cos (c+d x))^{2/3}}-\frac{3 (A (3 m+4)+3 C m+C) \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (3 m+1);\frac{1}{6} (3 m+7);\cos ^2(c+d x)\right )}{d (3 m+1) (3 m+4) \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.104, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {20, 3014, 2643} \[ \frac{3 C \sin (c+d x) \cos ^{m+1}(c+d x)}{d (3 m+4) (b \cos (c+d x))^{2/3}}-\frac{3 (A (3 m+4)+3 C m+C) \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (3 m+1);\frac{1}{6} (3 m+7);\cos ^2(c+d x)\right )}{d (3 m+1) (3 m+4) \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 20
Rule 3014
Rule 2643
Rubi steps
\begin{align*} \int \frac{\cos ^m(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{2/3}} \, dx &=\frac{\cos ^{\frac{2}{3}}(c+d x) \int \cos ^{-\frac{2}{3}+m}(c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx}{(b \cos (c+d x))^{2/3}}\\ &=\frac{3 C \cos ^{1+m}(c+d x) \sin (c+d x)}{d (4+3 m) (b \cos (c+d x))^{2/3}}+\frac{\left (\left (C \left (\frac{1}{3}+m\right )+A \left (\frac{4}{3}+m\right )\right ) \cos ^{\frac{2}{3}}(c+d x)\right ) \int \cos ^{-\frac{2}{3}+m}(c+d x) \, dx}{\left (\frac{4}{3}+m\right ) (b \cos (c+d x))^{2/3}}\\ &=\frac{3 C \cos ^{1+m}(c+d x) \sin (c+d x)}{d (4+3 m) (b \cos (c+d x))^{2/3}}-\frac{3 (C+3 C m+A (4+3 m)) \cos ^{1+m}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (1+3 m);\frac{1}{6} (7+3 m);\cos ^2(c+d x)\right ) \sin (c+d x)}{d (1+3 m) (4+3 m) (b \cos (c+d x))^{2/3} \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.241487, size = 142, normalized size = 0.99 \[ -\frac{3 \sqrt{\sin ^2(c+d x)} \csc (c+d x) \cos ^{m+1}(c+d x) \left (A (3 m+7) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (3 m+1);\frac{1}{6} (3 m+7);\cos ^2(c+d x)\right )+C (3 m+1) \cos ^2(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (3 m+7);\frac{1}{6} (3 m+13);\cos ^2(c+d x)\right )\right )}{d (3 m+1) (3 m+7) (b \cos (c+d x))^{2/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.283, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \cos \left ( dx+c \right ) \right ) ^{m} \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 )^{m}}{\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}} \cos \left (d x + c\right )^{m}}{b \cos \left (d x + c\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (A + C \cos ^{2}{\left (c + d x \right )}\right ) \cos ^{m}{\left (c + d x \right )}}{\left (b \cos{\left (c + d x \right )}\right )^{\frac{2}{3}}}\, dx \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 )^{m}}{\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]