Optimal. Leaf size=145 \[ \frac{3 A b^2 \sin (c+d x)}{2 d (b \cos (c+d x))^{2/3}}+\frac{3 (A-2 C) \sin (c+d x) (b \cos (c+d x))^{4/3} \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right )}{8 d \sqrt{\sin ^2(c+d x)}}-\frac{3 b B \sin (c+d x) \sqrt [3]{b \cos (c+d x)} \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right )}{d \sqrt{\sin ^2(c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.193268, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.098, Rules used = {16, 3021, 2748, 2643} \[ \frac{3 A b^2 \sin (c+d x)}{2 d (b \cos (c+d x))^{2/3}}+\frac{3 (A-2 C) \sin (c+d x) (b \cos (c+d x))^{4/3} \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right )}{8 d \sqrt{\sin ^2(c+d x)}}-\frac{3 b B \sin (c+d x) \sqrt [3]{b \cos (c+d x)} \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right )}{d \sqrt{\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 16
Rule 3021
Rule 2748
Rule 2643
Rubi steps
\begin{align*} \int (b \cos (c+d x))^{4/3} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx &=b^3 \int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(b \cos (c+d x))^{5/3}} \, dx\\ &=\frac{3 A b^2 \sin (c+d x)}{2 d (b \cos (c+d x))^{2/3}}+\frac{3}{2} \int \frac{\frac{2 b^2 B}{3}-\frac{1}{3} b^2 (A-2 C) \cos (c+d x)}{(b \cos (c+d x))^{2/3}} \, dx\\ &=\frac{3 A b^2 \sin (c+d x)}{2 d (b \cos (c+d x))^{2/3}}+\left (b^2 B\right ) \int \frac{1}{(b \cos (c+d x))^{2/3}} \, dx-\frac{1}{2} (b (A-2 C)) \int \sqrt [3]{b \cos (c+d x)} \, dx\\ &=\frac{3 A b^2 \sin (c+d x)}{2 d (b \cos (c+d x))^{2/3}}-\frac{3 b B \sqrt [3]{b \cos (c+d x)} \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right ) \sin (c+d x)}{d \sqrt{\sin ^2(c+d x)}}+\frac{3 (A-2 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)}{8 d \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.229061, size = 117, normalized size = 0.81 \[ -\frac{3 b^2 \sqrt{\sin ^2(c+d x)} \csc (c+d x) \left (\cos (c+d x) \left (4 B \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right )+C \cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right )\right )-2 A \, _2F_1\left (-\frac{1}{3},\frac{1}{2};\frac{2}{3};\cos ^2(c+d x)\right )\right )}{4 d (b \cos (c+d x))^{2/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.421, size = 0, normalized size = 0. \begin{align*} \int \left ( b\cos \left ( dx+c \right ) \right ) ^{{\frac{4}{3}}} \left ( A+B\cos \left ( dx+c \right ) +C \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{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{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac{4}{3}} \sec \left (d x + c\right )^{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 ({\left (C b \cos \left (d x + c\right )^{3} + B b \cos \left (d x + c\right )^{2} + A b \cos \left (d x + c\right )\right )} \left (b \cos \left (d x + c\right )\right )^{\frac{1}{3}} \sec \left (d x + c\right )^{3}, 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{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac{4}{3}} \sec \left (d x + c\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]