Optimal. Leaf size=145 \[ -\frac{3 (A+4 C) \sin (c+d x) (b \cos (c+d x))^{2/3} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right )}{8 b d \sqrt{\sin ^2(c+d x)}}+\frac{3 A b \sin (c+d x)}{4 d (b \cos (c+d x))^{4/3}}+\frac{3 B \sin (c+d x) \, _2F_1\left (-\frac{1}{6},\frac{1}{2};\frac{5}{6};\cos ^2(c+d x)\right )}{d \sqrt{\sin ^2(c+d x)} \sqrt [3]{b \cos (c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.174072, 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} \[ -\frac{3 (A+4 C) \sin (c+d x) (b \cos (c+d x))^{2/3} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right )}{8 b d \sqrt{\sin ^2(c+d x)}}+\frac{3 A b \sin (c+d x)}{4 d (b \cos (c+d x))^{4/3}}+\frac{3 B \sin (c+d x) \, _2F_1\left (-\frac{1}{6},\frac{1}{2};\frac{5}{6};\cos ^2(c+d x)\right )}{d \sqrt{\sin ^2(c+d x)} \sqrt [3]{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 16
Rule 3021
Rule 2748
Rule 2643
Rubi steps
\begin{align*} \int \frac{\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{\sqrt [3]{b \cos (c+d x)}} \, dx &=b^2 \int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(b \cos (c+d x))^{7/3}} \, dx\\ &=\frac{3 A b \sin (c+d x)}{4 d (b \cos (c+d x))^{4/3}}+\frac{3 \int \frac{\frac{4 b^2 B}{3}+\frac{1}{3} b^2 (A+4 C) \cos (c+d x)}{(b \cos (c+d x))^{4/3}} \, dx}{4 b}\\ &=\frac{3 A b \sin (c+d x)}{4 d (b \cos (c+d x))^{4/3}}+(b B) \int \frac{1}{(b \cos (c+d x))^{4/3}} \, dx+\frac{1}{4} (A+4 C) \int \frac{1}{\sqrt [3]{b \cos (c+d x)}} \, dx\\ &=\frac{3 A b \sin (c+d x)}{4 d (b \cos (c+d x))^{4/3}}+\frac{3 B \, _2F_1\left (-\frac{1}{6},\frac{1}{2};\frac{5}{6};\cos ^2(c+d x)\right ) \sin (c+d x)}{d \sqrt [3]{b \cos (c+d x)} \sqrt{\sin ^2(c+d x)}}-\frac{3 (A+4 C) (b \cos (c+d x))^{2/3} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right ) \sin (c+d x)}{8 b d \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [B] time = 6.30457, size = 699, normalized size = 4.82 \[ \frac{4 B \csc (c) \cos ^{\frac{7}{3}}(c+d x) \left (A \sec ^2(c+d x)+B \sec (c+d x)+C\right ) \left (\frac{\tan (c) \sin \left (\tan ^{-1}(\tan (c))+d x\right ) \, _2F_1\left (-\frac{1}{2},-\frac{1}{6};\frac{5}{6};\cos ^2\left (d x+\tan ^{-1}(\tan (c))\right )\right )}{\sqrt{\tan ^2(c)+1} \sqrt{1-\cos \left (\tan ^{-1}(\tan (c))+d x\right )} \sqrt{\cos \left (\tan ^{-1}(\tan (c))+d x\right )+1} \sqrt [3]{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left (\tan ^{-1}(\tan (c))+d x\right )}}-\frac{\frac{\tan (c) \sin \left (\tan ^{-1}(\tan (c))+d x\right )}{\sqrt{\tan ^2(c)+1}}+\frac{3 \cos ^2(c) \sqrt{\tan ^2(c)+1} \cos \left (\tan ^{-1}(\tan (c))+d x\right )}{2 \left (\sin ^2(c)+\cos ^2(c)\right )}}{\sqrt [3]{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left (\tan ^{-1}(\tan (c))+d x\right )}}\right )}{d \sqrt [3]{b \cos (c+d x)} (2 A+2 B \cos (c+d x)+C \cos (2 c+2 d x)+C)}-\frac{A \cos ^{\frac{7}{3}}(c+d x) \cos \left (d x-\tan ^{-1}(\cot (c))\right ) \sin \left (d x-\tan ^{-1}(\cot (c))\right ) \left (A \sec ^2(c+d x)+B \sec (c+d x)+C\right ) \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{3}{2};\cos ^2\left (d x-\tan ^{-1}(\cot (c))\right )\right )}{2 d \sqrt [3]{b \cos (c+d x)} \sqrt [3]{\cos (c) \cos (d x)-\sin (c) \sin (d x)} \sqrt [3]{\sin ^2\left (d x-\tan ^{-1}(\cot (c))\right )} (2 A+2 B \cos (c+d x)+C \cos (2 c+2 d x)+C)}-\frac{2 C \cos ^{\frac{7}{3}}(c+d x) \cos \left (d x-\tan ^{-1}(\cot (c))\right ) \sin \left (d x-\tan ^{-1}(\cot (c))\right ) \left (A \sec ^2(c+d x)+B \sec (c+d x)+C\right ) \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{3}{2};\cos ^2\left (d x-\tan ^{-1}(\cot (c))\right )\right )}{d \sqrt [3]{b \cos (c+d x)} \sqrt [3]{\cos (c) \cos (d x)-\sin (c) \sin (d x)} \sqrt [3]{\sin ^2\left (d x-\tan ^{-1}(\cot (c))\right )} (2 A+2 B \cos (c+d x)+C \cos (2 c+2 d x)+C)}+\frac{\cos ^3(c+d x) \left (A \sec ^2(c+d x)+B \sec (c+d x)+C\right ) \left (\frac{3 \sec (c) \sec (c+d x) (A \sin (c)+4 B \sin (d x))}{2 d}+\frac{3 A \sec (c) \sin (d x) \sec ^2(c+d x)}{2 d}+\frac{6 B \csc (c) \sec (c)}{d}\right )}{\sqrt [3]{b \cos (c+d x)} (2 A+2 B \cos (c+d x)+C \cos (2 c+2 d x)+C)} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.375, size = 0, normalized size = 0. \begin{align*} \int{ \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 ) ^{2}{\frac{1}{\sqrt [3]{b\cos \left ( dx+c \right ) }}}}\, 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} + B \cos \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{2}}{\left (b \cos \left (d x + c\right )\right )^{\frac{1}{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} + B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac{2}{3}} \sec \left (d x + c\right )^{2}}{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(-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} + B \cos \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{2}}{\left (b \cos \left (d x + c\right )\right )^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]