Optimal. Leaf size=147 \[ -\frac {3 (A+4 C) \sin (c+d x) (b \cos (c+d x))^{2/3} \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {4}{3};\cos ^2(c+d x)\right )}{8 b^2 d \sqrt {\sin ^2(c+d x)}}+\frac {3 A \sin (c+d x)}{4 d (b \cos (c+d x))^{4/3}}+\frac {3 B \sin (c+d x) \, _2F_1\left (-\frac {1}{6},\frac {1}{2};\frac {5}{6};\cos ^2(c+d x)\right )}{b d \sqrt {\sin ^2(c+d x)} \sqrt [3]{b \cos (c+d x)}} \]
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Rubi [A] time = 0.17, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {16, 3021, 2748, 2643} \[ -\frac {3 (A+4 C) \sin (c+d x) (b \cos (c+d x))^{2/3} \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {4}{3};\cos ^2(c+d x)\right )}{8 b^2 d \sqrt {\sin ^2(c+d x)}}+\frac {3 A \sin (c+d x)}{4 d (b \cos (c+d x))^{4/3}}+\frac {3 B \sin (c+d x) \, _2F_1\left (-\frac {1}{6},\frac {1}{2};\frac {5}{6};\cos ^2(c+d x)\right )}{b d \sqrt {\sin ^2(c+d x)} \sqrt [3]{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 2643
Rule 2748
Rule 3021
Rubi steps
\begin {align*} \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x)}{(b \cos (c+d x))^{4/3}} \, dx &=b \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 \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^2}\\ &=\frac {3 A \sin (c+d x)}{4 d (b \cos (c+d x))^{4/3}}+B \int \frac {1}{(b \cos (c+d x))^{4/3}} \, dx+\frac {(A+4 C) \int \frac {1}{\sqrt [3]{b \cos (c+d x)}} \, dx}{4 b}\\ &=\frac {3 A \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)}{b 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^2 d \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [B] time = 6.29, size = 703, normalized size = 4.78 \[ \frac {\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)}}{b} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.11, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}} \sec \left (d x + c\right )}{b^{2} \cos \left (d x + c\right )^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )}{\left (b \cos \left (d x + c\right )\right )^{\frac {4}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.48, size = 0, normalized size = 0.00 \[ \int \frac {\left (A +B \cos \left (d x +c \right )+C \left (\cos ^{2}\left (d x +c \right )\right )\right ) \sec \left (d x +c \right )}{\left (b \cos \left (d x +c \right )\right )^{\frac {4}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )}{\left (b \cos \left (d x + c\right )\right )^{\frac {4}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A}{\cos \left (c+d\,x\right )\,{\left (b\,\cos \left (c+d\,x\right )\right )}^{4/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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