Optimal. Leaf size=149 \[ \frac {3 A b^2 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/3}}+\frac {3 (4 A+7 C) \sin (c+d x) \, _2F_1\left (-\frac {1}{6},\frac {1}{2};\frac {5}{6};\cos ^2(c+d x)\right )}{7 d \sqrt {\sin ^2(c+d x)} \sqrt [3]{b \cos (c+d x)}}+\frac {3 b B \sin (c+d x) \, _2F_1\left (-\frac {2}{3},\frac {1}{2};\frac {1}{3};\cos ^2(c+d x)\right )}{4 d \sqrt {\sin ^2(c+d x)} (b \cos (c+d x))^{4/3}} \]
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Rubi [A] time = 0.18, antiderivative size = 149, normalized size of antiderivative = 1.00, 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} \[ \frac {3 A b^2 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/3}}+\frac {3 (4 A+7 C) \sin (c+d x) \, _2F_1\left (-\frac {1}{6},\frac {1}{2};\frac {5}{6};\cos ^2(c+d x)\right )}{7 d \sqrt {\sin ^2(c+d x)} \sqrt [3]{b \cos (c+d x)}}+\frac {3 b B \sin (c+d x) \, _2F_1\left (-\frac {2}{3},\frac {1}{2};\frac {1}{3};\cos ^2(c+d x)\right )}{4 d \sqrt {\sin ^2(c+d x)} (b \cos (c+d x))^{4/3}} \]
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 ^3(c+d x)}{\sqrt [3]{b \cos (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))^{10/3}} \, dx\\ &=\frac {3 A b^2 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/3}}+\frac {3}{7} \int \frac {\frac {7 b^2 B}{3}+\frac {1}{3} b^2 (4 A+7 C) \cos (c+d x)}{(b \cos (c+d x))^{7/3}} \, dx\\ &=\frac {3 A b^2 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/3}}+\left (b^2 B\right ) \int \frac {1}{(b \cos (c+d x))^{7/3}} \, dx+\frac {1}{7} (b (4 A+7 C)) \int \frac {1}{(b \cos (c+d x))^{4/3}} \, dx\\ &=\frac {3 A b^2 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/3}}+\frac {3 b B \, _2F_1\left (-\frac {2}{3},\frac {1}{2};\frac {1}{3};\cos ^2(c+d x)\right ) \sin (c+d x)}{4 d (b \cos (c+d x))^{4/3} \sqrt {\sin ^2(c+d x)}}+\frac {3 (4 A+7 C) \, _2F_1\left (-\frac {1}{6},\frac {1}{2};\frac {5}{6};\cos ^2(c+d x)\right ) \sin (c+d x)}{7 d \sqrt [3]{b \cos (c+d x)} \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 118, normalized size = 0.79 \[ \frac {3 b^2 \sqrt {\sin ^2(c+d x)} \csc (c+d x) \left (4 A \, _2F_1\left (-\frac {7}{6},\frac {1}{2};-\frac {1}{6};\cos ^2(c+d x)\right )+7 \cos (c+d x) \left (B \, _2F_1\left (-\frac {2}{3},\frac {1}{2};\frac {1}{3};\cos ^2(c+d x)\right )+4 C \cos (c+d x) \, _2F_1\left (-\frac {1}{6},\frac {1}{2};\frac {5}{6};\cos ^2(c+d x)\right )\right )\right )}{28 d (b \cos (c+d x))^{7/3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}} \sec \left (d x + c\right )^{3}}{b \cos \left (d x + c\right )}, x\right ) \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{3}}{\left (b \cos \left (d x + c\right )\right )^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.62, size = 0, normalized size = 0.00 \[ \int \frac {\left (A +B \cos \left (d x +c \right )+C \left (\cos ^{2}\left (d x +c \right )\right )\right ) \left (\sec ^{3}\left (d x +c \right )\right )}{\left (b \cos \left (d x +c \right )\right )^{\frac {1}{3}}}\, dx \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{3}}{\left (b \cos \left (d x + c\right )\right )^{\frac {1}{3}}}\,{d x} \]
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 \[ \int \frac {C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A}{{\cos \left (c+d\,x\right )}^3\,{\left (b\,\cos \left (c+d\,x\right )\right )}^{1/3}} \,d x \]
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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