Optimal. Leaf size=117 \[ \frac {3 A b^2 \sin (c+d x) \, _2F_1\left (-\frac {4}{3},\frac {1}{2};-\frac {1}{3};\cos ^2(c+d x)\right )}{8 d \sqrt {\sin ^2(c+d x)} (b \cos (c+d x))^{8/3}}+\frac {3 b B \sin (c+d x) \, _2F_1\left (-\frac {5}{6},\frac {1}{2};\frac {1}{6};\cos ^2(c+d x)\right )}{5 d \sqrt {\sin ^2(c+d x)} (b \cos (c+d x))^{5/3}} \]
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Rubi [A] time = 0.10, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {16, 2748, 2643} \[ \frac {3 A b^2 \sin (c+d x) \, _2F_1\left (-\frac {4}{3},\frac {1}{2};-\frac {1}{3};\cos ^2(c+d x)\right )}{8 d \sqrt {\sin ^2(c+d x)} (b \cos (c+d x))^{8/3}}+\frac {3 b B \sin (c+d x) \, _2F_1\left (-\frac {5}{6},\frac {1}{2};\frac {1}{6};\cos ^2(c+d x)\right )}{5 d \sqrt {\sin ^2(c+d x)} (b \cos (c+d x))^{5/3}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 2643
Rule 2748
Rubi steps
\begin {align*} \int \frac {(A+B \cos (c+d x)) \sec ^3(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx &=b^3 \int \frac {A+B \cos (c+d x)}{(b \cos (c+d x))^{11/3}} \, dx\\ &=\left (A b^3\right ) \int \frac {1}{(b \cos (c+d x))^{11/3}} \, dx+\left (b^2 B\right ) \int \frac {1}{(b \cos (c+d x))^{8/3}} \, dx\\ &=\frac {3 A b^2 \, _2F_1\left (-\frac {4}{3},\frac {1}{2};-\frac {1}{3};\cos ^2(c+d x)\right ) \sin (c+d x)}{8 d (b \cos (c+d x))^{8/3} \sqrt {\sin ^2(c+d x)}}+\frac {3 b B \, _2F_1\left (-\frac {5}{6},\frac {1}{2};\frac {1}{6};\cos ^2(c+d x)\right ) \sin (c+d x)}{5 d (b \cos (c+d x))^{5/3} \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 89, normalized size = 0.76 \[ \frac {3 b^2 \sqrt {\sin ^2(c+d x)} \csc (c+d x) \left (5 A \, _2F_1\left (-\frac {4}{3},\frac {1}{2};-\frac {1}{3};\cos ^2(c+d x)\right )+8 B \cos (c+d x) \, _2F_1\left (-\frac {5}{6},\frac {1}{2};\frac {1}{6};\cos ^2(c+d x)\right )\right )}{40 d (b \cos (c+d x))^{8/3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.09, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {1}{3}} \sec \left (d x + c\right )^{3}}{b \cos \left (d x + c\right )}, x\right ) \]
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 \[ \int \frac {{\left (B \cos \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{3}}{\left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.28, size = 0, normalized size = 0.00 \[ \int \frac {\left (A +B \cos \left (d x +c \right )\right ) \left (\sec ^{3}\left (d x +c \right )\right )}{\left (b \cos \left (d x +c \right )\right )^{\frac {2}{3}}}\, dx \]
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 \[ \int \frac {{\left (B \cos \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{3}}{\left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}}}\,{d x} \]
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 \[ \int \frac {A+B\,\cos \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^3\,{\left (b\,\cos \left (c+d\,x\right )\right )}^{2/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (A + B \cos {\left (c + d x \right )}\right ) \sec ^{3}{\left (c + d x \right )}}{\left (b \cos {\left (c + d x \right )}\right )^{\frac {2}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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