Optimal. Leaf size=116 \[ -\frac {3 A \sin (c+d x) (b \cos (c+d x))^{4/3} \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};\cos ^2(c+d x)\right )}{4 d \sqrt {\sin ^2(c+d x)}}-\frac {3 B \sin (c+d x) (b \cos (c+d x))^{7/3} \, _2F_1\left (\frac {1}{2},\frac {7}{6};\frac {13}{6};\cos ^2(c+d x)\right )}{7 b d \sqrt {\sin ^2(c+d x)}} \]
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Rubi [A] time = 0.08, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {16, 2748, 2643} \[ -\frac {3 A \sin (c+d x) (b \cos (c+d x))^{4/3} \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};\cos ^2(c+d x)\right )}{4 d \sqrt {\sin ^2(c+d x)}}-\frac {3 B \sin (c+d x) (b \cos (c+d x))^{7/3} \, _2F_1\left (\frac {1}{2},\frac {7}{6};\frac {13}{6};\cos ^2(c+d x)\right )}{7 b d \sqrt {\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 2643
Rule 2748
Rubi steps
\begin {align*} \int (b \cos (c+d x))^{4/3} (A+B \cos (c+d x)) \sec (c+d x) \, dx &=b \int \sqrt [3]{b \cos (c+d x)} (A+B \cos (c+d x)) \, dx\\ &=(A b) \int \sqrt [3]{b \cos (c+d x)} \, dx+B \int (b \cos (c+d x))^{4/3} \, dx\\ &=-\frac {3 A (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)}{4 d \sqrt {\sin ^2(c+d x)}}-\frac {3 B (b \cos (c+d x))^{7/3} \, _2F_1\left (\frac {1}{2},\frac {7}{6};\frac {13}{6};\cos ^2(c+d x)\right ) \sin (c+d x)}{7 b d \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 87, normalized size = 0.75 \[ -\frac {3 b \sqrt {\sin ^2(c+d x)} \cot (c+d x) \sqrt [3]{b \cos (c+d x)} \left (7 A \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};\cos ^2(c+d x)\right )+4 B \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {7}{6};\frac {13}{6};\cos ^2(c+d x)\right )\right )}{28 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B b \cos \left (d x + c\right )^{2} + A b \cos \left (d x + c\right )\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {1}{3}} \sec \left (d x + c\right ), x\right ) \]
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 \[ \int {\left (B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {4}{3}} \sec \left (d x + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.23, size = 0, normalized size = 0.00 \[ \int \left (b \cos \left (d x +c \right )\right )^{\frac {4}{3}} \left (A +B \cos \left (d x +c \right )\right ) \sec \left (d x +c \right )\, dx \]
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 \[ \int {\left (B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {4}{3}} \sec \left (d x + c\right )\,{d x} \]
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 \[ \int \frac {{\left (b\,\cos \left (c+d\,x\right )\right )}^{4/3}\,\left (A+B\,\cos \left (c+d\,x\right )\right )}{\cos \left (c+d\,x\right )} \,d x \]
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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