Optimal. Leaf size=167 \[ \frac {3 A b \sin (c+d x) \sqrt [3]{b \sec (c+d x)} \sec ^m(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (-3 m-1);\frac {1}{6} (5-3 m);\cos ^2(c+d x)\right )}{d (3 m+1) \sqrt {\sin ^2(c+d x)}}+\frac {3 b B \sin (c+d x) \sqrt [3]{b \sec (c+d x)} \sec ^{m+1}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (-3 m-4);\frac {1}{6} (2-3 m);\cos ^2(c+d x)\right )}{d (3 m+4) \sqrt {\sin ^2(c+d x)}} \]
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Rubi [A] time = 0.12, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {20, 3787, 3772, 2643} \[ \frac {3 A b \sin (c+d x) \sqrt [3]{b \sec (c+d x)} \sec ^m(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (-3 m-1);\frac {1}{6} (5-3 m);\cos ^2(c+d x)\right )}{d (3 m+1) \sqrt {\sin ^2(c+d x)}}+\frac {3 b B \sin (c+d x) \sqrt [3]{b \sec (c+d x)} \sec ^{m+1}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (-3 m-4);\frac {1}{6} (2-3 m);\cos ^2(c+d x)\right )}{d (3 m+4) \sqrt {\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 20
Rule 2643
Rule 3772
Rule 3787
Rubi steps
\begin {align*} \int \sec ^m(c+d x) (b \sec (c+d x))^{4/3} (A+B \sec (c+d x)) \, dx &=\frac {\left (b \sqrt [3]{b \sec (c+d x)}\right ) \int \sec ^{\frac {4}{3}+m}(c+d x) (A+B \sec (c+d x)) \, dx}{\sqrt [3]{\sec (c+d x)}}\\ &=\frac {\left (A b \sqrt [3]{b \sec (c+d x)}\right ) \int \sec ^{\frac {4}{3}+m}(c+d x) \, dx}{\sqrt [3]{\sec (c+d x)}}+\frac {\left (b B \sqrt [3]{b \sec (c+d x)}\right ) \int \sec ^{\frac {7}{3}+m}(c+d x) \, dx}{\sqrt [3]{\sec (c+d x)}}\\ &=\left (A b \cos ^{\frac {1}{3}+m}(c+d x) \sec ^m(c+d x) \sqrt [3]{b \sec (c+d x)}\right ) \int \cos ^{-\frac {4}{3}-m}(c+d x) \, dx+\left (b B \cos ^{\frac {1}{3}+m}(c+d x) \sec ^m(c+d x) \sqrt [3]{b \sec (c+d x)}\right ) \int \cos ^{-\frac {7}{3}-m}(c+d x) \, dx\\ &=\frac {3 A b \, _2F_1\left (\frac {1}{2},\frac {1}{6} (-1-3 m);\frac {1}{6} (5-3 m);\cos ^2(c+d x)\right ) \sec ^m(c+d x) \sqrt [3]{b \sec (c+d x)} \sin (c+d x)}{d (1+3 m) \sqrt {\sin ^2(c+d x)}}+\frac {3 b B \, _2F_1\left (\frac {1}{2},\frac {1}{6} (-4-3 m);\frac {1}{6} (2-3 m);\cos ^2(c+d x)\right ) \sec ^{1+m}(c+d x) \sqrt [3]{b \sec (c+d x)} \sin (c+d x)}{d (4+3 m) \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.39, size = 140, normalized size = 0.84 \[ \frac {3 \sqrt {-\tan ^2(c+d x)} \csc (c+d x) (b \sec (c+d x))^{4/3} \sec ^m(c+d x) \left (A (3 m+7) \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (3 m+4);\frac {m}{2}+\frac {5}{3};\sec ^2(c+d x)\right )+B (3 m+4) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (3 m+7);\frac {1}{6} (3 m+13);\sec ^2(c+d x)\right )\right )}{d (3 m+4) (3 m+7)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B b \sec \left (d x + c\right )^{2} + A b \sec \left (d x + c\right )\right )} \left (b \sec \left (d x + c\right )\right )^{\frac {1}{3}} \sec \left (d x + c\right )^{m}, x\right ) \]
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 \[ \int {\left (B \sec \left (d x + c\right ) + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac {4}{3}} \sec \left (d x + c\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.74, size = 0, normalized size = 0.00 \[ \int \left (\sec ^{m}\left (d x +c \right )\right ) \left (b \sec \left (d x +c \right )\right )^{\frac {4}{3}} \left (A +B \sec \left (d x +c \right )\right )\, dx \]
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 \[ \int {\left (B \sec \left (d x + c\right ) + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac {4}{3}} \sec \left (d x + c\right )^{m}\,{d x} \]
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 \[ \int \left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^{4/3}\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^m \,d x \]
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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