Optimal. Leaf size=82 \[ -\frac{3 \sin (c+d x) \sqrt [3]{b \sec (c+d x)} \sec ^{m-1}(c+d x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{6} (2-3 m),\frac{1}{6} (8-3 m),\cos ^2(c+d x)\right )}{d (2-3 m) \sqrt{\sin ^2(c+d x)}} \]
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Rubi [A] time = 0.0406077, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {20, 3772, 2643} \[ -\frac{3 \sin (c+d x) \sqrt [3]{b \sec (c+d x)} \sec ^{m-1}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (2-3 m);\frac{1}{6} (8-3 m);\cos ^2(c+d x)\right )}{d (2-3 m) \sqrt{\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 20
Rule 3772
Rule 2643
Rubi steps
\begin{align*} \int \sec ^m(c+d x) \sqrt [3]{b \sec (c+d x)} \, dx &=\frac{\sqrt [3]{b \sec (c+d x)} \int \sec ^{\frac{1}{3}+m}(c+d x) \, dx}{\sqrt [3]{\sec (c+d x)}}\\ &=\left (\cos ^{\frac{1}{3}+m}(c+d x) \sec ^m(c+d x) \sqrt [3]{b \sec (c+d x)}\right ) \int \cos ^{-\frac{1}{3}-m}(c+d x) \, dx\\ &=-\frac{3 \, _2F_1\left (\frac{1}{2},\frac{1}{6} (2-3 m);\frac{1}{6} (8-3 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 (2-3 m) \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.140574, size = 83, normalized size = 1.01 \[ \frac{\sqrt{-\tan ^2(c+d x)} \csc (c+d x) \sqrt [3]{b \sec (c+d x)} \sec ^{m-1}(c+d x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{2} \left (m+\frac{1}{3}\right ),\frac{1}{2} \left (m+\frac{7}{3}\right ),\sec ^2(c+d x)\right )}{d \left (m+\frac{1}{3}\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.086, size = 0, normalized size = 0. \begin{align*} \int \left ( \sec \left ( dx+c \right ) \right ) ^{m}\sqrt [3]{b\sec \left ( dx+c \right ) }\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \sec \left (d x + c\right )\right )^{\frac{1}{3}} \sec \left (d x + c\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (b \sec \left (d x + c\right )\right )^{\frac{1}{3}} \sec \left (d x + c\right )^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt [3]{b \sec{\left (c + d x \right )}} \sec ^{m}{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \sec \left (d x + c\right )\right )^{\frac{1}{3}} \sec \left (d x + c\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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