Optimal. Leaf size=56 \[ -\frac {3 \sin (c+d x) \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {7}{6};\cos ^2(c+d x)\right )}{b d \sqrt {\sin ^2(c+d x)} \sqrt [3]{b \sec (c+d x)}} \]
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Rubi [A] time = 0.03, antiderivative size = 56, normalized size of antiderivative = 1.00, 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 = {16, 3772, 2643} \[ -\frac {3 \sin (c+d x) \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {7}{6};\cos ^2(c+d x)\right )}{b d \sqrt {\sin ^2(c+d x)} \sqrt [3]{b \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 2643
Rule 3772
Rubi steps
\begin {align*} \int \frac {\sec ^2(c+d x)}{(b \sec (c+d x))^{4/3}} \, dx &=\frac {\int (b \sec (c+d x))^{2/3} \, dx}{b^2}\\ &=\frac {\left (\left (\frac {\cos (c+d x)}{b}\right )^{2/3} (b \sec (c+d x))^{2/3}\right ) \int \frac {1}{\left (\frac {\cos (c+d x)}{b}\right )^{2/3}} \, dx}{b^2}\\ &=-\frac {3 \cos (c+d x) \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {7}{6};\cos ^2(c+d x)\right ) (b \sec (c+d x))^{2/3} \sin (c+d x)}{b^2 d \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 60, normalized size = 1.07 \[ \frac {3 \sqrt {-\tan ^2(c+d x)} \cot (c+d x) (b \sec (c+d x))^{2/3} \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {4}{3};\sec ^2(c+d x)\right )}{2 b^2 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.87, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (b \sec \left (d x + c\right )\right )^{\frac {2}{3}}}{b^{2}}, x\right ) \]
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 \[ \int \frac {\sec \left (d x + c\right )^{2}}{\left (b \sec \left (d x + c\right )\right )^{\frac {4}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.44, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^{2}\left (d x +c \right )}{\left (b \sec \left (d x +c \right )\right )^{\frac {4}{3}}}\, dx \]
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 \[ \int \frac {\sec \left (d x + c\right )^{2}}{\left (b \sec \left (d x + c\right )\right )^{\frac {4}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\cos \left (c+d\,x\right )}^2\,{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^{4/3}} \,d x \]
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 \[ \int \frac {\sec ^{2}{\left (c + d x \right )}}{\left (b \sec {\left (c + d x \right )}\right )^{\frac {4}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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