Optimal. Leaf size=82 \[ -\frac {3 \, _2F_1\left (\frac {1}{2},\frac {1}{6} (5-3 m);\frac {1}{6} (11-3 m);\cos ^2(c+d x)\right ) \sec ^{-1+m}(c+d x) \sin (c+d x)}{d (5-3 m) (b \sec (c+d x))^{2/3} \sqrt {\sin ^2(c+d x)}} \]
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Rubi [A]
time = 0.03, antiderivative size = 82, 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 = {20, 3857, 2722}
\begin {gather*} -\frac {3 \sin (c+d x) \sec ^{m-1}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (5-3 m);\frac {1}{6} (11-3 m);\cos ^2(c+d x)\right )}{d (5-3 m) \sqrt {\sin ^2(c+d x)} (b \sec (c+d x))^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 20
Rule 2722
Rule 3857
Rubi steps
\begin {align*} \int \frac {\sec ^m(c+d x)}{(b \sec (c+d x))^{2/3}} \, dx &=\frac {\sec ^{\frac {2}{3}}(c+d x) \int \sec ^{-\frac {2}{3}+m}(c+d x) \, dx}{(b \sec (c+d x))^{2/3}}\\ &=\frac {\left (\cos ^{\frac {1}{3}+m}(c+d x) \sec ^{1+m}(c+d x)\right ) \int \cos ^{\frac {2}{3}-m}(c+d x) \, dx}{(b \sec (c+d x))^{2/3}}\\ &=-\frac {3 \, _2F_1\left (\frac {1}{2},\frac {1}{6} (5-3 m);\frac {1}{6} (11-3 m);\cos ^2(c+d x)\right ) \sec ^{-1+m}(c+d x) \sin (c+d x)}{d (5-3 m) (b \sec (c+d x))^{2/3} \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 83, normalized size = 1.01 \begin {gather*} \frac {\csc (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{2} \left (-\frac {2}{3}+m\right );\frac {1}{2} \left (\frac {4}{3}+m\right );\sec ^2(c+d x)\right ) \sec ^{-1+m}(c+d x) \sqrt {-\tan ^2(c+d x)}}{d \left (-\frac {2}{3}+m\right ) (b \sec (c+d x))^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.44, size = 0, normalized size = 0.00 \[\int \frac {\sec ^{m}\left (d x +c \right )}{\left (b \sec \left (d x +c \right )\right )^{\frac {2}{3}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\sec ^{m}{\left (c + d x \right )}}{\left (b \sec {\left (c + d x \right )}\right )^{\frac {2}{3}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^m}{{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^{2/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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