Integrand size = 25, antiderivative size = 25 \[ \int \frac {A+B \sec (c+d x)}{(a+b \sec (c+d x))^{2/3}} \, dx=\frac {\sqrt {2} B \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},\frac {2}{3},\frac {3}{2},\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right ) \left (\frac {a+b \sec (c+d x)}{a+b}\right )^{2/3} \tan (c+d x)}{d \sqrt {1+\sec (c+d x)} (a+b \sec (c+d x))^{2/3}}+A \text {Int}\left (\frac {1}{(a+b \sec (c+d x))^{2/3}},x\right ) \]
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Not integrable
Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {A+B \sec (c+d x)}{(a+b \sec (c+d x))^{2/3}} \, dx=\int \frac {A+B \sec (c+d x)}{(a+b \sec (c+d x))^{2/3}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = A \int \frac {1}{(a+b \sec (c+d x))^{2/3}} \, dx+B \int \frac {\sec (c+d x)}{(a+b \sec (c+d x))^{2/3}} \, dx \\ & = A \int \frac {1}{(a+b \sec (c+d x))^{2/3}} \, dx-\frac {(B \tan (c+d x)) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} \sqrt {1+x} (a+b x)^{2/3}} \, dx,x,\sec (c+d x)\right )}{d \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}} \\ & = A \int \frac {1}{(a+b \sec (c+d x))^{2/3}} \, dx-\frac {\left (B \left (-\frac {a+b \sec (c+d x)}{-a-b}\right )^{2/3} \tan (c+d x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} \sqrt {1+x} \left (-\frac {a}{-a-b}-\frac {b x}{-a-b}\right )^{2/3}} \, dx,x,\sec (c+d x)\right )}{d \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)} (a+b \sec (c+d x))^{2/3}} \\ & = \frac {\sqrt {2} B \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},\frac {2}{3},\frac {3}{2},\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right ) \left (\frac {a+b \sec (c+d x)}{a+b}\right )^{2/3} \tan (c+d x)}{d \sqrt {1+\sec (c+d x)} (a+b \sec (c+d x))^{2/3}}+A \int \frac {1}{(a+b \sec (c+d x))^{2/3}} \, dx \\ \end{align*}
Not integrable
Time = 30.48 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {A+B \sec (c+d x)}{(a+b \sec (c+d x))^{2/3}} \, dx=\int \frac {A+B \sec (c+d x)}{(a+b \sec (c+d x))^{2/3}} \, dx \]
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Not integrable
Time = 0.56 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92
\[\int \frac {A +B \sec \left (d x +c \right )}{\left (a +b \sec \left (d x +c \right )\right )^{\frac {2}{3}}}d x\]
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Timed out. \[ \int \frac {A+B \sec (c+d x)}{(a+b \sec (c+d x))^{2/3}} \, dx=\text {Timed out} \]
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Not integrable
Time = 1.13 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {A+B \sec (c+d x)}{(a+b \sec (c+d x))^{2/3}} \, dx=\int \frac {A + B \sec {\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{\frac {2}{3}}}\, dx \]
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Not integrable
Time = 1.48 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {A+B \sec (c+d x)}{(a+b \sec (c+d x))^{2/3}} \, dx=\int { \frac {B \sec \left (d x + c\right ) + A}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {2}{3}}} \,d x } \]
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Not integrable
Time = 2.06 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {A+B \sec (c+d x)}{(a+b \sec (c+d x))^{2/3}} \, dx=\int { \frac {B \sec \left (d x + c\right ) + A}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {2}{3}}} \,d x } \]
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Not integrable
Time = 16.56 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {A+B \sec (c+d x)}{(a+b \sec (c+d x))^{2/3}} \, dx=\int \frac {A+\frac {B}{\cos \left (c+d\,x\right )}}{{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{2/3}} \,d x \]
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