Integrand size = 27, antiderivative size = 27 \[ \int \frac {A+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{2/3}} \, dx=\frac {\sqrt {2} C \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-\frac {1}{3},\frac {3}{2},\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right ) \sqrt [3]{a+b \sec (c+d x)} \tan (c+d x)}{b d \sqrt {1+\sec (c+d x)} \sqrt [3]{\frac {a+b \sec (c+d x)}{a+b}}}-\frac {\sqrt {2} a C \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},\frac {2}{3},\frac {3}{2},\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right ) \left (\frac {a+b \sec (c+d x)}{a+b}\right )^{2/3} \tan (c+d x)}{b d \sqrt {1+\sec (c+d x)} (a+b \sec (c+d x))^{2/3}}+A \text {Int}\left (\frac {1}{(a+b \sec (c+d x))^{2/3}},x\right ) \]
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Not integrable
Time = 0.36 (sec) , antiderivative size = 27, 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 = {} \[ \int \frac {A+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{2/3}} \, dx=\int \frac {A+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{2/3}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {A b-a C \sec (c+d x)}{(a+b \sec (c+d x))^{2/3}} \, dx}{b}+\frac {C \int \sec (c+d x) \sqrt [3]{a+b \sec (c+d x)} \, dx}{b} \\ & = A \int \frac {1}{(a+b \sec (c+d x))^{2/3}} \, dx-\frac {(a C) \int \frac {\sec (c+d x)}{(a+b \sec (c+d x))^{2/3}} \, dx}{b}-\frac {(C \tan (c+d x)) \text {Subst}\left (\int \frac {\sqrt [3]{a+b x}}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,\sec (c+d x)\right )}{b 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 {(a C \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 )}{b d \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}}-\frac {\left (C \sqrt [3]{a+b \sec (c+d x)} \tan (c+d x)\right ) \text {Subst}\left (\int \frac {\sqrt [3]{-\frac {a}{-a-b}-\frac {b x}{-a-b}}}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,\sec (c+d x)\right )}{b d \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)} \sqrt [3]{-\frac {a+b \sec (c+d x)}{-a-b}}} \\ & = \frac {\sqrt {2} C \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-\frac {1}{3},\frac {3}{2},\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right ) \sqrt [3]{a+b \sec (c+d x)} \tan (c+d x)}{b d \sqrt {1+\sec (c+d x)} \sqrt [3]{\frac {a+b \sec (c+d x)}{a+b}}}+A \int \frac {1}{(a+b \sec (c+d x))^{2/3}} \, dx+\frac {\left (a C \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 )}{b d \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)} (a+b \sec (c+d x))^{2/3}} \\ & = \frac {\sqrt {2} C \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-\frac {1}{3},\frac {3}{2},\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right ) \sqrt [3]{a+b \sec (c+d x)} \tan (c+d x)}{b d \sqrt {1+\sec (c+d x)} \sqrt [3]{\frac {a+b \sec (c+d x)}{a+b}}}-\frac {\sqrt {2} a C \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)}{b 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*}
Timed out. \[ \int \frac {A+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{2/3}} \, dx=\text {\$Aborted} \]
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Not integrable
Time = 0.12 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93
\[\int \frac {A +C \sec \left (d x +c \right )^{2}}{\left (a +b \sec \left (d x +c \right )\right )^{\frac {2}{3}}}d x\]
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Timed out. \[ \int \frac {A+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{2/3}} \, dx=\text {Timed out} \]
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Not integrable
Time = 0.96 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {A+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{2/3}} \, dx=\int \frac {A + C \sec ^{2}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{\frac {2}{3}}}\, dx \]
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Not integrable
Time = 11.76 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {A+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{2/3}} \, dx=\int { \frac {C \sec \left (d x + c\right )^{2} + A}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {2}{3}}} \,d x } \]
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Not integrable
Time = 2.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {A+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{2/3}} \, dx=\int { \frac {C \sec \left (d x + c\right )^{2} + A}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {2}{3}}} \,d x } \]
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Not integrable
Time = 19.40 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {A+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{2/3}} \, dx=\int \frac {A+\frac {C}{{\cos \left (c+d\,x\right )}^2}}{{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{2/3}} \,d x \]
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