Integrand size = 27, antiderivative size = 27 \[ \int (a+b \sec (c+d x))^{2/3} \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {\sqrt {2} (a+b) C \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-\frac {5}{3},\frac {3}{2},\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right ) (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{b d \sqrt {1+\sec (c+d x)} \left (\frac {a+b \sec (c+d x)}{a+b}\right )^{2/3}}-\frac {\sqrt {2} a C \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-\frac {2}{3},\frac {3}{2},\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right ) (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{b d \sqrt {1+\sec (c+d x)} \left (\frac {a+b \sec (c+d x)}{a+b}\right )^{2/3}}+A \text {Int}\left ((a+b \sec (c+d x))^{2/3},x\right ) \]
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Not integrable
Time = 0.39 (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 = {} \[ \int (a+b \sec (c+d x))^{2/3} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int (a+b \sec (c+d x))^{2/3} \left (A+C \sec ^2(c+d x)\right ) \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\int (a+b \sec (c+d x))^{2/3} (A b-a C \sec (c+d x)) \, dx}{b}+\frac {C \int \sec (c+d x) (a+b \sec (c+d x))^{5/3} \, dx}{b} \\ & = A \int (a+b \sec (c+d x))^{2/3} \, dx-\frac {(a C) \int \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 {(a+b x)^{5/3}}{\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 (a+b \sec (c+d x))^{2/3} \, dx+\frac {(a C \tan (c+d x)) \text {Subst}\left (\int \frac {(a+b x)^{2/3}}{\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)}}+\frac {\left ((-a-b) C (a+b \sec (c+d x))^{2/3} \tan (c+d x)\right ) \text {Subst}\left (\int \frac {\left (-\frac {a}{-a-b}-\frac {b x}{-a-b}\right )^{5/3}}{\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)} \left (-\frac {a+b \sec (c+d x)}{-a-b}\right )^{2/3}} \\ & = \frac {\sqrt {2} (a+b) C \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-\frac {5}{3},\frac {3}{2},\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right ) (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{b d \sqrt {1+\sec (c+d x)} \left (\frac {a+b \sec (c+d x)}{a+b}\right )^{2/3}}+A \int (a+b \sec (c+d x))^{2/3} \, dx+\frac {\left (a C (a+b \sec (c+d x))^{2/3} \tan (c+d x)\right ) \text {Subst}\left (\int \frac {\left (-\frac {a}{-a-b}-\frac {b x}{-a-b}\right )^{2/3}}{\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)} \left (-\frac {a+b \sec (c+d x)}{-a-b}\right )^{2/3}} \\ & = \frac {\sqrt {2} (a+b) C \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-\frac {5}{3},\frac {3}{2},\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right ) (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{b d \sqrt {1+\sec (c+d x)} \left (\frac {a+b \sec (c+d x)}{a+b}\right )^{2/3}}-\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 ) (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{b d \sqrt {1+\sec (c+d x)} \left (\frac {a+b \sec (c+d x)}{a+b}\right )^{2/3}}+A \int (a+b \sec (c+d x))^{2/3} \, dx \\ \end{align*}
Timed out. \[ \int (a+b \sec (c+d x))^{2/3} \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {\$Aborted} \]
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Not integrable
Time = 0.13 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93
\[\int \left (a +b \sec \left (d x +c \right )\right )^{\frac {2}{3}} \left (A +C \sec \left (d x +c \right )^{2}\right )d x\]
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Timed out. \[ \int (a+b \sec (c+d x))^{2/3} \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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Not integrable
Time = 6.29 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int (a+b \sec (c+d x))^{2/3} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int \left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right )^{\frac {2}{3}}\, dx \]
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Not integrable
Time = 7.04 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int (a+b \sec (c+d x))^{2/3} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \,d x } \]
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Not integrable
Time = 1.56 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int (a+b \sec (c+d x))^{2/3} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \,d x } \]
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Not integrable
Time = 20.79 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int (a+b \sec (c+d x))^{2/3} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int \left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{2/3} \,d x \]
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