Integrand size = 25, antiderivative size = 25 \[ \int \sqrt [3]{a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx=\frac {\sqrt {2} B \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-\frac {1}{3},\frac {3}{2},\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right ) \sqrt [3]{a+b \sec (c+d x)} \tan (c+d x)}{d \sqrt {1+\sec (c+d x)} \sqrt [3]{\frac {a+b \sec (c+d x)}{a+b}}}+A \text {Int}\left (\sqrt [3]{a+b \sec (c+d x)},x\right ) \]
[Out]
Not integrable
Time = 0.24 (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 = {} \[ \int \sqrt [3]{a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx=\int \sqrt [3]{a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx \]
[In]
[Out]
Rubi steps \begin{align*} \text {integral}& = A \int \sqrt [3]{a+b \sec (c+d x)} \, dx+B \int \sec (c+d x) \sqrt [3]{a+b \sec (c+d x)} \, dx \\ & = A \int \sqrt [3]{a+b \sec (c+d x)} \, dx-\frac {(B \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 )}{d \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}} \\ & = A \int \sqrt [3]{a+b \sec (c+d x)} \, dx-\frac {\left (B \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 )}{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} B \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)}{d \sqrt {1+\sec (c+d x)} \sqrt [3]{\frac {a+b \sec (c+d x)}{a+b}}}+A \int \sqrt [3]{a+b \sec (c+d x)} \, dx \\ \end{align*}
Not integrable
Time = 122.43 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \sqrt [3]{a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx=\int \sqrt [3]{a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx \]
[In]
[Out]
Not integrable
Time = 0.62 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92
\[\int \left (a +b \sec \left (d x +c \right )\right )^{\frac {1}{3}} \left (A +B \sec \left (d x +c \right )\right )d x\]
[In]
[Out]
Timed out. \[ \int \sqrt [3]{a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx=\text {Timed out} \]
[In]
[Out]
Not integrable
Time = 1.44 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \sqrt [3]{a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx=\int \left (A + B \sec {\left (c + d x \right )}\right ) \sqrt [3]{a + b \sec {\left (c + d x \right )}}\, dx \]
[In]
[Out]
Not integrable
Time = 1.43 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \sqrt [3]{a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx=\int { {\left (B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {1}{3}} \,d x } \]
[In]
[Out]
Not integrable
Time = 1.07 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \sqrt [3]{a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx=\int { {\left (B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {1}{3}} \,d x } \]
[In]
[Out]
Not integrable
Time = 17.99 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \sqrt [3]{a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx=\int \left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{1/3} \,d x \]
[In]
[Out]