\(\int \frac {A+B \sec (c+d x)}{\sqrt [3]{a+b \sec (c+d x)}} \, dx\) [476]

Optimal result

Integrand size = 25, antiderivative size = 25 \[ \int \frac {A+B \sec (c+d x)}{\sqrt [3]{a+b \sec (c+d x)}} \, dx=\frac {\sqrt {2} B \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},\frac {1}{3},\frac {3}{2},\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right ) \sqrt [3]{\frac {a+b \sec (c+d x)}{a+b}} \tan (c+d x)}{d \sqrt {1+\sec (c+d x)} \sqrt [3]{a+b \sec (c+d x)}}+A \text {Int}\left (\frac {1}{\sqrt [3]{a+b \sec (c+d x)}},x\right ) \]

[Out]

B*AppellF1(1/2,1/3,1/2,3/2,b*(1-sec(d*x+c))/(a+b),1/2-1/2*sec(d*x+c))*((a+b*sec(d*x+c))/(a+b))^(1/3)*2^(1/2)*t
an(d*x+c)/d/(a+b*sec(d*x+c))^(1/3)/(1+sec(d*x+c))^(1/2)+A*Unintegrable(1/(a+b*sec(d*x+c))^(1/3),x)

Rubi [N/A]

Not integrable

Time = 0.26 (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 = {} \[ \int \frac {A+B \sec (c+d x)}{\sqrt [3]{a+b \sec (c+d x)}} \, dx=\int \frac {A+B \sec (c+d x)}{\sqrt [3]{a+b \sec (c+d x)}} \, dx \]

[In]

Int[(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^(1/3),x]

[Out]

(Sqrt[2]*B*AppellF1[1/2, 1/2, 1/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c +
d*x])/(a + b))^(1/3)*Tan[c + d*x])/(d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(1/3)) + A*Defer[Int][(a + b
*Sec[c + d*x])^(-1/3), x]

Rubi steps \begin{align*} \text {integral}& = A \int \frac {1}{\sqrt [3]{a+b \sec (c+d x)}} \, dx+B \int \frac {\sec (c+d x)}{\sqrt [3]{a+b \sec (c+d x)}} \, dx \\ & = A \int \frac {1}{\sqrt [3]{a+b \sec (c+d x)}} \, dx-\frac {(B \tan (c+d x)) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} \sqrt {1+x} \sqrt [3]{a+b x}} \, dx,x,\sec (c+d x)\right )}{d \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}} \\ & = A \int \frac {1}{\sqrt [3]{a+b \sec (c+d x)}} \, dx-\frac {\left (B \sqrt [3]{-\frac {a+b \sec (c+d x)}{-a-b}} \tan (c+d x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} \sqrt {1+x} \sqrt [3]{-\frac {a}{-a-b}-\frac {b x}{-a-b}}} \, dx,x,\sec (c+d x)\right )}{d \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)} \sqrt [3]{a+b \sec (c+d x)}} \\ & = \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]{\frac {a+b \sec (c+d x)}{a+b}} \tan (c+d x)}{d \sqrt {1+\sec (c+d x)} \sqrt [3]{a+b \sec (c+d x)}}+A \int \frac {1}{\sqrt [3]{a+b \sec (c+d x)}} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 34.37 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {A+B \sec (c+d x)}{\sqrt [3]{a+b \sec (c+d x)}} \, dx=\int \frac {A+B \sec (c+d x)}{\sqrt [3]{a+b \sec (c+d x)}} \, dx \]

[In]

Integrate[(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^(1/3),x]

[Out]

Integrate[(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^(1/3), x]

Maple [N/A] (verified)

Not integrable

Time = 0.61 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92

\[\int \frac {A +B \sec \left (d x +c \right )}{\left (a +b \sec \left (d x +c \right )\right )^{\frac {1}{3}}}d x\]

[In]

int((A+B*sec(d*x+c))/(a+b*sec(d*x+c))^(1/3),x)

[Out]

int((A+B*sec(d*x+c))/(a+b*sec(d*x+c))^(1/3),x)

Fricas [F(-1)]

Timed out. \[ \int \frac {A+B \sec (c+d x)}{\sqrt [3]{a+b \sec (c+d x)}} \, dx=\text {Timed out} \]

[In]

integrate((A+B*sec(d*x+c))/(a+b*sec(d*x+c))^(1/3),x, algorithm="fricas")

[Out]

Timed out

Sympy [N/A]

Not integrable

Time = 1.05 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {A+B \sec (c+d x)}{\sqrt [3]{a+b \sec (c+d x)}} \, dx=\int \frac {A + B \sec {\left (c + d x \right )}}{\sqrt [3]{a + b \sec {\left (c + d x \right )}}}\, dx \]

[In]

integrate((A+B*sec(d*x+c))/(a+b*sec(d*x+c))**(1/3),x)

[Out]

Integral((A + B*sec(c + d*x))/(a + b*sec(c + d*x))**(1/3), x)

Maxima [N/A]

Not integrable

Time = 1.49 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {A+B \sec (c+d x)}{\sqrt [3]{a+b \sec (c+d x)}} \, dx=\int { \frac {B \sec \left (d x + c\right ) + A}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {1}{3}}} \,d x } \]

[In]

integrate((A+B*sec(d*x+c))/(a+b*sec(d*x+c))^(1/3),x, algorithm="maxima")

[Out]

integrate((B*sec(d*x + c) + A)/(b*sec(d*x + c) + a)^(1/3), x)

Giac [N/A]

Not integrable

Time = 1.20 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {A+B \sec (c+d x)}{\sqrt [3]{a+b \sec (c+d x)}} \, dx=\int { \frac {B \sec \left (d x + c\right ) + A}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {1}{3}}} \,d x } \]

[In]

integrate((A+B*sec(d*x+c))/(a+b*sec(d*x+c))^(1/3),x, algorithm="giac")

[Out]

integrate((B*sec(d*x + c) + A)/(b*sec(d*x + c) + a)^(1/3), x)

Mupad [N/A]

Not integrable

Time = 17.19 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {A+B \sec (c+d x)}{\sqrt [3]{a+b \sec (c+d x)}} \, dx=\int \frac {A+\frac {B}{\cos \left (c+d\,x\right )}}{{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{1/3}} \,d x \]

[In]

int((A + B/cos(c + d*x))/(a + b/cos(c + d*x))^(1/3),x)

[Out]

int((A + B/cos(c + d*x))/(a + b/cos(c + d*x))^(1/3), x)