3.221 \(\int \frac{\sqrt [3]{a+b \sec (e+f x)}}{\sqrt [3]{c+d \sec (e+f x)}} \, dx\)

Optimal. Leaf size=88 \[ \frac{\sqrt [3]{a+b \sec (e+f x)} \sqrt [3]{c \cos (e+f x)+d} \text{Unintegrable}\left (\frac{\sqrt [3]{a \cos (e+f x)+b}}{\sqrt [3]{c \cos (e+f x)+d}},x\right )}{\sqrt [3]{a \cos (e+f x)+b} \sqrt [3]{c+d \sec (e+f x)}} \]

[Out]

((d + c*Cos[e + f*x])^(1/3)*(a + b*Sec[e + f*x])^(1/3)*Unintegrable[(b + a*Cos[e + f*x])^(1/3)/(d + c*Cos[e +
f*x])^(1/3), x])/((b + a*Cos[e + f*x])^(1/3)*(c + d*Sec[e + f*x])^(1/3))

________________________________________________________________________________________

Rubi [A]  time = 0.190879, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\sqrt [3]{a+b \sec (e+f x)}}{\sqrt [3]{c+d \sec (e+f x)}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

((d + c*Cos[e + f*x])^(1/3)*(a + b*Sec[e + f*x])^(1/3)*Defer[Int][(b + a*Cos[e + f*x])^(1/3)/(d + c*Cos[e + f*
x])^(1/3), x])/((b + a*Cos[e + f*x])^(1/3)*(c + d*Sec[e + f*x])^(1/3))

Rubi steps

\begin{align*} \int \frac{\sqrt [3]{a+b \sec (e+f x)}}{\sqrt [3]{c+d \sec (e+f x)}} \, dx &=\frac{\left (\sqrt [3]{d+c \cos (e+f x)} \sqrt [3]{a+b \sec (e+f x)}\right ) \int \frac{\sqrt [3]{b+a \cos (e+f x)}}{\sqrt [3]{d+c \cos (e+f x)}} \, dx}{\sqrt [3]{b+a \cos (e+f x)} \sqrt [3]{c+d \sec (e+f x)}}\\ \end{align*}

Mathematica [A]  time = 2.15962, size = 0, normalized size = 0. \[ \int \frac{\sqrt [3]{a+b \sec (e+f x)}}{\sqrt [3]{c+d \sec (e+f x)}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.257, size = 0, normalized size = 0. \begin{align*} \int{\sqrt [3]{a+b\sec \left ( fx+e \right ) }{\frac{1}{\sqrt [3]{c+d\sec \left ( fx+e \right ) }}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac{1}{3}}}{{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac{1}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt [3]{a + b \sec{\left (e + f x \right )}}}{\sqrt [3]{c + d \sec{\left (e + f x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(f*x+e))**(1/3)/(c+d*sec(f*x+e))**(1/3),x)

[Out]

Integral((a + b*sec(e + f*x))**(1/3)/(c + d*sec(e + f*x))**(1/3), x)

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac{1}{3}}}{{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac{1}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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