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

Optimal. Leaf size=89 \[ \frac {\sqrt [3]{a+b \sec (e+f x)} \sqrt [3]{c \cos (e+f x)+d} \text {Int}\left (\frac {\sqrt [3]{a \cos (e+f x)+b}}{\sqrt [3]{c \cos (e+f x)+d}},x\right )}{\sqrt [3]{a \cos (e+f x)+b} \sqrt [3]{c+d \sec (e+f x)}} \]

[Out]

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

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Rubi [A]  time = 0.19, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [A]  time = 2.37, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt [3]{a+b \sec (e+f x)}}{\sqrt [3]{c+d \sec (e+f x)}} \, dx \]

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]

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac {1}{3}}}{{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac {1}{3}}}\,{d x} \]

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)

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maple [A]  time = 1.60, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \sec \left (f x +e \right )\right )^{\frac {1}{3}}}{\left (c +d \sec \left (f x +e \right )\right )^{\frac {1}{3}}}\, dx \]

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)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac {1}{3}}}{{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac {1}{3}}}\,{d x} \]

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)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+\frac {b}{\cos \left (e+f\,x\right )}\right )}^{1/3}}{{\left (c+\frac {d}{\cos \left (e+f\,x\right )}\right )}^{1/3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt [3]{a + b \sec {\left (e + f x \right )}}}{\sqrt [3]{c + d \sec {\left (e + f x \right )}}}\, dx \]

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)

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