3.222 \(\int \frac {\sqrt [3]{a+b \sec (e+f x)}}{(c+d \sec (e+f x))^{4/3}} \, dx\)

Optimal. Leaf size=32 \[ \text {Int}\left (\frac {\sqrt [3]{a+b \sec (e+f x)}}{(c+d \sec (e+f x))^{4/3}},x\right ) \]

[Out]

Unintegrable((a+b*sec(f*x+e))^(1/3)/(c+d*sec(f*x+e))^(4/3),x)

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Rubi [A]  time = 0.09, 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 = {} \[ \int \frac {\sqrt [3]{a+b \sec (e+f x)}}{(c+d \sec (e+f x))^{4/3}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*Sec[e + f*x])^(1/3)/(c + d*Sec[e + f*x])^(4/3),x]

[Out]

Defer[Int][(a + b*Sec[e + f*x])^(1/3)/(c + d*Sec[e + f*x])^(4/3), x]

Rubi steps

\begin {align*} \int \frac {\sqrt [3]{a+b \sec (e+f x)}}{(c+d \sec (e+f x))^{4/3}} \, dx &=\int \frac {\sqrt [3]{a+b \sec (e+f x)}}{(c+d \sec (e+f x))^{4/3}} \, dx\\ \end {align*}

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Mathematica [A]  time = 52.11, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt [3]{a+b \sec (e+f x)}}{(c+d \sec (e+f x))^{4/3}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*Sec[e + f*x])^(1/3)/(c + d*Sec[e + f*x])^(4/3),x]

[Out]

Integrate[(a + b*Sec[e + f*x])^(1/3)/(c + d*Sec[e + f*x])^(4/3), 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))^(4/3),x, algorithm="fricas")

[Out]

Timed out

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac {1}{3}}}{{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac {4}{3}}}\,{d x} \]

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))^(4/3),x, algorithm="giac")

[Out]

integrate((b*sec(f*x + e) + a)^(1/3)/(d*sec(f*x + e) + c)^(4/3), x)

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maple [A]  time = 1.33, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \sec \left (f x +e \right )\right )^{\frac {1}{3}}}{\left (c +d \sec \left (f x +e \right )\right )^{\frac {4}{3}}}\, dx \]

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))^(4/3),x)

[Out]

int((a+b*sec(f*x+e))^(1/3)/(c+d*sec(f*x+e))^(4/3),x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac {1}{3}}}{{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac {4}{3}}}\,{d x} \]

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))^(4/3),x, algorithm="maxima")

[Out]

integrate((b*sec(f*x + e) + a)^(1/3)/(d*sec(f*x + e) + c)^(4/3), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {{\left (a+\frac {b}{\cos \left (e+f\,x\right )}\right )}^{1/3}}{{\left (c+\frac {d}{\cos \left (e+f\,x\right )}\right )}^{4/3}} \,d x \]

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))^(4/3),x)

[Out]

int((a + b/cos(e + f*x))^(1/3)/(c + d/cos(e + f*x))^(4/3), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt [3]{a + b \sec {\left (e + f x \right )}}}{\left (c + d \sec {\left (e + f x \right )}\right )^{\frac {4}{3}}}\, dx \]

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))**(4/3),x)

[Out]

Integral((a + b*sec(e + f*x))**(1/3)/(c + d*sec(e + f*x))**(4/3), x)

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