3.224 \(\int \frac{(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{2/3}} \, dx\)

Optimal. Leaf size=88 \[ \frac{(a+b \sec (e+f x))^{2/3} (c \cos (e+f x)+d)^{2/3} \text{Unintegrable}\left (\frac{(a \cos (e+f x)+b)^{2/3}}{(c \cos (e+f x)+d)^{2/3}},x\right )}{(a \cos (e+f x)+b)^{2/3} (c+d \sec (e+f x))^{2/3}} \]

[Out]

((d + c*Cos[e + f*x])^(2/3)*(a + b*Sec[e + f*x])^(2/3)*Unintegrable[(b + a*Cos[e + f*x])^(2/3)/(d + c*Cos[e +
f*x])^(2/3), x])/((b + a*Cos[e + f*x])^(2/3)*(c + d*Sec[e + f*x])^(2/3))

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Rubi [A]  time = 0.220201, 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 = {} \[ \int \frac{(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{2/3}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*Sec[e + f*x])^(2/3)/(c + d*Sec[e + f*x])^(2/3),x]

[Out]

((d + c*Cos[e + f*x])^(2/3)*(a + b*Sec[e + f*x])^(2/3)*Defer[Int][(b + a*Cos[e + f*x])^(2/3)/(d + c*Cos[e + f*
x])^(2/3), x])/((b + a*Cos[e + f*x])^(2/3)*(c + d*Sec[e + f*x])^(2/3))

Rubi steps

\begin{align*} \int \frac{(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{2/3}} \, dx &=\frac{\left ((d+c \cos (e+f x))^{2/3} (a+b \sec (e+f x))^{2/3}\right ) \int \frac{(b+a \cos (e+f x))^{2/3}}{(d+c \cos (e+f x))^{2/3}} \, dx}{(b+a \cos (e+f x))^{2/3} (c+d \sec (e+f x))^{2/3}}\\ \end{align*}

Mathematica [A]  time = 2.17062, size = 0, normalized size = 0. \[ \int \frac{(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{2/3}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*Sec[e + f*x])^(2/3)/(c + d*Sec[e + f*x])^(2/3),x]

[Out]

Integrate[(a + b*Sec[e + f*x])^(2/3)/(c + d*Sec[e + f*x])^(2/3), x]

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Maple [A]  time = 0.227, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b\sec \left ( fx+e \right ) \right ) ^{{\frac{2}{3}}} \left ( c+d\sec \left ( fx+e \right ) \right ) ^{-{\frac{2}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sec(f*x+e))^(2/3)/(c+d*sec(f*x+e))^(2/3),x)

[Out]

int((a+b*sec(f*x+e))^(2/3)/(c+d*sec(f*x+e))^(2/3),x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac{2}{3}}}{{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac{2}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(f*x+e))^(2/3)/(c+d*sec(f*x+e))^(2/3),x, algorithm="maxima")

[Out]

integrate((b*sec(f*x + e) + a)^(2/3)/(d*sec(f*x + e) + c)^(2/3), x)

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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))^(2/3)/(c+d*sec(f*x+e))^(2/3),x, algorithm="fricas")

[Out]

Timed out

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \sec{\left (e + f x \right )}\right )^{\frac{2}{3}}}{\left (c + d \sec{\left (e + f x \right )}\right )^{\frac{2}{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(f*x+e))**(2/3)/(c+d*sec(f*x+e))**(2/3),x)

[Out]

Integral((a + b*sec(e + f*x))**(2/3)/(c + d*sec(e + f*x))**(2/3), x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac{2}{3}}}{{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac{2}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(f*x+e))^(2/3)/(c+d*sec(f*x+e))^(2/3),x, algorithm="giac")

[Out]

integrate((b*sec(f*x + e) + a)^(2/3)/(d*sec(f*x + e) + c)^(2/3), x)