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

Optimal. Leaf size=89 \[ \frac {(a+b \sec (e+f x))^{2/3} (c \cos (e+f x)+d)^{2/3} \text {Int}\left (\frac {(a \cos (e+f x)+b)^{2/3}}{(c \cos (e+f x)+d)^{2/3}},x\right )}{(a \cos (e+f x)+b)^{2/3} (c+d \sec (e+f x))^{2/3}} \]

[Out]

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

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Rubi [A]  time = 0.22, 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 = {} \[ \int \frac {(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{2/3}} \, dx \]

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*}

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Mathematica [A]  time = 2.61, size = 0, normalized size = 0.00 \[ \int \frac {(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{2/3}} \, dx \]

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

[Out]

Timed out

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac {2}{3}}}{{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac {2}{3}}}\,{d x} \]

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)

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maple [A]  time = 1.45, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \sec \left (f x +e \right )\right )^{\frac {2}{3}}}{\left (c +d \sec \left (f x +e \right )\right )^{\frac {2}{3}}}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac {2}{3}}}{{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac {2}{3}}}\,{d x} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+\frac {b}{\cos \left (e+f\,x\right )}\right )}^{2/3}}{{\left (c+\frac {d}{\cos \left (e+f\,x\right )}\right )}^{2/3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \sec {\left (e + f x \right )}\right )^{\frac {2}{3}}}{\left (c + d \sec {\left (e + f x \right )}\right )^{\frac {2}{3}}}\, dx \]

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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