3.3.27 \(\int \frac {(a+b \sec (e+f x))^{4/3}}{(c+d \sec (e+f x))^{4/3}} \, dx\) [227]

Optimal. Leaf size=89 \[ \frac {(d+c \cos (e+f x))^{4/3} (a+b \sec (e+f x))^{4/3} \text {Int}\left (\frac {(b+a \cos (e+f x))^{4/3}}{(d+c \cos (e+f x))^{4/3}},x\right )}{(b+a \cos (e+f x))^{4/3} (c+d \sec (e+f x))^{4/3}} \]

[Out]

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

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Rubi [A]
time = 0.14, 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 = {} \begin {gather*} \int \frac {(a+b \sec (e+f x))^{4/3}}{(c+d \sec (e+f x))^{4/3}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 56.51, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b \sec (e+f x))^{4/3}}{(c+d \sec (e+f x))^{4/3}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.15, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \sec \left (f x +e \right )\right )^{\frac {4}{3}}}{\left (c +d \sec \left (f x +e \right )\right )^{\frac {4}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(f*x+e))^(4/3)/(c+d*sec(f*x+e))^(4/3),x, algorithm="maxima")

[Out]

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

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(f*x+e))^(4/3)/(c+d*sec(f*x+e))^(4/3),x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(f*x+e))^(4/3)/(c+d*sec(f*x+e))^(4/3),x, algorithm="giac")

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+\frac {b}{\cos \left (e+f\,x\right )}\right )}^{4/3}}{{\left (c+\frac {d}{\cos \left (e+f\,x\right )}\right )}^{4/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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