3.705 \(\int \frac {1}{(a+b \sec (c+d x))^{2/3}} \, dx\)

Optimal. Leaf size=17 \[ \text {Int}\left (\frac {1}{(a+b \sec (c+d x))^{2/3}},x\right ) \]

[Out]

Unintegrable(1/(a+b*sec(d*x+c))^(2/3),x)

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Rubi [A]  time = 0.01, 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 = {} \[ \int \frac {1}{(a+b \sec (c+d x))^{2/3}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*Sec[c + d*x])^(-2/3),x]

[Out]

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

Rubi steps

\begin {align*} \int \frac {1}{(a+b \sec (c+d x))^{2/3}} \, dx &=\int \frac {1}{(a+b \sec (c+d x))^{2/3}} \, dx\\ \end {align*}

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Mathematica [A]  time = 1.06, size = 0, normalized size = 0.00 \[ \int \frac {1}{(a+b \sec (c+d x))^{2/3}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Integrate[(a + b*Sec[c + d*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(1/(a+b*sec(d*x+c))^(2/3),x, algorithm="fricas")

[Out]

Timed out

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sec(d*x+c))^(2/3),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.73, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a +b \sec \left (d x +c \right )\right )^{\frac {2}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*sec(d*x+c))^(2/3),x)

[Out]

int(1/(a+b*sec(d*x+c))^(2/3),x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sec(d*x+c))^(2/3),x, algorithm="maxima")

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.06 \[ \int \frac {1}{{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{2/3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b/cos(c + d*x))^(2/3),x)

[Out]

int(1/(a + b/cos(c + d*x))^(2/3), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a + b \sec {\left (c + d x \right )}\right )^{\frac {2}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sec(d*x+c))**(2/3),x)

[Out]

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

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