3.8.12 \(\int \frac {1}{(a+b \sec (c+d x))^{5/3}} \, dx\) [712]

Optimal. Leaf size=17 \[ \text {Int}\left (\frac {1}{(a+b \sec (c+d x))^{5/3}},x\right ) \]

[Out]

Unintegrable(1/(a+b*sec(d*x+c))^(5/3),x)

________________________________________________________________________________________

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 = {} \begin {gather*} \int \frac {1}{(a+b \sec (c+d x))^{5/3}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(a + b*Sec[c + d*x])^(-5/3),x]

[Out]

Defer[Int][(a + b*Sec[c + d*x])^(-5/3), x]

Rubi steps

\begin {align*} \int \frac {1}{(a+b \sec (c+d x))^{5/3}} \, dx &=\int \frac {1}{(a+b \sec (c+d x))^{5/3}} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 49.96, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(a+b \sec (c+d x))^{5/3}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(a + b*Sec[c + d*x])^(-5/3),x]

[Out]

Integrate[(a + b*Sec[c + d*x])^(-5/3), x]

________________________________________________________________________________________

Maple [A]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a +b \sec \left (d x +c \right )\right )^{\frac {5}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*sec(d*x+c))^(5/3),x)

[Out]

int(1/(a+b*sec(d*x+c))^(5/3),x)

________________________________________________________________________________________

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(1/(a+b*sec(d*x+c))^(5/3),x, algorithm="maxima")

[Out]

integrate((b*sec(d*x + c) + a)^(-5/3), x)

________________________________________________________________________________________

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(1/(a+b*sec(d*x+c))^(5/3),x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b \sec {\left (c + d x \right )}\right )^{\frac {5}{3}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sec(d*x+c))**(5/3),x)

[Out]

Integral((a + b*sec(c + d*x))**(-5/3), x)

________________________________________________________________________________________

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(1/(a+b*sec(d*x+c))^(5/3),x, algorithm="giac")

[Out]

integrate((b*sec(d*x + c) + a)^(-5/3), x)

________________________________________________________________________________________

Mupad [A]
time = 0.00, size = -1, normalized size = -0.06 \begin {gather*} \int \frac {1}{{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b/cos(c + d*x))^(5/3),x)

[Out]

int(1/(a + b/cos(c + d*x))^(5/3), x)

________________________________________________________________________________________