3.754 \(\int \frac{1}{\sec ^{\frac{4}{3}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx\)

Optimal. Leaf size=27 \[ \text{Unintegrable}\left (\frac{1}{\sec ^{\frac{4}{3}}(c+d x) \sqrt{a+b \sec (c+d x)}},x\right ) \]

[Out]

Unintegrable[1/(Sec[c + d*x]^(4/3)*Sqrt[a + b*Sec[c + d*x]]), x]

________________________________________________________________________________________

Rubi [A]  time = 0.0528158, 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 = {} \[ \int \frac{1}{\sec ^{\frac{4}{3}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/(Sec[c + d*x]^(4/3)*Sqrt[a + b*Sec[c + d*x]]),x]

[Out]

Defer[Int][1/(Sec[c + d*x]^(4/3)*Sqrt[a + b*Sec[c + d*x]]), x]

Rubi steps

\begin{align*} \int \frac{1}{\sec ^{\frac{4}{3}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx &=\int \frac{1}{\sec ^{\frac{4}{3}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx\\ \end{align*}

Mathematica [A]  time = 27.0593, size = 0, normalized size = 0. \[ \int \frac{1}{\sec ^{\frac{4}{3}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/(Sec[c + d*x]^(4/3)*Sqrt[a + b*Sec[c + d*x]]),x]

[Out]

Integrate[1/(Sec[c + d*x]^(4/3)*Sqrt[a + b*Sec[c + d*x]]), x]

________________________________________________________________________________________

Maple [A]  time = 0.255, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \sec \left ( dx+c \right ) \right ) ^{-{\frac{4}{3}}}{\frac{1}{\sqrt{a+b\sec \left ( dx+c \right ) }}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/sec(d*x+c)^(4/3)/(a+b*sec(d*x+c))^(1/2),x)

[Out]

int(1/sec(d*x+c)^(4/3)/(a+b*sec(d*x+c))^(1/2),x)

________________________________________________________________________________________

Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b \sec \left (d x + c\right ) + a} \sec \left (d x + c\right )^{\frac{4}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sec(d*x+c)^(4/3)/(a+b*sec(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(b*sec(d*x + c) + a)*sec(d*x + c)^(4/3)), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sec(d*x+c)^(4/3)/(a+b*sec(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(b*sec(d*x + c) + a)*sec(d*x + c)^(2/3)/(b*sec(d*x + c)^3 + a*sec(d*x + c)^2), x)

________________________________________________________________________________________

Sympy [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(1/sec(d*x+c)**(4/3)/(a+b*sec(d*x+c))**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b \sec \left (d x + c\right ) + a} \sec \left (d x + c\right )^{\frac{4}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sec(d*x+c)^(4/3)/(a+b*sec(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(b*sec(d*x + c) + a)*sec(d*x + c)^(4/3)), x)