3.722 \(\int \frac{\sqrt{a+b \sec (c+d x)}}{\sqrt [3]{\sec (c+d x)}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.052302, 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 = {} \[ \int \frac{\sqrt{a+b \sec (c+d x)}}{\sqrt [3]{\sec (c+d x)}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 7.81607, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+b \sec (c+d x)}}{\sqrt [3]{\sec (c+d x)}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.308, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{a+b\sec \left ( dx+c \right ) }{\frac{1}{\sqrt [3]{\sec \left ( dx+c \right ) }}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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{1}{3}}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a + b*sec(c + d*x))/sec(c + d*x)**(1/3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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