3.703 \(\int \frac{1}{\sqrt [3]{a+b \sec (c+d x)}} \, dx\)

Optimal. Leaf size=16 \[ \text{Unintegrable}\left (\frac{1}{\sqrt [3]{a+b \sec (c+d x)}},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0107156, 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 = {} \[ \int \frac{1}{\sqrt [3]{a+b \sec (c+d x)}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*Sec[c + d*x])^(-1/3),x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.9116, size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt [3]{a+b \sec (c+d x)}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.114, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt [3]{a+b\sec \left ( dx+c \right ) }}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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