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

Optimal. Leaf size=27 \[ \text{Unintegrable}\left (\frac{1}{\sqrt [3]{\sec (c+d x)} (a+b \sec (c+d x))^{3/2}},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0579604, 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 = {} \[ \int \frac{1}{\sqrt [3]{\sec (c+d x)} (a+b \sec (c+d x))^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/(Sec[c + d*x]^(1/3)*(a + b*Sec[c + d*x])^(3/2)),x]

[Out]

Defer[Int][1/(Sec[c + d*x]^(1/3)*(a + b*Sec[c + d*x])^(3/2)), x]

Rubi steps

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

Mathematica [A]  time = 37.2718, size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt [3]{\sec (c+d x)} (a+b \sec (c+d x))^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/(Sec[c + d*x]^(1/3)*(a + b*Sec[c + d*x])^(3/2)),x]

[Out]

Integrate[1/(Sec[c + d*x]^(1/3)*(a + b*Sec[c + d*x])^(3/2)), x]

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Maple [A]  time = 0.194, size = 0, normalized size = 0. \begin{align*} \int{{\frac{1}{\sqrt [3]{\sec \left ( dx+c \right ) }}} \left ( a+b\sec \left ( dx+c \right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(1/((b*sec(d*x + c) + a)^(3/2)*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{2}{3}}}{b^{2} \sec \left (d x + c\right )^{3} + 2 \, a b \sec \left (d x + c\right )^{2} + a^{2} \sec \left (d x + c\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b \sec{\left (c + d x \right )}\right )^{\frac{3}{2}} \sqrt [3]{\sec{\left (c + d x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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