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

Optimal. Leaf size=28 \[ \text {Int}\left (\frac {1}{\sec ^{\frac {2}{3}}(c+d x) (a+b \sec (c+d x))^{3/2}},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.06, 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 = {} \[ \int \frac {1}{\sec ^{\frac {2}{3}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 42.31, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sec ^{\frac {2}{3}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \sec \left (d x + c\right ) + a} \sec \left (d x + c\right )^{\frac {1}{3}}}{b^{2} \sec \left (d x + c\right )^{3} + 2 \, a b \sec \left (d x + c\right )^{2} + a^{2} \sec \left (d x + c\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(b*sec(d*x + c) + a)*sec(d*x + c)^(1/3)/(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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sec \left (d x + c\right )^{\frac {2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maple [A]  time = 1.44, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sec \left (d x +c \right )^{\frac {2}{3}} \left (a +b \sec \left (d x +c \right )\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sec \left (d x + c\right )^{\frac {2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {1}{{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{3/2}\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{2/3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a + b \sec {\left (c + d x \right )}\right )^{\frac {3}{2}} \sec ^{\frac {2}{3}}{\left (c + d x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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