3.1.50 \(\int \frac {1}{x^2 (a+b \sec (c+d \sqrt {x}))^2} \, dx\) [50]
Optimal. Leaf size=23 \[ \text {Int}\left (\frac {1}{x^2 \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2},x\right ) \]
[Out]
Unintegrable(1/x^2/(a+b*sec(c+d*x^(1/2)))^2,x)
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Rubi [A]
time = 0.02, 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 = {}
\begin {gather*} \int \frac {1}{x^2 \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[1/(x^2*(a + b*Sec[c + d*Sqrt[x]])^2),x]
[Out]
Defer[Int][1/(x^2*(a + b*Sec[c + d*Sqrt[x]])^2), x]
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx &=\int \frac {1}{x^2 \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx\\ \end {align*}
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Mathematica [A]
time = 25.85, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^2 \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Integrate[1/(x^2*(a + b*Sec[c + d*Sqrt[x]])^2),x]
[Out]
Integrate[1/(x^2*(a + b*Sec[c + d*Sqrt[x]])^2), x]
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Maple [A]
time = 0.66, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{2} \left (a +b \sec \left (c +d \sqrt {x}\right )\right )^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/x^2/(a+b*sec(c+d*x^(1/2)))^2,x)
[Out]
int(1/x^2/(a+b*sec(c+d*x^(1/2)))^2,x)
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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^2/(a+b*sec(c+d*x^(1/2)))^2,x, algorithm="maxima")
[Out]
((a^8*d*cos(2*d*sqrt(x) + 2*c)^2 + a^8*d*sin(2*d*sqrt(x) + 2*c)^2 + (a^4*b^4*cos(2*c)^2 + a^4*b^4*sin(2*c)^2)*
d*cos(2*d*sqrt(x))^2 + 4*((a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*cos(c)^2 + (a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*sin(c)^2)
*d*cos(d*sqrt(x))^2 + 4*(a^7*b - 2*a^5*b^3 + a^3*b^5)*d*cos(d*sqrt(x))*cos(c) + (a^4*b^4*cos(2*c)^2 + a^4*b^4*
sin(2*c)^2)*d*sin(2*d*sqrt(x))^2 + 4*((a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*cos(c)^2 + (a^6*b^2 - 2*a^4*b^4 + a^2*b^
6)*sin(c)^2)*d*sin(d*sqrt(x))^2 - 4*(a^7*b - 2*a^5*b^3 + a^3*b^5)*d*sin(d*sqrt(x))*sin(c) + (a^8 - 2*a^6*b^2 +
a^4*b^4)*d - 2*(2*((a^5*b^3 - a^3*b^5)*cos(2*c)*cos(c) + (a^5*b^3 - a^3*b^5)*sin(2*c)*sin(c))*d*cos(d*sqrt(x)
) + (a^6*b^2 - a^4*b^4)*d*cos(2*c) + 2*((a^5*b^3 - a^3*b^5)*cos(c)*sin(2*c) - (a^5*b^3 - a^3*b^5)*cos(2*c)*sin
(c))*d*sin(d*sqrt(x)))*cos(2*d*sqrt(x)) - 2*(a^6*b^2*d*cos(2*d*sqrt(x))*cos(2*c) - a^6*b^2*d*sin(2*d*sqrt(x))*
sin(2*c) - 2*(a^7*b - a^5*b^3)*d*cos(d*sqrt(x))*cos(c) + 2*(a^7*b - a^5*b^3)*d*sin(d*sqrt(x))*sin(c) - (a^8 -
a^6*b^2)*d)*cos(2*d*sqrt(x) + 2*c) + 2*(2*((a^5*b^3 - a^3*b^5)*cos(c)*sin(2*c) - (a^5*b^3 - a^3*b^5)*cos(2*c)*
sin(c))*d*cos(d*sqrt(x)) - 2*((a^5*b^3 - a^3*b^5)*cos(2*c)*cos(c) + (a^5*b^3 - a^3*b^5)*sin(2*c)*sin(c))*d*sin
(d*sqrt(x)) + (a^6*b^2 - a^4*b^4)*d*sin(2*c))*sin(2*d*sqrt(x)) - 2*(a^6*b^2*d*cos(2*c)*sin(2*d*sqrt(x)) + a^6*
b^2*d*cos(2*d*sqrt(x))*sin(2*c) - 2*(a^7*b - a^5*b^3)*d*cos(c)*sin(d*sqrt(x)) - 2*(a^7*b - a^5*b^3)*d*cos(d*sq
rt(x))*sin(c))*sin(2*d*sqrt(x) + 2*c))*x^2*integrate(-2*(((2*a^5*b - a^3*b^3)*d*cos(2*d*sqrt(x) + 2*c)*cos(d*s
qrt(x) + c) + (2*a^5*b - a^3*b^3)*d*sin(2*d*sqrt(x) + 2*c)*sin(d*sqrt(x) + c) - ((2*a^3*b^3 - a*b^5)*d*cos(2*d
*sqrt(x))*cos(2*c) - 2*(2*a^4*b^2 - 3*a^2*b^4 + b^6)*d*cos(d*sqrt(x))*cos(c) - (2*a^3*b^3 - a*b^5)*d*sin(2*d*s
qrt(x))*sin(2*c) + 2*(2*a^4*b^2 - 3*a^2*b^4 + b^6)*d*sin(d*sqrt(x))*sin(c) - (2*a^5*b - 3*a^3*b^3 + a*b^5)*d)*
cos(d*sqrt(x) + c) - ((2*a^3*b^3 - a*b^5)*d*cos(2*c)*sin(2*d*sqrt(x)) - 2*(2*a^4*b^2 - 3*a^2*b^4 + b^6)*d*cos(
c)*sin(d*sqrt(x)) + (2*a^3*b^3 - a*b^5)*d*cos(2*d*sqrt(x))*sin(2*c) - 2*(2*a^4*b^2 - 3*a^2*b^4 + b^6)*d*cos(d*
sqrt(x))*sin(c))*sin(d*sqrt(x) + c))*x + 3*(a^2*b^4*cos(2*c)*sin(2*d*sqrt(x)) + a^3*b^3*cos(2*d*sqrt(x) + 2*c)
*sin(d*sqrt(x) + c) + a^2*b^4*cos(2*d*sqrt(x))*sin(2*c) - 2*(a^3*b^3 - a*b^5)*cos(c)*sin(d*sqrt(x)) - 2*(a^3*b
^3 - a*b^5)*cos(d*sqrt(x))*sin(c) + (a*b^5*cos(2*c)*sin(2*d*sqrt(x)) + a*b^5*cos(2*d*sqrt(x))*sin(2*c) - 2*(a^
2*b^4 - b^6)*cos(c)*sin(d*sqrt(x)) - 2*(a^2*b^4 - b^6)*cos(d*sqrt(x))*sin(c))*cos(d*sqrt(x) + c) - (a^3*b^3*co
s(d*sqrt(x) + c) + a^4*b^2)*sin(2*d*sqrt(x) + 2*c) - (a*b^5*cos(2*d*sqrt(x))*cos(2*c) - a*b^5*sin(2*d*sqrt(x))
*sin(2*c) - a^3*b^3 + a*b^5 - 2*(a^2*b^4 - b^6)*cos(d*sqrt(x))*cos(c) + 2*(a^2*b^4 - b^6)*sin(d*sqrt(x))*sin(c
))*sin(d*sqrt(x) + c))*sqrt(x))/((a^8*d*cos(2*d*sqrt(x) + 2*c)^2 + a^8*d*sin(2*d*sqrt(x) + 2*c)^2 + (a^4*b^4*c
os(2*c)^2 + a^4*b^4*sin(2*c)^2)*d*cos(2*d*sqrt(x))^2 + 4*((a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*cos(c)^2 + (a^6*b^2
- 2*a^4*b^4 + a^2*b^6)*sin(c)^2)*d*cos(d*sqrt(x))^2 + 4*(a^7*b - 2*a^5*b^3 + a^3*b^5)*d*cos(d*sqrt(x))*cos(c)
+ (a^4*b^4*cos(2*c)^2 + a^4*b^4*sin(2*c)^2)*d*sin(2*d*sqrt(x))^2 + 4*((a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*cos(c)^2
+ (a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*sin(c)^2)*d*sin(d*sqrt(x))^2 - 4*(a^7*b - 2*a^5*b^3 + a^3*b^5)*d*sin(d*sqrt
(x))*sin(c) + (a^8 - 2*a^6*b^2 + a^4*b^4)*d - 2*(2*((a^5*b^3 - a^3*b^5)*cos(2*c)*cos(c) + (a^5*b^3 - a^3*b^5)*
sin(2*c)*sin(c))*d*cos(d*sqrt(x)) + (a^6*b^2 - a^4*b^4)*d*cos(2*c) + 2*((a^5*b^3 - a^3*b^5)*cos(c)*sin(2*c) -
(a^5*b^3 - a^3*b^5)*cos(2*c)*sin(c))*d*sin(d*sqrt(x)))*cos(2*d*sqrt(x)) - 2*(a^6*b^2*d*cos(2*d*sqrt(x))*cos(2*
c) - a^6*b^2*d*sin(2*d*sqrt(x))*sin(2*c) - 2*(a^7*b - a^5*b^3)*d*cos(d*sqrt(x))*cos(c) + 2*(a^7*b - a^5*b^3)*d
*sin(d*sqrt(x))*sin(c) - (a^8 - a^6*b^2)*d)*cos(2*d*sqrt(x) + 2*c) + 2*(2*((a^5*b^3 - a^3*b^5)*cos(c)*sin(2*c)
- (a^5*b^3 - a^3*b^5)*cos(2*c)*sin(c))*d*cos(d*sqrt(x)) - 2*((a^5*b^3 - a^3*b^5)*cos(2*c)*cos(c) + (a^5*b^3 -
a^3*b^5)*sin(2*c)*sin(c))*d*sin(d*sqrt(x)) + (a^6*b^2 - a^4*b^4)*d*sin(2*c))*sin(2*d*sqrt(x)) - 2*(a^6*b^2*d*
cos(2*c)*sin(2*d*sqrt(x)) + a^6*b^2*d*cos(2*d*sqrt(x))*sin(2*c) - 2*(a^7*b - a^5*b^3)*d*cos(c)*sin(d*sqrt(x))
- 2*(a^7*b - a^5*b^3)*d*cos(d*sqrt(x))*sin(c))*sin(2*d*sqrt(x) + 2*c))*x^3), x) - ((a^6 - a^4*b^2)*d*cos(2*d*s
qrt(x) + 2*c)^2 - (a^4*b^2 - a^2*b^4)*d*cos(2*d*sqrt(x))*cos(2*c) + 2*(a^5*b - 2*a^3*b^3 + a*b^5)*d*cos(d*sqrt
(x))*cos(c) + (a^6 - a^4*b^2)*d*sin(2*d*sqrt(x) + 2*c)^2 + (a^4*b^2 - a^2*b^4)*d*sin(2*d*sqrt(x))*sin(2*c) - 2
*(a^5*b - 2*a^3*b^3 + a*b^5)*d*sin(d*sqrt(x))*sin(c) + (a^6 - 2*a^4*b^2 + a^2*b^4)*d - ((a^4*b^2 - a^2*b^4)*d*
cos(2*d*sqrt(x))*cos(2*c) - 2*(a^5*b - 2*a^3*b^3 + a*b^5)*d*cos(d*sqrt(x))*cos(c) - (a^4*b^2 - a^2*b^4)*d*sin(
2*d*sqrt(x))*sin(2*c) + 2*(a^5*b - 2*a^3*b^3 + a*b^5)*d*sin(d*sqrt(x))*sin(c) - 2*(a^5*b - a^3*b^3)*d*cos(d*sq
rt(x) + c) - (2*a^6 - 3*a^4*b^2 + a^2*b^4)*d)*cos(2*d*sqrt(x) + 2*c) - 2*((a^3*b^3 - a*b^5)*d*cos(2*d*sqrt(x))
*cos(2*c) - 2*(a^4*b^2 - 2*a^2*b^4 + b^6)*d*cos...
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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^2/(a+b*sec(c+d*x^(1/2)))^2,x, algorithm="fricas")
[Out]
integral(1/(b^2*x^2*sec(d*sqrt(x) + c)^2 + 2*a*b*x^2*sec(d*sqrt(x) + c) + a^2*x^2), x)
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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{2} \left (a + b \sec {\left (c + d \sqrt {x} \right )}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x**2/(a+b*sec(c+d*x**(1/2)))**2,x)
[Out]
Integral(1/(x**2*(a + b*sec(c + d*sqrt(x)))**2), x)
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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^2/(a+b*sec(c+d*x^(1/2)))^2,x, algorithm="giac")
[Out]
integrate(1/((b*sec(d*sqrt(x) + c) + a)^2*x^2), x)
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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {1}{x^2\,{\left (a+\frac {b}{\cos \left (c+d\,\sqrt {x}\right )}\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(x^2*(a + b/cos(c + d*x^(1/2)))^2),x)
[Out]
int(1/(x^2*(a + b/cos(c + d*x^(1/2)))^2), x)
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