\(\int \frac {1}{x (a+b \csc (c+d \sqrt {x}))^2} \, dx\) [49]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {1}{x \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2} \, dx=\text {Int}\left (\frac {1}{x \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{x \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2} \, dx=\int \frac {1}{x \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2} \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 61.39 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{x \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2} \, dx=\int \frac {1}{x \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.40 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90

\[\int \frac {1}{x \left (a +b \csc \left (c +d \sqrt {x}\right )\right )^{2}}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.90 \[ \int \frac {1}{x \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \csc \left (d \sqrt {x} + c\right ) + a\right )}^{2} x} \,d x } \]

[In]

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

[Out]

integral(1/(b^2*x*csc(d*sqrt(x) + c)^2 + 2*a*b*x*csc(d*sqrt(x) + c) + a^2*x), x)

Sympy [N/A]

Not integrable

Time = 3.88 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {1}{x \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2} \, dx=\int \frac {1}{x \left (a + b \csc {\left (c + d \sqrt {x} \right )}\right )^{2}}\, dx \]

[In]

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

[Out]

Integral(1/(x*(a + b*csc(c + d*sqrt(x)))**2), x)

Maxima [N/A]

Not integrable

Time = 13.44 (sec) , antiderivative size = 4411, normalized size of antiderivative = 220.55 \[ \int \frac {1}{x \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \csc \left (d \sqrt {x} + c\right ) + a\right )}^{2} x} \,d x } \]

[In]

integrate(1/x/(a+b*csc(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 + (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^7*b - 2*a^5*b^3 +
a^3*b^5)*d*cos(c)*sin(d*sqrt(x)) + 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*cos(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(c)*sin(2*c) - (a^5*b^3 - a^3*b^5)*cos(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(2*c)*cos(c) + (a^5*b^3 - a^3*b^5)*sin(2*c)*si
n(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(c)*sin(d*sqrt(x)) + 2*(a^7*b - a^5*b^3)*d*cos(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(2*c)*cos(c) + (a^5*b^3 - a^3*b^5)*sin(2*c)
*sin(c))*d*cos(d*sqrt(x)) + 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*si
n(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(d*sqrt(x))*cos(c) + 2*(a^7*b - a^5*b^3)*d*sin(d*s
qrt(x))*sin(c))*sin(2*d*sqrt(x) + 2*c))*x*integrate(2*(((2*a^5*b - a^3*b^3)*d*cos(d*sqrt(x) + c)*sin(2*d*sqrt(
x) + 2*c) - (2*a^5*b - a^3*b^3)*d*cos(2*d*sqrt(x) + 2*c)*sin(d*sqrt(x) + 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*cos(2*c)*sin(2*d*sqrt(x)) - (2*a^3*b^3 - a*b^5)*d*cos(2*d*sqr
t(x))*sin(2*c) - 2*(2*a^4*b^2 - 3*a^2*b^4 + b^6)*d*sin(d*sqrt(x))*sin(c))*cos(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(c)*sin(d*sqrt(x)) - (2*a^3*b^3 - a*b
^5)*d*sin(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) + (2*a^5*b - 3*a^3*b
^3 + a*b^5)*d)*sin(d*sqrt(x) + c))*x + (a^3*b^3*cos(2*d*sqrt(x) + 2*c)*cos(d*sqrt(x) + c) - a^2*b^4*cos(2*c)*s
in(2*d*sqrt(x)) - a^2*b^4*cos(2*d*sqrt(x))*sin(2*c) + 2*(a^3*b^3 - a*b^5)*cos(d*sqrt(x))*cos(c) - 2*(a^3*b^3 -
a*b^5)*sin(d*sqrt(x))*sin(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(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*sin(d*sqrt(x) + c) + a^4*b^2)*sin(2*d*sqrt(x) + 2*c) - (a*b^5*cos(2*c)*sin(2*d*sqrt(x)) + a*b^5*co
s(2*d*sqrt(x))*sin(2*c) - 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))*s
in(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*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 + (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^7*b - 2*a^5*b^3 + a^3*b^5)*d*cos(c)*sin(d*sqrt(x)) + 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*cos(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(c)*sin(2*c) - (a^5*b^3 - a^3*b^5)*cos(
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(2*c)*cos(c) + (a^5
*b^3 - a^3*b^5)*sin(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(c)*sin(d*sqrt(x)) + 2*(a^7*b - a^5*b^3)*d*cos
(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(2*c)*cos(c) + (
a^5*b^3 - a^3*b^5)*sin(2*c)*sin(c))*d*cos(d*sqrt(x)) + 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)) + (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(d*sqrt(x))*cos(c) + 2*
(a^7*b - a^5*b^3)*d*sin(d*sqrt(x))*sin(c))*sin(2*d*sqrt(x) + 2*c))*x^2), x) - (a^6*d*cos(2*d*sqrt(x) + 2*c)^2
+ a^6*d*sin(2*d*sqrt(x) + 2*c)^2 + (a^2*b^4*cos(2*c)^2 + a^2*b^4*sin(2*c)^2)*d*cos(2*d*sqrt(x))^2 + 4*((a^4*b^
2 - 2*a^2*b^4 + b^6)*cos(c)^2 + (a^4*b^2 - 2*a^2*b^4 + b^6)*sin(c)^2)*d*cos(d*sqrt(x))^2 + (a^2*b^4*cos(2*c)^2
+ a^2*b^4*sin(2*c)^2)*d*sin(2*d*sqrt(x))^2 + 4*(a^5*b - 2*a^3*b^3 + a*b^5)*d*cos(c)*sin(d*sqrt(x)) + 4*((a^4*
b^2 - 2*a^2*b^4 + b^6)*cos(c)^2 + (a^4*b^2 - 2*a^2*b^4 + b^6)*sin(c)^2)*d*sin(d*sqrt(x))^2 + 4*(a^5*b - 2*a^3*
b^3 + a*b^5)*d*cos(d*sqrt(x))*sin(c) + (a^6 - 2*a^4*b^2 + a^2*b^4)*d - 2*(2*((a^3*b^3 - a*b^5)*cos(c)*sin(2*c)
- (a^3*b^3 - a*b^5)*cos(2*c)*sin(c))*d*cos(d*sqrt(x)) - (a^4*b^2 - a^2*b^4)*d*cos(2*c) - 2*((a^3*b^3 - a*b^5)
*cos(2*c)*cos(c) + (a^3*b^3 - a*b^5)*sin(2*c)*sin(c))*d*sin(d*sqrt(x)))*cos(2*d*sqrt(x)) - 2*(a^4*b^2*d*cos(2*
d*sqrt(x))*cos(2*c) - a^4*b^2*d*sin(2*d*sqrt(x))*sin(2*c) + 2*(a^5*b - a^3*b^3)*d*cos(c)*sin(d*sqrt(x)) + 2*(a
^5*b - a^3*b^3)*d*cos(d*sqrt(x))*sin(c) + (a^6 - a^4*b^2)*d)*cos(2*d*sqrt(x) + 2*c) - 2*(2*((a^3*b^3 - a*b^5)*
cos(2*c)*cos(c) + (a^3*b^3 - a*b^5)*sin(2*c)*sin(c))*d*cos(d*sqrt(x)) + 2*((a^3*b^3 - a*b^5)*cos(c)*sin(2*c) -
(a^3*b^3 - a*b^5)*cos(2*c)*sin(c))*d*sin(d*sqrt(x)) + (a^4*b^2 - a^2*b^4)*d*sin(2*c))*sin(2*d*sqrt(x)) - 2*(a
^4*b^2*d*cos(2*c)*sin(2*d*sqrt(x)) + a^4*b^2*d*cos(2*d*sqrt(x))*sin(2*c) - 2*(a^5*b - a^3*b^3)*d*cos(d*sqrt(x)
)*cos(c) + 2*(a^5*b - a^3*b^3)*d*sin(d*sqrt(x))*sin(c))*sin(2*d*sqrt(x) + 2*c))*x*log(x) + 4*(a^3*b^3*cos(2*d*
sqrt(x) + 2*c)*cos(d*sqrt(x) + c) - a^2*b^4*cos(2*c)*sin(2*d*sqrt(x)) - a^2*b^4*cos(2*d*sqrt(x))*sin(2*c) + 2*
(a^3*b^3 - a*b^5)*cos(d*sqrt(x))*cos(c) - 2*(a^3*b^3 - a*b^5)*sin(d*sqrt(x))*sin(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(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*sin(d*sqrt(x) + c) + a^4*b^2)*sin(2*d*sqrt(x
) + 2*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(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*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 + (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^7*b - 2*a^5*b^3 + a^3*b^5)*d*cos(c)*sin(d*sqrt
(x)) + 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*cos(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(c)*sin(2*c) - (a^5*b^3 - a^3*b^5)*cos(2*c)*sin(c))*d*cos(d*sqrt(x)) - (a^6*b^2 - a^4*b^4)*d*co
s(2*c) - 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)))*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(c)*sin(d*sqrt(x)) + 2*(a^7*b - a^5*b^3)*d*cos(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(2*c)*cos(c) + (a^5*b^3 - a^3*b^5)*sin(2*c)*sin(c))*d*cos(d*sqrt(x)) + 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)) + (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(d*sqrt(x))*cos(c) + 2*(a^7*b - a^5*b^3)*d*sin(d*sqrt(x))*sin(c))*sin(2*d*sqrt(
x) + 2*c))*x)

Giac [N/A]

Not integrable

Time = 0.67 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \csc \left (d \sqrt {x} + c\right ) + a\right )}^{2} x} \,d x } \]

[In]

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

[Out]

integrate(1/((b*csc(d*sqrt(x) + c) + a)^2*x), x)

Mupad [N/A]

Not integrable

Time = 18.15 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{x \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2} \, dx=\int \frac {1}{x\,{\left (a+\frac {b}{\sin \left (c+d\,\sqrt {x}\right )}\right )}^2} \,d x \]

[In]

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

[Out]

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