∫ 1_______ x3(a+b csc(c+dx2))2 dx [30]

3.1.30 \(\int \frac {1}{x^3 (a+b \csc (c+d x^2))^2} \, dx\) [30]

Optimal. Leaf size=21 \[ \text {Int}\left (\frac {1}{x^3 \left (a+b \csc \left (c+d x^2\right )\right )^2},x\right ) \]

[Out]

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

________________________________________________________________________________________

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^3 \left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 17.02, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^3 \left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{3} \left (a +b \csc \left (d \,x^{2}+c \right )\right )^{2}}\, dx\]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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^3/(a+b*csc(d*x^2+c))^2,x, algorithm="maxima")

[Out]

-1/2*((a^4 - a^2*b^2)*d*x^2 - ((a^4 - a^2*b^2)*d*x^2*cos(2*c) - 2*a^2*b^2*sin(2*c))*cos(2*d*x^2) + ((a^4 - a^2

*b^2)*d*x^2*cos(2*d*x^2)*cos(2*c) - 2*(a^3*b - a*b^3)*d*x^2*cos(c)*sin(d*x^2) - (a^4 - a^2*b^2)*d*x^2*sin(2*d*

x^2)*sin(2*c) - 2*(a^3*b - a*b^3)*d*x^2*cos(d*x^2)*sin(c) - (a^4 - a^2*b^2)*d*x^2)*cos(2*d*x^2 + 2*c) - 2*(a*b

^3 - (a*b^3*cos(2*c) + (a^3*b - a*b^3)*d*x^2*sin(2*c))*cos(2*d*x^2) - 2*((a^2*b^2 - b^4)*d*x^2*cos(c) - b^4*si

n(c))*cos(d*x^2) - ((a^3*b - a*b^3)*d*x^2*cos(2*c) - a*b^3*sin(2*c))*sin(2*d*x^2) + 2*(b^4*cos(c) + (a^2*b^2 -

b^4)*d*x^2*sin(c))*sin(d*x^2))*cos(d*x^2 + c) + 2*(2*a*b^3*cos(c) + (a^3*b - a*b^3)*d*x^2*sin(c))*cos(d*x^2)

+ 2*(((a^6 - a^4*b^2)*cos(2*c)^2 + (a^6 - a^4*b^2)*sin(2*c)^2)*d*x^4*cos(2*d*x^2)^2 + 4*((a^4*b^2 - a^2*b^4)*c

os(c)^2 + (a^4*b^2 - a^2*b^4)*sin(c)^2)*d*x^4*cos(d*x^2)^2 + ((a^6 - a^4*b^2)*cos(2*c)^2 + (a^6 - a^4*b^2)*sin

(2*c)^2)*d*x^4*sin(2*d*x^2)^2 + 4*(a^5*b - a^3*b^3)*d*x^4*cos(c)*sin(d*x^2) + 4*((a^4*b^2 - a^2*b^4)*cos(c)^2

+ (a^4*b^2 - a^2*b^4)*sin(c)^2)*d*x^4*sin(d*x^2)^2 + 4*(a^5*b - a^3*b^3)*d*x^4*cos(d*x^2)*sin(c) + (a^6 - a^4*

b^2)*d*x^4 + 2*(2*((a^5*b - a^3*b^3)*cos(c)*sin(2*c) - (a^5*b - a^3*b^3)*cos(2*c)*sin(c))*d*x^4*cos(d*x^2) - (

a^6 - a^4*b^2)*d*x^4*cos(2*c) - 2*((a^5*b - a^3*b^3)*cos(2*c)*cos(c) + (a^5*b - a^3*b^3)*sin(2*c)*sin(c))*d*x^

4*sin(d*x^2))*cos(2*d*x^2) + 2*(2*((a^5*b - a^3*b^3)*cos(2*c)*cos(c) + (a^5*b - a^3*b^3)*sin(2*c)*sin(c))*d*x^

4*cos(d*x^2) + 2*((a^5*b - a^3*b^3)*cos(c)*sin(2*c) - (a^5*b - a^3*b^3)*cos(2*c)*sin(c))*d*x^4*sin(d*x^2) + (a

^6 - a^4*b^2)*d*x^4*sin(2*c))*sin(2*d*x^2))*integrate(-2*(2*a^2*b^4*cos(2*c)*sin(2*d*x^2) + 2*a^2*b^4*cos(2*d*

x^2)*sin(2*c) - 4*(a^3*b^3 - a*b^5)*cos(d*x^2)*cos(c) + 4*(a^3*b^3 - a*b^5)*sin(d*x^2)*sin(c) - (2*a^3*b^3*cos

(d*x^2 + c) - (2*a^5*b - a^3*b^3)*d*x^2*sin(d*x^2 + c))*cos(2*d*x^2 + 2*c) + (2*a^3*b^3 - 2*a*b^5 + (2*a*b^5*c

os(2*c) + (2*a^3*b^3 - a*b^5)*d*x^2*sin(2*c))*cos(2*d*x^2) - 2*((2*a^4*b^2 - 3*a^2*b^4 + b^6)*d*x^2*cos(c) - 2

*(a^2*b^4 - b^6)*sin(c))*cos(d*x^2) - (2*a*b^5*sin(2*c) - (2*a^3*b^3 - a*b^5)*d*x^2*cos(2*c))*sin(2*d*x^2) + 2

*((2*a^4*b^2 - 3*a^2*b^4 + b^6)*d*x^2*sin(c) + 2*(a^2*b^4 - b^6)*cos(c))*sin(d*x^2))*cos(d*x^2 + c) - (2*a^3*b

^3*sin(d*x^2 + c) + 2*a^4*b^2 + (2*a^5*b - a^3*b^3)*d*x^2*cos(d*x^2 + c))*sin(2*d*x^2 + 2*c) - ((2*a^5*b - 3*a

^3*b^3 + a*b^5)*d*x^2 - (2*a*b^5*sin(2*c) - (2*a^3*b^3 - a*b^5)*d*x^2*cos(2*c))*cos(2*d*x^2) + 2*((2*a^4*b^2 -

3*a^2*b^4 + b^6)*d*x^2*sin(c) + 2*(a^2*b^4 - b^6)*cos(c))*cos(d*x^2) - (2*a*b^5*cos(2*c) + (2*a^3*b^3 - a*b^5

)*d*x^2*sin(2*c))*sin(2*d*x^2) + 2*((2*a^4*b^2 - 3*a^2*b^4 + b^6)*d*x^2*cos(c) - 2*(a^2*b^4 - b^6)*sin(c))*sin

(d*x^2))*sin(d*x^2 + c))/(a^8*d*x^5*cos(2*d*x^2 + 2*c)^2 + a^8*d*x^5*sin(2*d*x^2 + 2*c)^2 + (a^4*b^4*cos(2*c)^

2 + a^4*b^4*sin(2*c)^2)*d*x^5*cos(2*d*x^2)^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*x^5*cos(d*x^2)^2 + (a^4*b^4*cos(2*c)^2 + a^4*b^4*sin(2*c)^2)*d*x^5*sin(2*d*x^2)^2 +

4*(a^7*b - 2*a^5*b^3 + a^3*b^5)*d*x^5*cos(c)*sin(d*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*x^5*sin(d*x^2)^2 + 4*(a^7*b - 2*a^5*b^3 + a^3*b^5)*d*x^5*cos(d*x^2)*sin

2*c)*sin(c))*d*x^5*cos(d*x^2) - (a^6*b^2 - a^4*b^4)*d*x^5*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*x^5*sin(d*x^2))*cos(2*d*x^2) - 2*(a^6*b^2*d*x^5*cos(2*d*x^2)*cos(2*c) -

a^6*b^2*d*x^5*sin(2*d*x^2)*sin(2*c) + 2*(a^7*b - a^5*b^3)*d*x^5*cos(c)*sin(d*x^2) + 2*(a^7*b - a^5*b^3)*d*x^5

*cos(d*x^2)*sin(c) + (a^8 - a^6*b^2)*d*x^5)*cos(2*d*x^2 + 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*x^5*cos(d*x^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*x^5*sin(d*x^2) + (a^6*b^2 - a^4*b^4)*d*x^5*sin(2*c))*sin(2*d*x^2) - 2*(a^6*b^2*d*x^5*

cos(2*c)*sin(2*d*x^2) + a^6*b^2*d*x^5*cos(2*d*x^2)*sin(2*c) - 2*(a^7*b - a^5*b^3)*d*x^5*cos(d*x^2)*cos(c) + 2*

(a^7*b - a^5*b^3)*d*x^5*sin(d*x^2)*sin(c))*sin(2*d*x^2 + 2*c)), x) + (2*a^2*b^2*cos(2*c) + (a^4 - a^2*b^2)*d*x

^2*sin(2*c))*sin(2*d*x^2) + (2*(a^3*b - a*b^3)*d*x^2*cos(d*x^2)*cos(c) + (a^4 - a^2*b^2)*d*x^2*cos(2*c)*sin(2*

d*x^2) + (a^4 - a^2*b^2)*d*x^2*cos(2*d*x^2)*sin(2*c) - 2*(a^3*b - a*b^3)*d*x^2*sin(d*x^2)*sin(c))*sin(2*d*x^2

+ 2*c) + 2*((a^3*b - a*b^3)*d*x^2 - ((a^3*b - a*b^3)*d*x^2*cos(2*c) - a*b^3*sin(2*c))*cos(2*d*x^2) + 2*(b^4*co

s(c) + (a^2*b^2 - b^4)*d*x^2*sin(c))*cos(d*x^2) + (a*b^3*cos(2*c) + (a^3*b - a*b^3)*d*x^2*sin(2*c))*sin(2*d*x^

2) + 2*((a^2*b^2 - b^4)*d*x^2*cos(c) - b^4*sin(c))*sin(d*x^2))*sin(d*x^2 + c) + 2*((a^3*b - a*b^3)*d*x^2*cos(c

) - 2*a*b^3*sin(c))*sin(d*x^2))/(((a^6 - a^4*b^2)*cos(2*c)^2 + (a^6 - a^4*b^2)*sin(2*c)^2)*d*x^4*cos(2*d*x^2)^

2 + 4*((a^4*b^2 - a^2*b^4)*cos(c)^2 + (a^4*b^2 - a^2*b^4)*sin(c)^2)*d*x^4*cos(d*x^2)^2 + ((a^6 - a^4*b^2)*cos(

2*c)^2 + (a^6 - a^4*b^2)*sin(2*c)^2)*d*x^4*sin(...

________________________________________________________________________________________

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

[Out]

integral(1/(b^2*x^3*csc(d*x^2 + c)^2 + 2*a*b*x^3*csc(d*x^2 + c) + a^2*x^3), x)

________________________________________________________________________________________

Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{3} \left (a + b \csc {\left (c + d x^{2} \right )}\right )^{2}}\, dx \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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^3/(a+b*csc(d*x^2+c))^2,x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

Mupad [A]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {1}{x^3\,{\left (a+\frac {b}{\sin \left (d\,x^2+c\right )}\right )}^2} \,d x \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

[next] [prev] [prev-tail] [front] [up]