3.94 \(\int \frac {1}{x^2 (a+b \sin (c+d x^3))^2} \, dx\)
Optimal. Leaf size=21 \[ \text {Int}\left (\frac {1}{x^2 \left (a+b \sin \left (c+d x^3\right )\right )^2},x\right ) \]
[Out]
Unintegrable(1/x^2/(a+b*sin(d*x^3+c))^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 =
{} \[ \int \frac {1}{x^2 \left (a+b \sin \left (c+d x^3\right )\right )^2} \, dx \]
Verification is Not applicable to the result.
[In]
Int[1/(x^2*(a + b*Sin[c + d*x^3])^2),x]
[Out]
Defer[Int][1/(x^2*(a + b*Sin[c + d*x^3])^2), x]
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (a+b \sin \left (c+d x^3\right )\right )^2} \, dx &=\int \frac {1}{x^2 \left (a+b \sin \left (c+d x^3\right )\right )^2} \, dx\\ \end {align*}
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Mathematica [A] time = 11.68, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^2 \left (a+b \sin \left (c+d x^3\right )\right )^2} \, dx \]
Verification is Not applicable to the result.
[In]
Integrate[1/(x^2*(a + b*Sin[c + d*x^3])^2),x]
[Out]
Integrate[1/(x^2*(a + b*Sin[c + d*x^3])^2), x]
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fricas [A] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {1}{b^{2} x^{2} \cos \left (d x^{3} + c\right )^{2} - 2 \, a b x^{2} \sin \left (d x^{3} + c\right ) - {\left (a^{2} + b^{2}\right )} x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^2/(a+b*sin(d*x^3+c))^2,x, algorithm="fricas")
[Out]
integral(-1/(b^2*x^2*cos(d*x^3 + c)^2 - 2*a*b*x^2*sin(d*x^3 + c) - (a^2 + b^2)*x^2), x)
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b \sin \left (d x^{3} + c\right ) + a\right )}^{2} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^2/(a+b*sin(d*x^3+c))^2,x, algorithm="giac")
[Out]
integrate(1/((b*sin(d*x^3 + c) + a)^2*x^2), x)
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maple [A] time = 1.25, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \left (a +b \sin \left (d \,x^{3}+c \right )\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/x^2/(a+b*sin(d*x^3+c))^2,x)
[Out]
int(1/x^2/(a+b*sin(d*x^3+c))^2,x)
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^2/(a+b*sin(d*x^3+c))^2,x, algorithm="maxima")
[Out]
1/3*(4*a*b*cos(d*x^3)*cos(c) + 2*b^2*cos(2*c)*sin(2*d*x^3) + 2*b^2*cos(2*d*x^3)*sin(2*c) - 4*a*b*sin(d*x^3)*si
n(c) + 2*(a*b*cos(2*d*x^3)*cos(2*c) - 2*a^2*cos(c)*sin(d*x^3) - a*b*sin(2*d*x^3)*sin(2*c) - 2*a^2*cos(d*x^3)*s
in(c) - a*b)*cos(d*x^3 + c) - 3*(((a^2*b^2 - b^4)*cos(2*c)^2 + (a^2*b^2 - b^4)*sin(2*c)^2)*d*x^4*cos(2*d*x^3)^
2 + 4*((a^4 - a^2*b^2)*cos(c)^2 + (a^4 - a^2*b^2)*sin(c)^2)*d*x^4*cos(d*x^3)^2 + ((a^2*b^2 - b^4)*cos(2*c)^2 +
(a^2*b^2 - b^4)*sin(2*c)^2)*d*x^4*sin(2*d*x^3)^2 + 4*(a^3*b - a*b^3)*d*x^4*cos(c)*sin(d*x^3) + 4*((a^4 - a^2*
b^2)*cos(c)^2 + (a^4 - a^2*b^2)*sin(c)^2)*d*x^4*sin(d*x^3)^2 + 4*(a^3*b - a*b^3)*d*x^4*cos(d*x^3)*sin(c) + (a^
2*b^2 - b^4)*d*x^4 + 2*(2*((a^3*b - a*b^3)*cos(c)*sin(2*c) - (a^3*b - a*b^3)*cos(2*c)*sin(c))*d*x^4*cos(d*x^3)
- (a^2*b^2 - b^4)*d*x^4*cos(2*c) - 2*((a^3*b - a*b^3)*cos(2*c)*cos(c) + (a^3*b - a*b^3)*sin(2*c)*sin(c))*d*x^
4*sin(d*x^3))*cos(2*d*x^3) + 2*(2*((a^3*b - a*b^3)*cos(2*c)*cos(c) + (a^3*b - a*b^3)*sin(2*c)*sin(c))*d*x^4*co
s(d*x^3) + 2*((a^3*b - a*b^3)*cos(c)*sin(2*c) - (a^3*b - a*b^3)*cos(2*c)*sin(c))*d*x^4*sin(d*x^3) + (a^2*b^2 -
b^4)*d*x^4*sin(2*c))*sin(2*d*x^3))*integrate(-2/3*(8*a*b*cos(d*x^3)*cos(c) + 4*b^2*cos(2*c)*sin(2*d*x^3) + 4*
b^2*cos(2*d*x^3)*sin(2*c) - 8*a*b*sin(d*x^3)*sin(c) - (4*a*b - (3*a*b*d*x^3*sin(2*c) + 4*a*b*cos(2*c))*cos(2*d
*x^3) - 2*(3*a^2*d*x^3*cos(c) - 4*a^2*sin(c))*cos(d*x^3) - (3*a*b*d*x^3*cos(2*c) - 4*a*b*sin(2*c))*sin(2*d*x^3
) + 2*(3*a^2*d*x^3*sin(c) + 4*a^2*cos(c))*sin(d*x^3))*cos(d*x^3 + c) + (3*a*b*d*x^3 - (3*a*b*d*x^3*cos(2*c) -
4*a*b*sin(2*c))*cos(2*d*x^3) + 2*(3*a^2*d*x^3*sin(c) + 4*a^2*cos(c))*cos(d*x^3) + (3*a*b*d*x^3*sin(2*c) + 4*a*
b*cos(2*c))*sin(2*d*x^3) + 2*(3*a^2*d*x^3*cos(c) - 4*a^2*sin(c))*sin(d*x^3))*sin(d*x^3 + c))/(((a^2*b^2 - b^4)
*cos(2*c)^2 + (a^2*b^2 - b^4)*sin(2*c)^2)*d*x^5*cos(2*d*x^3)^2 + 4*((a^4 - a^2*b^2)*cos(c)^2 + (a^4 - a^2*b^2)
*sin(c)^2)*d*x^5*cos(d*x^3)^2 + ((a^2*b^2 - b^4)*cos(2*c)^2 + (a^2*b^2 - b^4)*sin(2*c)^2)*d*x^5*sin(2*d*x^3)^2
+ 4*(a^3*b - a*b^3)*d*x^5*cos(c)*sin(d*x^3) + 4*((a^4 - a^2*b^2)*cos(c)^2 + (a^4 - a^2*b^2)*sin(c)^2)*d*x^5*s
in(d*x^3)^2 + 4*(a^3*b - a*b^3)*d*x^5*cos(d*x^3)*sin(c) + (a^2*b^2 - b^4)*d*x^5 + 2*(2*((a^3*b - a*b^3)*cos(c)
*sin(2*c) - (a^3*b - a*b^3)*cos(2*c)*sin(c))*d*x^5*cos(d*x^3) - (a^2*b^2 - b^4)*d*x^5*cos(2*c) - 2*((a^3*b - a
*b^3)*cos(2*c)*cos(c) + (a^3*b - a*b^3)*sin(2*c)*sin(c))*d*x^5*sin(d*x^3))*cos(2*d*x^3) + 2*(2*((a^3*b - a*b^3
)*cos(2*c)*cos(c) + (a^3*b - a*b^3)*sin(2*c)*sin(c))*d*x^5*cos(d*x^3) + 2*((a^3*b - a*b^3)*cos(c)*sin(2*c) - (
a^3*b - a*b^3)*cos(2*c)*sin(c))*d*x^5*sin(d*x^3) + (a^2*b^2 - b^4)*d*x^5*sin(2*c))*sin(2*d*x^3)), x) + 2*(2*a^
2*cos(d*x^3)*cos(c) + a*b*cos(2*c)*sin(2*d*x^3) + a*b*cos(2*d*x^3)*sin(2*c) - 2*a^2*sin(d*x^3)*sin(c))*sin(d*x
^3 + c))/(((a^2*b^2 - b^4)*cos(2*c)^2 + (a^2*b^2 - b^4)*sin(2*c)^2)*d*x^4*cos(2*d*x^3)^2 + 4*((a^4 - a^2*b^2)*
cos(c)^2 + (a^4 - a^2*b^2)*sin(c)^2)*d*x^4*cos(d*x^3)^2 + ((a^2*b^2 - b^4)*cos(2*c)^2 + (a^2*b^2 - b^4)*sin(2*
c)^2)*d*x^4*sin(2*d*x^3)^2 + 4*(a^3*b - a*b^3)*d*x^4*cos(c)*sin(d*x^3) + 4*((a^4 - a^2*b^2)*cos(c)^2 + (a^4 -
a^2*b^2)*sin(c)^2)*d*x^4*sin(d*x^3)^2 + 4*(a^3*b - a*b^3)*d*x^4*cos(d*x^3)*sin(c) + (a^2*b^2 - b^4)*d*x^4 + 2*
(2*((a^3*b - a*b^3)*cos(c)*sin(2*c) - (a^3*b - a*b^3)*cos(2*c)*sin(c))*d*x^4*cos(d*x^3) - (a^2*b^2 - b^4)*d*x^
4*cos(2*c) - 2*((a^3*b - a*b^3)*cos(2*c)*cos(c) + (a^3*b - a*b^3)*sin(2*c)*sin(c))*d*x^4*sin(d*x^3))*cos(2*d*x
^3) + 2*(2*((a^3*b - a*b^3)*cos(2*c)*cos(c) + (a^3*b - a*b^3)*sin(2*c)*sin(c))*d*x^4*cos(d*x^3) + 2*((a^3*b -
a*b^3)*cos(c)*sin(2*c) - (a^3*b - a*b^3)*cos(2*c)*sin(c))*d*x^4*sin(d*x^3) + (a^2*b^2 - b^4)*d*x^4*sin(2*c))*s
in(2*d*x^3))
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mupad [A] time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {1}{x^2\,{\left (a+b\,\sin \left (d\,x^3+c\right )\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(x^2*(a + b*sin(c + d*x^3))^2),x)
[Out]
int(1/(x^2*(a + b*sin(c + d*x^3))^2), x)
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \left (a + b \sin {\left (c + d x^{3} \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x**2/(a+b*sin(d*x**3+c))**2,x)
[Out]
Integral(1/(x**2*(a + b*sin(c + d*x**3))**2), x)
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