3.1.92 \(\int \frac {1}{x^4 (a+b \sin (c+d x^3))^2} \, dx\) [92]
Optimal. Leaf size=21 \[ \text {Int}\left (\frac {1}{x^4 \left (a+b \sin \left (c+d x^3\right )\right )^2},x\right ) \]
[Out]
Unintegrable(1/x^4/(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 = {}
\begin {gather*} \int \frac {1}{x^4 \left (a+b \sin \left (c+d x^3\right )\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[1/(x^4*(a + b*Sin[c + d*x^3])^2),x]
[Out]
Defer[Int][1/(x^4*(a + b*Sin[c + d*x^3])^2), x]
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (a+b \sin \left (c+d x^3\right )\right )^2} \, dx &=\int \frac {1}{x^4 \left (a+b \sin \left (c+d x^3\right )\right )^2} \, dx\\ \end {align*}
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Mathematica [A]
time = 8.08, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^4 \left (a+b \sin \left (c+d x^3\right )\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Integrate[1/(x^4*(a + b*Sin[c + d*x^3])^2),x]
[Out]
Integrate[1/(x^4*(a + b*Sin[c + d*x^3])^2), x]
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Maple [A]
time = 0.21, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{4} \left (a +b \sin \left (d \,x^{3}+c \right )\right )^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/x^4/(a+b*sin(d*x^3+c))^2,x)
[Out]
int(1/x^4/(a+b*sin(d*x^3+c))^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^4/(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^6*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^6*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^6*sin(2*d*x^3)^2 + 4*(a^3*b - a*b^3)*d*x^6*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^6*sin(d*x^3)^2 + 4*(a^3*b - a*b^3)*d*x^6*cos(d*x^3)*sin(c) + (a^
2*b^2 - b^4)*d*x^6 + 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^6*cos(d*x^3)
- (a^2*b^2 - b^4)*d*x^6*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^
6*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^6*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^6*sin(d*x^3) + (a^2*b^2 -
b^4)*d*x^6*sin(2*c))*sin(2*d*x^3))*integrate(-2*(2*b^4*cos(2*c)*sin(2*d*x^3) + 2*b^4*cos(2*d*x^3)*sin(2*c) -
4*(a^3*b - a*b^3)*cos(d*x^3)*cos(c) + 4*(a^3*b - a*b^3)*sin(d*x^3)*sin(c) + (a^3*b*d*x^3*sin(d*x^3 + c) - 2*a^
3*b*cos(d*x^3 + c))*cos(2*d*x^3 + 2*c) + (2*a^3*b - 2*a*b^3 + (a*b^3*d*x^3*sin(2*c) + 2*a*b^3*cos(2*c))*cos(2*
d*x^3) - 2*((a^4 - a^2*b^2)*d*x^3*cos(c) - 2*(a^4 - a^2*b^2)*sin(c))*cos(d*x^3) + (a*b^3*d*x^3*cos(2*c) - 2*a*
b^3*sin(2*c))*sin(2*d*x^3) + 2*((a^4 - a^2*b^2)*d*x^3*sin(c) + 2*(a^4 - a^2*b^2)*cos(c))*sin(d*x^3))*cos(d*x^3
+ c) - (a^3*b*d*x^3*cos(d*x^3 + c) + 2*a^3*b*sin(d*x^3 + c) + 2*a^2*b^2)*sin(2*d*x^3 + 2*c) - ((a^3*b - a*b^3
)*d*x^3 + (a*b^3*d*x^3*cos(2*c) - 2*a*b^3*sin(2*c))*cos(2*d*x^3) + 2*((a^4 - a^2*b^2)*d*x^3*sin(c) + 2*(a^4 -
a^2*b^2)*cos(c))*cos(d*x^3) - (a*b^3*d*x^3*sin(2*c) + 2*a*b^3*cos(2*c))*sin(2*d*x^3) + 2*((a^4 - a^2*b^2)*d*x^
3*cos(c) - 2*(a^4 - a^2*b^2)*sin(c))*sin(d*x^3))*sin(d*x^3 + c))/(a^4*b^2*d*x^7*cos(2*d*x^3 + 2*c)^2 + a^4*b^2
*d*x^7*sin(2*d*x^3 + 2*c)^2 + (b^6*cos(2*c)^2 + b^6*sin(2*c)^2)*d*x^7*cos(2*d*x^3)^2 + 4*((a^6 - 2*a^4*b^2 + a
^2*b^4)*cos(c)^2 + (a^6 - 2*a^4*b^2 + a^2*b^4)*sin(c)^2)*d*x^7*cos(d*x^3)^2 + (b^6*cos(2*c)^2 + b^6*sin(2*c)^2
)*d*x^7*sin(2*d*x^3)^2 + 4*(a^5*b - 2*a^3*b^3 + a*b^5)*d*x^7*cos(c)*sin(d*x^3) + 4*((a^6 - 2*a^4*b^2 + a^2*b^4
)*cos(c)^2 + (a^6 - 2*a^4*b^2 + a^2*b^4)*sin(c)^2)*d*x^7*sin(d*x^3)^2 + 4*(a^5*b - 2*a^3*b^3 + a*b^5)*d*x^7*co
s(d*x^3)*sin(c) + (a^4*b^2 - 2*a^2*b^4 + b^6)*d*x^7 - 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*x^7*cos(d*x^3) - (a^2*b^4 - b^6)*d*x^7*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*x^7*sin(d*x^3))*cos(2*d*x^3) - 2*(a^2*b^4*d*x^7*cos(2*d*x^3)*cos(2*c) -
a^2*b^4*d*x^7*sin(2*d*x^3)*sin(2*c) + 2*(a^5*b - a^3*b^3)*d*x^7*cos(c)*sin(d*x^3) + 2*(a^5*b - a^3*b^3)*d*x^7
*cos(d*x^3)*sin(c) + (a^4*b^2 - a^2*b^4)*d*x^7)*cos(2*d*x^3 + 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*x^7*cos(d*x^3) + 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*x^7*sin(d*x^3) + (a^2*b^4 - b^6)*d*x^7*sin(2*c))*sin(2*d*x^3) - 2*(a^2*b^4*d*x^7*cos(2*c)
*sin(2*d*x^3) + a^2*b^4*d*x^7*cos(2*d*x^3)*sin(2*c) - 2*(a^5*b - a^3*b^3)*d*x^7*cos(d*x^3)*cos(c) + 2*(a^5*b -
a^3*b^3)*d*x^7*sin(d*x^3)*sin(c))*sin(2*d*x^3 + 2*c)), 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^6*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^6
*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^6*sin(2*d*x^3)^2 + 4*(a^3*b - a*
b^3)*d*x^6*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^6*sin(d*x^3)^2 + 4*
(a^3*b - a*b^3)*d*x^6*cos(d*x^3)*sin(c) + (a^2*b^2 - b^4)*d*x^6 + 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^6*cos(d*x^3) - (a^2*b^2 - b^4)*d*x^6*cos(2*c) - 2*((a^3*b - a*b^3)*cos(2*c)*c
os(c) + (a^3*b - a*b^3)*sin(2*c)*sin(c))*d*x^6*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^6*cos(d*x^3) + 2*((a^3*b - a*b^3)*cos(c)*sin(2*c) - (a^3*b - a*b^3)*c
os(2*c)*sin(c))*d*x^6*sin(d*x^3) + (a^2*b^2 - b^4)*d*x^6*sin(2*c))*sin(2*d*x^3))
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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^4/(a+b*sin(d*x^3+c))^2,x, algorithm="fricas")
[Out]
integral(-1/(b^2*x^4*cos(d*x^3 + c)^2 - 2*a*b*x^4*sin(d*x^3 + c) - (a^2 + b^2)*x^4), x)
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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{4} \left (a + b \sin {\left (c + d x^{3} \right )}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x**4/(a+b*sin(d*x**3+c))**2,x)
[Out]
Integral(1/(x**4*(a + b*sin(c + d*x**3))**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^4/(a+b*sin(d*x^3+c))^2,x, algorithm="giac")
[Out]
integrate(1/((b*sin(d*x^3 + c) + a)^2*x^4), x)
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Mupad [A]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {1}{x^4\,{\left (a+b\,\sin \left (d\,x^3+c\right )\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(x^4*(a + b*sin(c + d*x^3))^2),x)
[Out]
int(1/(x^4*(a + b*sin(c + d*x^3))^2), x)
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