3.1.96 \(\int \frac {1}{x^3 (a+b \sin (c+d x^3))^2} \, dx\) [96]

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

[Out]

Unintegrable(1/x^3/(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^3 \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^3*(a + b*Sin[c + d*x^3])^2),x]

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 7.74, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^3 \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^3*(a + b*Sin[c + d*x^3])^2),x]

[Out]

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

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Maple [A]
time = 0.16, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{3} \left (a +b \sin \left (d \,x^{3}+c \right )\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/x^3/(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^3/(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^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*sin(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*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^5*sin(d*x^3) + (a^2*b^2 -
b^4)*d*x^5*sin(2*c))*sin(2*d*x^3))*integrate(-2/3*(10*a*b*cos(d*x^3)*cos(c) + 5*b^2*cos(2*c)*sin(2*d*x^3) + 5
*b^2*cos(2*d*x^3)*sin(2*c) - 10*a*b*sin(d*x^3)*sin(c) - (5*a*b - (3*a*b*d*x^3*sin(2*c) + 5*a*b*cos(2*c))*cos(2
*d*x^3) - 2*(3*a^2*d*x^3*cos(c) - 5*a^2*sin(c))*cos(d*x^3) - (3*a*b*d*x^3*cos(2*c) - 5*a*b*sin(2*c))*sin(2*d*x
^3) + 2*(3*a^2*d*x^3*sin(c) + 5*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)
- 5*a*b*sin(2*c))*cos(2*d*x^3) + 2*(3*a^2*d*x^3*sin(c) + 5*a^2*cos(c))*cos(d*x^3) + (3*a*b*d*x^3*sin(2*c) + 5*
a*b*cos(2*c))*sin(2*d*x^3) + 2*(3*a^2*d*x^3*cos(c) - 5*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^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*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^6*sin(d*x^3) + (a^2*b^2 - b^4)*d*x^6*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^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*sin(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))

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

[Out]

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

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

[Out]

Integral(1/(x**3*(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^3/(a+b*sin(d*x^3+c))^2,x, algorithm="giac")

[Out]

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

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

[Out]

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

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