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

Optimal. Leaf size=20 \[ \text{Unintegrable}\left (\frac{1}{x^2 \left (a+b \sin \left (c+d x^3\right )\right )^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0242327, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., 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*}

Mathematica [A]  time = 10.6589, size = 0, normalized size = 0. \[ \int \frac{1}{x^2 \left (a+b \sin \left (c+d x^3\right )\right )^2} \, dx \]

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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Maple [A]  time = 0.884, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2} \left ( a+b\sin \left ( d{x}^{3}+c \right ) \right ) ^{2}}}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

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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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}

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

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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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \sin \left (d x^{3} + c\right ) + a\right )}^{2} x^{2}}\,{d x} \end{align*}

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)