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

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

[Out]

Unintegrable(1/x/(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 = {} \[ \int \frac {1}{x \left (a+b \sin \left (c+d x^3\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/(x*(a + b*Sin[c + d*x^3])^2),x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 11.04, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (a+b \sin \left (c+d x^3\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {1}{b^{2} x \cos \left (d x^{3} + c\right )^{2} - 2 \, a b x \sin \left (d x^{3} + c\right ) - {\left (a^{2} + b^{2}\right )} x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 1.09, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (a +b \sin \left (d \,x^{3}+c \right )\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a + b*sin(c + d*x^3))^2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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