3.3.82 \(\int \frac {x^2}{(a+b \sin (c+d (f+g x)^n))^2} \, dx\) [282]

Optimal. Leaf size=25 \[ \text {Int}\left (\frac {x^2}{\left (a+b \sin \left (c+d (f+g x)^n\right )\right )^2},x\right ) \]

[Out]

Unintegrable(x^2/(a+b*sin(c+d*(g*x+f)^n))^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 {x^2}{\left (a+b \sin \left (c+d (f+g x)^n\right )\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[x^2/(a + b*Sin[c + d*(f + g*x)^n])^2,x]

[Out]

Defer[Int][x^2/(a + b*Sin[c + d*(f + g*x)^n])^2, x]

Rubi steps

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

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Mathematica [F]
time = 180.09, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[x^2/(a + b*Sin[c + d*(f + g*x)^n])^2,x]

[Out]

$Aborted

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Maple [A]
time = 0.27, size = 0, normalized size = 0.00 \[\int \frac {x^{2}}{\left (a +b \sin \left (c +d \left (g x +f \right )^{n}\right )\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a+b*sin(c+d*(g*x+f)^n))^2,x)

[Out]

int(x^2/(a+b*sin(c+d*(g*x+f)^n))^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(x^2/(a+b*sin(c+d*(g*x+f)^n))^2,x, algorithm="maxima")

[Out]

(2*(a*b*g*x^3 + a*b*f*x^2)*cos(2*(g*x + f)^n*d + 2*c)*cos((g*x + f)^n*d + c) + 2*(a*b*g*x^3 + a*b*f*x^2)*cos((
g*x + f)^n*d + c) - ((a^2*b^2 - b^4)*(g*x + f)^n*d*g*n*cos(2*(g*x + f)^n*d + 2*c)^2 + 4*(a^4 - a^2*b^2)*(g*x +
f)^n*d*g*n*cos((g*x + f)^n*d + c)^2 + 4*(a^3*b - a*b^3)*(g*x + f)^n*d*g*n*cos((g*x + f)^n*d + c)*sin(2*(g*x +
f)^n*d + 2*c) + (a^2*b^2 - b^4)*(g*x + f)^n*d*g*n*sin(2*(g*x + f)^n*d + 2*c)^2 + 4*(a^4 - a^2*b^2)*(g*x + f)^
n*d*g*n*sin((g*x + f)^n*d + c)^2 + 4*(a^3*b - a*b^3)*(g*x + f)^n*d*g*n*sin((g*x + f)^n*d + c) + (a^2*b^2 - b^4
)*(g*x + f)^n*d*g*n - 2*(2*(a^3*b - a*b^3)*(g*x + f)^n*d*g*n*sin((g*x + f)^n*d + c) + (a^2*b^2 - b^4)*(g*x + f
)^n*d*g*n)*cos(2*(g*x + f)^n*d + 2*c))*integrate(-2*(2*(g*x + f)^n*a^2*d*g*n*x^2*cos((g*x + f)^n*d + c)^2 + 2*
(g*x + f)^n*a^2*d*g*n*x^2*sin((g*x + f)^n*d + c)^2 + (g*x + f)^n*a*b*d*g*n*x^2*sin((g*x + f)^n*d + c) - ((g*x
+ f)^n*a*b*d*g*n*x^2*sin((g*x + f)^n*d + c) + (2*a*b*f*x - (a*b*g*n - 3*a*b*g)*x^2)*cos((g*x + f)^n*d + c))*co
s(2*(g*x + f)^n*d + 2*c) - (2*a*b*f*x - (a*b*g*n - 3*a*b*g)*x^2)*cos((g*x + f)^n*d + c) + ((g*x + f)^n*a*b*d*g
*n*x^2*cos((g*x + f)^n*d + c) - 2*b^2*f*x + (b^2*g*n - 3*b^2*g)*x^2 - (2*a*b*f*x - (a*b*g*n - 3*a*b*g)*x^2)*si
n((g*x + f)^n*d + c))*sin(2*(g*x + f)^n*d + 2*c))/((a^2*b^2 - b^4)*(g*x + f)^n*d*g*n*cos(2*(g*x + f)^n*d + 2*c
)^2 + 4*(a^4 - a^2*b^2)*(g*x + f)^n*d*g*n*cos((g*x + f)^n*d + c)^2 + 4*(a^3*b - a*b^3)*(g*x + f)^n*d*g*n*cos((
g*x + f)^n*d + c)*sin(2*(g*x + f)^n*d + 2*c) + (a^2*b^2 - b^4)*(g*x + f)^n*d*g*n*sin(2*(g*x + f)^n*d + 2*c)^2
+ 4*(a^4 - a^2*b^2)*(g*x + f)^n*d*g*n*sin((g*x + f)^n*d + c)^2 + 4*(a^3*b - a*b^3)*(g*x + f)^n*d*g*n*sin((g*x
+ f)^n*d + c) + (a^2*b^2 - b^4)*(g*x + f)^n*d*g*n - 2*(2*(a^3*b - a*b^3)*(g*x + f)^n*d*g*n*sin((g*x + f)^n*d +
c) + (a^2*b^2 - b^4)*(g*x + f)^n*d*g*n)*cos(2*(g*x + f)^n*d + 2*c)), x) + 2*(b^2*g*x^3 + b^2*f*x^2 + (a*b*g*x
^3 + a*b*f*x^2)*sin((g*x + f)^n*d + c))*sin(2*(g*x + f)^n*d + 2*c))/((a^2*b^2 - b^4)*(g*x + f)^n*d*g*n*cos(2*(
g*x + f)^n*d + 2*c)^2 + 4*(a^4 - a^2*b^2)*(g*x + f)^n*d*g*n*cos((g*x + f)^n*d + c)^2 + 4*(a^3*b - a*b^3)*(g*x
+ f)^n*d*g*n*cos((g*x + f)^n*d + c)*sin(2*(g*x + f)^n*d + 2*c) + (a^2*b^2 - b^4)*(g*x + f)^n*d*g*n*sin(2*(g*x
+ f)^n*d + 2*c)^2 + 4*(a^4 - a^2*b^2)*(g*x + f)^n*d*g*n*sin((g*x + f)^n*d + c)^2 + 4*(a^3*b - a*b^3)*(g*x + f)
^n*d*g*n*sin((g*x + f)^n*d + c) + (a^2*b^2 - b^4)*(g*x + f)^n*d*g*n - 2*(2*(a^3*b - a*b^3)*(g*x + f)^n*d*g*n*s
in((g*x + f)^n*d + c) + (a^2*b^2 - b^4)*(g*x + f)^n*d*g*n)*cos(2*(g*x + f)^n*d + 2*c))

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

[Out]

integral(-x^2/(b^2*cos((g*x + f)^n*d + c)^2 - 2*a*b*sin((g*x + f)^n*d + c) - a^2 - b^2), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(a+b*sin(c+d*(g*x+f)**n))**2,x)

[Out]

Timed out

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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(x^2/(a+b*sin(c+d*(g*x+f)^n))^2,x, algorithm="giac")

[Out]

integrate(x^2/(b*sin((g*x + f)^n*d + c) + a)^2, x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {x^2}{{\left (a+b\,\sin \left (c+d\,{\left (f+g\,x\right )}^n\right )\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a + b*sin(c + d*(f + g*x)^n))^2,x)

[Out]

int(x^2/(a + b*sin(c + d*(f + g*x)^n))^2, x)

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