3.3.84 \(\int \frac {1}{(a+b \sin (c+d (f+g x)^n))^2} \, dx\) [284]

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

[Out]

Unintegrable(1/(a+b*sin(c+d*(g*x+f)^n))^2,x)

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Rubi [A]
time = 0.00, 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}{\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[(a + b*Sin[c + d*(f + g*x)^n])^(-2),x]

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 7.47, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\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]

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

[Out]

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

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Maple [A]
time = 0.26, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a +b \sin \left (c +d \left (g x +f \right )^{n}\right )\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(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(1/(a+b*sin(c+d*(g*x+f)^n))^2,x, algorithm="maxima")

[Out]

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

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

[Out]

integral(-1/(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(1/(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(1/(a+b*sin(c+d*(g*x+f)^n))^2,x, algorithm="giac")

[Out]

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

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

[Out]

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

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