3.284 \(\int \frac {1}{(a+b \sin (c+d (f+g x)^n))^2} \, dx\)
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 = 0.01, 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 =
{} \[ \int \frac {1}{\left (a+b \sin \left (c+d (f+g x)^n\right )\right )^2} \, dx \]
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 = 11.58, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a+b \sin \left (c+d (f+g x)^n\right )\right )^2} \, dx \]
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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fricas [A] time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {1}{b^{2} \cos \left ({\left (g x + f\right )}^{n} d + c\right )^{2} - 2 \, a b \sin \left ({\left (g x + f\right )}^{n} d + c\right ) - a^{2} - b^{2}}, x\right ) \]
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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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b \sin \left ({\left (g x + f\right )}^{n} d + c\right ) + a\right )}^{2}}\,{d x} \]
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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maple [A] time = 2.10, 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 \[ \text {result too large to display} \]
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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mupad [A] time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {1}{{\left (a+b\,\sin \left (c+d\,{\left (f+g\,x\right )}^n\right )\right )}^2} \,d x \]
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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sympy [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/(a+b*sin(c+d*(g*x+f)**n))**2,x)
[Out]
Timed out
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