3.3.85 \(\int \frac {1}{x (a+b \sin (c+d (f+g x)^n))^2} \, dx\) [285]
Optimal. Leaf size=25 \[ \text {Int}\left (\frac {1}{x \left (a+b \sin \left (c+d (f+g x)^n\right )\right )^2},x\right ) \]
[Out]
Unintegrable(1/x/(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 {1}{x \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[1/(x*(a + b*Sin[c + d*(f + g*x)^n])^2),x]
[Out]
Defer[Int][1/(x*(a + b*Sin[c + d*(f + g*x)^n])^2), x]
Rubi steps
\begin {align*} \int \frac {1}{x \left (a+b \sin \left (c+d (f+g x)^n\right )\right )^2} \, dx &=\int \frac {1}{x \left (a+b \sin \left (c+d (f+g x)^n\right )\right )^2} \, dx\\ \end {align*}
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Mathematica [A]
time = 137.98, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \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[1/(x*(a + b*Sin[c + d*(f + g*x)^n])^2),x]
[Out]
Integrate[1/(x*(a + b*Sin[c + d*(f + g*x)^n])^2), x]
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Maple [A]
time = 0.16, size = 0, normalized size = 0.00 \[\int \frac {1}{x \left (a +b \sin \left (c +d \left (g x +f \right )^{n}\right )\right )^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/x/(a+b*sin(c+d*(g*x+f)^n))^2,x)
[Out]
int(1/x/(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/x/(a+b*sin(c+d*(g*x+f)^n))^2,x, algorithm="maxima")
[Out]
(2*(a^3*b*g*x + a^3*b*f)*cos(2*(g*x + f)^n*d + 2*c)*cos((g*x + f)^n*d + c) - 2*(b^4*g*x*sin(2*c) + b^4*f*sin(2
*c))*cos(2*(g*x + f)^n*d) - 2*((a^3*b - a*b^3)*g*x + (a^3*b - a*b^3)*f + (a*b^3*g*x*cos(2*c) + a*b^3*f*cos(2*c
))*cos(2*(g*x + f)^n*d) + 2*((a^4 - a^2*b^2)*g*x*sin(c) + (a^4 - a^2*b^2)*f*sin(c))*cos((g*x + f)^n*d) - (a*b^
3*g*x*sin(2*c) + a*b^3*f*sin(2*c))*sin(2*(g*x + f)^n*d) + 2*((a^4 - a^2*b^2)*g*x*cos(c) + (a^4 - a^2*b^2)*f*co
s(c))*sin((g*x + f)^n*d))*cos((g*x + f)^n*d + c) + 4*((a^3*b - a*b^3)*g*x*cos(c) + (a^3*b - a*b^3)*f*cos(c))*c
os((g*x + f)^n*d) + ((g*x + f)^n*a^4*b^2*d*g*n*x*cos(2*(g*x + f)^n*d + 2*c)^2 + (g*x + f)^n*a^4*b^2*d*g*n*x*si
n(2*(g*x + f)^n*d + 2*c)^2 + (b^6*cos(2*c)^2 + b^6*sin(2*c)^2)*(g*x + f)^n*d*g*n*x*cos(2*(g*x + f)^n*d)^2 + 4*
((a^6 - 2*a^4*b^2 + a^2*b^4)*cos(c)^2 + (a^6 - 2*a^4*b^2 + a^2*b^4)*sin(c)^2)*(g*x + f)^n*d*g*n*x*cos((g*x + f
)^n*d)^2 + (b^6*cos(2*c)^2 + b^6*sin(2*c)^2)*(g*x + f)^n*d*g*n*x*sin(2*(g*x + f)^n*d)^2 + 4*(a^5*b - 2*a^3*b^3
+ a*b^5)*(g*x + f)^n*d*g*n*x*cos(c)*sin((g*x + f)^n*d) + 4*((a^6 - 2*a^4*b^2 + a^2*b^4)*cos(c)^2 + (a^6 - 2*a
^4*b^2 + a^2*b^4)*sin(c)^2)*(g*x + f)^n*d*g*n*x*sin((g*x + f)^n*d)^2 + 4*(a^5*b - 2*a^3*b^3 + a*b^5)*(g*x + f)
^n*d*g*n*x*cos((g*x + f)^n*d)*sin(c) + (a^4*b^2 - 2*a^2*b^4 + b^6)*(g*x + f)^n*d*g*n*x - 2*(2*((a^3*b^3 - a*b^
5)*cos(c)*sin(2*c) - (a^3*b^3 - a*b^5)*cos(2*c)*sin(c))*(g*x + f)^n*d*g*n*x*cos((g*x + f)^n*d) - (a^2*b^4 - b^
6)*(g*x + f)^n*d*g*n*x*cos(2*c) - 2*((a^3*b^3 - a*b^5)*cos(2*c)*cos(c) + (a^3*b^3 - a*b^5)*sin(2*c)*sin(c))*(g
*x + f)^n*d*g*n*x*sin((g*x + f)^n*d))*cos(2*(g*x + f)^n*d) - 2*((g*x + f)^n*a^2*b^4*d*g*n*x*cos(2*(g*x + f)^n*
d)*cos(2*c) - (g*x + f)^n*a^2*b^4*d*g*n*x*sin(2*(g*x + f)^n*d)*sin(2*c) + 2*(a^5*b - a^3*b^3)*(g*x + f)^n*d*g*
n*x*cos(c)*sin((g*x + f)^n*d) + 2*(a^5*b - a^3*b^3)*(g*x + f)^n*d*g*n*x*cos((g*x + f)^n*d)*sin(c) + (a^4*b^2 -
a^2*b^4)*(g*x + f)^n*d*g*n*x)*cos(2*(g*x + f)^n*d + 2*c) - 2*(2*((a^3*b^3 - a*b^5)*cos(2*c)*cos(c) + (a^3*b^3
- a*b^5)*sin(2*c)*sin(c))*(g*x + f)^n*d*g*n*x*cos((g*x + f)^n*d) + 2*((a^3*b^3 - a*b^5)*cos(c)*sin(2*c) - (a^
3*b^3 - a*b^5)*cos(2*c)*sin(c))*(g*x + f)^n*d*g*n*x*sin((g*x + f)^n*d) + (a^2*b^4 - b^6)*(g*x + f)^n*d*g*n*x*s
in(2*c))*sin(2*(g*x + f)^n*d) - 2*((g*x + f)^n*a^2*b^4*d*g*n*x*cos(2*c)*sin(2*(g*x + f)^n*d) + (g*x + f)^n*a^2
*b^4*d*g*n*x*cos(2*(g*x + f)^n*d)*sin(2*c) - 2*(a^5*b - a^3*b^3)*(g*x + f)^n*d*g*n*x*cos((g*x + f)^n*d)*cos(c)
+ 2*(a^5*b - a^3*b^3)*(g*x + f)^n*d*g*n*x*sin((g*x + f)^n*d)*sin(c))*sin(2*(g*x + f)^n*d + 2*c))*integrate(-2
*((b^4*g*n*x*sin(2*c) + b^4*f*sin(2*c))*cos(2*(g*x + f)^n*d) + ((g*x + f)^n*a^3*b*d*g*n*x*sin((g*x + f)^n*d +
c) - (a^3*b*g*n*x + a^3*b*f)*cos((g*x + f)^n*d + c))*cos(2*(g*x + f)^n*d + 2*c) + ((a^3*b - a*b^3)*g*n*x + (a^
3*b - a*b^3)*f + ((g*x + f)^n*a*b^3*d*g*n*x*sin(2*c) + a*b^3*g*n*x*cos(2*c) + a*b^3*f*cos(2*c))*cos(2*(g*x + f
)^n*d) - 2*((a^4 - a^2*b^2)*(g*x + f)^n*d*g*n*x*cos(c) - (a^4 - a^2*b^2)*g*n*x*sin(c) - (a^4 - a^2*b^2)*f*sin(
c))*cos((g*x + f)^n*d) + ((g*x + f)^n*a*b^3*d*g*n*x*cos(2*c) - a*b^3*g*n*x*sin(2*c) - a*b^3*f*sin(2*c))*sin(2*
(g*x + f)^n*d) + 2*((a^4 - a^2*b^2)*(g*x + f)^n*d*g*n*x*sin(c) + (a^4 - a^2*b^2)*g*n*x*cos(c) + (a^4 - a^2*b^2
)*f*cos(c))*sin((g*x + f)^n*d))*cos((g*x + f)^n*d + c) - 2*((a^3*b - a*b^3)*g*n*x*cos(c) + (a^3*b - a*b^3)*f*c
os(c))*cos((g*x + f)^n*d) + (b^4*g*n*x*cos(2*c) + b^4*f*cos(2*c))*sin(2*(g*x + f)^n*d) - ((g*x + f)^n*a^3*b*d*
g*n*x*cos((g*x + f)^n*d + c) + a^2*b^2*g*n*x + a^2*b^2*f + (a^3*b*g*n*x + a^3*b*f)*sin((g*x + f)^n*d + c))*sin
(2*(g*x + f)^n*d + 2*c) - ((a^3*b - a*b^3)*(g*x + f)^n*d*g*n*x + ((g*x + f)^n*a*b^3*d*g*n*x*cos(2*c) - a*b^3*g
*n*x*sin(2*c) - a*b^3*f*sin(2*c))*cos(2*(g*x + f)^n*d) + 2*((a^4 - a^2*b^2)*(g*x + f)^n*d*g*n*x*sin(c) + (a^4
- a^2*b^2)*g*n*x*cos(c) + (a^4 - a^2*b^2)*f*cos(c))*cos((g*x + f)^n*d) - ((g*x + f)^n*a*b^3*d*g*n*x*sin(2*c) +
a*b^3*g*n*x*cos(2*c) + a*b^3*f*cos(2*c))*sin(2*(g*x + f)^n*d) + 2*((a^4 - a^2*b^2)*(g*x + f)^n*d*g*n*x*cos(c)
- (a^4 - a^2*b^2)*g*n*x*sin(c) - (a^4 - a^2*b^2)*f*sin(c))*sin((g*x + f)^n*d))*sin((g*x + f)^n*d + c) + 2*((a
^3*b - a*b^3)*g*n*x*sin(c) + (a^3*b - a*b^3)*f*sin(c))*sin((g*x + f)^n*d))/((g*x + f)^n*a^4*b^2*d*g*n*x^2*cos(
2*(g*x + f)^n*d + 2*c)^2 + (g*x + f)^n*a^4*b^2*d*g*n*x^2*sin(2*(g*x + f)^n*d + 2*c)^2 + (b^6*cos(2*c)^2 + b^6*
sin(2*c)^2)*(g*x + f)^n*d*g*n*x^2*cos(2*(g*x + f)^n*d)^2 + 4*((a^6 - 2*a^4*b^2 + a^2*b^4)*cos(c)^2 + (a^6 - 2*
a^4*b^2 + a^2*b^4)*sin(c)^2)*(g*x + f)^n*d*g*n*x^2*cos((g*x + f)^n*d)^2 + (b^6*cos(2*c)^2 + b^6*sin(2*c)^2)*(g
*x + f)^n*d*g*n*x^2*sin(2*(g*x + f)^n*d)^2 + 4*(a^5*b - 2*a^3*b^3 + a*b^5)*(g*x + f)^n*d*g*n*x^2*cos(c)*sin((g
*x + f)^n*d) + 4*((a^6 - 2*a^4*b^2 + a^2*b^4)*cos(c)^2 + (a^6 - 2*a^4*b^2 + a^2*b^4)*sin(c)^2)*(g*x + f)^n*d*g
*n*x^2*sin((g*x + f)^n*d)^2 + 4*(a^5*b - 2*a^3*b^3 + a*b^5)*(g*x + f)^n*d*g*n*x^2*cos((g*x + f)^n*d)*sin(c) +
(a^4*b^2 - 2*a^2*b^4 + b^6)*(g*x + f)^n*d*g*n*x^2 - 2*(2*((a^3*b^3 - a*b^5)*cos(c)*sin(2*c) - (a^3*b^3 - a*b^5
)*cos(2*c)*sin(c))*(g*x + f)^n*d*g*n*x^2*cos((g...
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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/x/(a+b*sin(c+d*(g*x+f)^n))^2,x, algorithm="fricas")
[Out]
integral(-1/(b^2*x*cos((g*x + f)^n*d + c)^2 - 2*a*b*x*sin((g*x + f)^n*d + c) - (a^2 + b^2)*x), 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/x/(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/x/(a+b*sin(c+d*(g*x+f)^n))^2,x, algorithm="giac")
[Out]
integrate(1/((b*sin((g*x + f)^n*d + c) + a)^2*x), x)
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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {1}{x\,{\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/(x*(a + b*sin(c + d*(f + g*x)^n))^2),x)
[Out]
int(1/(x*(a + b*sin(c + d*(f + g*x)^n))^2), x)
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