3.1.49 \(\int \frac {1}{(a+b \sin (c+d x^2))^2} \, dx\) [49]
Optimal. Leaf size=17 \[ \text {Int}\left (\frac {1}{\left (a+b \sin \left (c+d x^2\right )\right )^2},x\right ) \]
[Out]
Unintegrable(1/(a+b*sin(d*x^2+c))^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 x^2\right )\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[(a + b*Sin[c + d*x^2])^(-2),x]
[Out]
Defer[Int][(a + b*Sin[c + d*x^2])^(-2), x]
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \sin \left (c+d x^2\right )\right )^2} \, dx &=\int \frac {1}{\left (a+b \sin \left (c+d x^2\right )\right )^2} \, dx\\ \end {align*}
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Mathematica [A]
time = 3.21, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a+b \sin \left (c+d x^2\right )\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Integrate[(a + b*Sin[c + d*x^2])^(-2),x]
[Out]
Integrate[(a + b*Sin[c + d*x^2])^(-2), x]
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Maple [A]
time = 0.21, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a +b \sin \left (d \,x^{2}+c \right )\right )^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(a+b*sin(d*x^2+c))^2,x)
[Out]
int(1/(a+b*sin(d*x^2+c))^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(d*x^2+c))^2,x, algorithm="maxima")
[Out]
(a^3*b*cos(2*d*x^2 + 2*c)*cos(d*x^2 + c) - b^4*cos(2*c)*sin(2*d*x^2) - b^4*cos(2*d*x^2)*sin(2*c) + 2*(a^3*b -
a*b^3)*cos(d*x^2)*cos(c) - 2*(a^3*b - a*b^3)*sin(d*x^2)*sin(c) - (a*b^3*cos(2*d*x^2)*cos(2*c) - a*b^3*sin(2*d*
x^2)*sin(2*c) + a^3*b - a*b^3 + 2*(a^4 - a^2*b^2)*cos(c)*sin(d*x^2) + 2*(a^4 - a^2*b^2)*cos(d*x^2)*sin(c))*cos
(d*x^2 + c) + (a^4*b^2*d*x*cos(2*d*x^2 + 2*c)^2 + a^4*b^2*d*x*sin(2*d*x^2 + 2*c)^2 + (b^6*cos(2*c)^2 + b^6*sin
(2*c)^2)*d*x*cos(2*d*x^2)^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)*
d*x*cos(d*x^2)^2 + (b^6*cos(2*c)^2 + b^6*sin(2*c)^2)*d*x*sin(2*d*x^2)^2 + 4*(a^5*b - 2*a^3*b^3 + a*b^5)*d*x*co
s(c)*sin(d*x^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)*d*x*sin(d*x^
2)^2 + 4*(a^5*b - 2*a^3*b^3 + a*b^5)*d*x*cos(d*x^2)*sin(c) + (a^4*b^2 - 2*a^2*b^4 + b^6)*d*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))*d*x*cos(d*x^2) - (a^2*b^4 - b^6)*d*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))*d*x*sin(d*x^2))*cos(2*d*x^2) - 2*(a^
2*b^4*d*x*cos(2*d*x^2)*cos(2*c) - a^2*b^4*d*x*sin(2*d*x^2)*sin(2*c) + 2*(a^5*b - a^3*b^3)*d*x*cos(c)*sin(d*x^2
) + 2*(a^5*b - a^3*b^3)*d*x*cos(d*x^2)*sin(c) + (a^4*b^2 - a^2*b^4)*d*x)*cos(2*d*x^2 + 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))*d*x*cos(d*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))*d*x*sin(d*x^2) + (a^2*b^4 - b^6)*d*x*sin(2*c))*sin(2*d*x^2) - 2*(a^2
*b^4*d*x*cos(2*c)*sin(2*d*x^2) + a^2*b^4*d*x*cos(2*d*x^2)*sin(2*c) - 2*(a^5*b - a^3*b^3)*d*x*cos(d*x^2)*cos(c)
+ 2*(a^5*b - a^3*b^3)*d*x*sin(d*x^2)*sin(c))*sin(2*d*x^2 + 2*c))*integrate(-(b^4*cos(2*c)*sin(2*d*x^2) + b^4*
cos(2*d*x^2)*sin(2*c) - 2*(a^3*b - a*b^3)*cos(d*x^2)*cos(c) + 2*(a^3*b - a*b^3)*sin(d*x^2)*sin(c) + (2*a^3*b*d
*x^2*sin(d*x^2 + c) - a^3*b*cos(d*x^2 + c))*cos(2*d*x^2 + 2*c) + (a^3*b - a*b^3 + (2*a*b^3*d*x^2*sin(2*c) + a*
b^3*cos(2*c))*cos(2*d*x^2) - 2*(2*(a^4 - a^2*b^2)*d*x^2*cos(c) - (a^4 - a^2*b^2)*sin(c))*cos(d*x^2) + (2*a*b^3
*d*x^2*cos(2*c) - a*b^3*sin(2*c))*sin(2*d*x^2) + 2*(2*(a^4 - a^2*b^2)*d*x^2*sin(c) + (a^4 - a^2*b^2)*cos(c))*s
in(d*x^2))*cos(d*x^2 + c) - (2*a^3*b*d*x^2*cos(d*x^2 + c) + a^3*b*sin(d*x^2 + c) + a^2*b^2)*sin(2*d*x^2 + 2*c)
- (2*(a^3*b - a*b^3)*d*x^2 + (2*a*b^3*d*x^2*cos(2*c) - a*b^3*sin(2*c))*cos(2*d*x^2) + 2*(2*(a^4 - a^2*b^2)*d*
x^2*sin(c) + (a^4 - a^2*b^2)*cos(c))*cos(d*x^2) - (2*a*b^3*d*x^2*sin(2*c) + a*b^3*cos(2*c))*sin(2*d*x^2) + 2*(
2*(a^4 - a^2*b^2)*d*x^2*cos(c) - (a^4 - a^2*b^2)*sin(c))*sin(d*x^2))*sin(d*x^2 + c))/(a^4*b^2*d*x^2*cos(2*d*x^
2 + 2*c)^2 + a^4*b^2*d*x^2*sin(2*d*x^2 + 2*c)^2 + (b^6*cos(2*c)^2 + b^6*sin(2*c)^2)*d*x^2*cos(2*d*x^2)^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)*d*x^2*cos(d*x^2)^2 + (b^6*cos(2*c
)^2 + b^6*sin(2*c)^2)*d*x^2*sin(2*d*x^2)^2 + 4*(a^5*b - 2*a^3*b^3 + a*b^5)*d*x^2*cos(c)*sin(d*x^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)*d*x^2*sin(d*x^2)^2 + 4*(a^5*b - 2*a^3*b
^3 + a*b^5)*d*x^2*cos(d*x^2)*sin(c) + (a^4*b^2 - 2*a^2*b^4 + b^6)*d*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))*d*x^2*cos(d*x^2) - (a^2*b^4 - b^6)*d*x^2*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))*d*x^2*sin(d*x^2))*cos(2*d*x^2) - 2*(a^2*b^4*d*x^2*cos
(2*d*x^2)*cos(2*c) - a^2*b^4*d*x^2*sin(2*d*x^2)*sin(2*c) + 2*(a^5*b - a^3*b^3)*d*x^2*cos(c)*sin(d*x^2) + 2*(a^
5*b - a^3*b^3)*d*x^2*cos(d*x^2)*sin(c) + (a^4*b^2 - a^2*b^4)*d*x^2)*cos(2*d*x^2 + 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))*d*x^2*cos(d*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))*d*x^2*sin(d*x^2) + (a^2*b^4 - b^6)*d*x^2*sin(2*c))*sin(2*d*x^2) - 2*(a^
2*b^4*d*x^2*cos(2*c)*sin(2*d*x^2) + a^2*b^4*d*x^2*cos(2*d*x^2)*sin(2*c) - 2*(a^5*b - a^3*b^3)*d*x^2*cos(d*x^2)
*cos(c) + 2*(a^5*b - a^3*b^3)*d*x^2*sin(d*x^2)*sin(c))*sin(2*d*x^2 + 2*c)), x) + (a^3*b*sin(d*x^2 + c) + a^2*b
^2)*sin(2*d*x^2 + 2*c) - (a*b^3*cos(2*c)*sin(2*d*x^2) + a*b^3*cos(2*d*x^2)*sin(2*c) - 2*(a^4 - a^2*b^2)*cos(d*
x^2)*cos(c) + 2*(a^4 - a^2*b^2)*sin(d*x^2)*sin(c))*sin(d*x^2 + c))/(a^4*b^2*d*x*cos(2*d*x^2 + 2*c)^2 + a^4*b^2
*d*x*sin(2*d*x^2 + 2*c)^2 + (b^6*cos(2*c)^2 + b^6*sin(2*c)^2)*d*x*cos(2*d*x^2)^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)*d*x*cos(d*x^2)^2 + (b^6*cos(2*c)^2 + b^6*sin(2*c)^2)*d*x*
sin(2*d*x^2)^2 + 4*(a^5*b - 2*a^3*b^3 + a*b^5)*d*x*cos(c)*sin(d*x^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)*d*x*sin(d*x^2)^2 + 4*(a^5*b - 2*a^3*b^3 + a*b^5)*d*x*cos(d*x^2)*sin(c
) + (a^4*b^2 - 2*a^2*b^4 + b^6)*d*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))*d*x*cos(d*x^2) - (a^2*b^4 - b^6)*d*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))*d*x*sin(d*x^2))*cos(2*d*x^2) -...
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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(d*x^2+c))^2,x, algorithm="fricas")
[Out]
integral(-1/(b^2*cos(d*x^2 + c)^2 - 2*a*b*sin(d*x^2 + c) - a^2 - b^2), x)
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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b \sin {\left (c + d x^{2} \right )}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*sin(d*x**2+c))**2,x)
[Out]
Integral((a + b*sin(c + d*x**2))**(-2), x)
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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(d*x^2+c))^2,x, algorithm="giac")
[Out]
integrate((b*sin(d*x^2 + c) + a)^(-2), x)
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Mupad [A]
time = 0.00, size = -1, normalized size = -0.06 \begin {gather*} \int \frac {1}{{\left (a+b\,\sin \left (d\,x^2+c\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*x^2))^2,x)
[Out]
int(1/(a + b*sin(c + d*x^2))^2, x)
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