3.1.22 \(\int \frac {1}{(a+b \tan (c+d x^2))^2} \, dx\) [22]
Optimal. Leaf size=17 \[ \text {Int}\left (\frac {1}{\left (a+b \tan \left (c+d x^2\right )\right )^2},x\right ) \]
[Out]
Unintegrable(1/(a+b*tan(d*x^2+c))^2,x)
________________________________________________________________________________________
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 \tan \left (c+d x^2\right )\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[(a + b*Tan[c + d*x^2])^(-2),x]
[Out]
Defer[Int][(a + b*Tan[c + d*x^2])^(-2), x]
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx &=\int \frac {1}{\left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 5.48, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Integrate[(a + b*Tan[c + d*x^2])^(-2),x]
[Out]
Integrate[(a + b*Tan[c + d*x^2])^(-2), x]
________________________________________________________________________________________
Maple [A]
time = 0.33, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a +b \tan \left (d \,x^{2}+c \right )\right )^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(a+b*tan(d*x^2+c))^2,x)
[Out]
int(1/(a+b*tan(d*x^2+c))^2,x)
________________________________________________________________________________________
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*tan(d*x^2+c))^2,x, algorithm="maxima")
[Out]
((a^6 + a^4*b^2)*d*x^2*cos(2*d*x^2 + 2*c)^2 + (a^6 + a^4*b^2)*d*x^2*sin(2*d*x^2 + 2*c)^2 + (a^6 + a^4*b^2 - a^
2*b^4 - b^6)*d*x^2 - (b^6*sin(2*c) + ((4*a^4*b^2 + 5*a^2*b^4 - b^6)*cos(2*c) - 2*(a^5*b - 2*a*b^5)*sin(2*c))*d
*x^2 + 2*(a^3*b^3 + a*b^5)*cos(2*c))*cos(2*d*x^2) - (((a^2*b^4 + b^6)*cos(2*c) - 2*(a^5*b + 2*a^3*b^3 + a*b^5)
*sin(2*c))*d*x^2*cos(2*d*x^2) - (2*(a^5*b + 2*a^3*b^3 + a*b^5)*cos(2*c) + (a^2*b^4 + b^6)*sin(2*c))*d*x^2*sin(
2*d*x^2) - (2*a^6 + 2*a^4*b^2 + 3*a^2*b^4 + b^6)*d*x^2)*cos(2*d*x^2 + 2*c) - (a^8*d*x*cos(2*d*x^2 + 2*c)^2 + a
^8*d*x*sin(2*d*x^2 + 2*c)^2 + ((4*a^6*b^2 + 8*a^4*b^4 + 4*a^2*b^6 + b^8)*cos(2*c)^2 + (4*a^6*b^2 + 8*a^4*b^4 +
4*a^2*b^6 + b^8)*sin(2*c)^2)*d*x*cos(2*d*x^2)^2 + ((4*a^6*b^2 + 8*a^4*b^4 + 4*a^2*b^6 + b^8)*cos(2*c)^2 + (4*
a^6*b^2 + 8*a^4*b^4 + 4*a^2*b^6 + b^8)*sin(2*c)^2)*d*x*sin(2*d*x^2)^2 - 2*((a^4*b^4 + 2*a^2*b^6 + b^8)*cos(2*c
) - 2*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*sin(2*c))*d*x*cos(2*d*x^2) + 2*(2*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5
+ a*b^7)*cos(2*c) + (a^4*b^4 + 2*a^2*b^6 + b^8)*sin(2*c))*d*x*sin(2*d*x^2) + (a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4
*a^2*b^6 + b^8)*d*x - 2*((a^4*b^4*cos(2*c) - 2*(a^7*b + a^5*b^3)*sin(2*c))*d*x*cos(2*d*x^2) - (a^4*b^4*sin(2*c
) + 2*(a^7*b + a^5*b^3)*cos(2*c))*d*x*sin(2*d*x^2) - (a^8 + 2*a^6*b^2 + a^4*b^4)*d*x)*cos(2*d*x^2 + 2*c) - 2*(
(a^4*b^4*sin(2*c) + 2*(a^7*b + a^5*b^3)*cos(2*c))*d*x*cos(2*d*x^2) + (a^4*b^4*cos(2*c) - 2*(a^7*b + a^5*b^3)*s
in(2*c))*d*x*sin(2*d*x^2))*sin(2*d*x^2 + 2*c))*integrate(((b^6*sin(2*c) - 4*(a*b^5*sin(2*c) + 2*(a^4*b^2 + a^2
*b^4)*cos(2*c))*d*x^2 + 2*(a^3*b^3 + a*b^5)*cos(2*c))*cos(2*d*x^2) + (b^6*cos(2*c) - 4*(a*b^5*cos(2*c) - 2*(a^
4*b^2 + a^2*b^4)*sin(2*c))*d*x^2 - 2*(a^3*b^3 + a*b^5)*sin(2*c))*sin(2*d*x^2) + (4*a^5*b*d*x^2 - a^4*b^2)*sin(
2*d*x^2 + 2*c))/(a^8*d*x^2*cos(2*d*x^2 + 2*c)^2 + a^8*d*x^2*sin(2*d*x^2 + 2*c)^2 + ((4*a^6*b^2 + 8*a^4*b^4 + 4
*a^2*b^6 + b^8)*cos(2*c)^2 + (4*a^6*b^2 + 8*a^4*b^4 + 4*a^2*b^6 + b^8)*sin(2*c)^2)*d*x^2*cos(2*d*x^2)^2 + ((4*
a^6*b^2 + 8*a^4*b^4 + 4*a^2*b^6 + b^8)*cos(2*c)^2 + (4*a^6*b^2 + 8*a^4*b^4 + 4*a^2*b^6 + b^8)*sin(2*c)^2)*d*x^
2*sin(2*d*x^2)^2 - 2*((a^4*b^4 + 2*a^2*b^6 + b^8)*cos(2*c) - 2*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*sin(2*c
))*d*x^2*cos(2*d*x^2) + 2*(2*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*cos(2*c) + (a^4*b^4 + 2*a^2*b^6 + b^8)*si
n(2*c))*d*x^2*sin(2*d*x^2) + (a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d*x^2 - 2*((a^4*b^4*cos(2*c) - 2*
(a^7*b + a^5*b^3)*sin(2*c))*d*x^2*cos(2*d*x^2) - (a^4*b^4*sin(2*c) + 2*(a^7*b + a^5*b^3)*cos(2*c))*d*x^2*sin(2
*d*x^2) - (a^8 + 2*a^6*b^2 + a^4*b^4)*d*x^2)*cos(2*d*x^2 + 2*c) - 2*((a^4*b^4*sin(2*c) + 2*(a^7*b + a^5*b^3)*c
os(2*c))*d*x^2*cos(2*d*x^2) + (a^4*b^4*cos(2*c) - 2*(a^7*b + a^5*b^3)*sin(2*c))*d*x^2*sin(2*d*x^2))*sin(2*d*x^
2 + 2*c)), x) - (b^6*cos(2*c) - (2*(a^5*b - 2*a*b^5)*cos(2*c) + (4*a^4*b^2 + 5*a^2*b^4 - b^6)*sin(2*c))*d*x^2
- 2*(a^3*b^3 + a*b^5)*sin(2*c))*sin(2*d*x^2) + (2*a^5*b*d*x^2 + a^4*b^2 - (2*(a^5*b + 2*a^3*b^3 + a*b^5)*cos(2
*c) + (a^2*b^4 + b^6)*sin(2*c))*d*x^2*cos(2*d*x^2) - ((a^2*b^4 + b^6)*cos(2*c) - 2*(a^5*b + 2*a^3*b^3 + a*b^5)
*sin(2*c))*d*x^2*sin(2*d*x^2))*sin(2*d*x^2 + 2*c))/(a^8*d*x*cos(2*d*x^2 + 2*c)^2 + a^8*d*x*sin(2*d*x^2 + 2*c)^
2 + ((4*a^6*b^2 + 8*a^4*b^4 + 4*a^2*b^6 + b^8)*cos(2*c)^2 + (4*a^6*b^2 + 8*a^4*b^4 + 4*a^2*b^6 + b^8)*sin(2*c)
^2)*d*x*cos(2*d*x^2)^2 + ((4*a^6*b^2 + 8*a^4*b^4 + 4*a^2*b^6 + b^8)*cos(2*c)^2 + (4*a^6*b^2 + 8*a^4*b^4 + 4*a^
2*b^6 + b^8)*sin(2*c)^2)*d*x*sin(2*d*x^2)^2 - 2*((a^4*b^4 + 2*a^2*b^6 + b^8)*cos(2*c) - 2*(a^7*b + 3*a^5*b^3 +
3*a^3*b^5 + a*b^7)*sin(2*c))*d*x*cos(2*d*x^2) + 2*(2*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*cos(2*c) + (a^4*
b^4 + 2*a^2*b^6 + b^8)*sin(2*c))*d*x*sin(2*d*x^2) + (a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d*x - 2*((
a^4*b^4*cos(2*c) - 2*(a^7*b + a^5*b^3)*sin(2*c))*d*x*cos(2*d*x^2) - (a^4*b^4*sin(2*c) + 2*(a^7*b + a^5*b^3)*co
s(2*c))*d*x*sin(2*d*x^2) - (a^8 + 2*a^6*b^2 + a^4*b^4)*d*x)*cos(2*d*x^2 + 2*c) - 2*((a^4*b^4*sin(2*c) + 2*(a^7
*b + a^5*b^3)*cos(2*c))*d*x*cos(2*d*x^2) + (a^4*b^4*cos(2*c) - 2*(a^7*b + a^5*b^3)*sin(2*c))*d*x*sin(2*d*x^2))
*sin(2*d*x^2 + 2*c))
________________________________________________________________________________________
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*tan(d*x^2+c))^2,x, algorithm="fricas")
[Out]
integral(1/(b^2*tan(d*x^2 + c)^2 + 2*a*b*tan(d*x^2 + c) + a^2), x)
________________________________________________________________________________________
Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b \tan {\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*tan(d*x**2+c))**2,x)
[Out]
Integral((a + b*tan(c + d*x**2))**(-2), x)
________________________________________________________________________________________
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*tan(d*x^2+c))^2,x, algorithm="giac")
[Out]
integrate((b*tan(d*x^2 + c) + a)^(-2), x)
________________________________________________________________________________________
Mupad [A]
time = 0.00, size = -1, normalized size = -0.06 \begin {gather*} \int \frac {1}{{\left (a+b\,\mathrm {tan}\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*tan(c + d*x^2))^2,x)
[Out]
int(1/(a + b*tan(c + d*x^2))^2, x)
________________________________________________________________________________________