3.473 \(\int \frac {1}{(a+b \tan (c+d x))^2} \, dx\)

Optimal. Leaf size=82 \[ -\frac {b}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac {2 a b \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^2}+\frac {x \left (a^2-b^2\right )}{\left (a^2+b^2\right )^2} \]

[Out]

(a^2-b^2)*x/(a^2+b^2)^2+2*a*b*ln(a*cos(d*x+c)+b*sin(d*x+c))/(a^2+b^2)^2/d-b/(a^2+b^2)/d/(a+b*tan(d*x+c))

________________________________________________________________________________________

Rubi [A]  time = 0.08, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3483, 3531, 3530} \[ -\frac {b}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac {2 a b \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^2}+\frac {x \left (a^2-b^2\right )}{\left (a^2+b^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Tan[c + d*x])^(-2),x]

[Out]

((a^2 - b^2)*x)/(a^2 + b^2)^2 + (2*a*b*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^2*d) - b/((a^2 + b^2
)*d*(a + b*Tan[c + d*x]))

Rule 3483

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(a + b*Tan[c + d*x])^(n + 1))/(d*(n + 1)
*(a^2 + b^2)), x] + Dist[1/(a^2 + b^2), Int[(a - b*Tan[c + d*x])*(a + b*Tan[c + d*x])^(n + 1), x], x] /; FreeQ
[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1]

Rule 3530

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c*Log[Re
moveContent[a*Cos[e + f*x] + b*Sin[e + f*x], x]])/(b*f), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d,
0] && NeQ[a^2 + b^2, 0] && EqQ[a*c + b*d, 0]

Rule 3531

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[((a*c +
b*d)*x)/(a^2 + b^2), x] + Dist[(b*c - a*d)/(a^2 + b^2), Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x], x]
/; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[a*c + b*d, 0]

Rubi steps

\begin {align*} \int \frac {1}{(a+b \tan (c+d x))^2} \, dx &=-\frac {b}{\left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {\int \frac {a-b \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{a^2+b^2}\\ &=\frac {\left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^2}-\frac {b}{\left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {(2 a b) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{\left (a^2+b^2\right )^2}\\ &=\frac {\left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^2}+\frac {2 a b \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac {b}{\left (a^2+b^2\right ) d (a+b \tan (c+d x))}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 1.32, size = 106, normalized size = 1.29 \[ \frac {\frac {2 b \left (2 a \log (a+b \tan (c+d x))-\frac {a^2+b^2}{a+b \tan (c+d x)}\right )}{\left (a^2+b^2\right )^2}-\frac {i \log (-\tan (c+d x)+i)}{(a+i b)^2}+\frac {i \log (\tan (c+d x)+i)}{(a-i b)^2}}{2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Tan[c + d*x])^(-2),x]

[Out]

(((-I)*Log[I - Tan[c + d*x]])/(a + I*b)^2 + (I*Log[I + Tan[c + d*x]])/(a - I*b)^2 + (2*b*(2*a*Log[a + b*Tan[c
+ d*x]] - (a^2 + b^2)/(a + b*Tan[c + d*x])))/(a^2 + b^2)^2)/(2*d)

________________________________________________________________________________________

fricas [A]  time = 0.44, size = 154, normalized size = 1.88 \[ -\frac {b^{3} - {\left (a^{3} - a b^{2}\right )} d x - {\left (a b^{2} \tan \left (d x + c\right ) + a^{2} b\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - {\left (a b^{2} + {\left (a^{2} b - b^{3}\right )} d x\right )} \tan \left (d x + c\right )}{{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} d \tan \left (d x + c\right ) + {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*tan(d*x+c))^2,x, algorithm="fricas")

[Out]

-(b^3 - (a^3 - a*b^2)*d*x - (a*b^2*tan(d*x + c) + a^2*b)*log((b^2*tan(d*x + c)^2 + 2*a*b*tan(d*x + c) + a^2)/(
tan(d*x + c)^2 + 1)) - (a*b^2 + (a^2*b - b^3)*d*x)*tan(d*x + c))/((a^4*b + 2*a^2*b^3 + b^5)*d*tan(d*x + c) + (
a^5 + 2*a^3*b^2 + a*b^4)*d)

________________________________________________________________________________________

giac [A]  time = 0.49, size = 159, normalized size = 1.94 \[ \frac {\frac {2 \, a b^{2} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{4} b + 2 \, a^{2} b^{3} + b^{5}} - \frac {a b \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {{\left (a^{2} - b^{2}\right )} {\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {2 \, a b^{2} \tan \left (d x + c\right ) + 3 \, a^{2} b + b^{3}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*tan(d*x+c))^2,x, algorithm="giac")

[Out]

(2*a*b^2*log(abs(b*tan(d*x + c) + a))/(a^4*b + 2*a^2*b^3 + b^5) - a*b*log(tan(d*x + c)^2 + 1)/(a^4 + 2*a^2*b^2
+ b^4) + (a^2 - b^2)*(d*x + c)/(a^4 + 2*a^2*b^2 + b^4) - (2*a*b^2*tan(d*x + c) + 3*a^2*b + b^3)/((a^4 + 2*a^2
*b^2 + b^4)*(b*tan(d*x + c) + a)))/d

________________________________________________________________________________________

maple [A]  time = 0.16, size = 130, normalized size = 1.59 \[ -\frac {b}{\left (a^{2}+b^{2}\right ) d \left (a +b \tan \left (d x +c \right )\right )}+\frac {2 a b \ln \left (a +b \tan \left (d x +c \right )\right )}{d \left (a^{2}+b^{2}\right )^{2}}+\frac {\arctan \left (\tan \left (d x +c \right )\right ) a^{2}}{d \left (a^{2}+b^{2}\right )^{2}}-\frac {\arctan \left (\tan \left (d x +c \right )\right ) b^{2}}{d \left (a^{2}+b^{2}\right )^{2}}-\frac {a b \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d \left (a^{2}+b^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*tan(d*x+c))^2,x)

[Out]

-b/(a^2+b^2)/d/(a+b*tan(d*x+c))+2/d*a*b/(a^2+b^2)^2*ln(a+b*tan(d*x+c))+1/d/(a^2+b^2)^2*arctan(tan(d*x+c))*a^2-
1/d/(a^2+b^2)^2*arctan(tan(d*x+c))*b^2-1/d/(a^2+b^2)^2*a*b*ln(1+tan(d*x+c)^2)

________________________________________________________________________________________

maxima [A]  time = 0.72, size = 131, normalized size = 1.60 \[ \frac {\frac {2 \, a b \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {a b \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {{\left (a^{2} - b^{2}\right )} {\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {b}{a^{3} + a b^{2} + {\left (a^{2} b + b^{3}\right )} \tan \left (d x + c\right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*tan(d*x+c))^2,x, algorithm="maxima")

[Out]

(2*a*b*log(b*tan(d*x + c) + a)/(a^4 + 2*a^2*b^2 + b^4) - a*b*log(tan(d*x + c)^2 + 1)/(a^4 + 2*a^2*b^2 + b^4) +
(a^2 - b^2)*(d*x + c)/(a^4 + 2*a^2*b^2 + b^4) - b/(a^3 + a*b^2 + (a^2*b + b^3)*tan(d*x + c)))/d

________________________________________________________________________________________

mupad [B]  time = 4.08, size = 121, normalized size = 1.48 \[ \frac {2\,a\,b\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}{d\,{\left (a^2+b^2\right )}^2}-\frac {b}{d\,\left (a^2+b^2\right )\,\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}{2\,d\,\left (-a^2\,1{}\mathrm {i}+2\,a\,b+b^2\,1{}\mathrm {i}\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (-a^2+a\,b\,2{}\mathrm {i}+b^2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*tan(c + d*x))^2,x)

[Out]

(2*a*b*log(a + b*tan(c + d*x)))/(d*(a^2 + b^2)^2) - log(tan(c + d*x) - 1i)/(2*d*(2*a*b - a^2*1i + b^2*1i)) - b
/(d*(a^2 + b^2)*(a + b*tan(c + d*x))) - (log(tan(c + d*x) + 1i)*1i)/(2*d*(a*b*2i - a^2 + b^2))

________________________________________________________________________________________

sympy [A]  time = 1.58, size = 1260, normalized size = 15.37 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*tan(d*x+c))**2,x)

[Out]

Piecewise((zoo*x/tan(c)**2, Eq(a, 0) & Eq(b, 0) & Eq(d, 0)), (d*x*tan(c + d*x)**2/(-4*b**2*d*tan(c + d*x)**2 +
8*I*b**2*d*tan(c + d*x) + 4*b**2*d) - 2*I*d*x*tan(c + d*x)/(-4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*
x) + 4*b**2*d) - d*x/(-4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) + 4*b**2*d) + tan(c + d*x)/(-4*b**2*
d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) + 4*b**2*d) - 2*I/(-4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c +
d*x) + 4*b**2*d), Eq(a, -I*b)), (d*x*tan(c + d*x)**2/(-4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) + 4*
b**2*d) + 2*I*d*x*tan(c + d*x)/(-4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) + 4*b**2*d) - d*x/(-4*b**2
*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) + 4*b**2*d) + tan(c + d*x)/(-4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*
d*tan(c + d*x) + 4*b**2*d) + 2*I/(-4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) + 4*b**2*d), Eq(a, I*b))
, (x/(a + b*tan(c))**2, Eq(d, 0)), (x/a**2, Eq(b, 0)), (a**3*d*x/(a**5*d + a**4*b*d*tan(c + d*x) + 2*a**3*b**2
*d + 2*a**2*b**3*d*tan(c + d*x) + a*b**4*d + b**5*d*tan(c + d*x)) + a**2*b*d*x*tan(c + d*x)/(a**5*d + a**4*b*d
*tan(c + d*x) + 2*a**3*b**2*d + 2*a**2*b**3*d*tan(c + d*x) + a*b**4*d + b**5*d*tan(c + d*x)) + 2*a**2*b*log(a/
b + tan(c + d*x))/(a**5*d + a**4*b*d*tan(c + d*x) + 2*a**3*b**2*d + 2*a**2*b**3*d*tan(c + d*x) + a*b**4*d + b*
*5*d*tan(c + d*x)) - a**2*b*log(tan(c + d*x)**2 + 1)/(a**5*d + a**4*b*d*tan(c + d*x) + 2*a**3*b**2*d + 2*a**2*
b**3*d*tan(c + d*x) + a*b**4*d + b**5*d*tan(c + d*x)) - a**2*b/(a**5*d + a**4*b*d*tan(c + d*x) + 2*a**3*b**2*d
+ 2*a**2*b**3*d*tan(c + d*x) + a*b**4*d + b**5*d*tan(c + d*x)) - a*b**2*d*x/(a**5*d + a**4*b*d*tan(c + d*x) +
2*a**3*b**2*d + 2*a**2*b**3*d*tan(c + d*x) + a*b**4*d + b**5*d*tan(c + d*x)) + 2*a*b**2*log(a/b + tan(c + d*x
))*tan(c + d*x)/(a**5*d + a**4*b*d*tan(c + d*x) + 2*a**3*b**2*d + 2*a**2*b**3*d*tan(c + d*x) + a*b**4*d + b**5
*d*tan(c + d*x)) - a*b**2*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(a**5*d + a**4*b*d*tan(c + d*x) + 2*a**3*b**2*
d + 2*a**2*b**3*d*tan(c + d*x) + a*b**4*d + b**5*d*tan(c + d*x)) - b**3*d*x*tan(c + d*x)/(a**5*d + a**4*b*d*ta
n(c + d*x) + 2*a**3*b**2*d + 2*a**2*b**3*d*tan(c + d*x) + a*b**4*d + b**5*d*tan(c + d*x)) - b**3/(a**5*d + a**
4*b*d*tan(c + d*x) + 2*a**3*b**2*d + 2*a**2*b**3*d*tan(c + d*x) + a*b**4*d + b**5*d*tan(c + d*x)), True))

________________________________________________________________________________________