\(\int \frac {\tan ^2(c+d x)}{(a+a \sec (c+d x))^2} \, dx\) [82]

Optimal result

Integrand size = 21, antiderivative size = 33 \[ \int \frac {\tan ^2(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {x}{a^2}+\frac {2 \tan (c+d x)}{a d (a+a \sec (c+d x))} \]

[Out]

-x/a^2+2*tan(d*x+c)/a/d/(a+a*sec(d*x+c))

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3973, 3971, 3554, 8, 2686, 3852} \[ \int \frac {\tan ^2(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {2 \cot (c+d x)}{a^2 d}+\frac {2 \csc (c+d x)}{a^2 d}-\frac {x}{a^2} \]

[In]

Int[Tan[c + d*x]^2/(a + a*Sec[c + d*x])^2,x]

[Out]

-(x/a^2) - (2*Cot[c + d*x])/(a^2*d) + (2*Csc[c + d*x])/(a^2*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2686

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 3554

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d*x])^(n - 1)/(d*(n - 1))), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 3971

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Int[ExpandI
ntegrand[(e*Cot[c + d*x])^m, (a + b*Csc[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0]

Rule 3973

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Dist[a^(2*n
)/e^(2*n), Int[(e*Cot[c + d*x])^(m + 2*n)/(-a + b*Csc[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && E
qQ[a^2 - b^2, 0] && ILtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \cot ^2(c+d x) (-a+a \sec (c+d x))^2 \, dx}{a^4} \\ & = \frac {\int \left (a^2 \cot ^2(c+d x)-2 a^2 \cot (c+d x) \csc (c+d x)+a^2 \csc ^2(c+d x)\right ) \, dx}{a^4} \\ & = \frac {\int \cot ^2(c+d x) \, dx}{a^2}+\frac {\int \csc ^2(c+d x) \, dx}{a^2}-\frac {2 \int \cot (c+d x) \csc (c+d x) \, dx}{a^2} \\ & = -\frac {\cot (c+d x)}{a^2 d}-\frac {\int 1 \, dx}{a^2}-\frac {\text {Subst}(\int 1 \, dx,x,\cot (c+d x))}{a^2 d}+\frac {2 \text {Subst}(\int 1 \, dx,x,\csc (c+d x))}{a^2 d} \\ & = -\frac {x}{a^2}-\frac {2 \cot (c+d x)}{a^2 d}+\frac {2 \csc (c+d x)}{a^2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.27 \[ \int \frac {\tan ^2(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {-\frac {2 \arctan \left (\tan \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d}+\frac {2 \tan \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}}{a^2} \]

[In]

Integrate[Tan[c + d*x]^2/(a + a*Sec[c + d*x])^2,x]

[Out]

((-2*ArcTan[Tan[c/2 + (d*x)/2]])/d + (2*Tan[c/2 + (d*x)/2])/d)/a^2

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.62 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.91

method	result	size
risch	\(-\frac {x}{a^{2}}+\frac {4 i}{a^{2} d \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}\)	\(30\)
derivativedivides	\(\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2} d}\)	\(31\)
default	\(\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2} d}\)	\(31\)

[In]

int(tan(d*x+c)^2/(a+a*sec(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

-x/a^2+4*I/a^2/d/(exp(I*(d*x+c))+1)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.27 \[ \int \frac {\tan ^2(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {d x \cos \left (d x + c\right ) + d x - 2 \, \sin \left (d x + c\right )}{a^{2} d \cos \left (d x + c\right ) + a^{2} d} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {\tan ^2(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {\int \frac {\tan ^{2}{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]

[In]

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

[Out]

Integral(tan(c + d*x)**2/(sec(c + d*x)**2 + 2*sec(c + d*x) + 1), x)/a**2

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.48 \[ \int \frac {\tan ^2(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {2 \, {\left (\frac {\arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} - \frac {\sin \left (d x + c\right )}{a^{2} {\left (\cos \left (d x + c\right ) + 1\right )}}\right )}}{d} \]

[In]

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

[Out]

-2*(arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^2 - sin(d*x + c)/(a^2*(cos(d*x + c) + 1)))/d

Giac [A] (verification not implemented)

none

Time = 0.51 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.88 \[ \int \frac {\tan ^2(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {\frac {d x + c}{a^{2}} - \frac {2 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{2}}}{d} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 13.99 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.67 \[ \int \frac {\tan ^2(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {2\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\frac {d\,x}{2}\right )}{a^2\,d} \]

[In]

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

[Out]

(2*(tan(c/2 + (d*x)/2) - (d*x)/2))/(a^2*d)