\(\int (a+a \tan ^2(c+d x))^2 \, dx\) [181]

Optimal result

Integrand size = 14, antiderivative size = 32 \[ \int \left (a+a \tan ^2(c+d x)\right )^2 \, dx=\frac {a^2 \tan (c+d x)}{d}+\frac {a^2 \tan ^3(c+d x)}{3 d} \]

[Out]

a^2*tan(d*x+c)/d+1/3*a^2*tan(d*x+c)^3/d

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3738, 12, 3852} \[ \int \left (a+a \tan ^2(c+d x)\right )^2 \, dx=\frac {a^2 \tan ^3(c+d x)}{3 d}+\frac {a^2 \tan (c+d x)}{d} \]

[In]

Int[(a + a*Tan[c + d*x]^2)^2,x]

[Out]

(a^2*Tan[c + d*x])/d + (a^2*Tan[c + d*x]^3)/(3*d)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3738

Int[(u_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[ActivateTrig[u*(a*sec[e + f*x]^2)^p]
, x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a, b]

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]

Rubi steps \begin{align*} \text {integral}& = \int a^2 \sec ^4(c+d x) \, dx \\ & = a^2 \int \sec ^4(c+d x) \, dx \\ & = -\frac {a^2 \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{d} \\ & = \frac {a^2 \tan (c+d x)}{d}+\frac {a^2 \tan ^3(c+d x)}{3 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \left (a+a \tan ^2(c+d x)\right )^2 \, dx=\frac {a^2 \left (\tan (c+d x)+\frac {1}{3} \tan ^3(c+d x)\right )}{d} \]

[In]

Integrate[(a + a*Tan[c + d*x]^2)^2,x]

[Out]

(a^2*(Tan[c + d*x] + Tan[c + d*x]^3/3))/d

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.78

method	result	size
derivativedivides	\(\frac {a^{2} \left (\frac {\tan \left (d x +c \right )^{3}}{3}+\tan \left (d x +c \right )\right )}{d}\)	\(25\)
default	\(\frac {a^{2} \left (\frac {\tan \left (d x +c \right )^{3}}{3}+\tan \left (d x +c \right )\right )}{d}\)	\(25\)
parallelrisch	\(\frac {a^{2} \tan \left (d x +c \right )^{3}+3 a^{2} \tan \left (d x +c \right )}{3 d}\)	\(30\)
norman	\(\frac {a^{2} \tan \left (d x +c \right )}{d}+\frac {a^{2} \tan \left (d x +c \right )^{3}}{3 d}\)	\(31\)
risch	\(\frac {4 i a^{2} \left (3 \,{\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}\)	\(36\)
parts	\(x \,a^{2}+\frac {a^{2} \left (\frac {\tan \left (d x +c \right )^{3}}{3}-\tan \left (d x +c \right )+\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}+\frac {2 a^{2} \left (\tan \left (d x +c \right )-\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\)	\(64\)

[In]

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

[Out]

1/d*a^2*(1/3*tan(d*x+c)^3+tan(d*x+c))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.91 \[ \int \left (a+a \tan ^2(c+d x)\right )^2 \, dx=\frac {a^{2} \tan \left (d x + c\right )^{3} + 3 \, a^{2} \tan \left (d x + c\right )}{3 \, d} \]

[In]

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

[Out]

1/3*(a^2*tan(d*x + c)^3 + 3*a^2*tan(d*x + c))/d

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16 \[ \int \left (a+a \tan ^2(c+d x)\right )^2 \, dx=\begin {cases} \frac {a^{2} \tan ^{3}{\left (c + d x \right )}}{3 d} + \frac {a^{2} \tan {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a \tan ^{2}{\left (c \right )} + a\right )^{2} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((a**2*tan(c + d*x)**3/(3*d) + a**2*tan(c + d*x)/d, Ne(d, 0)), (x*(a*tan(c)**2 + a)**2, True))

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.84 \[ \int \left (a+a \tan ^2(c+d x)\right )^2 \, dx=a^{2} x + \frac {{\left (\tan \left (d x + c\right )^{3} + 3 \, d x + 3 \, c - 3 \, \tan \left (d x + c\right )\right )} a^{2}}{3 \, d} - \frac {2 \, {\left (d x + c - \tan \left (d x + c\right )\right )} a^{2}}{d} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 133 vs. \(2 (30) = 60\).

Time = 0.44 (sec) , antiderivative size = 133, normalized size of antiderivative = 4.16 \[ \int \left (a+a \tan ^2(c+d x)\right )^2 \, dx=-\frac {3 \, a^{2} \tan \left (d x\right )^{3} \tan \left (c\right )^{2} + 3 \, a^{2} \tan \left (d x\right )^{2} \tan \left (c\right )^{3} + a^{2} \tan \left (d x\right )^{3} - 3 \, a^{2} \tan \left (d x\right )^{2} \tan \left (c\right ) - 3 \, a^{2} \tan \left (d x\right ) \tan \left (c\right )^{2} + a^{2} \tan \left (c\right )^{3} + 3 \, a^{2} \tan \left (d x\right ) + 3 \, a^{2} \tan \left (c\right )}{3 \, {\left (d \tan \left (d x\right )^{3} \tan \left (c\right )^{3} - 3 \, d \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + 3 \, d \tan \left (d x\right ) \tan \left (c\right ) - d\right )}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 11.39 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.75 \[ \int \left (a+a \tan ^2(c+d x)\right )^2 \, dx=\frac {a^2\,\mathrm {tan}\left (c+d\,x\right )\,\left ({\mathrm {tan}\left (c+d\,x\right )}^2+3\right )}{3\,d} \]

[In]

int((a + a*tan(c + d*x)^2)^2,x)

[Out]

(a^2*tan(c + d*x)*(tan(c + d*x)^2 + 3))/(3*d)