3.2 \(\int \tan ^2(c+d x) \, dx\)

Optimal. Leaf size=14 \[ \frac{\tan (c+d x)}{d}-x \]

[Out]

-x + Tan[c + d*x]/d

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Rubi [A]  time = 0.0076875, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3473, 8} \[ \frac{\tan (c+d x)}{d}-x \]

Antiderivative was successfully verified.

[In]

Int[Tan[c + d*x]^2,x]

[Out]

-x + Tan[c + d*x]/d

Rule 3473

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 8

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

Rubi steps

\begin{align*} \int \tan ^2(c+d x) \, dx &=\frac{\tan (c+d x)}{d}-\int 1 \, dx\\ &=-x+\frac{\tan (c+d x)}{d}\\ \end{align*}

Mathematica [A]  time = 0.0063154, size = 23, normalized size = 1.64 \[ \frac{\tan (c+d x)}{d}-\frac{\tan ^{-1}(\tan (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[c + d*x]^2,x]

[Out]

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

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Maple [A]  time = 0.003, size = 24, normalized size = 1.7 \begin{align*}{\frac{\tan \left ( dx+c \right ) }{d}}-{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

tan(d*x+c)/d-1/d*arctan(tan(d*x+c))

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Maxima [A]  time = 1.38102, size = 24, normalized size = 1.71 \begin{align*} -\frac{d x + c - \tan \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(d*x + c - tan(d*x + c))/d

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Fricas [A]  time = 1.89881, size = 34, normalized size = 2.43 \begin{align*} -\frac{d x - \tan \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(d*x - tan(d*x + c))/d

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Sympy [A]  time = 0.169093, size = 15, normalized size = 1.07 \begin{align*} \begin{cases} - x + \frac{\tan{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \tan ^{2}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.59345, size = 305, normalized size = 21.79 \begin{align*} \frac{\pi - 4 \, d x \tan \left (d x\right ) \tan \left (c\right ) - \pi \mathrm{sgn}\left (2 \, \tan \left (d x\right )^{2} \tan \left (c\right ) + 2 \, \tan \left (d x\right ) \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right ) - 2 \, \tan \left (c\right )\right ) \tan \left (d x\right ) \tan \left (c\right ) - \pi \tan \left (d x\right ) \tan \left (c\right ) + 2 \, \arctan \left (\frac{\tan \left (d x\right ) \tan \left (c\right ) - 1}{\tan \left (d x\right ) + \tan \left (c\right )}\right ) \tan \left (d x\right ) \tan \left (c\right ) + 2 \, \arctan \left (\frac{\tan \left (d x\right ) + \tan \left (c\right )}{\tan \left (d x\right ) \tan \left (c\right ) - 1}\right ) \tan \left (d x\right ) \tan \left (c\right ) + 4 \, d x + \pi \mathrm{sgn}\left (2 \, \tan \left (d x\right )^{2} \tan \left (c\right ) + 2 \, \tan \left (d x\right ) \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right ) - 2 \, \tan \left (c\right )\right ) - 2 \, \arctan \left (\frac{\tan \left (d x\right ) \tan \left (c\right ) - 1}{\tan \left (d x\right ) + \tan \left (c\right )}\right ) - 2 \, \arctan \left (\frac{\tan \left (d x\right ) + \tan \left (c\right )}{\tan \left (d x\right ) \tan \left (c\right ) - 1}\right ) - 4 \, \tan \left (d x\right ) - 4 \, \tan \left (c\right )}{4 \,{\left (d \tan \left (d x\right ) \tan \left (c\right ) - d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(pi - 4*d*x*tan(d*x)*tan(c) - pi*sgn(2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 - 2*tan(d*x) - 2*tan(c))*ta
n(d*x)*tan(c) - pi*tan(d*x)*tan(c) + 2*arctan((tan(d*x)*tan(c) - 1)/(tan(d*x) + tan(c)))*tan(d*x)*tan(c) + 2*a
rctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(d*x)*tan(c) + 4*d*x + pi*sgn(2*tan(d*x)^2*tan(c) + 2*tan(
d*x)*tan(c)^2 - 2*tan(d*x) - 2*tan(c)) - 2*arctan((tan(d*x)*tan(c) - 1)/(tan(d*x) + tan(c))) - 2*arctan((tan(d
*x) + tan(c))/(tan(d*x)*tan(c) - 1)) - 4*tan(d*x) - 4*tan(c))/(d*tan(d*x)*tan(c) - d)