\(\int (a+b \tan ^2(e+f x)) \, dx\) [39]

Optimal result

Integrand size = 12, antiderivative size = 19 \[ \int \left (a+b \tan ^2(e+f x)\right ) \, dx=a x-b x+\frac {b \tan (e+f x)}{f} \]

[Out]

a*x-b*x+b*tan(f*x+e)/f

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3554, 8} \[ \int \left (a+b \tan ^2(e+f x)\right ) \, dx=a x+\frac {b \tan (e+f x)}{f}-b x \]

[In]

Int[a + b*Tan[e + f*x]^2,x]

[Out]

a*x - b*x + (b*Tan[e + f*x])/f

Rule 8

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

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]

Rubi steps \begin{align*} \text {integral}& = a x+b \int \tan ^2(e+f x) \, dx \\ & = a x+\frac {b \tan (e+f x)}{f}-b \int 1 \, dx \\ & = a x-b x+\frac {b \tan (e+f x)}{f} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.47 \[ \int \left (a+b \tan ^2(e+f x)\right ) \, dx=a x-\frac {b \arctan (\tan (e+f x))}{f}+\frac {b \tan (e+f x)}{f} \]

[In]

Integrate[a + b*Tan[e + f*x]^2,x]

[Out]

a*x - (b*ArcTan[Tan[e + f*x]])/f + (b*Tan[e + f*x])/f

Maple [A] (verified)

Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05

method	result	size
norman	\(\left (a -b \right ) x +\frac {b \tan \left (f x +e \right )}{f}\)	\(20\)
parallelrisch	\(-\frac {b \left (f x -\tan \left (f x +e \right )\right )}{f}+a x\)	\(23\)
default	\(a x +\frac {b \left (\tan \left (f x +e \right )-\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}\)	\(26\)
parts	\(a x +\frac {b \left (\tan \left (f x +e \right )-\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}\)	\(26\)
derivativedivides	\(\frac {b \tan \left (f x +e \right )+\left (a -b \right ) \arctan \left (\tan \left (f x +e \right )\right )}{f}\)	\(27\)
risch	\(a x -b x +\frac {2 i b}{f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}\)	\(29\)

[In]

int(a+b*tan(f*x+e)^2,x,method=_RETURNVERBOSE)

[Out]

(a-b)*x+b*tan(f*x+e)/f

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \left (a+b \tan ^2(e+f x)\right ) \, dx=\frac {{\left (a - b\right )} f x + b \tan \left (f x + e\right )}{f} \]

[In]

integrate(a+b*tan(f*x+e)^2,x, algorithm="fricas")

[Out]

((a - b)*f*x + b*tan(f*x + e))/f

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \left (a+b \tan ^2(e+f x)\right ) \, dx=a x + b \left (\begin {cases} - x + \frac {\tan {\left (e + f x \right )}}{f} & \text {for}\: f \neq 0 \\x \tan ^{2}{\left (e \right )} & \text {otherwise} \end {cases}\right ) \]

[In]

integrate(a+b*tan(f*x+e)**2,x)

[Out]

a*x + b*Piecewise((-x + tan(e + f*x)/f, Ne(f, 0)), (x*tan(e)**2, True))

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21 \[ \int \left (a+b \tan ^2(e+f x)\right ) \, dx=a x - \frac {{\left (f x + e - \tan \left (f x + e\right )\right )} b}{f} \]

[In]

integrate(a+b*tan(f*x+e)^2,x, algorithm="maxima")

[Out]

a*x - (f*x + e - tan(f*x + e))*b/f

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 231 vs. \(2 (19) = 38\).

Time = 0.36 (sec) , antiderivative size = 231, normalized size of antiderivative = 12.16 \[ \int \left (a+b \tan ^2(e+f x)\right ) \, dx=a x + \frac {{\left (\pi - 4 \, f x \tan \left (f x\right ) \tan \left (e\right ) - \pi \mathrm {sgn}\left (2 \, \tan \left (f x\right )^{2} \tan \left (e\right ) + 2 \, \tan \left (f x\right ) \tan \left (e\right )^{2} - 2 \, \tan \left (f x\right ) - 2 \, \tan \left (e\right )\right ) \tan \left (f x\right ) \tan \left (e\right ) - \pi \tan \left (f x\right ) \tan \left (e\right ) + 2 \, \arctan \left (\frac {\tan \left (f x\right ) \tan \left (e\right ) - 1}{\tan \left (f x\right ) + \tan \left (e\right )}\right ) \tan \left (f x\right ) \tan \left (e\right ) + 2 \, \arctan \left (\frac {\tan \left (f x\right ) + \tan \left (e\right )}{\tan \left (f x\right ) \tan \left (e\right ) - 1}\right ) \tan \left (f x\right ) \tan \left (e\right ) + 4 \, f x + \pi \mathrm {sgn}\left (2 \, \tan \left (f x\right )^{2} \tan \left (e\right ) + 2 \, \tan \left (f x\right ) \tan \left (e\right )^{2} - 2 \, \tan \left (f x\right ) - 2 \, \tan \left (e\right )\right ) - 2 \, \arctan \left (\frac {\tan \left (f x\right ) \tan \left (e\right ) - 1}{\tan \left (f x\right ) + \tan \left (e\right )}\right ) - 2 \, \arctan \left (\frac {\tan \left (f x\right ) + \tan \left (e\right )}{\tan \left (f x\right ) \tan \left (e\right ) - 1}\right ) - 4 \, \tan \left (f x\right ) - 4 \, \tan \left (e\right )\right )} b}{4 \, {\left (f \tan \left (f x\right ) \tan \left (e\right ) - f\right )}} \]

[In]

integrate(a+b*tan(f*x+e)^2,x, algorithm="giac")

[Out]

a*x + 1/4*(pi - 4*f*x*tan(f*x)*tan(e) - pi*sgn(2*tan(f*x)^2*tan(e) + 2*tan(f*x)*tan(e)^2 - 2*tan(f*x) - 2*tan(
e))*tan(f*x)*tan(e) - pi*tan(f*x)*tan(e) + 2*arctan((tan(f*x)*tan(e) - 1)/(tan(f*x) + tan(e)))*tan(f*x)*tan(e)
+ 2*arctan((tan(f*x) + tan(e))/(tan(f*x)*tan(e) - 1))*tan(f*x)*tan(e) + 4*f*x + pi*sgn(2*tan(f*x)^2*tan(e) +
2*tan(f*x)*tan(e)^2 - 2*tan(f*x) - 2*tan(e)) - 2*arctan((tan(f*x)*tan(e) - 1)/(tan(f*x) + tan(e))) - 2*arctan(
(tan(f*x) + tan(e))/(tan(f*x)*tan(e) - 1)) - 4*tan(f*x) - 4*tan(e))*b/(f*tan(f*x)*tan(e) - f)

Mupad [B] (verification not implemented)

Time = 10.29 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \left (a+b \tan ^2(e+f x)\right ) \, dx=\frac {b\,\mathrm {tan}\left (e+f\,x\right )+f\,x\,\left (a-b\right )}{f} \]

[In]

int(a + b*tan(e + f*x)^2,x)

[Out]

(b*tan(e + f*x) + f*x*(a - b))/f