\(\int \frac {\tan ^2(c+d x) (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx\) [299]

Optimal result

Integrand size = 34, antiderivative size = 16 \[ \int \frac {\tan ^2(c+d x) (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=-B x+\frac {B \tan (c+d x)}{d} \]

[Out]

-B*x+B*tan(d*x+c)/d

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {21, 3554, 8} \[ \int \frac {\tan ^2(c+d x) (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=\frac {B \tan (c+d x)}{d}-B x \]

[In]

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

[Out]

-(B*x) + (B*Tan[c + d*x])/d

Rule 8

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

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*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}& = B \int \tan ^2(c+d x) \, dx \\ & = \frac {B \tan (c+d x)}{d}-B \int 1 \, dx \\ & = -B x+\frac {B \tan (c+d x)}{d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.56 \[ \int \frac {\tan ^2(c+d x) (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=B \left (-\frac {\arctan (\tan (c+d x))}{d}+\frac {\tan (c+d x)}{d}\right ) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06

method	result	size
norman	\(-B x +\frac {B \tan \left (d x +c \right )}{d}\)	\(17\)
parallelrisch	\(-\frac {B d x -B \tan \left (d x +c \right )}{d}\)	\(20\)
derivativedivides	\(\frac {B \left (\tan \left (d x +c \right )-\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\)	\(22\)
default	\(\frac {B \left (\tan \left (d x +c \right )-\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\)	\(22\)
risch	\(-B x +\frac {2 i B}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}\)	\(26\)

[In]

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

[Out]

-B*x+B*tan(d*x+c)/d

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.19 \[ \int \frac {\tan ^2(c+d x) (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=-\frac {B d x - B \tan \left (d x + c\right )}{d} \]

[In]

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

[Out]

-(B*d*x - B*tan(d*x + c))/d

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (12) = 24\).

Time = 0.40 (sec) , antiderivative size = 36, normalized size of antiderivative = 2.25 \[ \int \frac {\tan ^2(c+d x) (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=\begin {cases} - B x + \frac {B \tan {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\\frac {x \left (B a + B b \tan {\left (c \right )}\right ) \tan ^{2}{\left (c \right )}}{a + b \tan {\left (c \right )}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.38 \[ \int \frac {\tan ^2(c+d x) (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=-\frac {{\left (d x + c\right )} B - B \tan \left (d x + c\right )}{d} \]

[In]

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

[Out]

-((d*x + c)*B - B*tan(d*x + c))/d

Giac [A] (verification not implemented)

none

Time = 0.44 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.38 \[ \int \frac {\tan ^2(c+d x) (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=-\frac {{\left (d x + c\right )} B - B \tan \left (d x + c\right )}{d} \]

[In]

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

[Out]

-((d*x + c)*B - B*tan(d*x + c))/d

Mupad [B] (verification not implemented)

Time = 7.98 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\tan ^2(c+d x) (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=\frac {B\,\mathrm {tan}\left (c+d\,x\right )}{d}-B\,x \]

[In]

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

[Out]

(B*tan(c + d*x))/d - B*x