\(\int \frac {a B+b B \tan (c+d x)}{a+b \tan (c+d x)} \, dx\) [301]

Optimal result

Integrand size = 26, antiderivative size = 3 \[ \int \frac {a B+b B \tan (c+d x)}{a+b \tan (c+d x)} \, dx=B x \]

[Out]

B*x

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 3, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {21, 8} \[ \int \frac {a B+b B \tan (c+d x)}{a+b \tan (c+d x)} \, dx=B x \]

[In]

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

[Out]

B*x

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])

Rubi steps \begin{align*} \text {integral}& = B \int 1 \, dx \\ & = B x \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 3, normalized size of antiderivative = 1.00 \[ \int \frac {a B+b B \tan (c+d x)}{a+b \tan (c+d x)} \, dx=B x \]

[In]

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

[Out]

B*x

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.33

method	result	size
default	\(B x\)	\(4\)
norman	\(B x\)	\(4\)
risch	\(B x\)	\(4\)
derivativedivides	\(\frac {B \arctan \left (\tan \left (d x +c \right )\right )}{d}\)	\(13\)

[In]

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

[Out]

B*x

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 3, normalized size of antiderivative = 1.00 \[ \int \frac {a B+b B \tan (c+d x)}{a+b \tan (c+d x)} \, dx=B x \]

[In]

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

[Out]

B*x

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 2, normalized size of antiderivative = 0.67 \[ \int \frac {a B+b B \tan (c+d x)}{a+b \tan (c+d x)} \, dx=B x \]

[In]

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

[Out]

B*x

Maxima [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.36 (sec) , antiderivative size = 10, normalized size of antiderivative = 3.33 \[ \int \frac {a B+b B \tan (c+d x)}{a+b \tan (c+d x)} \, dx=\frac {{\left (d x + c\right )} B}{d} \]

[In]

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

[Out]

(d*x + c)*B/d

Giac [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.32 (sec) , antiderivative size = 10, normalized size of antiderivative = 3.33 \[ \int \frac {a B+b B \tan (c+d x)}{a+b \tan (c+d x)} \, dx=\frac {{\left (d x + c\right )} B}{d} \]

[In]

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

[Out]

(d*x + c)*B/d

Mupad [B] (verification not implemented)

Time = 7.31 (sec) , antiderivative size = 3, normalized size of antiderivative = 1.00 \[ \int \frac {a B+b B \tan (c+d x)}{a+b \tan (c+d x)} \, dx=B\,x \]

[In]

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

[Out]

B*x