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

Optimal. Leaf size=3 \[ B x \]

[Out]

B*x

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 3, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {21, 8} \[ B x \]

Antiderivative was successfully verified.

[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*} \int \frac {a B+b B \tan (c+d x)}{a+b \tan (c+d x)} \, dx &=B \int 1 \, dx\\ &=B x\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 3, normalized size = 1.00 \[ B x \]

Antiderivative was successfully verified.

[In]

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

[Out]

B*x

________________________________________________________________________________________

fricas [A]  time = 0.54, size = 3, normalized size = 1.00 \[ B x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*x

________________________________________________________________________________________

giac [C]  time = 0.22, size = 10, normalized size = 3.33 \[ \frac {{\left (d x + c\right )} B}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d*x + c)*B/d

________________________________________________________________________________________

maple [A]  time = 0.02, size = 4, normalized size = 1.33 \[ B x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*x

________________________________________________________________________________________

maxima [C]  time = 1.16, size = 10, normalized size = 3.33 \[ \frac {{\left (d x + c\right )} B}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d*x + c)*B/d

________________________________________________________________________________________

mupad [B]  time = 6.26, size = 3, normalized size = 1.00 \[ B\,x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*x

________________________________________________________________________________________

sympy [A]  time = 0.14, size = 2, normalized size = 0.67 \[ B x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*x

________________________________________________________________________________________