\(\int \frac {\frac {3 B}{b}+B \sin (x)}{3+b \sin (x)} \, dx\) [695]

Optimal result

Integrand size = 20, antiderivative size = 6 \[ \int \frac {\frac {3 B}{b}+B \sin (x)}{3+b \sin (x)} \, dx=\frac {B x}{b} \]

[Out]

B*x/b

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {21, 8} \[ \int \frac {\frac {3 B}{b}+B \sin (x)}{3+b \sin (x)} \, dx=\frac {B x}{b} \]

[In]

Int[((a*B)/b + B*Sin[x])/(a + b*Sin[x]),x]

[Out]

(B*x)/b

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}& = \frac {B \int 1 \, dx}{b} \\ & = \frac {B x}{b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00 \[ \int \frac {\frac {3 B}{b}+B \sin (x)}{3+b \sin (x)} \, dx=\frac {B x}{b} \]

[In]

Integrate[((3*B)/b + B*Sin[x])/(3 + b*Sin[x]),x]

[Out]

(B*x)/b

Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.17

method	result	size
default	\(\frac {B x}{b}\)	\(7\)
risch	\(\frac {B x}{b}\)	\(7\)
norman	\(\frac {\frac {B x}{b}+\frac {B x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{b}}{1+\tan ^{2}\left (\frac {x}{2}\right )}\)	\(31\)

[In]

int((a*B/b+B*sin(x))/(a+b*sin(x)),x,method=_RETURNVERBOSE)

[Out]

B*x/b

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00 \[ \int \frac {\frac {3 B}{b}+B \sin (x)}{3+b \sin (x)} \, dx=\frac {B x}{b} \]

[In]

integrate((a*B/b+B*sin(x))/(a+b*sin(x)),x, algorithm="fricas")

[Out]

B*x/b

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.50 \[ \int \frac {\frac {3 B}{b}+B \sin (x)}{3+b \sin (x)} \, dx=\frac {B x}{b} \]

[In]

integrate((a*B/b+B*sin(x))/(a+b*sin(x)),x)

[Out]

B*x/b

Maxima [F(-2)]

Exception generated. \[ \int \frac {\frac {3 B}{b}+B \sin (x)}{3+b \sin (x)} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((a*B/b+B*sin(x))/(a+b*sin(x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?`
for more de

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00 \[ \int \frac {\frac {3 B}{b}+B \sin (x)}{3+b \sin (x)} \, dx=\frac {B x}{b} \]

[In]

integrate((a*B/b+B*sin(x))/(a+b*sin(x)),x, algorithm="giac")

[Out]

B*x/b

Mupad [B] (verification not implemented)

Time = 7.45 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00 \[ \int \frac {\frac {3 B}{b}+B \sin (x)}{3+b \sin (x)} \, dx=\frac {B\,x}{b} \]

[In]

int((B*sin(x) + (B*a)/b)/(a + b*sin(x)),x)

[Out]

(B*x)/b