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

Optimal. Leaf size=6 \[ \frac {B x}{b} \]

[Out]

B*x/b

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Rubi [A]
time = 0.00, antiderivative size = 6, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {21, 8} \begin {gather*} \frac {B x}{b} \end {gather*}

Antiderivative was successfully verified.

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

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Mathematica [A]
time = 0.00, size = 6, normalized size = 1.00 \begin {gather*} \frac {B x}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(B*x)/b

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Maple [A]
time = 0.05, size = 7, normalized size = 1.17

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*x/b

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]
time = 0.32, size = 6, normalized size = 1.00 \begin {gather*} \frac {B x}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*x/b

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Sympy [A]
time = 0.14, size = 3, normalized size = 0.50 \begin {gather*} \frac {B x}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*x/b

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Giac [A]
time = 0.44, size = 6, normalized size = 1.00 \begin {gather*} \frac {B x}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*x/b

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Mupad [B]
time = 7.70, size = 6, normalized size = 1.00 \begin {gather*} \frac {B\,x}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(B*x)/b

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