3.169 \(\int \frac{\text{b1} \cos (x)+\text{a1} \sin (x)}{b \cos (x)+a \sin (x)} \, dx\)

Optimal. Leaf size=48 \[ \frac{x (a \text{a1}+b \text{b1})}{a^2+b^2}-\frac{(\text{a1} b-a \text{b1}) \log (a \sin (x)+b \cos (x))}{a^2+b^2} \]

[Out]

((a*a1 + b*b1)*x)/(a^2 + b^2) - ((a1*b - a*b1)*Log[b*Cos[x] + a*Sin[x]])/(a^2 + b^2)

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Rubi [A]  time = 0.0343966, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {3133} \[ \frac{x (a \text{a1}+b \text{b1})}{a^2+b^2}-\frac{(\text{a1} b-a \text{b1}) \log (a \sin (x)+b \cos (x))}{a^2+b^2} \]

Antiderivative was successfully verified.

[In]

Int[(b1*Cos[x] + a1*Sin[x])/(b*Cos[x] + a*Sin[x]),x]

[Out]

((a*a1 + b*b1)*x)/(a^2 + b^2) - ((a1*b - a*b1)*Log[b*Cos[x] + a*Sin[x]])/(a^2 + b^2)

Rule 3133

Int[((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_)])/((a_.) + cos[(d_.) + (e_.)*(x_)]*(
b_.) + (c_.)*sin[(d_.) + (e_.)*(x_)]), x_Symbol] :> Simp[((b*B + c*C)*x)/(b^2 + c^2), x] + Simp[((c*B - b*C)*L
og[a + b*Cos[d + e*x] + c*Sin[d + e*x]])/(e*(b^2 + c^2)), x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && NeQ[b^2
+ c^2, 0] && EqQ[A*(b^2 + c^2) - a*(b*B + c*C), 0]

Rubi steps

\begin{align*} \int \frac{\text{b1} \cos (x)+\text{a1} \sin (x)}{b \cos (x)+a \sin (x)} \, dx &=\frac{(a \text{a1}+b \text{b1}) x}{a^2+b^2}-\frac{(\text{a1} b-a \text{b1}) \log (b \cos (x)+a \sin (x))}{a^2+b^2}\\ \end{align*}

Mathematica [A]  time = 0.0948063, size = 39, normalized size = 0.81 \[ \frac{x (a \text{a1}+b \text{b1})+(a \text{b1}-\text{a1} b) \log (a \sin (x)+b \cos (x))}{a^2+b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(b1*Cos[x] + a1*Sin[x])/(b*Cos[x] + a*Sin[x]),x]

[Out]

((a*a1 + b*b1)*x + (-(a1*b) + a*b1)*Log[b*Cos[x] + a*Sin[x]])/(a^2 + b^2)

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Maple [B]  time = 0.079, size = 111, normalized size = 2.3 \begin{align*} -{\frac{\ln \left ( \left ( \tan \left ( x \right ) \right ) ^{2}+1 \right ) a{\it b1}}{2\,{a}^{2}+2\,{b}^{2}}}+{\frac{\ln \left ( \left ( \tan \left ( x \right ) \right ) ^{2}+1 \right ){\it a1}\,b}{2\,{a}^{2}+2\,{b}^{2}}}+{\frac{\arctan \left ( \tan \left ( x \right ) \right ) a{\it a1}}{{a}^{2}+{b}^{2}}}+{\frac{\arctan \left ( \tan \left ( x \right ) \right ) b{\it b1}}{{a}^{2}+{b}^{2}}}+{\frac{\ln \left ( a\tan \left ( x \right ) +b \right ) a{\it b1}}{{a}^{2}+{b}^{2}}}-{\frac{\ln \left ( a\tan \left ( x \right ) +b \right ){\it a1}\,b}{{a}^{2}+{b}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b1*cos(x)+a1*sin(x))/(b*cos(x)+a*sin(x)),x)

[Out]

-1/2/(a^2+b^2)*ln(tan(x)^2+1)*a*b1+1/2/(a^2+b^2)*ln(tan(x)^2+1)*a1*b+1/(a^2+b^2)*arctan(tan(x))*a*a1+1/(a^2+b^
2)*arctan(tan(x))*b*b1+1/(a^2+b^2)*ln(a*tan(x)+b)*a*b1-1/(a^2+b^2)*ln(a*tan(x)+b)*a1*b

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Maxima [B]  time = 1.44219, size = 244, normalized size = 5.08 \begin{align*} a_{1}{\left (\frac{2 \, a \arctan \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a^{2} + b^{2}} - \frac{b \log \left (-b - \frac{2 \, a \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{b \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}\right )}{a^{2} + b^{2}} + \frac{b \log \left (\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right )}{a^{2} + b^{2}}\right )} + b_{1}{\left (\frac{2 \, b \arctan \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a^{2} + b^{2}} + \frac{a \log \left (-b - \frac{2 \, a \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{b \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}\right )}{a^{2} + b^{2}} - \frac{a \log \left (\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right )}{a^{2} + b^{2}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b1*cos(x)+a1*sin(x))/(b*cos(x)+a*sin(x)),x, algorithm="maxima")

[Out]

a1*(2*a*arctan(sin(x)/(cos(x) + 1))/(a^2 + b^2) - b*log(-b - 2*a*sin(x)/(cos(x) + 1) + b*sin(x)^2/(cos(x) + 1)
^2)/(a^2 + b^2) + b*log(sin(x)^2/(cos(x) + 1)^2 + 1)/(a^2 + b^2)) + b1*(2*b*arctan(sin(x)/(cos(x) + 1))/(a^2 +
b^2) + a*log(-b - 2*a*sin(x)/(cos(x) + 1) + b*sin(x)^2/(cos(x) + 1)^2)/(a^2 + b^2) - a*log(sin(x)^2/(cos(x) +
1)^2 + 1)/(a^2 + b^2))

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Fricas [A]  time = 1.28448, size = 144, normalized size = 3. \begin{align*} \frac{2 \,{\left (a a_{1} + b b_{1}\right )} x -{\left (a_{1} b - a b_{1}\right )} \log \left (2 \, a b \cos \left (x\right ) \sin \left (x\right ) -{\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + a^{2}\right )}{2 \,{\left (a^{2} + b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b1*cos(x)+a1*sin(x))/(b*cos(x)+a*sin(x)),x, algorithm="fricas")

[Out]

1/2*(2*(a*a1 + b*b1)*x - (a1*b - a*b1)*log(2*a*b*cos(x)*sin(x) - (a^2 - b^2)*cos(x)^2 + a^2))/(a^2 + b^2)

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Sympy [A]  time = 2.49333, size = 360, normalized size = 7.5 \begin{align*} \begin{cases} \tilde{\infty } \left (- a_{1} \log{\left (\cos{\left (x \right )} \right )} + b_{1} x\right ) & \text{for}\: a = 0 \wedge b = 0 \\- \frac{a_{1} x \sin{\left (x \right )}}{2 i b \sin{\left (x \right )} - 2 b \cos{\left (x \right )}} - \frac{i a_{1} x \cos{\left (x \right )}}{2 i b \sin{\left (x \right )} - 2 b \cos{\left (x \right )}} + \frac{i a_{1} \sin{\left (x \right )}}{2 i b \sin{\left (x \right )} - 2 b \cos{\left (x \right )}} + \frac{i b_{1} x \sin{\left (x \right )}}{2 i b \sin{\left (x \right )} - 2 b \cos{\left (x \right )}} - \frac{b_{1} x \cos{\left (x \right )}}{2 i b \sin{\left (x \right )} - 2 b \cos{\left (x \right )}} - \frac{b_{1} \sin{\left (x \right )}}{2 i b \sin{\left (x \right )} - 2 b \cos{\left (x \right )}} & \text{for}\: a = - i b \\\frac{a_{1} x \sin{\left (x \right )}}{2 i b \sin{\left (x \right )} + 2 b \cos{\left (x \right )}} - \frac{i a_{1} x \cos{\left (x \right )}}{2 i b \sin{\left (x \right )} + 2 b \cos{\left (x \right )}} + \frac{i a_{1} \sin{\left (x \right )}}{2 i b \sin{\left (x \right )} + 2 b \cos{\left (x \right )}} + \frac{i b_{1} x \sin{\left (x \right )}}{2 i b \sin{\left (x \right )} + 2 b \cos{\left (x \right )}} + \frac{b_{1} x \cos{\left (x \right )}}{2 i b \sin{\left (x \right )} + 2 b \cos{\left (x \right )}} + \frac{b_{1} \sin{\left (x \right )}}{2 i b \sin{\left (x \right )} + 2 b \cos{\left (x \right )}} & \text{for}\: a = i b \\\frac{- a_{1} \log{\left (\cos{\left (x \right )} \right )} + b_{1} x}{b} & \text{for}\: a = 0 \\\frac{a a_{1} x}{a^{2} + b^{2}} + \frac{a b_{1} \log{\left (\sin{\left (x \right )} + \frac{b \cos{\left (x \right )}}{a} \right )}}{a^{2} + b^{2}} - \frac{a_{1} b \log{\left (\sin{\left (x \right )} + \frac{b \cos{\left (x \right )}}{a} \right )}}{a^{2} + b^{2}} + \frac{b b_{1} x}{a^{2} + b^{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b1*cos(x)+a1*sin(x))/(b*cos(x)+a*sin(x)),x)

[Out]

Piecewise((zoo*(-a1*log(cos(x)) + b1*x), Eq(a, 0) & Eq(b, 0)), (-a1*x*sin(x)/(2*I*b*sin(x) - 2*b*cos(x)) - I*a
1*x*cos(x)/(2*I*b*sin(x) - 2*b*cos(x)) + I*a1*sin(x)/(2*I*b*sin(x) - 2*b*cos(x)) + I*b1*x*sin(x)/(2*I*b*sin(x)
- 2*b*cos(x)) - b1*x*cos(x)/(2*I*b*sin(x) - 2*b*cos(x)) - b1*sin(x)/(2*I*b*sin(x) - 2*b*cos(x)), Eq(a, -I*b))
, (a1*x*sin(x)/(2*I*b*sin(x) + 2*b*cos(x)) - I*a1*x*cos(x)/(2*I*b*sin(x) + 2*b*cos(x)) + I*a1*sin(x)/(2*I*b*si
n(x) + 2*b*cos(x)) + I*b1*x*sin(x)/(2*I*b*sin(x) + 2*b*cos(x)) + b1*x*cos(x)/(2*I*b*sin(x) + 2*b*cos(x)) + b1*
sin(x)/(2*I*b*sin(x) + 2*b*cos(x)), Eq(a, I*b)), ((-a1*log(cos(x)) + b1*x)/b, Eq(a, 0)), (a*a1*x/(a**2 + b**2)
+ a*b1*log(sin(x) + b*cos(x)/a)/(a**2 + b**2) - a1*b*log(sin(x) + b*cos(x)/a)/(a**2 + b**2) + b*b1*x/(a**2 +
b**2), True))

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Giac [A]  time = 1.11823, size = 104, normalized size = 2.17 \begin{align*} \frac{{\left (a a_{1} + b b_{1}\right )} x}{a^{2} + b^{2}} + \frac{{\left (a_{1} b - a b_{1}\right )} \log \left (\tan \left (x\right )^{2} + 1\right )}{2 \,{\left (a^{2} + b^{2}\right )}} - \frac{{\left (a a_{1} b - a^{2} b_{1}\right )} \log \left ({\left | a \tan \left (x\right ) + b \right |}\right )}{a^{3} + a b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b1*cos(x)+a1*sin(x))/(b*cos(x)+a*sin(x)),x, algorithm="giac")

[Out]

(a*a1 + b*b1)*x/(a^2 + b^2) + 1/2*(a1*b - a*b1)*log(tan(x)^2 + 1)/(a^2 + b^2) - (a*a1*b - a^2*b1)*log(abs(a*ta
n(x) + b))/(a^3 + a*b^2)