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

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Rubi [A]  time = 0.03, antiderivative size = 48, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.13, 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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fricas [A]  time = 0.44, size = 60, normalized size = 1.25 \[ \frac {2 \, {\left (a a_{1} + b b_{1}\right )} x - {\left (a_{1} b - a b_{1}\right )} \log \left (2 \, a b \cos \relax (x) \sin \relax (x) - {\left (a^{2} - b^{2}\right )} \cos \relax (x)^{2} + a^{2}\right )}{2 \, {\left (a^{2} + b^{2}\right )}} \]

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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giac [A]  time = 0.05, size = 77, normalized size = 1.60 \[ \frac {{\left (a a_{1} + b b_{1}\right )} x}{a^{2} + b^{2}} + \frac {{\left (a_{1} b - a b_{1}\right )} \log \left (\tan \relax (x)^{2} + 1\right )}{2 \, {\left (a^{2} + b^{2}\right )}} - \frac {{\left (a a_{1} b - a^{2} b_{1}\right )} \log \left ({\left | a \tan \relax (x) + b \right |}\right )}{a^{3} + a b^{2}} \]

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)

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maple [B]  time = 0.13, size = 111, normalized size = 2.31 \[ \frac {a \mathit {a1} \arctan \left (\tan \relax (x )\right )}{a^{2}+b^{2}}-\frac {a \mathit {b1} \ln \left (\tan ^{2}\relax (x )+1\right )}{2 \left (a^{2}+b^{2}\right )}+\frac {a \mathit {b1} \ln \left (a \tan \relax (x )+b \right )}{a^{2}+b^{2}}+\frac {\mathit {a1} b \ln \left (\tan ^{2}\relax (x )+1\right )}{2 a^{2}+2 b^{2}}-\frac {\mathit {a1} b \ln \left (a \tan \relax (x )+b \right )}{a^{2}+b^{2}}+\frac {b \mathit {b1} \arctan \left (\tan \relax (x )\right )}{a^{2}+b^{2}} \]

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

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maxima [B]  time = 1.30, size = 181, normalized size = 3.77 \[ a_{1} {\left (\frac {2 \, a \arctan \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right )}{a^{2} + b^{2}} - \frac {b \log \left (-b - \frac {2 \, a \sin \relax (x)}{\cos \relax (x) + 1} + \frac {b \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}}\right )}{a^{2} + b^{2}} + \frac {b \log \left (\frac {\sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + 1\right )}{a^{2} + b^{2}}\right )} + b_{1} {\left (\frac {2 \, b \arctan \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right )}{a^{2} + b^{2}} + \frac {a \log \left (-b - \frac {2 \, a \sin \relax (x)}{\cos \relax (x) + 1} + \frac {b \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}}\right )}{a^{2} + b^{2}} - \frac {a \log \left (\frac {\sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + 1\right )}{a^{2} + b^{2}}\right )} \]

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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mupad [B]  time = 10.61, size = 2034, normalized size = 42.38 \[ \text {result too large to display} \]

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]

(2*atan((tan(x/2)*(((((a*a1 + b*b1)^3*(96*a^4*b + 96*a^2*b^3))/(a^2 + b^2)^3 + ((((a*a1 + b*b1)*(32*a^2*a1*b^2
- ((2*a*b1 - 2*a1*b)*(96*a^4*b + 96*a^2*b^3))/(2*(a^2 + b^2)) - 64*a1*b^4 + 128*a*b^3*b1 + 32*a^3*b*b1))/(a^2
+ b^2) - ((a*a1 + b*b1)*(2*a*b1 - 2*a1*b)*(96*a^4*b + 96*a^2*b^3))/(2*(a^2 + b^2)^2))*(2*a*b1 - 2*a1*b))/(2*(
a^2 + b^2)) - ((a*a1 + b*b1)*(32*b^3*b1^2 - ((2*a*b1 - 2*a1*b)*(32*a^2*a1*b^2 - ((2*a*b1 - 2*a1*b)*(96*a^4*b +
96*a^2*b^3))/(2*(a^2 + b^2)) - 64*a1*b^4 + 128*a*b^3*b1 + 32*a^3*b*b1))/(2*(a^2 + b^2)) + 64*a^2*a1^2*b - 96*
a^2*b*b1^2 + 192*a*a1*b^2*b1))/(a^2 + b^2))*(a^4*a1^2 + 4*a1^2*b^4 - 4*a^4*b1^2 - b^4*b1^2 - 13*a^2*a1^2*b^2 +
13*a^2*b^2*b1^2 - 18*a*a1*b^3*b1 + 18*a^3*a1*b*b1))/((a^2 + b^2)^2*(a^2*a1^2 + 4*a^2*b1^2 + 4*a1^2*b^2 + b^2*
b1^2 - 6*a*a1*b*b1)^2) - ((((((a*a1 + b*b1)*(32*a^2*a1*b^2 - ((2*a*b1 - 2*a1*b)*(96*a^4*b + 96*a^2*b^3))/(2*(a
^2 + b^2)) - 64*a1*b^4 + 128*a*b^3*b1 + 32*a^3*b*b1))/(a^2 + b^2) - ((a*a1 + b*b1)*(2*a*b1 - 2*a1*b)*(96*a^4*b
+ 96*a^2*b^3))/(2*(a^2 + b^2)^2))*(a*a1 + b*b1))/(a^2 + b^2) - 32*a1*b^2*b1^2 - 64*a1^3*b^2 + ((2*a*b1 - 2*a1
*b)*(32*b^3*b1^2 - ((2*a*b1 - 2*a1*b)*(32*a^2*a1*b^2 - ((2*a*b1 - 2*a1*b)*(96*a^4*b + 96*a^2*b^3))/(2*(a^2 + b
^2)) - 64*a1*b^4 + 128*a*b^3*b1 + 32*a^3*b*b1))/(2*(a^2 + b^2)) + 64*a^2*a1^2*b - 96*a^2*b*b1^2 + 192*a*a1*b^2
*b1))/(2*(a^2 + b^2)) + 32*a*b*b1^3 - ((a*a1 + b*b1)^2*(2*a*b1 - 2*a1*b)*(96*a^4*b + 96*a^2*b^3))/(2*(a^2 + b^
2)^3) + 64*a*a1^2*b*b1)*(12*a*a1^2*b^3 - 6*a^3*a1^2*b - 6*a*b^3*b1^2 + 12*a^3*b*b1^2 + 4*a^4*a1*b1 + 4*a1*b^4*
b1 - 28*a^2*a1*b^2*b1))/((a^2 + b^2)^2*(a^2*a1^2 + 4*a^2*b1^2 + 4*a1^2*b^2 + b^2*b1^2 - 6*a*a1*b*b1)^2))*(a^4
+ b^4 + 2*a^2*b^2))/(32*b^2*b1 + 32*a*a1*b) + ((a^4 + b^4 + 2*a^2*b^2)*(32*a1^2*b^2*b1 + ((((a*a1 + b*b1)*(((2
*a*b1 - 2*a1*b)*(96*a*b^4 + 96*a^3*b^2))/(2*(a^2 + b^2)) - 32*b^4*b1 + 64*a^2*b^2*b1 - 64*a*a1*b^3 + 32*a^3*a1
*b))/(a^2 + b^2) + ((a*a1 + b*b1)*(2*a*b1 - 2*a1*b)*(96*a*b^4 + 96*a^3*b^2))/(2*(a^2 + b^2)^2))*(a*a1 + b*b1))
/(a^2 + b^2) - ((2*a*b1 - 2*a1*b)*(((2*a*b1 - 2*a1*b)*(((2*a*b1 - 2*a1*b)*(96*a*b^4 + 96*a^3*b^2))/(2*(a^2 + b
^2)) - 32*b^4*b1 + 64*a^2*b^2*b1 - 64*a*a1*b^3 + 32*a^3*a1*b))/(2*(a^2 + b^2)) - 32*a*a1^2*b^2 - 32*a*b^2*b1^2
+ 64*a1*b^3*b1 + 64*a^2*a1*b*b1))/(2*(a^2 + b^2)) + ((a*a1 + b*b1)^2*(2*a*b1 - 2*a1*b)*(96*a*b^4 + 96*a^3*b^2
))/(2*(a^2 + b^2)^3) - 32*a*a1*b*b1^2)*(12*a*a1^2*b^3 - 6*a^3*a1^2*b - 6*a*b^3*b1^2 + 12*a^3*b*b1^2 + 4*a^4*a1
*b1 + 4*a1*b^4*b1 - 28*a^2*a1*b^2*b1))/((32*b^2*b1 + 32*a*a1*b)*(a^2 + b^2)^2*(a^2*a1^2 + 4*a^2*b1^2 + 4*a1^2*
b^2 + b^2*b1^2 - 6*a*a1*b*b1)^2) - ((a^4 + b^4 + 2*a^2*b^2)*(((((a*a1 + b*b1)*(((2*a*b1 - 2*a1*b)*(96*a*b^4 +
96*a^3*b^2))/(2*(a^2 + b^2)) - 32*b^4*b1 + 64*a^2*b^2*b1 - 64*a*a1*b^3 + 32*a^3*a1*b))/(a^2 + b^2) + ((a*a1 +
b*b1)*(2*a*b1 - 2*a1*b)*(96*a*b^4 + 96*a^3*b^2))/(2*(a^2 + b^2)^2))*(2*a*b1 - 2*a1*b))/(2*(a^2 + b^2)) - ((a*a
1 + b*b1)^3*(96*a*b^4 + 96*a^3*b^2))/(a^2 + b^2)^3 + ((a*a1 + b*b1)*(((2*a*b1 - 2*a1*b)*(((2*a*b1 - 2*a1*b)*(9
6*a*b^4 + 96*a^3*b^2))/(2*(a^2 + b^2)) - 32*b^4*b1 + 64*a^2*b^2*b1 - 64*a*a1*b^3 + 32*a^3*a1*b))/(2*(a^2 + b^2
)) - 32*a*a1^2*b^2 - 32*a*b^2*b1^2 + 64*a1*b^3*b1 + 64*a^2*a1*b*b1))/(a^2 + b^2))*(a^4*a1^2 + 4*a1^2*b^4 - 4*a
^4*b1^2 - b^4*b1^2 - 13*a^2*a1^2*b^2 + 13*a^2*b^2*b1^2 - 18*a*a1*b^3*b1 + 18*a^3*a1*b*b1))/((32*b^2*b1 + 32*a*
a1*b)*(a^2 + b^2)^2*(a^2*a1^2 + 4*a^2*b1^2 + 4*a1^2*b^2 + b^2*b1^2 - 6*a*a1*b*b1)^2))*(a*a1 + b*b1))/(a^2 + b^
2) - (log(1/(cos(x) + 1))*(2*a*b1 - 2*a1*b))/(2*(a^2 + b^2)) + (log(b + 2*a*tan(x/2) - b*tan(x/2)^2)*(a*b1 - a
1*b))/(a^2 + b^2)

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sympy [A]  time = 1.00, size = 360, normalized size = 7.50 \[ \begin {cases} \tilde {\infty } \left (- a_{1} \log {\left (\cos {\relax (x )} \right )} + b_{1} x\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {- a_{1} \log {\left (\cos {\relax (x )} \right )} + b_{1} x}{b} & \text {for}\: a = 0 \\\frac {i a_{1} x \sin {\relax (x )}}{2 b \sin {\relax (x )} + 2 i b \cos {\relax (x )}} - \frac {a_{1} x \cos {\relax (x )}}{2 b \sin {\relax (x )} + 2 i b \cos {\relax (x )}} - \frac {i a_{1} \cos {\relax (x )}}{2 b \sin {\relax (x )} + 2 i b \cos {\relax (x )}} + \frac {b_{1} x \sin {\relax (x )}}{2 b \sin {\relax (x )} + 2 i b \cos {\relax (x )}} + \frac {i b_{1} x \cos {\relax (x )}}{2 b \sin {\relax (x )} + 2 i b \cos {\relax (x )}} + \frac {b_{1} \cos {\relax (x )}}{2 b \sin {\relax (x )} + 2 i b \cos {\relax (x )}} & \text {for}\: a = - i b \\- \frac {i a_{1} x \sin {\relax (x )}}{2 b \sin {\relax (x )} - 2 i b \cos {\relax (x )}} - \frac {a_{1} x \cos {\relax (x )}}{2 b \sin {\relax (x )} - 2 i b \cos {\relax (x )}} + \frac {i a_{1} \cos {\relax (x )}}{2 b \sin {\relax (x )} - 2 i b \cos {\relax (x )}} + \frac {b_{1} x \sin {\relax (x )}}{2 b \sin {\relax (x )} - 2 i b \cos {\relax (x )}} - \frac {i b_{1} x \cos {\relax (x )}}{2 b \sin {\relax (x )} - 2 i b \cos {\relax (x )}} + \frac {b_{1} \cos {\relax (x )}}{2 b \sin {\relax (x )} - 2 i b \cos {\relax (x )}} & \text {for}\: a = i b \\\frac {a a_{1} x}{a^{2} + b^{2}} + \frac {a b_{1} \log {\left (\sin {\relax (x )} + \frac {b \cos {\relax (x )}}{a} \right )}}{a^{2} + b^{2}} - \frac {a_{1} b \log {\left (\sin {\relax (x )} + \frac {b \cos {\relax (x )}}{a} \right )}}{a^{2} + b^{2}} + \frac {b b_{1} x}{a^{2} + b^{2}} & \text {otherwise} \end {cases} \]

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*log(cos(x)) + b1*x)/b, Eq(a, 0)), (I*a1*x
*sin(x)/(2*b*sin(x) + 2*I*b*cos(x)) - a1*x*cos(x)/(2*b*sin(x) + 2*I*b*cos(x)) - I*a1*cos(x)/(2*b*sin(x) + 2*I*
b*cos(x)) + b1*x*sin(x)/(2*b*sin(x) + 2*I*b*cos(x)) + I*b1*x*cos(x)/(2*b*sin(x) + 2*I*b*cos(x)) + b1*cos(x)/(2
*b*sin(x) + 2*I*b*cos(x)), Eq(a, -I*b)), (-I*a1*x*sin(x)/(2*b*sin(x) - 2*I*b*cos(x)) - a1*x*cos(x)/(2*b*sin(x)
- 2*I*b*cos(x)) + I*a1*cos(x)/(2*b*sin(x) - 2*I*b*cos(x)) + b1*x*sin(x)/(2*b*sin(x) - 2*I*b*cos(x)) - I*b1*x*
cos(x)/(2*b*sin(x) - 2*I*b*cos(x)) + b1*cos(x)/(2*b*sin(x) - 2*I*b*cos(x)), Eq(a, I*b)), (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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