3.2.69 \(\int \frac {\text {b1} \cos (x)+\text {a1} \sin (x)}{b \cos (x)+a \sin (x)} \, dx\) [169]

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

[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 = 0.02, 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 = {3212} \begin {gather*} \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} \end {gather*}

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 3212

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 = 0.09, size = 39, normalized size = 0.81 \begin {gather*} \frac {(a \text {a1}+b \text {b1}) x+(-\text {a1} b+a \text {b1}) \log (b \cos (x)+a \sin (x))}{a^2+b^2} \end {gather*}

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 [A]
time = 0.10, size = 66, normalized size = 1.38

method	result	size
default	\(\frac {\frac {\left (-a \mathit {b1} +\mathit {a1} b \right ) \ln \left (1+\tan ^{2}\left (x \right )\right )}{2}+\left (a \mathit {a1} +b \mathit {b1} \right ) \arctan \left (\tan \left (x \right )\right )}{a^{2}+b^{2}}+\frac {\left (a \mathit {b1} -\mathit {a1} b \right ) \ln \left (a \tan \left (x \right )+b \right )}{a^{2}+b^{2}}\)	\(66\)
norman	\(\frac {\frac {\left (a \mathit {a1} +b \mathit {b1} \right ) x}{a^{2}+b^{2}}+\frac {\left (a \mathit {a1} +b \mathit {b1} \right ) x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{a^{2}+b^{2}}}{1+\tan ^{2}\left (\frac {x}{2}\right )}+\frac {\left (a \mathit {b1} -\mathit {a1} b \right ) \ln \left (-b \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+2 a \tan \left (\frac {x}{2}\right )+b \right )}{a^{2}+b^{2}}-\frac {\left (a \mathit {b1} -\mathit {a1} b \right ) \ln \left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )}{a^{2}+b^{2}}\)	\(121\)
risch	\(\frac {i x \mathit {b1}}{i b +a}+\frac {x \mathit {a1}}{i b +a}-\frac {2 i x a \mathit {b1}}{a^{2}+b^{2}}+\frac {2 i x \mathit {a1} b}{a^{2}+b^{2}}+\frac {\ln \left ({\mathrm e}^{2 i x}+\frac {i b -a}{i b +a}\right ) a \mathit {b1}}{a^{2}+b^{2}}-\frac {\ln \left ({\mathrm e}^{2 i x}+\frac {i b -a}{i b +a}\right ) \mathit {a1} b}{a^{2}+b^{2}}\)	\(129\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 181 vs. \(2 (48) = 96\).
time = 1.39, size = 181, normalized size = 3.77 \begin {gather*} 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 {gather*}

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 = 0.69, size = 60, normalized size = 1.25 \begin {gather*} \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 {gather*}

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 [C] Result contains complex when optimal does not.
time = 0.39, size = 360, normalized size = 7.50 \begin {gather*} \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 {a_{1} \cos {\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 {i b_{1} \cos {\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 {a_{1} \cos {\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 {i b_{1} \cos {\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 + b_{1} \log {\left (\sin {\left (x \right )} \right )}}{a} & \text {for}\: b = 0 \\\frac {a a_{1} x}{a^{2} + b^{2}} + \frac {a b_{1} \log {\left (\frac {a \sin {\left (x \right )}}{b} + \cos {\left (x \right )} \right )}}{a^{2} + b^{2}} - \frac {a_{1} b \log {\left (\frac {a \sin {\left (x \right )}}{b} + \cos {\left (x \right )} \right )}}{a^{2} + b^{2}} + \frac {b b_{1} x}{a^{2} + b^{2}} & \text {otherwise} \end {cases} \end {gather*}

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

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Giac [A]
time = 0.48, size = 77, normalized size = 1.60 \begin {gather*} \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 {gather*}

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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Mupad [B]
time = 10.61, size = 2034, normalized size = 42.38 \begin {gather*} \text {Too large to display} \end {gather*}

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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