\(\int \frac {\sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx\) [9]

Optimal result

Integrand size = 16, antiderivative size = 68 \[ \int \frac {\sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx=-\frac {a^2 \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}-\frac {b \cos (x)}{a^2+b^2}-\frac {a \sin (x)}{a^2+b^2} \]

[Out]

-a^2*arctanh((b*cos(x)-a*sin(x))/(a^2+b^2)^(1/2))/(a^2+b^2)^(3/2)-b*cos(x)/(a^2+b^2)-a*sin(x)/(a^2+b^2)

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3178, 3153, 212, 2718} \[ \int \frac {\sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx=-\frac {a^2 \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}-\frac {a \sin (x)}{a^2+b^2}-\frac {b \cos (x)}{a^2+b^2} \]

[In]

Int[Sin[x]^2/(a*Cos[x] + b*Sin[x]),x]

[Out]

-((a^2*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(3/2)) - (b*Cos[x])/(a^2 + b^2) - (a*Sin[x]
)/(a^2 + b^2)

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3153

Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> Dist[-d^(-1), Subst[Int
[1/(a^2 + b^2 - x^2), x], x, b*Cos[c + d*x] - a*Sin[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2,
0]

Rule 3178

Int[sin[(c_.) + (d_.)*(x_)]^(m_)/(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)]), x_Symbol] :>
Simp[(-a)*(Sin[c + d*x]^(m - 1)/(d*(a^2 + b^2)*(m - 1))), x] + (Dist[a^2/(a^2 + b^2), Int[Sin[c + d*x]^(m - 2
)/(a*Cos[c + d*x] + b*Sin[c + d*x]), x], x] + Dist[b/(a^2 + b^2), Int[Sin[c + d*x]^(m - 1), x], x]) /; FreeQ[{
a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && GtQ[m, 1]

Rubi steps \begin{align*} \text {integral}& = -\frac {a \sin (x)}{a^2+b^2}+\frac {a^2 \int \frac {1}{a \cos (x)+b \sin (x)} \, dx}{a^2+b^2}+\frac {b \int \sin (x) \, dx}{a^2+b^2} \\ & = -\frac {b \cos (x)}{a^2+b^2}-\frac {a \sin (x)}{a^2+b^2}-\frac {a^2 \text {Subst}\left (\int \frac {1}{a^2+b^2-x^2} \, dx,x,b \cos (x)-a \sin (x)\right )}{a^2+b^2} \\ & = -\frac {a^2 \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}-\frac {b \cos (x)}{a^2+b^2}-\frac {a \sin (x)}{a^2+b^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.91 \[ \int \frac {\sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx=\frac {2 a^2 \text {arctanh}\left (\frac {-b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}-\frac {b \cos (x)+a \sin (x)}{a^2+b^2} \]

[In]

Integrate[Sin[x]^2/(a*Cos[x] + b*Sin[x]),x]

[Out]

(2*a^2*ArcTanh[(-b + a*Tan[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(3/2) - (b*Cos[x] + a*Sin[x])/(a^2 + b^2)

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.24

method	result	size
default	\(\frac {8 a^{2} \operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (4 a^{2}+4 b^{2}\right ) \sqrt {a^{2}+b^{2}}}+\frac {-2 a \tan \left (\frac {x}{2}\right )-2 b}{\left (a^{2}+b^{2}\right ) \left (1+\tan \left (\frac {x}{2}\right )^{2}\right )}\)	\(84\)
risch	\(\frac {i {\mathrm e}^{i x}}{-2 i b +2 a}-\frac {i {\mathrm e}^{-i x}}{2 \left (i b +a \right )}-\frac {a^{2} \ln \left ({\mathrm e}^{i x}-\frac {i a^{3}+i a \,b^{2}-a^{2} b -b^{3}}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}+\frac {a^{2} \ln \left ({\mathrm e}^{i x}+\frac {i a^{3}+i a \,b^{2}-a^{2} b -b^{3}}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\)	\(146\)

[In]

int(sin(x)^2/(a*cos(x)+b*sin(x)),x,method=_RETURNVERBOSE)

[Out]

8*a^2/(4*a^2+4*b^2)/(a^2+b^2)^(1/2)*arctanh(1/2*(2*a*tan(1/2*x)-2*b)/(a^2+b^2)^(1/2))+2/(a^2+b^2)*(-a*tan(1/2*
x)-b)/(1+tan(1/2*x)^2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 144 vs. \(2 (64) = 128\).

Time = 0.27 (sec) , antiderivative size = 144, normalized size of antiderivative = 2.12 \[ \int \frac {\sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx=\frac {\sqrt {a^{2} + b^{2}} a^{2} \log \left (-\frac {2 \, a b \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} - 2 \, a^{2} - b^{2} + 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cos \left (x\right ) - a \sin \left (x\right )\right )}}{2 \, a b \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + b^{2}}\right ) - 2 \, {\left (a^{2} b + b^{3}\right )} \cos \left (x\right ) - 2 \, {\left (a^{3} + a b^{2}\right )} \sin \left (x\right )}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}} \]

[In]

integrate(sin(x)^2/(a*cos(x)+b*sin(x)),x, algorithm="fricas")

[Out]

1/2*(sqrt(a^2 + b^2)*a^2*log(-(2*a*b*cos(x)*sin(x) + (a^2 - b^2)*cos(x)^2 - 2*a^2 - b^2 + 2*sqrt(a^2 + b^2)*(b
*cos(x) - a*sin(x)))/(2*a*b*cos(x)*sin(x) + (a^2 - b^2)*cos(x)^2 + b^2)) - 2*(a^2*b + b^3)*cos(x) - 2*(a^3 + a
*b^2)*sin(x))/(a^4 + 2*a^2*b^2 + b^4)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 76.32 (sec) , antiderivative size = 706, normalized size of antiderivative = 10.38 \[ \int \frac {\sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx=\begin {cases} \tilde {\infty } \cos {\left (x \right )} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {\cos {\left (x \right )}}{b} & \text {for}\: a = 0 \\- \frac {\sin ^{2}{\left (x \right )}}{3 i b \sin {\left (x \right )} + 3 b \cos {\left (x \right )}} - \frac {2 i \sin {\left (x \right )} \cos {\left (x \right )}}{3 i b \sin {\left (x \right )} + 3 b \cos {\left (x \right )}} - \frac {2 \cos ^{2}{\left (x \right )}}{3 i b \sin {\left (x \right )} + 3 b \cos {\left (x \right )}} & \text {for}\: a = - i b \\- \frac {\sin ^{2}{\left (x \right )}}{- 3 i b \sin {\left (x \right )} + 3 b \cos {\left (x \right )}} + \frac {2 i \sin {\left (x \right )} \cos {\left (x \right )}}{- 3 i b \sin {\left (x \right )} + 3 b \cos {\left (x \right )}} - \frac {2 \cos ^{2}{\left (x \right )}}{- 3 i b \sin {\left (x \right )} + 3 b \cos {\left (x \right )}} & \text {for}\: a = i b \\- \frac {a^{2} \log {\left (\tan {\left (\frac {x}{2} \right )} - \frac {b}{a} - \frac {\sqrt {a^{2} + b^{2}}}{a} \right )} \tan ^{2}{\left (\frac {x}{2} \right )}}{a^{2} \sqrt {a^{2} + b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} + a^{2} \sqrt {a^{2} + b^{2}} + b^{2} \sqrt {a^{2} + b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} + b^{2} \sqrt {a^{2} + b^{2}}} - \frac {a^{2} \log {\left (\tan {\left (\frac {x}{2} \right )} - \frac {b}{a} - \frac {\sqrt {a^{2} + b^{2}}}{a} \right )}}{a^{2} \sqrt {a^{2} + b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} + a^{2} \sqrt {a^{2} + b^{2}} + b^{2} \sqrt {a^{2} + b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} + b^{2} \sqrt {a^{2} + b^{2}}} + \frac {a^{2} \log {\left (\tan {\left (\frac {x}{2} \right )} - \frac {b}{a} + \frac {\sqrt {a^{2} + b^{2}}}{a} \right )} \tan ^{2}{\left (\frac {x}{2} \right )}}{a^{2} \sqrt {a^{2} + b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} + a^{2} \sqrt {a^{2} + b^{2}} + b^{2} \sqrt {a^{2} + b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} + b^{2} \sqrt {a^{2} + b^{2}}} + \frac {a^{2} \log {\left (\tan {\left (\frac {x}{2} \right )} - \frac {b}{a} + \frac {\sqrt {a^{2} + b^{2}}}{a} \right )}}{a^{2} \sqrt {a^{2} + b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} + a^{2} \sqrt {a^{2} + b^{2}} + b^{2} \sqrt {a^{2} + b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} + b^{2} \sqrt {a^{2} + b^{2}}} - \frac {2 a \sqrt {a^{2} + b^{2}} \tan {\left (\frac {x}{2} \right )}}{a^{2} \sqrt {a^{2} + b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} + a^{2} \sqrt {a^{2} + b^{2}} + b^{2} \sqrt {a^{2} + b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} + b^{2} \sqrt {a^{2} + b^{2}}} - \frac {2 b \sqrt {a^{2} + b^{2}}}{a^{2} \sqrt {a^{2} + b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} + a^{2} \sqrt {a^{2} + b^{2}} + b^{2} \sqrt {a^{2} + b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} + b^{2} \sqrt {a^{2} + b^{2}}} & \text {otherwise} \end {cases} \]

[In]

integrate(sin(x)**2/(a*cos(x)+b*sin(x)),x)

[Out]

Piecewise((zoo*cos(x), Eq(a, 0) & Eq(b, 0)), (-cos(x)/b, Eq(a, 0)), (-sin(x)**2/(3*I*b*sin(x) + 3*b*cos(x)) -
2*I*sin(x)*cos(x)/(3*I*b*sin(x) + 3*b*cos(x)) - 2*cos(x)**2/(3*I*b*sin(x) + 3*b*cos(x)), Eq(a, -I*b)), (-sin(x
)**2/(-3*I*b*sin(x) + 3*b*cos(x)) + 2*I*sin(x)*cos(x)/(-3*I*b*sin(x) + 3*b*cos(x)) - 2*cos(x)**2/(-3*I*b*sin(x
) + 3*b*cos(x)), Eq(a, I*b)), (-a**2*log(tan(x/2) - b/a - sqrt(a**2 + b**2)/a)*tan(x/2)**2/(a**2*sqrt(a**2 + b
**2)*tan(x/2)**2 + a**2*sqrt(a**2 + b**2) + b**2*sqrt(a**2 + b**2)*tan(x/2)**2 + b**2*sqrt(a**2 + b**2)) - a**
2*log(tan(x/2) - b/a - sqrt(a**2 + b**2)/a)/(a**2*sqrt(a**2 + b**2)*tan(x/2)**2 + a**2*sqrt(a**2 + b**2) + b**
2*sqrt(a**2 + b**2)*tan(x/2)**2 + b**2*sqrt(a**2 + b**2)) + a**2*log(tan(x/2) - b/a + sqrt(a**2 + b**2)/a)*tan
(x/2)**2/(a**2*sqrt(a**2 + b**2)*tan(x/2)**2 + a**2*sqrt(a**2 + b**2) + b**2*sqrt(a**2 + b**2)*tan(x/2)**2 + b
**2*sqrt(a**2 + b**2)) + a**2*log(tan(x/2) - b/a + sqrt(a**2 + b**2)/a)/(a**2*sqrt(a**2 + b**2)*tan(x/2)**2 +
a**2*sqrt(a**2 + b**2) + b**2*sqrt(a**2 + b**2)*tan(x/2)**2 + b**2*sqrt(a**2 + b**2)) - 2*a*sqrt(a**2 + b**2)*
tan(x/2)/(a**2*sqrt(a**2 + b**2)*tan(x/2)**2 + a**2*sqrt(a**2 + b**2) + b**2*sqrt(a**2 + b**2)*tan(x/2)**2 + b
**2*sqrt(a**2 + b**2)) - 2*b*sqrt(a**2 + b**2)/(a**2*sqrt(a**2 + b**2)*tan(x/2)**2 + a**2*sqrt(a**2 + b**2) +
b**2*sqrt(a**2 + b**2)*tan(x/2)**2 + b**2*sqrt(a**2 + b**2)), True))

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.56 \[ \int \frac {\sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx=-\frac {a^{2} \log \left (\frac {b - \frac {a \sin \left (x\right )}{\cos \left (x\right ) + 1} + \sqrt {a^{2} + b^{2}}}{b - \frac {a \sin \left (x\right )}{\cos \left (x\right ) + 1} - \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{2} + b^{2}\right )}^{\frac {3}{2}}} - \frac {2 \, {\left (b + \frac {a \sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}}{a^{2} + b^{2} + \frac {{\left (a^{2} + b^{2}\right )} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}} \]

[In]

integrate(sin(x)^2/(a*cos(x)+b*sin(x)),x, algorithm="maxima")

[Out]

-a^2*log((b - a*sin(x)/(cos(x) + 1) + sqrt(a^2 + b^2))/(b - a*sin(x)/(cos(x) + 1) - sqrt(a^2 + b^2)))/(a^2 + b
^2)^(3/2) - 2*(b + a*sin(x)/(cos(x) + 1))/(a^2 + b^2 + (a^2 + b^2)*sin(x)^2/(cos(x) + 1)^2)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.38 \[ \int \frac {\sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx=-\frac {a^{2} \log \left (\frac {{\left | 2 \, a \tan \left (\frac {1}{2} \, x\right ) - 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a \tan \left (\frac {1}{2} \, x\right ) - 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{2} + b^{2}\right )}^{\frac {3}{2}}} - \frac {2 \, {\left (a \tan \left (\frac {1}{2} \, x\right ) + b\right )}}{{\left (a^{2} + b^{2}\right )} {\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}} \]

[In]

integrate(sin(x)^2/(a*cos(x)+b*sin(x)),x, algorithm="giac")

[Out]

-a^2*log(abs(2*a*tan(1/2*x) - 2*b - 2*sqrt(a^2 + b^2))/abs(2*a*tan(1/2*x) - 2*b + 2*sqrt(a^2 + b^2)))/(a^2 + b
^2)^(3/2) - 2*(a*tan(1/2*x) + b)/((a^2 + b^2)*(tan(1/2*x)^2 + 1))

Mupad [B] (verification not implemented)

Time = 20.94 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.38 \[ \int \frac {\sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx=-\frac {\frac {2\,b}{a^2+b^2}+\frac {2\,a\,\mathrm {tan}\left (\frac {x}{2}\right )}{a^2+b^2}}{{\mathrm {tan}\left (\frac {x}{2}\right )}^2+1}-\frac {2\,a^2\,\mathrm {atanh}\left (\frac {2\,a^2\,b+2\,b^3-2\,a\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (a^2+b^2\right )}{2\,{\left (a^2+b^2\right )}^{3/2}}\right )}{{\left (a^2+b^2\right )}^{3/2}} \]

[In]

int(sin(x)^2/(a*cos(x) + b*sin(x)),x)

[Out]

- ((2*b)/(a^2 + b^2) + (2*a*tan(x/2))/(a^2 + b^2))/(tan(x/2)^2 + 1) - (2*a^2*atanh((2*a^2*b + 2*b^3 - 2*a*tan(
x/2)*(a^2 + b^2))/(2*(a^2 + b^2)^(3/2))))/(a^2 + b^2)^(3/2)