3.2.78 \(\int \frac {\sin ^2(x)}{a+b \sin (x)} \, dx\) [178]

Optimal. Leaf size=61 \[ -\frac {a x}{b^2}+\frac {2 a^2 \tan ^{-1}\left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{b^2 \sqrt {a^2-b^2}}-\frac {\cos (x)}{b} \]

[Out]

-a*x/b^2-cos(x)/b+2*a^2*arctan((b+a*tan(1/2*x))/(a^2-b^2)^(1/2))/b^2/(a^2-b^2)^(1/2)

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Rubi [A]
time = 0.08, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2825, 12, 2814, 2739, 632, 210} \begin {gather*} \frac {2 a^2 \text {ArcTan}\left (\frac {a \tan \left (\frac {x}{2}\right )+b}{\sqrt {a^2-b^2}}\right )}{b^2 \sqrt {a^2-b^2}}-\frac {a x}{b^2}-\frac {\cos (x)}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[x]^2/(a + b*Sin[x]),x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 210

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

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 2739

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[2*(e/d), Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[a^2 - b^2, 0]

Rule 2814

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[b*(x/d)
, x] - Dist[(b*c - a*d)/d, Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d
, 0]

Rule 2825

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

Rubi steps

\begin {align*} \int \frac {\sin ^2(x)}{a+b \sin (x)} \, dx &=-\frac {\cos (x)}{b}-\frac {\int \frac {a \sin (x)}{a+b \sin (x)} \, dx}{b}\\ &=-\frac {\cos (x)}{b}-\frac {a \int \frac {\sin (x)}{a+b \sin (x)} \, dx}{b}\\ &=-\frac {a x}{b^2}-\frac {\cos (x)}{b}+\frac {a^2 \int \frac {1}{a+b \sin (x)} \, dx}{b^2}\\ &=-\frac {a x}{b^2}-\frac {\cos (x)}{b}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{b^2}\\ &=-\frac {a x}{b^2}-\frac {\cos (x)}{b}-\frac {\left (4 a^2\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac {x}{2}\right )\right )}{b^2}\\ &=-\frac {a x}{b^2}+\frac {2 a^2 \tan ^{-1}\left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{b^2 \sqrt {a^2-b^2}}-\frac {\cos (x)}{b}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 56, normalized size = 0.92 \begin {gather*} -\frac {a x-\frac {2 a^2 \tan ^{-1}\left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+b \cos (x)}{b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]^2/(a + b*Sin[x]),x]

[Out]

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

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Maple [A]
time = 0.14, size = 71, normalized size = 1.16

method	result	size
default	\(\frac {2 a^{2} \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{b^{2} \sqrt {a^{2}-b^{2}}}-\frac {2 \left (\frac {b}{\tan ^{2}\left (\frac {x}{2}\right )+1}+a \arctan \left (\tan \left (\frac {x}{2}\right )\right )\right )}{b^{2}}\)	\(71\)
risch	\(-\frac {a x}{b^{2}}-\frac {{\mathrm e}^{i x}}{2 b}-\frac {{\mathrm e}^{-i x}}{2 b}+\frac {a^{2} \ln \left ({\mathrm e}^{i x}+\frac {i a \sqrt {-a^{2}+b^{2}}+a^{2}-b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, b^{2}}-\frac {a^{2} \ln \left ({\mathrm e}^{i x}+\frac {i a \sqrt {-a^{2}+b^{2}}-a^{2}+b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, b^{2}}\)	\(158\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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(sin(x)^2/(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 = 0.46, size = 231, normalized size = 3.79 \begin {gather*} \left [-\frac {\sqrt {-a^{2} + b^{2}} a^{2} \log \left (\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (x\right )^{2} - 2 \, a b \sin \left (x\right ) - a^{2} - b^{2} + 2 \, {\left (a \cos \left (x\right ) \sin \left (x\right ) + b \cos \left (x\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (x\right )^{2} - 2 \, a b \sin \left (x\right ) - a^{2} - b^{2}}\right ) + 2 \, {\left (a^{3} - a b^{2}\right )} x + 2 \, {\left (a^{2} b - b^{3}\right )} \cos \left (x\right )}{2 \, {\left (a^{2} b^{2} - b^{4}\right )}}, -\frac {\sqrt {a^{2} - b^{2}} a^{2} \arctan \left (-\frac {a \sin \left (x\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (x\right )}\right ) + {\left (a^{3} - a b^{2}\right )} x + {\left (a^{2} b - b^{3}\right )} \cos \left (x\right )}{a^{2} b^{2} - b^{4}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*(sqrt(-a^2 + b^2)*a^2*log(((2*a^2 - b^2)*cos(x)^2 - 2*a*b*sin(x) - a^2 - b^2 + 2*(a*cos(x)*sin(x) + b*co
s(x))*sqrt(-a^2 + b^2))/(b^2*cos(x)^2 - 2*a*b*sin(x) - a^2 - b^2)) + 2*(a^3 - a*b^2)*x + 2*(a^2*b - b^3)*cos(x
))/(a^2*b^2 - b^4), -(sqrt(a^2 - b^2)*a^2*arctan(-(a*sin(x) + b)/(sqrt(a^2 - b^2)*cos(x))) + (a^3 - a*b^2)*x +
(a^2*b - b^3)*cos(x))/(a^2*b^2 - b^4)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1192 vs. \(2 (49) = 98\).
time = 146.75, size = 1192, normalized size = 19.54 \begin {gather*} \begin {cases} \tilde {\infty } \cos {\left (x \right )} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {b x \tan ^{2}{\left (\frac {x}{2} \right )}}{b^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + b^{2} \tan {\left (\frac {x}{2} \right )} - b \sqrt {b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} - b \sqrt {b^{2}}} - \frac {b x}{b^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + b^{2} \tan {\left (\frac {x}{2} \right )} - b \sqrt {b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} - b \sqrt {b^{2}}} - \frac {2 b \tan {\left (\frac {x}{2} \right )}}{b^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + b^{2} \tan {\left (\frac {x}{2} \right )} - b \sqrt {b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} - b \sqrt {b^{2}}} + \frac {x \sqrt {b^{2}} \tan ^{3}{\left (\frac {x}{2} \right )}}{b^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + b^{2} \tan {\left (\frac {x}{2} \right )} - b \sqrt {b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} - b \sqrt {b^{2}}} + \frac {x \sqrt {b^{2}} \tan {\left (\frac {x}{2} \right )}}{b^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + b^{2} \tan {\left (\frac {x}{2} \right )} - b \sqrt {b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} - b \sqrt {b^{2}}} + \frac {2 \sqrt {b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )}}{b^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + b^{2} \tan {\left (\frac {x}{2} \right )} - b \sqrt {b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} - b \sqrt {b^{2}}} + \frac {4 \sqrt {b^{2}}}{b^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + b^{2} \tan {\left (\frac {x}{2} \right )} - b \sqrt {b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} - b \sqrt {b^{2}}} & \text {for}\: a = - \sqrt {b^{2}} \\- \frac {b x \tan ^{2}{\left (\frac {x}{2} \right )}}{b^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + b^{2} \tan {\left (\frac {x}{2} \right )} + b \sqrt {b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} + b \sqrt {b^{2}}} - \frac {b x}{b^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + b^{2} \tan {\left (\frac {x}{2} \right )} + b \sqrt {b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} + b \sqrt {b^{2}}} - \frac {2 b \tan {\left (\frac {x}{2} \right )}}{b^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + b^{2} \tan {\left (\frac {x}{2} \right )} + b \sqrt {b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} + b \sqrt {b^{2}}} - \frac {x \sqrt {b^{2}} \tan ^{3}{\left (\frac {x}{2} \right )}}{b^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + b^{2} \tan {\left (\frac {x}{2} \right )} + b \sqrt {b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} + b \sqrt {b^{2}}} - \frac {x \sqrt {b^{2}} \tan {\left (\frac {x}{2} \right )}}{b^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + b^{2} \tan {\left (\frac {x}{2} \right )} + b \sqrt {b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} + b \sqrt {b^{2}}} - \frac {2 \sqrt {b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )}}{b^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + b^{2} \tan {\left (\frac {x}{2} \right )} + b \sqrt {b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} + b \sqrt {b^{2}}} - \frac {4 \sqrt {b^{2}}}{b^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + b^{2} \tan {\left (\frac {x}{2} \right )} + b \sqrt {b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} + b \sqrt {b^{2}}} & \text {for}\: a = \sqrt {b^{2}} \\\frac {\frac {x \sin ^{2}{\left (x \right )}}{2} + \frac {x \cos ^{2}{\left (x \right )}}{2} - \frac {\sin {\left (x \right )} \cos {\left (x \right )}}{2}}{a} & \text {for}\: b = 0 \\- \frac {\cos {\left (x \right )}}{b} & \text {for}\: a = 0 \\\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 )}}{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 )}}{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 )}}{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 )}}{b^{2} \sqrt {- a^{2} + b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} + b^{2} \sqrt {- a^{2} + b^{2}}} - \frac {a x \sqrt {- a^{2} + b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )}}{b^{2} \sqrt {- a^{2} + b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} + b^{2} \sqrt {- a^{2} + b^{2}}} - \frac {a x \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}}}{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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*cos(x), Eq(a, 0) & Eq(b, 0)), (-b*x*tan(x/2)**2/(b**2*tan(x/2)**3 + b**2*tan(x/2) - b*sqrt(b**2
)*tan(x/2)**2 - b*sqrt(b**2)) - b*x/(b**2*tan(x/2)**3 + b**2*tan(x/2) - b*sqrt(b**2)*tan(x/2)**2 - b*sqrt(b**2
)) - 2*b*tan(x/2)/(b**2*tan(x/2)**3 + b**2*tan(x/2) - b*sqrt(b**2)*tan(x/2)**2 - b*sqrt(b**2)) + x*sqrt(b**2)*
tan(x/2)**3/(b**2*tan(x/2)**3 + b**2*tan(x/2) - b*sqrt(b**2)*tan(x/2)**2 - b*sqrt(b**2)) + x*sqrt(b**2)*tan(x/
2)/(b**2*tan(x/2)**3 + b**2*tan(x/2) - b*sqrt(b**2)*tan(x/2)**2 - b*sqrt(b**2)) + 2*sqrt(b**2)*tan(x/2)**2/(b*
*2*tan(x/2)**3 + b**2*tan(x/2) - b*sqrt(b**2)*tan(x/2)**2 - b*sqrt(b**2)) + 4*sqrt(b**2)/(b**2*tan(x/2)**3 + b
**2*tan(x/2) - b*sqrt(b**2)*tan(x/2)**2 - b*sqrt(b**2)), Eq(a, -sqrt(b**2))), (-b*x*tan(x/2)**2/(b**2*tan(x/2)
**3 + b**2*tan(x/2) + b*sqrt(b**2)*tan(x/2)**2 + b*sqrt(b**2)) - b*x/(b**2*tan(x/2)**3 + b**2*tan(x/2) + b*sqr
t(b**2)*tan(x/2)**2 + b*sqrt(b**2)) - 2*b*tan(x/2)/(b**2*tan(x/2)**3 + b**2*tan(x/2) + b*sqrt(b**2)*tan(x/2)**
2 + b*sqrt(b**2)) - x*sqrt(b**2)*tan(x/2)**3/(b**2*tan(x/2)**3 + b**2*tan(x/2) + b*sqrt(b**2)*tan(x/2)**2 + b*
sqrt(b**2)) - x*sqrt(b**2)*tan(x/2)/(b**2*tan(x/2)**3 + b**2*tan(x/2) + b*sqrt(b**2)*tan(x/2)**2 + b*sqrt(b**2
)) - 2*sqrt(b**2)*tan(x/2)**2/(b**2*tan(x/2)**3 + b**2*tan(x/2) + b*sqrt(b**2)*tan(x/2)**2 + b*sqrt(b**2)) - 4
*sqrt(b**2)/(b**2*tan(x/2)**3 + b**2*tan(x/2) + b*sqrt(b**2)*tan(x/2)**2 + b*sqrt(b**2)), Eq(a, sqrt(b**2))),
((x*sin(x)**2/2 + x*cos(x)**2/2 - sin(x)*cos(x)/2)/a, Eq(b, 0)), (-cos(x)/b, Eq(a, 0)), (a**2*log(tan(x/2) + b
/a - sqrt(-a**2 + b**2)/a)*tan(x/2)**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)/(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/(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)/(b**2*sqrt(-a**2 + b**2)*tan(x/2)**2 + b**2*sqr
t(-a**2 + b**2)) - a*x*sqrt(-a**2 + b**2)*tan(x/2)**2/(b**2*sqrt(-a**2 + b**2)*tan(x/2)**2 + b**2*sqrt(-a**2 +
b**2)) - a*x*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)/(b**2*sqrt(-a**2 + b**2)*tan(x/2)**2 + b**2*sqrt(-a**2 + b**2)), True))

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Giac [A]
time = 0.45, size = 77, normalized size = 1.26 \begin {gather*} \frac {2 \, {\left (\pi \left \lfloor \frac {x}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, x\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )} a^{2}}{\sqrt {a^{2} - b^{2}} b^{2}} - \frac {a x}{b^{2}} - \frac {2}{{\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(pi*floor(1/2*x/pi + 1/2)*sgn(a) + arctan((a*tan(1/2*x) + b)/sqrt(a^2 - b^2)))*a^2/(sqrt(a^2 - b^2)*b^2) - a
*x/b^2 - 2/((tan(1/2*x)^2 + 1)*b)

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Mupad [B]
time = 6.91, size = 623, normalized size = 10.21 \begin {gather*} -\frac {2}{b\,\left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )}-\frac {a\,x}{b^2}-\frac {a^2\,\mathrm {atan}\left (\frac {\frac {a^2\,\left (\frac {32\,a^4}{b}-\frac {32\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (2\,a^5\,b-2\,a^3\,b^3\right )}{b^3}+\frac {a^2\,\left (32\,a^2\,b^2+64\,a^3\,b\,\mathrm {tan}\left (\frac {x}{2}\right )+\frac {a^2\,\left (32\,a^2\,b^3+\frac {32\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (3\,a\,b^7-2\,a^3\,b^5\right )}{b^3}\right )}{b^2\,\sqrt {b^2-a^2}}\right )}{b^2\,\sqrt {b^2-a^2}}\right )\,1{}\mathrm {i}}{b^2\,\sqrt {b^2-a^2}}-\frac {a^2\,\left (\frac {32\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (2\,a^5\,b-2\,a^3\,b^3\right )}{b^3}-\frac {32\,a^4}{b}+\frac {a^2\,\left (32\,a^2\,b^2+64\,a^3\,b\,\mathrm {tan}\left (\frac {x}{2}\right )-\frac {a^2\,\left (32\,a^2\,b^3+\frac {32\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (3\,a\,b^7-2\,a^3\,b^5\right )}{b^3}\right )}{b^2\,\sqrt {b^2-a^2}}\right )}{b^2\,\sqrt {b^2-a^2}}\right )\,1{}\mathrm {i}}{b^2\,\sqrt {b^2-a^2}}}{\frac {128\,a^5\,\mathrm {tan}\left (\frac {x}{2}\right )}{b^3}+\frac {a^2\,\left (\frac {32\,a^4}{b}-\frac {32\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (2\,a^5\,b-2\,a^3\,b^3\right )}{b^3}+\frac {a^2\,\left (32\,a^2\,b^2+64\,a^3\,b\,\mathrm {tan}\left (\frac {x}{2}\right )+\frac {a^2\,\left (32\,a^2\,b^3+\frac {32\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (3\,a\,b^7-2\,a^3\,b^5\right )}{b^3}\right )}{b^2\,\sqrt {b^2-a^2}}\right )}{b^2\,\sqrt {b^2-a^2}}\right )}{b^2\,\sqrt {b^2-a^2}}+\frac {a^2\,\left (\frac {32\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (2\,a^5\,b-2\,a^3\,b^3\right )}{b^3}-\frac {32\,a^4}{b}+\frac {a^2\,\left (32\,a^2\,b^2+64\,a^3\,b\,\mathrm {tan}\left (\frac {x}{2}\right )-\frac {a^2\,\left (32\,a^2\,b^3+\frac {32\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (3\,a\,b^7-2\,a^3\,b^5\right )}{b^3}\right )}{b^2\,\sqrt {b^2-a^2}}\right )}{b^2\,\sqrt {b^2-a^2}}\right )}{b^2\,\sqrt {b^2-a^2}}}\right )\,2{}\mathrm {i}}{b^2\,\sqrt {b^2-a^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^2/(a + b*sin(x)),x)

[Out]

- 2/(b*(tan(x/2)^2 + 1)) - (a*x)/b^2 - (a^2*atan(((a^2*((32*a^4)/b - (32*tan(x/2)*(2*a^5*b - 2*a^3*b^3))/b^3 +
(a^2*(32*a^2*b^2 + 64*a^3*b*tan(x/2) + (a^2*(32*a^2*b^3 + (32*tan(x/2)*(3*a*b^7 - 2*a^3*b^5))/b^3))/(b^2*(b^2
- a^2)^(1/2))))/(b^2*(b^2 - a^2)^(1/2)))*1i)/(b^2*(b^2 - a^2)^(1/2)) - (a^2*((32*tan(x/2)*(2*a^5*b - 2*a^3*b^
3))/b^3 - (32*a^4)/b + (a^2*(32*a^2*b^2 + 64*a^3*b*tan(x/2) - (a^2*(32*a^2*b^3 + (32*tan(x/2)*(3*a*b^7 - 2*a^3
*b^5))/b^3))/(b^2*(b^2 - a^2)^(1/2))))/(b^2*(b^2 - a^2)^(1/2)))*1i)/(b^2*(b^2 - a^2)^(1/2)))/((128*a^5*tan(x/2
))/b^3 + (a^2*((32*a^4)/b - (32*tan(x/2)*(2*a^5*b - 2*a^3*b^3))/b^3 + (a^2*(32*a^2*b^2 + 64*a^3*b*tan(x/2) + (
a^2*(32*a^2*b^3 + (32*tan(x/2)*(3*a*b^7 - 2*a^3*b^5))/b^3))/(b^2*(b^2 - a^2)^(1/2))))/(b^2*(b^2 - a^2)^(1/2)))
)/(b^2*(b^2 - a^2)^(1/2)) + (a^2*((32*tan(x/2)*(2*a^5*b - 2*a^3*b^3))/b^3 - (32*a^4)/b + (a^2*(32*a^2*b^2 + 64
*a^3*b*tan(x/2) - (a^2*(32*a^2*b^3 + (32*tan(x/2)*(3*a*b^7 - 2*a^3*b^5))/b^3))/(b^2*(b^2 - a^2)^(1/2))))/(b^2*
(b^2 - a^2)^(1/2))))/(b^2*(b^2 - a^2)^(1/2))))*2i)/(b^2*(b^2 - a^2)^(1/2))

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