\(\int \frac {\sin ^3(x)}{a+b \cos (x)} \, dx\) [25]

Optimal result

Integrand size = 13, antiderivative size = 40 \[ \int \frac {\sin ^3(x)}{a+b \cos (x)} \, dx=-\frac {a \cos (x)}{b^2}+\frac {\cos ^2(x)}{2 b}+\frac {\left (a^2-b^2\right ) \log (a+b \cos (x))}{b^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2747, 711} \[ \int \frac {\sin ^3(x)}{a+b \cos (x)} \, dx=\frac {\left (a^2-b^2\right ) \log (a+b \cos (x))}{b^3}-\frac {a \cos (x)}{b^2}+\frac {\cos ^2(x)}{2 b} \]

[In]

Int[Sin[x]^3/(a + b*Cos[x]),x]

[Out]

-((a*Cos[x])/b^2) + Cos[x]^2/(2*b) + ((a^2 - b^2)*Log[a + b*Cos[x]])/b^3

Rule 711

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rule 2747

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00 \[ \int \frac {\sin ^3(x)}{a+b \cos (x)} \, dx=-\frac {a \cos (x)}{b^2}+\frac {\cos (2 x)}{4 b}+\frac {\left (a^2-b^2\right ) \log (a+b \cos (x))}{b^3} \]

[In]

Integrate[Sin[x]^3/(a + b*Cos[x]),x]

[Out]

-((a*Cos[x])/b^2) + Cos[2*x]/(4*b) + ((a^2 - b^2)*Log[a + b*Cos[x]])/b^3

Maple [A] (verified)

Time = 0.80 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.98

method	result	size
derivativedivides	\(-\frac {-\frac {b \left (\cos ^{2}\left (x \right )\right )}{2}+\cos \left (x \right ) a}{b^{2}}+\frac {\left (a^{2}-b^{2}\right ) \ln \left (a +\cos \left (x \right ) b \right )}{b^{3}}\)	\(39\)
default	\(-\frac {-\frac {b \left (\cos ^{2}\left (x \right )\right )}{2}+\cos \left (x \right ) a}{b^{2}}+\frac {\left (a^{2}-b^{2}\right ) \ln \left (a +\cos \left (x \right ) b \right )}{b^{3}}\)	\(39\)
parallelrisch	\(\frac {\left (a^{2}-b^{2}\right ) \ln \left (\frac {a +\cos \left (x \right ) b}{\cos \left (x \right )+1}\right )+\left (-a^{2}+b^{2}\right ) \ln \left (\frac {1}{\cos \left (x \right )+1}\right )-b \left (\cos \left (x \right ) a -\frac {b \cos \left (2 x \right )}{4}+a +\frac {b}{4}\right )}{b^{3}}\)	\(66\)
norman	\(\frac {\frac {2 a \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{b^{2}}-\frac {2 a -2 b}{3 b^{2}}+\frac {\left (4 a +2 b \right ) \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{3 b^{2}}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{3}}+\frac {\left (a -b \right ) \left (a +b \right ) \ln \left (a \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+a +b \right )}{b^{3}}-\frac {\left (a -b \right ) \left (a +b \right ) \ln \left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )}{b^{3}}\)	\(111\)
risch	\(-\frac {i a^{2} x}{b^{3}}+\frac {i x}{b}+\frac {{\mathrm e}^{2 i x}}{8 b}-\frac {a \,{\mathrm e}^{i x}}{2 b^{2}}-\frac {a \,{\mathrm e}^{-i x}}{2 b^{2}}+\frac {{\mathrm e}^{-2 i x}}{8 b}+\frac {\ln \left ({\mathrm e}^{2 i x}+\frac {2 a \,{\mathrm e}^{i x}}{b}+1\right ) a^{2}}{b^{3}}-\frac {\ln \left ({\mathrm e}^{2 i x}+\frac {2 a \,{\mathrm e}^{i x}}{b}+1\right )}{b}\)	\(111\)

[In]

int(sin(x)^3/(a+cos(x)*b),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.02 \[ \int \frac {\sin ^3(x)}{a+b \cos (x)} \, dx=\frac {b^{2} \cos \left (x\right )^{2} - 2 \, a b \cos \left (x\right ) + 2 \, {\left (a^{2} - b^{2}\right )} \log \left (-b \cos \left (x\right ) - a\right )}{2 \, b^{3}} \]

[In]

integrate(sin(x)^3/(a+b*cos(x)),x, algorithm="fricas")

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1421 vs. \(2 (34) = 68\).

Time = 173.47 (sec) , antiderivative size = 1421, normalized size of antiderivative = 35.52 \[ \int \frac {\sin ^3(x)}{a+b \cos (x)} \, dx=\text {Too large to display} \]

[In]

integrate(sin(x)**3/(a+b*cos(x)),x)

[Out]

Piecewise((zoo*(-log(tan(x/2) - 1)*tan(x/2)**4/(tan(x/2)**4 + 2*tan(x/2)**2 + 1) - 2*log(tan(x/2) - 1)*tan(x/2
)**2/(tan(x/2)**4 + 2*tan(x/2)**2 + 1) - log(tan(x/2) - 1)/(tan(x/2)**4 + 2*tan(x/2)**2 + 1) - log(tan(x/2) +
1)*tan(x/2)**4/(tan(x/2)**4 + 2*tan(x/2)**2 + 1) - 2*log(tan(x/2) + 1)*tan(x/2)**2/(tan(x/2)**4 + 2*tan(x/2)**
2 + 1) - log(tan(x/2) + 1)/(tan(x/2)**4 + 2*tan(x/2)**2 + 1) + log(tan(x/2)**2 + 1)*tan(x/2)**4/(tan(x/2)**4 +
2*tan(x/2)**2 + 1) + 2*log(tan(x/2)**2 + 1)*tan(x/2)**2/(tan(x/2)**4 + 2*tan(x/2)**2 + 1) + log(tan(x/2)**2 +
1)/(tan(x/2)**4 + 2*tan(x/2)**2 + 1) - 2*tan(x/2)**2/(tan(x/2)**4 + 2*tan(x/2)**2 + 1)), Eq(a, 0) & Eq(b, 0))
, (-4*tan(x/2)**2/(b*tan(x/2)**4 + 2*b*tan(x/2)**2 + b) - 2/(b*tan(x/2)**4 + 2*b*tan(x/2)**2 + b), Eq(a, b)),
((-sin(x)**2*cos(x) - 2*cos(x)**3/3)/a, Eq(b, 0)), (a**2*log(-sqrt(-a/(a - b) - b/(a - b)) + tan(x/2))*tan(x/2
)**4/(b**3*tan(x/2)**4 + 2*b**3*tan(x/2)**2 + b**3) + 2*a**2*log(-sqrt(-a/(a - b) - b/(a - b)) + tan(x/2))*tan
(x/2)**2/(b**3*tan(x/2)**4 + 2*b**3*tan(x/2)**2 + b**3) + a**2*log(-sqrt(-a/(a - b) - b/(a - b)) + tan(x/2))/(
b**3*tan(x/2)**4 + 2*b**3*tan(x/2)**2 + b**3) + a**2*log(sqrt(-a/(a - b) - b/(a - b)) + tan(x/2))*tan(x/2)**4/
(b**3*tan(x/2)**4 + 2*b**3*tan(x/2)**2 + b**3) + 2*a**2*log(sqrt(-a/(a - b) - b/(a - b)) + tan(x/2))*tan(x/2)*
*2/(b**3*tan(x/2)**4 + 2*b**3*tan(x/2)**2 + b**3) + a**2*log(sqrt(-a/(a - b) - b/(a - b)) + tan(x/2))/(b**3*ta
n(x/2)**4 + 2*b**3*tan(x/2)**2 + b**3) - a**2*log(tan(x/2)**2 + 1)*tan(x/2)**4/(b**3*tan(x/2)**4 + 2*b**3*tan(
x/2)**2 + b**3) - 2*a**2*log(tan(x/2)**2 + 1)*tan(x/2)**2/(b**3*tan(x/2)**4 + 2*b**3*tan(x/2)**2 + b**3) - a**
2*log(tan(x/2)**2 + 1)/(b**3*tan(x/2)**4 + 2*b**3*tan(x/2)**2 + b**3) - 2*a*b*tan(x/2)**2/(b**3*tan(x/2)**4 +
2*b**3*tan(x/2)**2 + b**3) - 2*a*b/(b**3*tan(x/2)**4 + 2*b**3*tan(x/2)**2 + b**3) - b**2*log(-sqrt(-a/(a - b)
- b/(a - b)) + tan(x/2))*tan(x/2)**4/(b**3*tan(x/2)**4 + 2*b**3*tan(x/2)**2 + b**3) - 2*b**2*log(-sqrt(-a/(a -
b) - b/(a - b)) + tan(x/2))*tan(x/2)**2/(b**3*tan(x/2)**4 + 2*b**3*tan(x/2)**2 + b**3) - b**2*log(-sqrt(-a/(a
- b) - b/(a - b)) + tan(x/2))/(b**3*tan(x/2)**4 + 2*b**3*tan(x/2)**2 + b**3) - b**2*log(sqrt(-a/(a - b) - b/(
a - b)) + tan(x/2))*tan(x/2)**4/(b**3*tan(x/2)**4 + 2*b**3*tan(x/2)**2 + b**3) - 2*b**2*log(sqrt(-a/(a - b) -
b/(a - b)) + tan(x/2))*tan(x/2)**2/(b**3*tan(x/2)**4 + 2*b**3*tan(x/2)**2 + b**3) - b**2*log(sqrt(-a/(a - b) -
b/(a - b)) + tan(x/2))/(b**3*tan(x/2)**4 + 2*b**3*tan(x/2)**2 + b**3) + b**2*log(tan(x/2)**2 + 1)*tan(x/2)**4
/(b**3*tan(x/2)**4 + 2*b**3*tan(x/2)**2 + b**3) + 2*b**2*log(tan(x/2)**2 + 1)*tan(x/2)**2/(b**3*tan(x/2)**4 +
2*b**3*tan(x/2)**2 + b**3) + b**2*log(tan(x/2)**2 + 1)/(b**3*tan(x/2)**4 + 2*b**3*tan(x/2)**2 + b**3) - 2*b**2
*tan(x/2)**2/(b**3*tan(x/2)**4 + 2*b**3*tan(x/2)**2 + b**3), True))

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.95 \[ \int \frac {\sin ^3(x)}{a+b \cos (x)} \, dx=\frac {b \cos \left (x\right )^{2} - 2 \, a \cos \left (x\right )}{2 \, b^{2}} + \frac {{\left (a^{2} - b^{2}\right )} \log \left (b \cos \left (x\right ) + a\right )}{b^{3}} \]

[In]

integrate(sin(x)^3/(a+b*cos(x)),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.98 \[ \int \frac {\sin ^3(x)}{a+b \cos (x)} \, dx=\frac {b \cos \left (x\right )^{2} - 2 \, a \cos \left (x\right )}{2 \, b^{2}} + \frac {{\left (a^{2} - b^{2}\right )} \log \left ({\left | b \cos \left (x\right ) + a \right |}\right )}{b^{3}} \]

[In]

integrate(sin(x)^3/(a+b*cos(x)),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.95 \[ \int \frac {\sin ^3(x)}{a+b \cos (x)} \, dx=\frac {{\cos \left (x\right )}^2}{2\,b}+\frac {\ln \left (a+b\,\cos \left (x\right )\right )\,\left (a^2-b^2\right )}{b^3}-\frac {a\,\cos \left (x\right )}{b^2} \]

[In]

int(sin(x)^3/(a + b*cos(x)),x)

[Out]

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