3.16 \(\int \frac{\sin ^4(x)}{a+b \cot (x)} \, dx\)

Optimal. Leaf size=120 \[ \frac{a x \left (10 a^2 b^2+3 a^4+15 b^4\right )}{8 \left (a^2+b^2\right )^3}-\frac{\sin ^4(x) (a \cot (x)+b)}{4 \left (a^2+b^2\right )}-\frac{\sin ^2(x) \left (a \left (3 a^2+7 b^2\right ) \cot (x)+4 b^3\right )}{8 \left (a^2+b^2\right )^2}-\frac{b^5 \log (a \sin (x)+b \cos (x))}{\left (a^2+b^2\right )^3} \]

[Out]

(a*(3*a^4 + 10*a^2*b^2 + 15*b^4)*x)/(8*(a^2 + b^2)^3) - (b^5*Log[b*Cos[x] + a*Sin[x]])/(a^2 + b^2)^3 - ((4*b^3
+ a*(3*a^2 + 7*b^2)*Cot[x])*Sin[x]^2)/(8*(a^2 + b^2)^2) - ((b + a*Cot[x])*Sin[x]^4)/(4*(a^2 + b^2))

________________________________________________________________________________________

Rubi [A]  time = 0.193016, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538, Rules used = {3506, 741, 823, 801, 635, 203, 260} \[ \frac{a x \left (10 a^2 b^2+3 a^4+15 b^4\right )}{8 \left (a^2+b^2\right )^3}-\frac{\sin ^4(x) (a \cot (x)+b)}{4 \left (a^2+b^2\right )}-\frac{\sin ^2(x) \left (a \left (3 a^2+7 b^2\right ) \cot (x)+4 b^3\right )}{8 \left (a^2+b^2\right )^2}-\frac{b^5 \log (a \sin (x)+b \cos (x))}{\left (a^2+b^2\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]^4/(a + b*Cot[x]),x]

[Out]

(a*(3*a^4 + 10*a^2*b^2 + 15*b^4)*x)/(8*(a^2 + b^2)^3) - (b^5*Log[b*Cos[x] + a*Sin[x]])/(a^2 + b^2)^3 - ((4*b^3
+ a*(3*a^2 + 7*b^2)*Cot[x])*Sin[x]^2)/(8*(a^2 + b^2)^2) - ((b + a*Cot[x])*Sin[x]^4)/(4*(a^2 + b^2))

Rule 3506

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[1/(b*f), Subst
[Int[(a + x)^n*(1 + x^2/b^2)^(m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x] && NeQ[a^2 + b
^2, 0] && IntegerQ[m/2]

Rule 741

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*(a*e + c*d*x)*(
a + c*x^2)^(p + 1))/(2*a*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[1/(2*a*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*
Simp[c*d^2*(2*p + 3) + a*e^2*(m + 2*p + 3) + c*e*d*(m + 2*p + 4)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a
, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 823

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(
m + 1)*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), x] + Di
st[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^2*
(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 801

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
(d + e*x)^m*(f + g*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

Rule 203

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

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

\begin{align*} \int \frac{\sin ^4(x)}{a+b \cot (x)} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{(a+x) \left (1+\frac{x^2}{b^2}\right )^3} \, dx,x,b \cot (x)\right )}{b}\\ &=-\frac{(b+a \cot (x)) \sin ^4(x)}{4 \left (a^2+b^2\right )}+\frac{b \operatorname{Subst}\left (\int \frac{-4-\frac{3 a^2}{b^2}-\frac{3 a x}{b^2}}{(a+x) \left (1+\frac{x^2}{b^2}\right )^2} \, dx,x,b \cot (x)\right )}{4 \left (a^2+b^2\right )}\\ &=-\frac{\left (4 b^3+a \left (3 a^2+7 b^2\right ) \cot (x)\right ) \sin ^2(x)}{8 \left (a^2+b^2\right )^2}-\frac{(b+a \cot (x)) \sin ^4(x)}{4 \left (a^2+b^2\right )}-\frac{b^5 \operatorname{Subst}\left (\int \frac{\frac{3 a^4+7 a^2 b^2+8 b^4}{b^6}+\frac{a \left (3 a^2+7 b^2\right ) x}{b^6}}{(a+x) \left (1+\frac{x^2}{b^2}\right )} \, dx,x,b \cot (x)\right )}{8 \left (a^2+b^2\right )^2}\\ &=-\frac{\left (4 b^3+a \left (3 a^2+7 b^2\right ) \cot (x)\right ) \sin ^2(x)}{8 \left (a^2+b^2\right )^2}-\frac{(b+a \cot (x)) \sin ^4(x)}{4 \left (a^2+b^2\right )}-\frac{b^5 \operatorname{Subst}\left (\int \left (\frac{8}{\left (a^2+b^2\right ) (a+x)}+\frac{3 a^5+10 a^3 b^2+15 a b^4-8 b^4 x}{b^4 \left (a^2+b^2\right ) \left (b^2+x^2\right )}\right ) \, dx,x,b \cot (x)\right )}{8 \left (a^2+b^2\right )^2}\\ &=-\frac{b^5 \log (a+b \cot (x))}{\left (a^2+b^2\right )^3}-\frac{\left (4 b^3+a \left (3 a^2+7 b^2\right ) \cot (x)\right ) \sin ^2(x)}{8 \left (a^2+b^2\right )^2}-\frac{(b+a \cot (x)) \sin ^4(x)}{4 \left (a^2+b^2\right )}-\frac{b \operatorname{Subst}\left (\int \frac{3 a^5+10 a^3 b^2+15 a b^4-8 b^4 x}{b^2+x^2} \, dx,x,b \cot (x)\right )}{8 \left (a^2+b^2\right )^3}\\ &=-\frac{b^5 \log (a+b \cot (x))}{\left (a^2+b^2\right )^3}-\frac{\left (4 b^3+a \left (3 a^2+7 b^2\right ) \cot (x)\right ) \sin ^2(x)}{8 \left (a^2+b^2\right )^2}-\frac{(b+a \cot (x)) \sin ^4(x)}{4 \left (a^2+b^2\right )}+\frac{b^5 \operatorname{Subst}\left (\int \frac{x}{b^2+x^2} \, dx,x,b \cot (x)\right )}{\left (a^2+b^2\right )^3}-\frac{\left (a b \left (3 a^4+10 a^2 b^2+15 b^4\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b^2+x^2} \, dx,x,b \cot (x)\right )}{8 \left (a^2+b^2\right )^3}\\ &=\frac{a \left (3 a^4+10 a^2 b^2+15 b^4\right ) x}{8 \left (a^2+b^2\right )^3}-\frac{b^5 \log (a+b \cot (x))}{\left (a^2+b^2\right )^3}-\frac{b^5 \log (\sin (x))}{\left (a^2+b^2\right )^3}-\frac{\left (4 b^3+a \left (3 a^2+7 b^2\right ) \cot (x)\right ) \sin ^2(x)}{8 \left (a^2+b^2\right )^2}-\frac{(b+a \cot (x)) \sin ^4(x)}{4 \left (a^2+b^2\right )}\\ \end{align*}

Mathematica [A]  time = 0.247769, size = 151, normalized size = 1.26 \[ \frac{40 a^3 b^2 x-24 a^3 b^2 \sin (2 x)+2 a^3 b^2 \sin (4 x)+4 b \left (4 a^2 b^2+a^4+3 b^4\right ) \cos (2 x)-b \left (a^2+b^2\right )^2 \cos (4 x)+12 a^5 x-8 a^5 \sin (2 x)+a^5 \sin (4 x)+60 a b^4 x-16 a b^4 \sin (2 x)+a b^4 \sin (4 x)-32 b^5 \log (a \sin (x)+b \cos (x))}{32 \left (a^2+b^2\right )^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]^4/(a + b*Cot[x]),x]

[Out]

(12*a^5*x + 40*a^3*b^2*x + 60*a*b^4*x + 4*b*(a^4 + 4*a^2*b^2 + 3*b^4)*Cos[2*x] - b*(a^2 + b^2)^2*Cos[4*x] - 32
*b^5*Log[b*Cos[x] + a*Sin[x]] - 8*a^5*Sin[2*x] - 24*a^3*b^2*Sin[2*x] - 16*a*b^4*Sin[2*x] + a^5*Sin[4*x] + 2*a^
3*b^2*Sin[4*x] + a*b^4*Sin[4*x])/(32*(a^2 + b^2)^3)

________________________________________________________________________________________

Maple [B]  time = 0.069, size = 407, normalized size = 3.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^4/(a+b*cot(x)),x)

[Out]

-7/4/(a^2+b^2)^3/(1+tan(x)^2)^2*tan(x)^3*a^3*b^2-9/8/(a^2+b^2)^3/(1+tan(x)^2)^2*tan(x)^3*a*b^4-5/8/(a^2+b^2)^3
/(1+tan(x)^2)^2*tan(x)^3*a^5+1/2/(a^2+b^2)^3/(1+tan(x)^2)^2*tan(x)^2*a^4*b+3/2/(a^2+b^2)^3/(1+tan(x)^2)^2*tan(
x)^2*a^2*b^3+1/(a^2+b^2)^3/(1+tan(x)^2)^2*tan(x)^2*b^5-3/8/(a^2+b^2)^3/(1+tan(x)^2)^2*tan(x)*a^5-5/4/(a^2+b^2)
^3/(1+tan(x)^2)^2*tan(x)*a^3*b^2-7/8/(a^2+b^2)^3/(1+tan(x)^2)^2*tan(x)*a*b^4+1/4/(a^2+b^2)^3/(1+tan(x)^2)^2*a^
4*b+1/(a^2+b^2)^3/(1+tan(x)^2)^2*a^2*b^3+3/4/(a^2+b^2)^3/(1+tan(x)^2)^2*b^5+1/2/(a^2+b^2)^3*b^5*ln(1+tan(x)^2)
+15/8/(a^2+b^2)^3*arctan(tan(x))*a*b^4+3/8/(a^2+b^2)^3*arctan(tan(x))*a^5+5/4/(a^2+b^2)^3*arctan(tan(x))*a^3*b
^2-1/(a^2+b^2)^3*b^5*ln(a*tan(x)+b)

________________________________________________________________________________________

Maxima [B]  time = 1.84045, size = 329, normalized size = 2.74 \begin{align*} -\frac{b^{5} \log \left (a \tan \left (x\right ) + b\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac{b^{5} \log \left (\tan \left (x\right )^{2} + 1\right )}{2 \,{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} + \frac{{\left (3 \, a^{5} + 10 \, a^{3} b^{2} + 15 \, a b^{4}\right )} x}{8 \,{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} - \frac{{\left (5 \, a^{3} + 9 \, a b^{2}\right )} \tan \left (x\right )^{3} - 2 \, a^{2} b - 6 \, b^{3} - 4 \,{\left (a^{2} b + 2 \, b^{3}\right )} \tan \left (x\right )^{2} +{\left (3 \, a^{3} + 7 \, a b^{2}\right )} \tan \left (x\right )}{8 \,{\left ({\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \tan \left (x\right )^{4} + a^{4} + 2 \, a^{2} b^{2} + b^{4} + 2 \,{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \tan \left (x\right )^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^4/(a+b*cot(x)),x, algorithm="maxima")

[Out]

-b^5*log(a*tan(x) + b)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + 1/2*b^5*log(tan(x)^2 + 1)/(a^6 + 3*a^4*b^2 + 3*a^
2*b^4 + b^6) + 1/8*(3*a^5 + 10*a^3*b^2 + 15*a*b^4)*x/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - 1/8*((5*a^3 + 9*a*b
^2)*tan(x)^3 - 2*a^2*b - 6*b^3 - 4*(a^2*b + 2*b^3)*tan(x)^2 + (3*a^3 + 7*a*b^2)*tan(x))/((a^4 + 2*a^2*b^2 + b^
4)*tan(x)^4 + a^4 + 2*a^2*b^2 + b^4 + 2*(a^4 + 2*a^2*b^2 + b^4)*tan(x)^2)

________________________________________________________________________________________

Fricas [A]  time = 1.99056, size = 416, normalized size = 3.47 \begin{align*} -\frac{4 \, b^{5} \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^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (x\right )^{4} - 4 \,{\left (a^{4} b + 3 \, a^{2} b^{3} + 2 \, b^{5}\right )} \cos \left (x\right )^{2} -{\left (3 \, a^{5} + 10 \, a^{3} b^{2} + 15 \, a b^{4}\right )} x -{\left (2 \,{\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \cos \left (x\right )^{3} -{\left (5 \, a^{5} + 14 \, a^{3} b^{2} + 9 \, a b^{4}\right )} \cos \left (x\right )\right )} \sin \left (x\right )}{8 \,{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^4/(a+b*cot(x)),x, algorithm="fricas")

[Out]

-1/8*(4*b^5*log(2*a*b*cos(x)*sin(x) - (a^2 - b^2)*cos(x)^2 + a^2) + 2*(a^4*b + 2*a^2*b^3 + b^5)*cos(x)^4 - 4*(
a^4*b + 3*a^2*b^3 + 2*b^5)*cos(x)^2 - (3*a^5 + 10*a^3*b^2 + 15*a*b^4)*x - (2*(a^5 + 2*a^3*b^2 + a*b^4)*cos(x)^
3 - (5*a^5 + 14*a^3*b^2 + 9*a*b^4)*cos(x))*sin(x))/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin ^{4}{\left (x \right )}}{a + b \cot{\left (x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)**4/(a+b*cot(x)),x)

[Out]

Integral(sin(x)**4/(a + b*cot(x)), x)

________________________________________________________________________________________

Giac [B]  time = 1.30127, size = 369, normalized size = 3.08 \begin{align*} -\frac{a b^{5} \log \left ({\left | a \tan \left (x\right ) + b \right |}\right )}{a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}} + \frac{b^{5} \log \left (\tan \left (x\right )^{2} + 1\right )}{2 \,{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} + \frac{{\left (3 \, a^{5} + 10 \, a^{3} b^{2} + 15 \, a b^{4}\right )} x}{8 \,{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} - \frac{6 \, b^{5} \tan \left (x\right )^{4} + 5 \, a^{5} \tan \left (x\right )^{3} + 14 \, a^{3} b^{2} \tan \left (x\right )^{3} + 9 \, a b^{4} \tan \left (x\right )^{3} - 4 \, a^{4} b \tan \left (x\right )^{2} - 12 \, a^{2} b^{3} \tan \left (x\right )^{2} + 4 \, b^{5} \tan \left (x\right )^{2} + 3 \, a^{5} \tan \left (x\right ) + 10 \, a^{3} b^{2} \tan \left (x\right ) + 7 \, a b^{4} \tan \left (x\right ) - 2 \, a^{4} b - 8 \, a^{2} b^{3}}{8 \,{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}{\left (\tan \left (x\right )^{2} + 1\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^4/(a+b*cot(x)),x, algorithm="giac")

[Out]

-a*b^5*log(abs(a*tan(x) + b))/(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6) + 1/2*b^5*log(tan(x)^2 + 1)/(a^6 + 3*a^4*b
^2 + 3*a^2*b^4 + b^6) + 1/8*(3*a^5 + 10*a^3*b^2 + 15*a*b^4)*x/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - 1/8*(6*b^5
*tan(x)^4 + 5*a^5*tan(x)^3 + 14*a^3*b^2*tan(x)^3 + 9*a*b^4*tan(x)^3 - 4*a^4*b*tan(x)^2 - 12*a^2*b^3*tan(x)^2 +
4*b^5*tan(x)^2 + 3*a^5*tan(x) + 10*a^3*b^2*tan(x) + 7*a*b^4*tan(x) - 2*a^4*b - 8*a^2*b^3)/((a^6 + 3*a^4*b^2 +
3*a^2*b^4 + b^6)*(tan(x)^2 + 1)^2)