3.10 \(\int \frac{\sin ^4(x)}{a+a \csc (x)} \, dx\)

Optimal. Leaf size=66 \[ \frac{15 x}{8 a}-\frac{4 \cos ^3(x)}{3 a}+\frac{4 \cos (x)}{a}-\frac{5 \sin ^3(x) \cos (x)}{4 a}-\frac{15 \sin (x) \cos (x)}{8 a}+\frac{\sin ^3(x) \cos (x)}{a \csc (x)+a} \]

[Out]

(15*x)/(8*a) + (4*Cos[x])/a - (4*Cos[x]^3)/(3*a) - (15*Cos[x]*Sin[x])/(8*a) - (5*Cos[x]*Sin[x]^3)/(4*a) + (Cos
[x]*Sin[x]^3)/(a + a*Csc[x])

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Rubi [A]  time = 0.0733532, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {3819, 3787, 2635, 8, 2633} \[ \frac{15 x}{8 a}-\frac{4 \cos ^3(x)}{3 a}+\frac{4 \cos (x)}{a}-\frac{5 \sin ^3(x) \cos (x)}{4 a}-\frac{15 \sin (x) \cos (x)}{8 a}+\frac{\sin ^3(x) \cos (x)}{a \csc (x)+a} \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]^4/(a + a*Csc[x]),x]

[Out]

(15*x)/(8*a) + (4*Cos[x])/a - (4*Cos[x]^3)/(3*a) - (15*Cos[x]*Sin[x])/(8*a) - (5*Cos[x]*Sin[x]^3)/(4*a) + (Cos
[x]*Sin[x]^3)/(a + a*Csc[x])

Rule 3819

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[(Cot[e + f*
x]*(d*Csc[e + f*x])^n)/(f*(a + b*Csc[e + f*x])), x] - Dist[1/a^2, Int[(d*Csc[e + f*x])^n*(a*(n - 1) - b*n*Csc[
e + f*x]), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[n, 0]

Rule 3787

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[a, Int[(d*
Csc[e + f*x])^n, x], x] + Dist[b/d, Int[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rubi steps

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

Mathematica [A]  time = 0.20905, size = 57, normalized size = 0.86 \[ \frac{168 \cos (x)-8 \cos (3 x)+3 \left (60 x-16 \sin (2 x)+\sin (4 x)-\frac{64 \sin \left (\frac{x}{2}\right )}{\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )}\right )}{96 a} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]^4/(a + a*Csc[x]),x]

[Out]

(168*Cos[x] - 8*Cos[3*x] + 3*(60*x - (64*Sin[x/2])/(Cos[x/2] + Sin[x/2]) - 16*Sin[2*x] + Sin[4*x]))/(96*a)

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Maple [B]  time = 0.046, size = 185, normalized size = 2.8 \begin{align*}{\frac{7}{4\,a} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{7} \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-4}}+2\,{\frac{ \left ( \tan \left ( x/2 \right ) \right ) ^{6}}{a \left ( \left ( \tan \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{4}}}+{\frac{15}{4\,a} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{5} \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-4}}+10\,{\frac{ \left ( \tan \left ( x/2 \right ) \right ) ^{4}}{a \left ( \left ( \tan \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{4}}}-{\frac{15}{4\,a} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{3} \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-4}}+{\frac{34}{3\,a} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2} \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-4}}-{\frac{7}{4\,a}\tan \left ({\frac{x}{2}} \right ) \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-4}}+{\frac{10}{3\,a} \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-4}}+{\frac{15}{4\,a}\arctan \left ( \tan \left ({\frac{x}{2}} \right ) \right ) }+2\,{\frac{1}{a \left ( \tan \left ( x/2 \right ) +1 \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

7/4/a/(tan(1/2*x)^2+1)^4*tan(1/2*x)^7+2/a/(tan(1/2*x)^2+1)^4*tan(1/2*x)^6+15/4/a/(tan(1/2*x)^2+1)^4*tan(1/2*x)
^5+10/a/(tan(1/2*x)^2+1)^4*tan(1/2*x)^4-15/4/a/(tan(1/2*x)^2+1)^4*tan(1/2*x)^3+34/3/a/(tan(1/2*x)^2+1)^4*tan(1
/2*x)^2-7/4/a/(tan(1/2*x)^2+1)^4*tan(1/2*x)+10/3/a/(tan(1/2*x)^2+1)^4+15/4/a*arctan(tan(1/2*x))+2/a/(tan(1/2*x
)+1)

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Maxima [B]  time = 1.47195, size = 311, normalized size = 4.71 \begin{align*} \frac{\frac{19 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{211 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{91 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac{219 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac{165 \, \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac{165 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + \frac{45 \, \sin \left (x\right )^{7}}{{\left (\cos \left (x\right ) + 1\right )}^{7}} + \frac{45 \, \sin \left (x\right )^{8}}{{\left (\cos \left (x\right ) + 1\right )}^{8}} + 64}{12 \,{\left (a + \frac{a \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{4 \, a \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{4 \, a \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac{6 \, a \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac{6 \, a \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac{4 \, a \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + \frac{4 \, a \sin \left (x\right )^{7}}{{\left (\cos \left (x\right ) + 1\right )}^{7}} + \frac{a \sin \left (x\right )^{8}}{{\left (\cos \left (x\right ) + 1\right )}^{8}} + \frac{a \sin \left (x\right )^{9}}{{\left (\cos \left (x\right ) + 1\right )}^{9}}\right )}} + \frac{15 \, \arctan \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{4 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(19*sin(x)/(cos(x) + 1) + 211*sin(x)^2/(cos(x) + 1)^2 + 91*sin(x)^3/(cos(x) + 1)^3 + 219*sin(x)^4/(cos(x)
+ 1)^4 + 165*sin(x)^5/(cos(x) + 1)^5 + 165*sin(x)^6/(cos(x) + 1)^6 + 45*sin(x)^7/(cos(x) + 1)^7 + 45*sin(x)^8
/(cos(x) + 1)^8 + 64)/(a + a*sin(x)/(cos(x) + 1) + 4*a*sin(x)^2/(cos(x) + 1)^2 + 4*a*sin(x)^3/(cos(x) + 1)^3 +
6*a*sin(x)^4/(cos(x) + 1)^4 + 6*a*sin(x)^5/(cos(x) + 1)^5 + 4*a*sin(x)^6/(cos(x) + 1)^6 + 4*a*sin(x)^7/(cos(x
) + 1)^7 + a*sin(x)^8/(cos(x) + 1)^8 + a*sin(x)^9/(cos(x) + 1)^9) + 15/4*arctan(sin(x)/(cos(x) + 1))/a

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Fricas [A]  time = 0.485014, size = 259, normalized size = 3.92 \begin{align*} -\frac{6 \, \cos \left (x\right )^{5} + 8 \, \cos \left (x\right )^{4} - 25 \, \cos \left (x\right )^{3} - 45 \,{\left (x + 1\right )} \cos \left (x\right ) - 48 \, \cos \left (x\right )^{2} -{\left (6 \, \cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{3} - 27 \, \cos \left (x\right )^{2} + 45 \, x + 21 \, \cos \left (x\right ) - 24\right )} \sin \left (x\right ) - 45 \, x - 24}{24 \,{\left (a \cos \left (x\right ) + a \sin \left (x\right ) + a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(6*cos(x)^5 + 8*cos(x)^4 - 25*cos(x)^3 - 45*(x + 1)*cos(x) - 48*cos(x)^2 - (6*cos(x)^4 - 2*cos(x)^3 - 27
*cos(x)^2 + 45*x + 21*cos(x) - 24)*sin(x) - 45*x - 24)/(a*cos(x) + a*sin(x) + a)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\sin ^{4}{\left (x \right )}}{\csc{\left (x \right )} + 1}\, dx}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)**4/(a+a*csc(x)),x)

[Out]

Integral(sin(x)**4/(csc(x) + 1), x)/a

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Giac [A]  time = 1.39204, size = 123, normalized size = 1.86 \begin{align*} \frac{15 \, x}{8 \, a} + \frac{2}{a{\left (\tan \left (\frac{1}{2} \, x\right ) + 1\right )}} + \frac{21 \, \tan \left (\frac{1}{2} \, x\right )^{7} + 24 \, \tan \left (\frac{1}{2} \, x\right )^{6} + 45 \, \tan \left (\frac{1}{2} \, x\right )^{5} + 120 \, \tan \left (\frac{1}{2} \, x\right )^{4} - 45 \, \tan \left (\frac{1}{2} \, x\right )^{3} + 136 \, \tan \left (\frac{1}{2} \, x\right )^{2} - 21 \, \tan \left (\frac{1}{2} \, x\right ) + 40}{12 \,{\left (\tan \left (\frac{1}{2} \, x\right )^{2} + 1\right )}^{4} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

15/8*x/a + 2/(a*(tan(1/2*x) + 1)) + 1/12*(21*tan(1/2*x)^7 + 24*tan(1/2*x)^6 + 45*tan(1/2*x)^5 + 120*tan(1/2*x)
^4 - 45*tan(1/2*x)^3 + 136*tan(1/2*x)^2 - 21*tan(1/2*x) + 40)/((tan(1/2*x)^2 + 1)^4*a)