3.449 \(\int \cot ^4(c+d x) \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=173 \[ \frac{61 a \cot (c+d x)}{64 d \sqrt{a \sin (c+d x)+a}}-\frac{67 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{64 d}-\frac{\cot (c+d x) \csc ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{4 d}-\frac{a \cot (c+d x) \csc ^2(c+d x)}{24 d \sqrt{a \sin (c+d x)+a}}+\frac{61 a \cot (c+d x) \csc (c+d x)}{96 d \sqrt{a \sin (c+d x)+a}} \]

[Out]

(-67*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(64*d) + (61*a*Cot[c + d*x])/(64*d*Sqrt
[a + a*Sin[c + d*x]]) + (61*a*Cot[c + d*x]*Csc[c + d*x])/(96*d*Sqrt[a + a*Sin[c + d*x]]) - (a*Cot[c + d*x]*Csc
[c + d*x]^2)/(24*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]])/(4*d)

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Rubi [A]  time = 0.538, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {2881, 2773, 206, 3044, 2980, 2772} \[ \frac{61 a \cot (c+d x)}{64 d \sqrt{a \sin (c+d x)+a}}-\frac{67 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{64 d}-\frac{\cot (c+d x) \csc ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{4 d}-\frac{a \cot (c+d x) \csc ^2(c+d x)}{24 d \sqrt{a \sin (c+d x)+a}}+\frac{61 a \cot (c+d x) \csc (c+d x)}{96 d \sqrt{a \sin (c+d x)+a}} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^4*Csc[c + d*x]*Sqrt[a + a*Sin[c + d*x]],x]

[Out]

(-67*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(64*d) + (61*a*Cot[c + d*x])/(64*d*Sqrt
[a + a*Sin[c + d*x]]) + (61*a*Cot[c + d*x]*Csc[c + d*x])/(96*d*Sqrt[a + a*Sin[c + d*x]]) - (a*Cot[c + d*x]*Csc
[c + d*x]^2)/(24*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]])/(4*d)

Rule 2881

Int[cos[(e_.) + (f_.)*(x_)]^4*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)
, x_Symbol] :> Dist[1/d^4, Int[(d*Sin[e + f*x])^(n + 4)*(a + b*Sin[e + f*x])^m, x], x] + Int[(d*Sin[e + f*x])^
n*(a + b*Sin[e + f*x])^m*(1 - 2*Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] &&
!IGtQ[m, 0]

Rule 2773

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

Rule 206

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

Rule 3044

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*s
in[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((c^2*C + A*d^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[
e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 - d^2)), x] + Dist[1/(b*d*(n + 1)*(c^2 - d^2)), Int[(a + b*Sin[e + f*x])^
m*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(a*d*m + b*c*(n + 1)) + c*C*(a*c*m + b*d*(n + 1)) - b*(A*d^2*(m + n +
2) + C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m}, x] && NeQ[b
*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &&  !LtQ[m, -2^(-1)] && (LtQ[n, -1] || EqQ[m + n + 2, 0
])

Rule 2980

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.
) + (f_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b^2*(B*c - A*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(n
+ 1)*(b*c + a*d)*Sqrt[a + b*Sin[e + f*x]]), x] + Dist[(A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)))/(2*d*(n + 1)
*(b*c + a*d)), Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, A
, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1]

Rule 2772

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp
[((b*c - a*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(n + 1))/(f*(n + 1)*(c^2 - d^2)*Sqrt[a + b*Sin[e + f*x]]), x]
+ Dist[((2*n + 3)*(b*c - a*d))/(2*b*(n + 1)*(c^2 - d^2)), Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n
+ 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &
& LtQ[n, -1] && NeQ[2*n + 3, 0] && IntegerQ[2*n]

Rubi steps

\begin{align*} \int \cot ^4(c+d x) \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx &=\int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx+\int \csc ^5(c+d x) \sqrt{a+a \sin (c+d x)} \left (1-2 \sin ^2(c+d x)\right ) \, dx\\ &=-\frac{\cot (c+d x) \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)}}{4 d}+\frac{\int \csc ^4(c+d x) \left (\frac{a}{2}-\frac{11}{2} a \sin (c+d x)\right ) \sqrt{a+a \sin (c+d x)} \, dx}{4 a}-\frac{(2 a) \operatorname{Subst}\left (\int \frac{1}{a-x^2} \, dx,x,\frac{a \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{d}\\ &=-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{d}-\frac{a \cot (c+d x) \csc ^2(c+d x)}{24 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)}}{4 d}-\frac{61}{48} \int \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{d}+\frac{61 a \cot (c+d x) \csc (c+d x)}{96 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc ^2(c+d x)}{24 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)}}{4 d}-\frac{61}{64} \int \csc ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{d}+\frac{61 a \cot (c+d x)}{64 d \sqrt{a+a \sin (c+d x)}}+\frac{61 a \cot (c+d x) \csc (c+d x)}{96 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc ^2(c+d x)}{24 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)}}{4 d}-\frac{61}{128} \int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{d}+\frac{61 a \cot (c+d x)}{64 d \sqrt{a+a \sin (c+d x)}}+\frac{61 a \cot (c+d x) \csc (c+d x)}{96 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc ^2(c+d x)}{24 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)}}{4 d}+\frac{(61 a) \operatorname{Subst}\left (\int \frac{1}{a-x^2} \, dx,x,\frac{a \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{64 d}\\ &=-\frac{67 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{64 d}+\frac{61 a \cot (c+d x)}{64 d \sqrt{a+a \sin (c+d x)}}+\frac{61 a \cot (c+d x) \csc (c+d x)}{96 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc ^2(c+d x)}{24 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)}}{4 d}\\ \end{align*}

Mathematica [B]  time = 2.66687, size = 367, normalized size = 2.12 \[ -\frac{\csc ^{13}\left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\sin (c+d x)+1)} \left (-442 \sin \left (\frac{1}{2} (c+d x)\right )-162 \sin \left (\frac{3}{2} (c+d x)\right )-122 \sin \left (\frac{5}{2} (c+d x)\right )+366 \sin \left (\frac{7}{2} (c+d x)\right )+442 \cos \left (\frac{1}{2} (c+d x)\right )-162 \cos \left (\frac{3}{2} (c+d x)\right )+122 \cos \left (\frac{5}{2} (c+d x)\right )+366 \cos \left (\frac{7}{2} (c+d x)\right )-804 \cos (2 (c+d x)) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+201 \cos (4 (c+d x)) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+603 \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+804 \cos (2 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-201 \cos (4 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-603 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )\right )}{192 d \left (\cot \left (\frac{1}{2} (c+d x)\right )+1\right ) \left (\csc ^2\left (\frac{1}{4} (c+d x)\right )-\sec ^2\left (\frac{1}{4} (c+d x)\right )\right )^4} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^4*Csc[c + d*x]*Sqrt[a + a*Sin[c + d*x]],x]

[Out]

-(Csc[(c + d*x)/2]^13*Sqrt[a*(1 + Sin[c + d*x])]*(442*Cos[(c + d*x)/2] - 162*Cos[(3*(c + d*x))/2] + 122*Cos[(5
*(c + d*x))/2] + 366*Cos[(7*(c + d*x))/2] + 603*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 804*Cos[2*(c +
d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 201*Cos[4*(c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c +
d*x)/2]] - 603*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 804*Cos[2*(c + d*x)]*Log[1 - Cos[(c + d*x)/2] +
Sin[(c + d*x)/2]] - 201*Cos[4*(c + d*x)]*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 442*Sin[(c + d*x)/2] -
162*Sin[(3*(c + d*x))/2] - 122*Sin[(5*(c + d*x))/2] + 366*Sin[(7*(c + d*x))/2]))/(192*d*(1 + Cot[(c + d*x)/2]
)*(Csc[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^4)

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Maple [A]  time = 1.003, size = 162, normalized size = 0.9 \begin{align*} -{\frac{1+\sin \left ( dx+c \right ) }{192\, \left ( \sin \left ( dx+c \right ) \right ) ^{4}\cos \left ( dx+c \right ) d}\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) } \left ( 201\,{\it Artanh} \left ({\frac{\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }}{\sqrt{a}}} \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{4}{a}^{4}+183\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{7/2}\sqrt{a}-671\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{5/2}{a}^{3/2}+737\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{3/2}{a}^{5/2}-201\,\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }{a}^{7/2} \right ){a}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{a+a\sin \left ( dx+c \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^4*csc(d*x+c)^5*(a+a*sin(d*x+c))^(1/2),x)

[Out]

-1/192*(1+sin(d*x+c))*(-a*(sin(d*x+c)-1))^(1/2)/a^(7/2)*(201*arctanh((-a*(sin(d*x+c)-1))^(1/2)/a^(1/2))*sin(d*
x+c)^4*a^4+183*(-a*(sin(d*x+c)-1))^(7/2)*a^(1/2)-671*(-a*(sin(d*x+c)-1))^(5/2)*a^(3/2)+737*(-a*(sin(d*x+c)-1))
^(3/2)*a^(5/2)-201*(-a*(sin(d*x+c)-1))^(1/2)*a^(7/2))/sin(d*x+c)^4/cos(d*x+c)/(a+a*sin(d*x+c))^(1/2)/d

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{4} \csc \left (d x + c\right )^{5}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^5*(a+a*sin(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(a*sin(d*x + c) + a)*cos(d*x + c)^4*csc(d*x + c)^5, x)

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Fricas [B]  time = 1.26237, size = 1141, normalized size = 6.6 \begin{align*} \frac{201 \,{\left (\cos \left (d x + c\right )^{5} + \cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} +{\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \sin \left (d x + c\right ) + \cos \left (d x + c\right ) + 1\right )} \sqrt{a} \log \left (\frac{a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - 4 \,{\left (\cos \left (d x + c\right )^{2} +{\left (\cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 3\right )} \sqrt{a \sin \left (d x + c\right ) + a} \sqrt{a} - 9 \, a \cos \left (d x + c\right ) +{\left (a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} +{\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1}\right ) - 4 \,{\left (183 \, \cos \left (d x + c\right )^{4} + 122 \, \cos \left (d x + c\right )^{3} - 188 \, \cos \left (d x + c\right )^{2} +{\left (183 \, \cos \left (d x + c\right )^{3} + 61 \, \cos \left (d x + c\right )^{2} - 127 \, \cos \left (d x + c\right ) - 53\right )} \sin \left (d x + c\right ) - 74 \, \cos \left (d x + c\right ) + 53\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{768 \,{\left (d \cos \left (d x + c\right )^{5} + d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{3} - 2 \, d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right ) +{\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right ) + d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^5*(a+a*sin(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

1/768*(201*(cos(d*x + c)^5 + cos(d*x + c)^4 - 2*cos(d*x + c)^3 - 2*cos(d*x + c)^2 + (cos(d*x + c)^4 - 2*cos(d*
x + c)^2 + 1)*sin(d*x + c) + cos(d*x + c) + 1)*sqrt(a)*log((a*cos(d*x + c)^3 - 7*a*cos(d*x + c)^2 - 4*(cos(d*x
+ c)^2 + (cos(d*x + c) + 3)*sin(d*x + c) - 2*cos(d*x + c) - 3)*sqrt(a*sin(d*x + c) + a)*sqrt(a) - 9*a*cos(d*x
+ c) + (a*cos(d*x + c)^2 + 8*a*cos(d*x + c) - a)*sin(d*x + c) - a)/(cos(d*x + c)^3 + cos(d*x + c)^2 + (cos(d*
x + c)^2 - 1)*sin(d*x + c) - cos(d*x + c) - 1)) - 4*(183*cos(d*x + c)^4 + 122*cos(d*x + c)^3 - 188*cos(d*x + c
)^2 + (183*cos(d*x + c)^3 + 61*cos(d*x + c)^2 - 127*cos(d*x + c) - 53)*sin(d*x + c) - 74*cos(d*x + c) + 53)*sq
rt(a*sin(d*x + c) + a))/(d*cos(d*x + c)^5 + d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^3 - 2*d*cos(d*x + c)^2 + d*cos
(d*x + c) + (d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 + d)*sin(d*x + c) + d)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**4*csc(d*x+c)**5*(a+a*sin(d*x+c))**(1/2),x)

[Out]

Timed out

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^5*(a+a*sin(d*x+c))^(1/2),x, algorithm="giac")

[Out]

Timed out