3.466 \(\int \frac{\cos ^3(c+d x) \cot (c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx\)

Optimal. Leaf size=130 \[ -\frac{2 \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt{a \sin (c+d x)+a}}+\frac{2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{15 a d}+\frac{32 \cos (c+d x)}{15 d \sqrt{a \sin (c+d x)+a}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{\sqrt{a} d} \]

[Out]

(-2*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(Sqrt[a]*d) + (32*Cos[c + d*x])/(15*d*Sqrt[a + a
*Sin[c + d*x]]) - (2*Cos[c + d*x]*Sin[c + d*x]^2)/(5*d*Sqrt[a + a*Sin[c + d*x]]) + (2*Cos[c + d*x]*Sqrt[a + a*
Sin[c + d*x]])/(15*a*d)

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Rubi [A]  time = 0.564465, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.345, Rules used = {2881, 2778, 2968, 3023, 2751, 2649, 206, 3046, 2985, 2773} \[ -\frac{2 \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt{a \sin (c+d x)+a}}+\frac{2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{15 a d}+\frac{32 \cos (c+d x)}{15 d \sqrt{a \sin (c+d x)+a}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{\sqrt{a} d} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]^3*Cot[c + d*x])/Sqrt[a + a*Sin[c + d*x]],x]

[Out]

(-2*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(Sqrt[a]*d) + (32*Cos[c + d*x])/(15*d*Sqrt[a + a
*Sin[c + d*x]]) - (2*Cos[c + d*x]*Sin[c + d*x]^2)/(5*d*Sqrt[a + a*Sin[c + d*x]]) + (2*Cos[c + d*x]*Sqrt[a + a*
Sin[c + d*x]])/(15*a*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 2778

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp
[(-2*d*Cos[e + f*x]*(c + d*Sin[e + f*x])^(n - 1))/(f*(2*n - 1)*Sqrt[a + b*Sin[e + f*x]]), x] - Dist[1/(b*(2*n
- 1)), Int[((c + d*Sin[e + f*x])^(n - 2)*Simp[a*c*d - b*(2*d^2*(n - 1) + c^2*(2*n - 1)) + d*(a*d - b*c*(4*n -
3))*Sin[e + f*x], x])/Sqrt[a + b*Sin[e + f*x]], 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] && GtQ[n, 1] && IntegerQ[2*n]

Rule 2968

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(
e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x
]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]

Rule 3023

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[1/(b*
(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x]
, x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rule 2751

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

Rule 2649

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[-2/d, Subst[Int[1/(2*a - x^2), x], x, (b*C
os[c + d*x])/Sqrt[a + b*Sin[c + d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^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 3046

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

Rule 2985

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])), x_Symbol] :> Dist[(A*b - a*B)/(b*c - a*d), Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] + Dist[(
B*c - A*d)/(b*c - a*d), Int[Sqrt[a + b*Sin[e + f*x]]/(c + d*Sin[e + f*x]), 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]

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]

Rubi steps

\begin{align*} \int \frac{\cos ^3(c+d x) \cot (c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx &=\int \frac{\sin ^3(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx+\int \frac{\csc (c+d x) \left (1-2 \sin ^2(c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx\\ &=\frac{4 \cos (c+d x)}{d \sqrt{a+a \sin (c+d x)}}-\frac{2 \cos (c+d x) \sin ^2(c+d x)}{5 d \sqrt{a+a \sin (c+d x)}}-\frac{\int \frac{\sin (c+d x) (-4 a+a \sin (c+d x))}{\sqrt{a+a \sin (c+d x)}} \, dx}{5 a}+\frac{2 \int \frac{\csc (c+d x) \left (\frac{a}{2}+a \sin (c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx}{a}\\ &=\frac{4 \cos (c+d x)}{d \sqrt{a+a \sin (c+d x)}}-\frac{2 \cos (c+d x) \sin ^2(c+d x)}{5 d \sqrt{a+a \sin (c+d x)}}-\frac{\int \frac{-4 a \sin (c+d x)+a \sin ^2(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx}{5 a}+\frac{\int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx}{a}+\int \frac{1}{\sqrt{a+a \sin (c+d x)}} \, dx\\ &=\frac{4 \cos (c+d x)}{d \sqrt{a+a \sin (c+d x)}}-\frac{2 \cos (c+d x) \sin ^2(c+d x)}{5 d \sqrt{a+a \sin (c+d x)}}+\frac{2 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{15 a d}-\frac{2 \int \frac{\frac{a^2}{2}-7 a^2 \sin (c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx}{15 a^2}-\frac{2 \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 \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,\frac{a \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{d}\\ &=-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{\sqrt{a} d}-\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a+a \sin (c+d x)}}\right )}{\sqrt{a} d}+\frac{32 \cos (c+d x)}{15 d \sqrt{a+a \sin (c+d x)}}-\frac{2 \cos (c+d x) \sin ^2(c+d x)}{5 d \sqrt{a+a \sin (c+d x)}}+\frac{2 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{15 a d}-\int \frac{1}{\sqrt{a+a \sin (c+d x)}} \, dx\\ &=-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{\sqrt{a} d}-\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a+a \sin (c+d x)}}\right )}{\sqrt{a} d}+\frac{32 \cos (c+d x)}{15 d \sqrt{a+a \sin (c+d x)}}-\frac{2 \cos (c+d x) \sin ^2(c+d x)}{5 d \sqrt{a+a \sin (c+d x)}}+\frac{2 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{15 a d}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,\frac{a \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{d}\\ &=-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{\sqrt{a} d}+\frac{32 \cos (c+d x)}{15 d \sqrt{a+a \sin (c+d x)}}-\frac{2 \cos (c+d x) \sin ^2(c+d x)}{5 d \sqrt{a+a \sin (c+d x)}}+\frac{2 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{15 a d}\\ \end{align*}

Mathematica [A]  time = 0.251687, size = 169, normalized size = 1.3 \[ \frac{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right ) \left (-60 \sin \left (\frac{1}{2} (c+d x)\right )+5 \sin \left (\frac{3}{2} (c+d x)\right )-3 \sin \left (\frac{5}{2} (c+d x)\right )+60 \cos \left (\frac{1}{2} (c+d x)\right )+5 \cos \left (\frac{3}{2} (c+d x)\right )+3 \cos \left (\frac{5}{2} (c+d x)\right )-30 \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+30 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )\right )}{30 d \sqrt{a (\sin (c+d x)+1)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]^3*Cot[c + d*x])/Sqrt[a + a*Sin[c + d*x]],x]

[Out]

((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(60*Cos[(c + d*x)/2] + 5*Cos[(3*(c + d*x))/2] + 3*Cos[(5*(c + d*x))/2]
- 30*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 30*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 60*Sin[(
c + d*x)/2] + 5*Sin[(3*(c + d*x))/2] - 3*Sin[(5*(c + d*x))/2]))/(30*d*Sqrt[a*(1 + Sin[c + d*x])])

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Maple [A]  time = 0.894, size = 123, normalized size = 1. \begin{align*} -{\frac{2+2\,\sin \left ( dx+c \right ) }{15\,{a}^{3}\cos \left ( dx+c \right ) d}\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) } \left ( 15\,{a}^{5/2}{\it Artanh} \left ({\frac{\sqrt{a-a\sin \left ( dx+c \right ) }}{\sqrt{a}}} \right ) +3\, \left ( a-a\sin \left ( dx+c \right ) \right ) ^{5/2}-5\, \left ( a-a\sin \left ( dx+c \right ) \right ) ^{3/2}a-15\,{a}^{2}\sqrt{a-a\sin \left ( dx+c \right ) } \right ){\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)/(a+a*sin(d*x+c))^(1/2),x)

[Out]

-2/15*(1+sin(d*x+c))*(-a*(sin(d*x+c)-1))^(1/2)*(15*a^(5/2)*arctanh((a-a*sin(d*x+c))^(1/2)/a^(1/2))+3*(a-a*sin(
d*x+c))^(5/2)-5*(a-a*sin(d*x+c))^(3/2)*a-15*a^2*(a-a*sin(d*x+c))^(1/2))/a^3/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 \frac{\cos \left (d x + c\right )^{4} \csc \left (d x + c\right )}{\sqrt{a \sin \left (d x + c\right ) + a}}\,{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)/(a+a*sin(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.25133, size = 761, normalized size = 5.85 \begin{align*} \frac{15 \, \sqrt{a}{\left (\cos \left (d x + c\right ) + \sin \left (d x + c\right ) + 1\right )} \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 (3 \, \cos \left (d x + c\right )^{3} + 4 \, \cos \left (d x + c\right )^{2} -{\left (3 \, \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) + 13\right )} \sin \left (d x + c\right ) + 14 \, \cos \left (d x + c\right ) + 13\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{30 \,{\left (a d \cos \left (d x + c\right ) + a d \sin \left (d x + c\right ) + a 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)/(a+a*sin(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

1/30*(15*sqrt(a)*(cos(d*x + c) + sin(d*x + c) + 1)*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*(3*cos(d*x + c)^3 + 4*cos(d*x + c)^2 - (3*cos(d*x + c)^2 - cos
(d*x + c) + 13)*sin(d*x + c) + 14*cos(d*x + c) + 13)*sqrt(a*sin(d*x + c) + a))/(a*d*cos(d*x + c) + a*d*sin(d*x
+ c) + a*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)/(a+a*sin(d*x+c))**(1/2),x)

[Out]

Timed out

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Giac [B]  time = 2.363, size = 498, normalized size = 3.83 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/120*(240*arctan(-(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))/sqrt(-a))/(sqrt(-a)*sgn
(tan(1/2*d*x + 1/2*c) + 1)) - 2*(((((17*sgn(tan(1/2*d*x + 1/2*c) + 1)*tan(1/2*d*x + 1/2*c)/a^7 - 15*sgn(tan(1/
2*d*x + 1/2*c) + 1)/a^7)*tan(1/2*d*x + 1/2*c) + 20*sgn(tan(1/2*d*x + 1/2*c) + 1)/a^7)*tan(1/2*d*x + 1/2*c) - 2
0*sgn(tan(1/2*d*x + 1/2*c) + 1)/a^7)*tan(1/2*d*x + 1/2*c) + 15*sgn(tan(1/2*d*x + 1/2*c) + 1)/a^7)*tan(1/2*d*x
+ 1/2*c) - 17*sgn(tan(1/2*d*x + 1/2*c) + 1)/a^7)/(a*tan(1/2*d*x + 1/2*c)^2 + a)^(5/2) - 15*log(abs(-sqrt(a)*ta
n(1/2*d*x + 1/2*c) + sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a)))*sgn(tan(1/2*d*x + 1/2*c) + 1)/a^(19/2) - (240*a^(19/
2)*arctan((sqrt(2)*sqrt(a) + sqrt(a))/sqrt(-a)) - 15*sqrt(-a)*log(sqrt(2)*sqrt(a) + sqrt(a)) + 26*sqrt(2)*sqrt
(-a))*sgn(tan(1/2*d*x + 1/2*c) + 1)/(sqrt(-a)*a^(19/2)))/d