3.487 \(\int \frac{\cot ^8(e+f x)}{(a-a \sin ^2(e+f x))^{3/2}} \, dx\)
Optimal. Leaf size=115 \[ -\frac{\cot (e+f x) \csc ^6(e+f x)}{7 a f \sqrt{a \cos ^2(e+f x)}}+\frac{2 \cot (e+f x) \csc ^4(e+f x)}{5 a f \sqrt{a \cos ^2(e+f x)}}-\frac{\cot (e+f x) \csc ^2(e+f x)}{3 a f \sqrt{a \cos ^2(e+f x)}} \]
[Out]
-(Cot[e + f*x]*Csc[e + f*x]^2)/(3*a*f*Sqrt[a*Cos[e + f*x]^2]) + (2*Cot[e + f*x]*Csc[e + f*x]^4)/(5*a*f*Sqrt[a*
Cos[e + f*x]^2]) - (Cot[e + f*x]*Csc[e + f*x]^6)/(7*a*f*Sqrt[a*Cos[e + f*x]^2])
________________________________________________________________________________________
Rubi [A] time = 0.151866, antiderivative size = 115, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used =
{3176, 3207, 2606, 270} \[ -\frac{\cot (e+f x) \csc ^6(e+f x)}{7 a f \sqrt{a \cos ^2(e+f x)}}+\frac{2 \cot (e+f x) \csc ^4(e+f x)}{5 a f \sqrt{a \cos ^2(e+f x)}}-\frac{\cot (e+f x) \csc ^2(e+f x)}{3 a f \sqrt{a \cos ^2(e+f x)}} \]
Antiderivative was successfully verified.
[In]
Int[Cot[e + f*x]^8/(a - a*Sin[e + f*x]^2)^(3/2),x]
[Out]
-(Cot[e + f*x]*Csc[e + f*x]^2)/(3*a*f*Sqrt[a*Cos[e + f*x]^2]) + (2*Cot[e + f*x]*Csc[e + f*x]^4)/(5*a*f*Sqrt[a*
Cos[e + f*x]^2]) - (Cot[e + f*x]*Csc[e + f*x]^6)/(7*a*f*Sqrt[a*Cos[e + f*x]^2])
Rule 3176
Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[ActivateTrig[u*(a*cos[e + f*x]^2)^p]
, x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0]
Rule 3207
Int[(u_.)*((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Di
st[((b*ff^n)^IntPart[p]*(b*Sin[e + f*x]^n)^FracPart[p])/(Sin[e + f*x]/ff)^(n*FracPart[p]), Int[ActivateTrig[u]
*(Sin[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] && !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |
| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig
]])
Rule 2606
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] && !(IntegerQ[m/2] && LtQ[0, m, n + 1])
Rule 270
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]
Rubi steps
\begin{align*} \int \frac{\cot ^8(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx &=\int \frac{\cot ^8(e+f x)}{\left (a \cos ^2(e+f x)\right )^{3/2}} \, dx\\ &=\frac{\cos (e+f x) \int \cot ^5(e+f x) \csc ^3(e+f x) \, dx}{a \sqrt{a \cos ^2(e+f x)}}\\ &=-\frac{\cos (e+f x) \operatorname{Subst}\left (\int x^2 \left (-1+x^2\right )^2 \, dx,x,\csc (e+f x)\right )}{a f \sqrt{a \cos ^2(e+f x)}}\\ &=-\frac{\cos (e+f x) \operatorname{Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\csc (e+f x)\right )}{a f \sqrt{a \cos ^2(e+f x)}}\\ &=-\frac{\cot (e+f x) \csc ^2(e+f x)}{3 a f \sqrt{a \cos ^2(e+f x)}}+\frac{2 \cot (e+f x) \csc ^4(e+f x)}{5 a f \sqrt{a \cos ^2(e+f x)}}-\frac{\cot (e+f x) \csc ^6(e+f x)}{7 a f \sqrt{a \cos ^2(e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.136375, size = 51, normalized size = 0.44 \[ -\frac{\cot ^3(e+f x) \left (15 \csc ^4(e+f x)-42 \csc ^2(e+f x)+35\right )}{105 f \left (a \cos ^2(e+f x)\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
Integrate[Cot[e + f*x]^8/(a - a*Sin[e + f*x]^2)^(3/2),x]
[Out]
-(Cot[e + f*x]^3*(35 - 42*Csc[e + f*x]^2 + 15*Csc[e + f*x]^4))/(105*f*(a*Cos[e + f*x]^2)^(3/2))
________________________________________________________________________________________
Maple [A] time = 0.731, size = 57, normalized size = 0.5 \begin{align*} -{\frac{\cos \left ( fx+e \right ) \left ( 35\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}-28\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}+8 \right ) }{105\,a \left ( \sin \left ( fx+e \right ) \right ) ^{7}f}{\frac{1}{\sqrt{a \left ( \cos \left ( fx+e \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cot(f*x+e)^8/(a-a*sin(f*x+e)^2)^(3/2),x)
[Out]
-1/105/a*cos(f*x+e)*(35*cos(f*x+e)^4-28*cos(f*x+e)^2+8)/sin(f*x+e)^7/(a*cos(f*x+e)^2)^(1/2)/f
________________________________________________________________________________________
Maxima [B] time = 1.82258, size = 2735, normalized size = 23.78 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(f*x+e)^8/(a-a*sin(f*x+e)^2)^(3/2),x, algorithm="maxima")
[Out]
-8/105*((35*sin(11*f*x + 11*e) + 28*sin(9*f*x + 9*e) + 114*sin(7*f*x + 7*e) + 28*sin(5*f*x + 5*e) + 35*sin(3*f
*x + 3*e))*cos(14*f*x + 14*e) - 7*(35*sin(11*f*x + 11*e) + 28*sin(9*f*x + 9*e) + 114*sin(7*f*x + 7*e) + 28*sin
(5*f*x + 5*e) + 35*sin(3*f*x + 3*e))*cos(12*f*x + 12*e) - 245*(3*sin(10*f*x + 10*e) - 5*sin(8*f*x + 8*e) + 5*s
in(6*f*x + 6*e) - 3*sin(4*f*x + 4*e) + sin(2*f*x + 2*e))*cos(11*f*x + 11*e) + 21*(28*sin(9*f*x + 9*e) + 114*si
n(7*f*x + 7*e) + 28*sin(5*f*x + 5*e) + 35*sin(3*f*x + 3*e))*cos(10*f*x + 10*e) + 196*(5*sin(8*f*x + 8*e) - 5*s
in(6*f*x + 6*e) + 3*sin(4*f*x + 4*e) - sin(2*f*x + 2*e))*cos(9*f*x + 9*e) - 35*(114*sin(7*f*x + 7*e) + 28*sin(
5*f*x + 5*e) + 35*sin(3*f*x + 3*e))*cos(8*f*x + 8*e) - 798*(5*sin(6*f*x + 6*e) - 3*sin(4*f*x + 4*e) + sin(2*f*
x + 2*e))*cos(7*f*x + 7*e) + 245*(4*sin(5*f*x + 5*e) + 5*sin(3*f*x + 3*e))*cos(6*f*x + 6*e) + 196*(3*sin(4*f*x
+ 4*e) - sin(2*f*x + 2*e))*cos(5*f*x + 5*e) - (35*cos(11*f*x + 11*e) + 28*cos(9*f*x + 9*e) + 114*cos(7*f*x +
7*e) + 28*cos(5*f*x + 5*e) + 35*cos(3*f*x + 3*e))*sin(14*f*x + 14*e) + 7*(35*cos(11*f*x + 11*e) + 28*cos(9*f*x
+ 9*e) + 114*cos(7*f*x + 7*e) + 28*cos(5*f*x + 5*e) + 35*cos(3*f*x + 3*e))*sin(12*f*x + 12*e) + 35*(21*cos(10
*f*x + 10*e) - 35*cos(8*f*x + 8*e) + 35*cos(6*f*x + 6*e) - 21*cos(4*f*x + 4*e) + 7*cos(2*f*x + 2*e) - 1)*sin(1
1*f*x + 11*e) - 21*(28*cos(9*f*x + 9*e) + 114*cos(7*f*x + 7*e) + 28*cos(5*f*x + 5*e) + 35*cos(3*f*x + 3*e))*si
n(10*f*x + 10*e) - 28*(35*cos(8*f*x + 8*e) - 35*cos(6*f*x + 6*e) + 21*cos(4*f*x + 4*e) - 7*cos(2*f*x + 2*e) +
1)*sin(9*f*x + 9*e) + 35*(114*cos(7*f*x + 7*e) + 28*cos(5*f*x + 5*e) + 35*cos(3*f*x + 3*e))*sin(8*f*x + 8*e) +
114*(35*cos(6*f*x + 6*e) - 21*cos(4*f*x + 4*e) + 7*cos(2*f*x + 2*e) - 1)*sin(7*f*x + 7*e) - 245*(4*cos(5*f*x
+ 5*e) + 5*cos(3*f*x + 3*e))*sin(6*f*x + 6*e) - 28*(21*cos(4*f*x + 4*e) - 7*cos(2*f*x + 2*e) + 1)*sin(5*f*x +
5*e) + 735*cos(3*f*x + 3*e)*sin(4*f*x + 4*e) + 35*(7*cos(2*f*x + 2*e) - 1)*sin(3*f*x + 3*e) - 735*cos(4*f*x +
4*e)*sin(3*f*x + 3*e) - 245*cos(3*f*x + 3*e)*sin(2*f*x + 2*e))*sqrt(a)/((a^2*cos(14*f*x + 14*e)^2 + 49*a^2*cos
(12*f*x + 12*e)^2 + 441*a^2*cos(10*f*x + 10*e)^2 + 1225*a^2*cos(8*f*x + 8*e)^2 + 1225*a^2*cos(6*f*x + 6*e)^2 +
441*a^2*cos(4*f*x + 4*e)^2 + 49*a^2*cos(2*f*x + 2*e)^2 + a^2*sin(14*f*x + 14*e)^2 + 49*a^2*sin(12*f*x + 12*e)
^2 + 441*a^2*sin(10*f*x + 10*e)^2 + 1225*a^2*sin(8*f*x + 8*e)^2 + 1225*a^2*sin(6*f*x + 6*e)^2 + 441*a^2*sin(4*
f*x + 4*e)^2 - 294*a^2*sin(4*f*x + 4*e)*sin(2*f*x + 2*e) + 49*a^2*sin(2*f*x + 2*e)^2 - 14*a^2*cos(2*f*x + 2*e)
+ a^2 - 2*(7*a^2*cos(12*f*x + 12*e) - 21*a^2*cos(10*f*x + 10*e) + 35*a^2*cos(8*f*x + 8*e) - 35*a^2*cos(6*f*x
+ 6*e) + 21*a^2*cos(4*f*x + 4*e) - 7*a^2*cos(2*f*x + 2*e) + a^2)*cos(14*f*x + 14*e) - 14*(21*a^2*cos(10*f*x +
10*e) - 35*a^2*cos(8*f*x + 8*e) + 35*a^2*cos(6*f*x + 6*e) - 21*a^2*cos(4*f*x + 4*e) + 7*a^2*cos(2*f*x + 2*e) -
a^2)*cos(12*f*x + 12*e) - 42*(35*a^2*cos(8*f*x + 8*e) - 35*a^2*cos(6*f*x + 6*e) + 21*a^2*cos(4*f*x + 4*e) - 7
*a^2*cos(2*f*x + 2*e) + a^2)*cos(10*f*x + 10*e) - 70*(35*a^2*cos(6*f*x + 6*e) - 21*a^2*cos(4*f*x + 4*e) + 7*a^
2*cos(2*f*x + 2*e) - a^2)*cos(8*f*x + 8*e) - 70*(21*a^2*cos(4*f*x + 4*e) - 7*a^2*cos(2*f*x + 2*e) + a^2)*cos(6
*f*x + 6*e) - 42*(7*a^2*cos(2*f*x + 2*e) - a^2)*cos(4*f*x + 4*e) - 14*(a^2*sin(12*f*x + 12*e) - 3*a^2*sin(10*f
*x + 10*e) + 5*a^2*sin(8*f*x + 8*e) - 5*a^2*sin(6*f*x + 6*e) + 3*a^2*sin(4*f*x + 4*e) - a^2*sin(2*f*x + 2*e))*
sin(14*f*x + 14*e) - 98*(3*a^2*sin(10*f*x + 10*e) - 5*a^2*sin(8*f*x + 8*e) + 5*a^2*sin(6*f*x + 6*e) - 3*a^2*si
n(4*f*x + 4*e) + a^2*sin(2*f*x + 2*e))*sin(12*f*x + 12*e) - 294*(5*a^2*sin(8*f*x + 8*e) - 5*a^2*sin(6*f*x + 6*
e) + 3*a^2*sin(4*f*x + 4*e) - a^2*sin(2*f*x + 2*e))*sin(10*f*x + 10*e) - 490*(5*a^2*sin(6*f*x + 6*e) - 3*a^2*s
in(4*f*x + 4*e) + a^2*sin(2*f*x + 2*e))*sin(8*f*x + 8*e) - 490*(3*a^2*sin(4*f*x + 4*e) - a^2*sin(2*f*x + 2*e))
*sin(6*f*x + 6*e))*f)
________________________________________________________________________________________
Fricas [A] time = 1.67666, size = 247, normalized size = 2.15 \begin{align*} \frac{{\left (35 \, \cos \left (f x + e\right )^{4} - 28 \, \cos \left (f x + e\right )^{2} + 8\right )} \sqrt{a \cos \left (f x + e\right )^{2}}}{105 \,{\left (a^{2} f \cos \left (f x + e\right )^{7} - 3 \, a^{2} f \cos \left (f x + e\right )^{5} + 3 \, a^{2} f \cos \left (f x + e\right )^{3} - a^{2} f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(f*x+e)^8/(a-a*sin(f*x+e)^2)^(3/2),x, algorithm="fricas")
[Out]
1/105*(35*cos(f*x + e)^4 - 28*cos(f*x + e)^2 + 8)*sqrt(a*cos(f*x + e)^2)/((a^2*f*cos(f*x + e)^7 - 3*a^2*f*cos(
f*x + e)^5 + 3*a^2*f*cos(f*x + e)^3 - a^2*f*cos(f*x + e))*sin(f*x + e))
________________________________________________________________________________________
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(cot(f*x+e)**8/(a-a*sin(f*x+e)**2)**(3/2),x)
[Out]
Timed out
________________________________________________________________________________________
Giac [A] time = 1.4991, size = 248, normalized size = 2.16 \begin{align*} \frac{\frac{525 \, \sqrt{a} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{6} + 35 \, \sqrt{a} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 63 \, \sqrt{a} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 15 \, \sqrt{a}}{a^{2} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 1\right ) \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{7}} + \frac{15 \, a^{\frac{25}{2}} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{7} - 63 \, a^{\frac{25}{2}} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} + 35 \, a^{\frac{25}{2}} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 525 \, a^{\frac{25}{2}} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )}{a^{14} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 1\right )}}{13440 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(f*x+e)^8/(a-a*sin(f*x+e)^2)^(3/2),x, algorithm="giac")
[Out]
1/13440*((525*sqrt(a)*tan(1/2*f*x + 1/2*e)^6 + 35*sqrt(a)*tan(1/2*f*x + 1/2*e)^4 - 63*sqrt(a)*tan(1/2*f*x + 1/
2*e)^2 + 15*sqrt(a))/(a^2*sgn(tan(1/2*f*x + 1/2*e)^4 - 1)*tan(1/2*f*x + 1/2*e)^7) + (15*a^(25/2)*tan(1/2*f*x +
1/2*e)^7 - 63*a^(25/2)*tan(1/2*f*x + 1/2*e)^5 + 35*a^(25/2)*tan(1/2*f*x + 1/2*e)^3 + 525*a^(25/2)*tan(1/2*f*x
+ 1/2*e))/(a^14*sgn(tan(1/2*f*x + 1/2*e)^4 - 1)))/f