3.17 \(\int \frac{\cot ^5(d+e x)}{\sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx\)

Optimal. Leaf size=182 \[ \frac{(b+2 c) \tanh ^{-1}\left (\frac{b+2 c \cot ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{4 c^{3/2} e}-\frac{\sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{2 c e}+\frac{\tanh ^{-1}\left (\frac{2 a+(b-2 c) \cot ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 e \sqrt{a-b+c}} \]

[Out]

ArcTanh[(2*a - b + (b - 2*c)*Cot[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])
]/(2*Sqrt[a - b + c]*e) + ((b + 2*c)*ArcTanh[(b + 2*c*Cot[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x]^2 + c
*Cot[d + e*x]^4])])/(4*c^(3/2)*e) - Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4]/(2*c*e)

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Rubi [A]  time = 0.388619, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3701, 1251, 1653, 843, 621, 206, 724} \[ \frac{(b+2 c) \tanh ^{-1}\left (\frac{b+2 c \cot ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{4 c^{3/2} e}-\frac{\sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{2 c e}+\frac{\tanh ^{-1}\left (\frac{2 a+(b-2 c) \cot ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 e \sqrt{a-b+c}} \]

Antiderivative was successfully verified.

[In]

Int[Cot[d + e*x]^5/Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4],x]

[Out]

ArcTanh[(2*a - b + (b - 2*c)*Cot[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])
]/(2*Sqrt[a - b + c]*e) + ((b + 2*c)*ArcTanh[(b + 2*c*Cot[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x]^2 + c
*Cot[d + e*x]^4])])/(4*c^(3/2)*e) - Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4]/(2*c*e)

Rule 3701

Int[cot[(d_.) + (e_.)*(x_)]^(m_.)*((a_.) + (b_.)*(cot[(d_.) + (e_.)*(x_)]*(f_.))^(n_.) + (c_.)*(cot[(d_.) + (e
_.)*(x_)]*(f_.))^(n2_.))^(p_), x_Symbol] :> -Dist[f/e, Subst[Int[((x/f)^m*(a + b*x^n + c*x^(2*n))^p)/(f^2 + x^
2), x], x, f*Cot[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]

Rule 1251

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

Rule 1653

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq
, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[(f*(d + e*x)^(m + q - 1)*(a + b*x + c*x^2)^(p + 1))/(c*e^(q - 1)*(
m + q + 2*p + 1)), x] + Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^
q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q
- 1) - c*d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p +
1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2
, 0] &&  !(IGtQ[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))

Rule 843

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 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 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{\cot ^5(d+e x)}{\sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{x^5}{\left (1+x^2\right ) \sqrt{a+b x^2+c x^4}} \, dx,x,\cot (d+e x)\right )}{e}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{x^2}{(1+x) \sqrt{a+b x+c x^2}} \, dx,x,\cot ^2(d+e x)\right )}{2 e}\\ &=-\frac{\sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{2 c e}-\frac{\operatorname{Subst}\left (\int \frac{-\frac{b}{2}-\frac{1}{2} (b+2 c) x}{(1+x) \sqrt{a+b x+c x^2}} \, dx,x,\cot ^2(d+e x)\right )}{2 c e}\\ &=-\frac{\sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{2 c e}-\frac{\operatorname{Subst}\left (\int \frac{1}{(1+x) \sqrt{a+b x+c x^2}} \, dx,x,\cot ^2(d+e x)\right )}{2 e}+\frac{(b+2 c) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x+c x^2}} \, dx,x,\cot ^2(d+e x)\right )}{4 c e}\\ &=-\frac{\sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{2 c e}+\frac{\operatorname{Subst}\left (\int \frac{1}{4 a-4 b+4 c-x^2} \, dx,x,\frac{2 a-b-(-b+2 c) \cot ^2(d+e x)}{\sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{e}+\frac{(b+2 c) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c \cot ^2(d+e x)}{\sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 c e}\\ &=\frac{\tanh ^{-1}\left (\frac{2 a-b+(b-2 c) \cot ^2(d+e x)}{2 \sqrt{a-b+c} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 \sqrt{a-b+c} e}+\frac{(b+2 c) \tanh ^{-1}\left (\frac{b+2 c \cot ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{4 c^{3/2} e}-\frac{\sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{2 c e}\\ \end{align*}

Mathematica [C]  time = 34.3707, size = 179905, normalized size = 988.49 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Cot[d + e*x]^5/Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4],x]

[Out]

Result too large to show

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Maple [A]  time = 0.117, size = 240, normalized size = 1.3 \begin{align*} -{\frac{1}{2\,ce}\sqrt{a+b \left ( \cot \left ( ex+d \right ) \right ) ^{2}+c \left ( \cot \left ( ex+d \right ) \right ) ^{4}}}+{\frac{b}{4\,e}\ln \left ({ \left ({\frac{b}{2}}+c \left ( \cot \left ( ex+d \right ) \right ) ^{2} \right ){\frac{1}{\sqrt{c}}}}+\sqrt{a+b \left ( \cot \left ( ex+d \right ) \right ) ^{2}+c \left ( \cot \left ( ex+d \right ) \right ) ^{4}} \right ){c}^{-{\frac{3}{2}}}}+{\frac{1}{2\,e}\ln \left ({ \left ({\frac{b}{2}}+c \left ( \cot \left ( ex+d \right ) \right ) ^{2} \right ){\frac{1}{\sqrt{c}}}}+\sqrt{a+b \left ( \cot \left ( ex+d \right ) \right ) ^{2}+c \left ( \cot \left ( ex+d \right ) \right ) ^{4}} \right ){\frac{1}{\sqrt{c}}}}+{\frac{1}{2\,e}\ln \left ({\frac{1}{ \left ( \cot \left ( ex+d \right ) \right ) ^{2}+1} \left ( 2\,a-2\,b+2\,c+ \left ( b-2\,c \right ) \left ( \left ( \cot \left ( ex+d \right ) \right ) ^{2}+1 \right ) +2\,\sqrt{a-b+c}\sqrt{ \left ( \left ( \cot \left ( ex+d \right ) \right ) ^{2}+1 \right ) ^{2}c+ \left ( b-2\,c \right ) \left ( \left ( \cot \left ( ex+d \right ) \right ) ^{2}+1 \right ) +a-b+c} \right ) } \right ){\frac{1}{\sqrt{a-b+c}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(e*x+d)^5/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2),x)

[Out]

-1/2*(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2)/c/e+1/4/e*b/c^(3/2)*ln((1/2*b+c*cot(e*x+d)^2)/c^(1/2)+(a+b*cot(e*
x+d)^2+c*cot(e*x+d)^4)^(1/2))+1/2/e*ln((1/2*b+c*cot(e*x+d)^2)/c^(1/2)+(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2))
/c^(1/2)+1/2/e/(a-b+c)^(1/2)*ln((2*a-2*b+2*c+(b-2*c)*(cot(e*x+d)^2+1)+2*(a-b+c)^(1/2)*((cot(e*x+d)^2+1)^2*c+(b
-2*c)*(cot(e*x+d)^2+1)+a-b+c)^(1/2))/(cot(e*x+d)^2+1))

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Maxima [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(e*x+d)^5/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [B]  time = 15.3129, size = 5068, normalized size = 27.85 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(e*x+d)^5/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2),x, algorithm="fricas")

[Out]

[1/8*(2*sqrt(a - b + c)*c^2*log(2*(a^2 - 2*a*b + b^2 + 2*(a - b)*c + c^2)*cos(2*e*x + 2*d)^2 + 2*a^2 - b^2 + 2
*c^2 + 2*((a - b + c)*cos(2*e*x + 2*d)^2 - (2*a - b)*cos(2*e*x + 2*d) + a - c)*sqrt(a - b + c)*sqrt(((a - b +
c)*cos(2*e*x + 2*d)^2 - 2*(a - c)*cos(2*e*x + 2*d) + a + b + c)/(cos(2*e*x + 2*d)^2 - 2*cos(2*e*x + 2*d) + 1))
- 4*(a^2 - a*b + b*c - c^2)*cos(2*e*x + 2*d)) + (a*b - b^2 + (2*a - b)*c + 2*c^2)*sqrt(c)*log(((b^2 + 4*(a -
2*b)*c + 8*c^2)*cos(2*e*x + 2*d)^2 + b^2 + 4*(a + 2*b)*c + 8*c^2 + 4*((b - 2*c)*cos(2*e*x + 2*d)^2 - 2*b*cos(2
*e*x + 2*d) + b + 2*c)*sqrt(c)*sqrt(((a - b + c)*cos(2*e*x + 2*d)^2 - 2*(a - c)*cos(2*e*x + 2*d) + a + b + c)/
(cos(2*e*x + 2*d)^2 - 2*cos(2*e*x + 2*d) + 1)) - 2*(b^2 + 4*a*c - 8*c^2)*cos(2*e*x + 2*d))/(cos(2*e*x + 2*d)^2
- 2*cos(2*e*x + 2*d) + 1)) - 4*((a - b)*c + c^2)*sqrt(((a - b + c)*cos(2*e*x + 2*d)^2 - 2*(a - c)*cos(2*e*x +
2*d) + a + b + c)/(cos(2*e*x + 2*d)^2 - 2*cos(2*e*x + 2*d) + 1)))/(((a - b)*c^2 + c^3)*e), 1/4*(sqrt(a - b +
c)*c^2*log(2*(a^2 - 2*a*b + b^2 + 2*(a - b)*c + c^2)*cos(2*e*x + 2*d)^2 + 2*a^2 - b^2 + 2*c^2 + 2*((a - b + c)
*cos(2*e*x + 2*d)^2 - (2*a - b)*cos(2*e*x + 2*d) + a - c)*sqrt(a - b + c)*sqrt(((a - b + c)*cos(2*e*x + 2*d)^2
- 2*(a - c)*cos(2*e*x + 2*d) + a + b + c)/(cos(2*e*x + 2*d)^2 - 2*cos(2*e*x + 2*d) + 1)) - 4*(a^2 - a*b + b*c
- c^2)*cos(2*e*x + 2*d)) + (a*b - b^2 + (2*a - b)*c + 2*c^2)*sqrt(-c)*arctan(-1/2*((b - 2*c)*cos(2*e*x + 2*d)
^2 - 2*b*cos(2*e*x + 2*d) + b + 2*c)*sqrt(-c)*sqrt(((a - b + c)*cos(2*e*x + 2*d)^2 - 2*(a - c)*cos(2*e*x + 2*d
) + a + b + c)/(cos(2*e*x + 2*d)^2 - 2*cos(2*e*x + 2*d) + 1))/(((a - b)*c + c^2)*cos(2*e*x + 2*d)^2 + (a + b)*
c + c^2 - 2*(a*c - c^2)*cos(2*e*x + 2*d))) - 2*((a - b)*c + c^2)*sqrt(((a - b + c)*cos(2*e*x + 2*d)^2 - 2*(a -
c)*cos(2*e*x + 2*d) + a + b + c)/(cos(2*e*x + 2*d)^2 - 2*cos(2*e*x + 2*d) + 1)))/(((a - b)*c^2 + c^3)*e), -1/
8*(4*sqrt(-a + b - c)*c^2*arctan(((a - b + c)*cos(2*e*x + 2*d)^2 - (2*a - b)*cos(2*e*x + 2*d) + a - c)*sqrt(-a
+ b - c)*sqrt(((a - b + c)*cos(2*e*x + 2*d)^2 - 2*(a - c)*cos(2*e*x + 2*d) + a + b + c)/(cos(2*e*x + 2*d)^2 -
2*cos(2*e*x + 2*d) + 1))/((a^2 - 2*a*b + b^2 + 2*(a - b)*c + c^2)*cos(2*e*x + 2*d)^2 + a^2 - b^2 + 2*a*c + c^
2 - 2*(a^2 - a*b + b*c - c^2)*cos(2*e*x + 2*d))) - (a*b - b^2 + (2*a - b)*c + 2*c^2)*sqrt(c)*log(((b^2 + 4*(a
- 2*b)*c + 8*c^2)*cos(2*e*x + 2*d)^2 + b^2 + 4*(a + 2*b)*c + 8*c^2 + 4*((b - 2*c)*cos(2*e*x + 2*d)^2 - 2*b*cos
(2*e*x + 2*d) + b + 2*c)*sqrt(c)*sqrt(((a - b + c)*cos(2*e*x + 2*d)^2 - 2*(a - c)*cos(2*e*x + 2*d) + a + b + c
)/(cos(2*e*x + 2*d)^2 - 2*cos(2*e*x + 2*d) + 1)) - 2*(b^2 + 4*a*c - 8*c^2)*cos(2*e*x + 2*d))/(cos(2*e*x + 2*d)
^2 - 2*cos(2*e*x + 2*d) + 1)) + 4*((a - b)*c + c^2)*sqrt(((a - b + c)*cos(2*e*x + 2*d)^2 - 2*(a - c)*cos(2*e*x
+ 2*d) + a + b + c)/(cos(2*e*x + 2*d)^2 - 2*cos(2*e*x + 2*d) + 1)))/(((a - b)*c^2 + c^3)*e), -1/4*(2*sqrt(-a
+ b - c)*c^2*arctan(((a - b + c)*cos(2*e*x + 2*d)^2 - (2*a - b)*cos(2*e*x + 2*d) + a - c)*sqrt(-a + b - c)*sqr
t(((a - b + c)*cos(2*e*x + 2*d)^2 - 2*(a - c)*cos(2*e*x + 2*d) + a + b + c)/(cos(2*e*x + 2*d)^2 - 2*cos(2*e*x
+ 2*d) + 1))/((a^2 - 2*a*b + b^2 + 2*(a - b)*c + c^2)*cos(2*e*x + 2*d)^2 + a^2 - b^2 + 2*a*c + c^2 - 2*(a^2 -
a*b + b*c - c^2)*cos(2*e*x + 2*d))) - (a*b - b^2 + (2*a - b)*c + 2*c^2)*sqrt(-c)*arctan(-1/2*((b - 2*c)*cos(2*
e*x + 2*d)^2 - 2*b*cos(2*e*x + 2*d) + b + 2*c)*sqrt(-c)*sqrt(((a - b + c)*cos(2*e*x + 2*d)^2 - 2*(a - c)*cos(2
*e*x + 2*d) + a + b + c)/(cos(2*e*x + 2*d)^2 - 2*cos(2*e*x + 2*d) + 1))/(((a - b)*c + c^2)*cos(2*e*x + 2*d)^2
+ (a + b)*c + c^2 - 2*(a*c - c^2)*cos(2*e*x + 2*d))) + 2*((a - b)*c + c^2)*sqrt(((a - b + c)*cos(2*e*x + 2*d)^
2 - 2*(a - c)*cos(2*e*x + 2*d) + a + b + c)/(cos(2*e*x + 2*d)^2 - 2*cos(2*e*x + 2*d) + 1)))/(((a - b)*c^2 + c^
3)*e)]

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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(cot(e*x+d)**5/(a+b*cot(e*x+d)**2+c*cot(e*x+d)**4)**(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(cot(e*x+d)^5/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2),x, algorithm="giac")

[Out]

Timed out