3.1.22 \(\int \cot ^5(d+e x) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx\) [22]

Optimal. Leaf size=270 \[ \frac {\sqrt {a-b+c} \tanh ^{-1}\left (\frac {2 a-b+(b-2 c) \cot ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 e}-\frac {\left (b^3+2 b^2 c-4 b (a-2 c) c-8 c^2 (a+2 c)\right ) \tanh ^{-1}\left (\frac {b+2 c \cot ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{32 c^{5/2} e}+\frac {\left ((b-2 c) (b+4 c)+2 c (b+2 c) \cot ^2(d+e x)\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{16 c^2 e}-\frac {\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}}{6 c e} \]

[Out]

-1/32*(b^3+2*b^2*c-4*b*(a-2*c)*c-8*c^2*(a+2*c))*arctanh(1/2*(b+2*c*cot(e*x+d)^2)/c^(1/2)/(a+b*cot(e*x+d)^2+c*c
ot(e*x+d)^4)^(1/2))/c^(5/2)/e-1/6*(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(3/2)/c/e+1/2*arctanh(1/2*(2*a-b+(b-2*c)*c
ot(e*x+d)^2)/(a-b+c)^(1/2)/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2))*(a-b+c)^(1/2)/e+1/16*((b-2*c)*(b+4*c)+2*c*
(b+2*c)*cot(e*x+d)^2)*(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2)/c^2/e

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Rubi [A]
time = 0.38, antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {3782, 1265, 1667, 828, 857, 635, 212, 738} \begin {gather*} -\frac {\left (-4 b c (a-2 c)-8 c^2 (a+2 c)+b^3+2 b^2 c\right ) \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 )}{32 c^{5/2} e}+\frac {\left (2 c (b+2 c) \cot ^2(d+e x)+(b-2 c) (b+4 c)\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{16 c^2 e}-\frac {\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}}{6 c e}+\frac {\sqrt {a-b+c} \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} \end {gather*}

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]

(Sqrt[a - b + c]*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*e) - ((b^3 + 2*b^2*c - 4*b*(a - 2*c)*c - 8*c^2*(a + 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])])/(32*c^(5/2)*e) + (((b - 2*c)*(b + 4*c) + 2*c*(
b + 2*c)*Cot[d + e*x]^2)*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])/(16*c^2*e) - (a + b*Cot[d + e*x]^2 + c
*Cot[d + e*x]^4)^(3/2)/(6*c*e)

Rule 212

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

Rule 635

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 738

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]

Rule 828

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^
2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Dist[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a
+ b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2*a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p -
c*d - 2*c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c^2*d^2*(1 + 2*p) - c*e*(b*
d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0
] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])
) &&  !ILtQ[m + 2*p, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 857

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 1265

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 1667

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 3782

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]

Rubi steps

\begin {align*} \int \cot ^5(d+e x) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx &=-\frac {\text {Subst}\left (\int \frac {x^5 \sqrt {a+b x^2+c x^4}}{1+x^2} \, dx,x,\cot (d+e x)\right )}{e}\\ &=-\frac {\text {Subst}\left (\int \frac {x^2 \sqrt {a+b x+c x^2}}{1+x} \, dx,x,\cot ^2(d+e x)\right )}{2 e}\\ &=-\frac {\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}}{6 c e}-\frac {\text {Subst}\left (\int \frac {\left (-\frac {3 b}{2}-\frac {3}{2} (b+2 c) x\right ) \sqrt {a+b x+c x^2}}{1+x} \, dx,x,\cot ^2(d+e x)\right )}{6 c e}\\ &=\frac {\left ((b-2 c) (b+4 c)+2 c (b+2 c) \cot ^2(d+e x)\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{16 c^2 e}-\frac {\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}}{6 c e}+\frac {\text {Subst}\left (\int \frac {-\frac {3}{4} (b-2 c) \left (b^2-4 a c+4 b c\right )-\frac {3}{4} \left (b^3+2 b^2 c-4 b (a-2 c) c-8 c^2 (a+2 c)\right ) x}{(1+x) \sqrt {a+b x+c x^2}} \, dx,x,\cot ^2(d+e x)\right )}{24 c^2 e}\\ &=\frac {\left ((b-2 c) (b+4 c)+2 c (b+2 c) \cot ^2(d+e x)\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{16 c^2 e}-\frac {\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}}{6 c e}-\frac {(a-b+c) \text {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 {\left (b^3+2 b^2 c-4 b (a-2 c) c-8 c^2 (a+2 c)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,\cot ^2(d+e x)\right )}{32 c^2 e}\\ &=\frac {\left ((b-2 c) (b+4 c)+2 c (b+2 c) \cot ^2(d+e x)\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{16 c^2 e}-\frac {\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}}{6 c e}+\frac {(a-b+c) \text {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 {\left (b^3+2 b^2 c-4 b (a-2 c) c-8 c^2 (a+2 c)\right ) \text {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 )}{16 c^2 e}\\ &=\frac {\sqrt {a-b+c} \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 e}-\frac {\left (b^3+2 b^2 c-4 b (a-2 c) c-8 c^2 (a+2 c)\right ) \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 )}{32 c^{5/2} e}+\frac {\left ((b-2 c) (b+4 c)+2 c (b+2 c) \cot ^2(d+e x)\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{16 c^2 e}-\frac {\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}}{6 c e}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 38.35, size = 539292, normalized size = 1997.38 \begin {gather*} \text {Result too large to show} \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. \(626\) vs. \(2(242)=484\).
time = 0.30, size = 627, normalized size = 2.32

method	result	size
derivativedivides	\(\frac {-\frac {\left (a +b \left (\cot ^{2}\left (e x +d \right )\right )+c \left (\cot ^{4}\left (e x +d \right )\right )\right )^{\frac {3}{2}}}{6 c}+\frac {b \left (\cot ^{2}\left (e x +d \right )\right ) \sqrt {a +b \left (\cot ^{2}\left (e x +d \right )\right )+c \left (\cot ^{4}\left (e x +d \right )\right )}}{8 c}+\frac {b^{2} \sqrt {a +b \left (\cot ^{2}\left (e x +d \right )\right )+c \left (\cot ^{4}\left (e x +d \right )\right )}}{16 c^{2}}+\frac {b \ln \left (\frac {\frac {b}{2}+c \left (\cot ^{2}\left (e x +d \right )\right )}{\sqrt {c}}+\sqrt {a +b \left (\cot ^{2}\left (e x +d \right )\right )+c \left (\cot ^{4}\left (e x +d \right )\right )}\right ) a}{8 c^{\frac {3}{2}}}-\frac {b^{3} \ln \left (\frac {\frac {b}{2}+c \left (\cot ^{2}\left (e x +d \right )\right )}{\sqrt {c}}+\sqrt {a +b \left (\cot ^{2}\left (e x +d \right )\right )+c \left (\cot ^{4}\left (e x +d \right )\right )}\right )}{32 c^{\frac {5}{2}}}+\frac {\left (b +2 c \left (\cot ^{2}\left (e x +d \right )\right )\right ) \sqrt {a +b \left (\cot ^{2}\left (e x +d \right )\right )+c \left (\cot ^{4}\left (e x +d \right )\right )}}{8 c}+\frac {\ln \left (\frac {\frac {b}{2}+c \left (\cot ^{2}\left (e x +d \right )\right )}{\sqrt {c}}+\sqrt {a +b \left (\cot ^{2}\left (e x +d \right )\right )+c \left (\cot ^{4}\left (e x +d \right )\right )}\right ) a}{4 \sqrt {c}}-\frac {\ln \left (\frac {\frac {b}{2}+c \left (\cot ^{2}\left (e x +d \right )\right )}{\sqrt {c}}+\sqrt {a +b \left (\cot ^{2}\left (e x +d \right )\right )+c \left (\cot ^{4}\left (e x +d \right )\right )}\right ) b^{2}}{16 c^{\frac {3}{2}}}-\frac {\sqrt {c \left (\cot ^{2}\left (e x +d \right )+1\right )^{2}+\left (b -2 c \right ) \left (\cot ^{2}\left (e x +d \right )+1\right )+a -b +c}}{2}-\frac {\ln \left (\frac {\frac {b}{2}-c +\left (\cot ^{2}\left (e x +d \right )+1\right ) c}{\sqrt {c}}+\sqrt {c \left (\cot ^{2}\left (e x +d \right )+1\right )^{2}+\left (b -2 c \right ) \left (\cot ^{2}\left (e x +d \right )+1\right )+a -b +c}\right ) b}{4 \sqrt {c}}+\frac {\ln \left (\frac {\frac {b}{2}-c +\left (\cot ^{2}\left (e x +d \right )+1\right ) c}{\sqrt {c}}+\sqrt {c \left (\cot ^{2}\left (e x +d \right )+1\right )^{2}+\left (b -2 c \right ) \left (\cot ^{2}\left (e x +d \right )+1\right )+a -b +c}\right ) \sqrt {c}}{2}+\frac {\sqrt {a -b +c}\, \ln \left (\frac {2 a -2 b +2 c +\left (b -2 c \right ) \left (\cot ^{2}\left (e x +d \right )+1\right )+2 \sqrt {a -b +c}\, \sqrt {c \left (\cot ^{2}\left (e x +d \right )+1\right )^{2}+\left (b -2 c \right ) \left (\cot ^{2}\left (e x +d \right )+1\right )+a -b +c}}{\cot ^{2}\left (e x +d \right )+1}\right )}{2}}{e}\)	\(627\)
default	\(\frac {-\frac {\left (a +b \left (\cot ^{2}\left (e x +d \right )\right )+c \left (\cot ^{4}\left (e x +d \right )\right )\right )^{\frac {3}{2}}}{6 c}+\frac {b \left (\cot ^{2}\left (e x +d \right )\right ) \sqrt {a +b \left (\cot ^{2}\left (e x +d \right )\right )+c \left (\cot ^{4}\left (e x +d \right )\right )}}{8 c}+\frac {b^{2} \sqrt {a +b \left (\cot ^{2}\left (e x +d \right )\right )+c \left (\cot ^{4}\left (e x +d \right )\right )}}{16 c^{2}}+\frac {b \ln \left (\frac {\frac {b}{2}+c \left (\cot ^{2}\left (e x +d \right )\right )}{\sqrt {c}}+\sqrt {a +b \left (\cot ^{2}\left (e x +d \right )\right )+c \left (\cot ^{4}\left (e x +d \right )\right )}\right ) a}{8 c^{\frac {3}{2}}}-\frac {b^{3} \ln \left (\frac {\frac {b}{2}+c \left (\cot ^{2}\left (e x +d \right )\right )}{\sqrt {c}}+\sqrt {a +b \left (\cot ^{2}\left (e x +d \right )\right )+c \left (\cot ^{4}\left (e x +d \right )\right )}\right )}{32 c^{\frac {5}{2}}}+\frac {\left (b +2 c \left (\cot ^{2}\left (e x +d \right )\right )\right ) \sqrt {a +b \left (\cot ^{2}\left (e x +d \right )\right )+c \left (\cot ^{4}\left (e x +d \right )\right )}}{8 c}+\frac {\ln \left (\frac {\frac {b}{2}+c \left (\cot ^{2}\left (e x +d \right )\right )}{\sqrt {c}}+\sqrt {a +b \left (\cot ^{2}\left (e x +d \right )\right )+c \left (\cot ^{4}\left (e x +d \right )\right )}\right ) a}{4 \sqrt {c}}-\frac {\ln \left (\frac {\frac {b}{2}+c \left (\cot ^{2}\left (e x +d \right )\right )}{\sqrt {c}}+\sqrt {a +b \left (\cot ^{2}\left (e x +d \right )\right )+c \left (\cot ^{4}\left (e x +d \right )\right )}\right ) b^{2}}{16 c^{\frac {3}{2}}}-\frac {\sqrt {c \left (\cot ^{2}\left (e x +d \right )+1\right )^{2}+\left (b -2 c \right ) \left (\cot ^{2}\left (e x +d \right )+1\right )+a -b +c}}{2}-\frac {\ln \left (\frac {\frac {b}{2}-c +\left (\cot ^{2}\left (e x +d \right )+1\right ) c}{\sqrt {c}}+\sqrt {c \left (\cot ^{2}\left (e x +d \right )+1\right )^{2}+\left (b -2 c \right ) \left (\cot ^{2}\left (e x +d \right )+1\right )+a -b +c}\right ) b}{4 \sqrt {c}}+\frac {\ln \left (\frac {\frac {b}{2}-c +\left (\cot ^{2}\left (e x +d \right )+1\right ) c}{\sqrt {c}}+\sqrt {c \left (\cot ^{2}\left (e x +d \right )+1\right )^{2}+\left (b -2 c \right ) \left (\cot ^{2}\left (e x +d \right )+1\right )+a -b +c}\right ) \sqrt {c}}{2}+\frac {\sqrt {a -b +c}\, \ln \left (\frac {2 a -2 b +2 c +\left (b -2 c \right ) \left (\cot ^{2}\left (e x +d \right )+1\right )+2 \sqrt {a -b +c}\, \sqrt {c \left (\cot ^{2}\left (e x +d \right )+1\right )^{2}+\left (b -2 c \right ) \left (\cot ^{2}\left (e x +d \right )+1\right )+a -b +c}}{\cot ^{2}\left (e x +d \right )+1}\right )}{2}}{e}\)	\(627\)

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,method=_RETURNVERBOSE)

[Out]

1/e*(-1/6*(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(3/2)/c+1/8*b/c*cot(e*x+d)^2*(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/
2)+1/16*b^2/c^2*(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2)+1/8*b/c^(3/2)*ln((1/2*b+c*cot(e*x+d)^2)/c^(1/2)+(a+b*c
ot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2))*a-1/32*b^3/c^(5/2)*ln((1/2*b+c*cot(e*x+d)^2)/c^(1/2)+(a+b*cot(e*x+d)^2+c*co
t(e*x+d)^4)^(1/2))+1/8*(b+2*c*cot(e*x+d)^2)/c*(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2)+1/4/c^(1/2)*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))*a-1/16/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))*b^2-1/2*(c*(cot(e*x+d)^2+1)^2+(b-2*c)*(cot(e*x+d)^2+1)+a-b+c)^(1/2)
-1/4*ln((1/2*b-c+(cot(e*x+d)^2+1)*c)/c^(1/2)+(c*(cot(e*x+d)^2+1)^2+(b-2*c)*(cot(e*x+d)^2+1)+a-b+c)^(1/2))/c^(1
/2)*b+1/2*ln((1/2*b-c+(cot(e*x+d)^2+1)*c)/c^(1/2)+(c*(cot(e*x+d)^2+1)^2+(b-2*c)*(cot(e*x+d)^2+1)+a-b+c)^(1/2))
*c^(1/2)+1/2*(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)*(c*(cot(e*x+d)^2+1)^2+(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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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]

integrate(sqrt(c*cot(x*e + d)^4 + b*cot(x*e + d)^2 + a)*cot(x*e + d)^5, x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 768 vs. \(2 (249) = 498\).
time = 8.56, size = 3147, normalized size = 11.66 \begin {gather*} \text {Too large to display} \end {gather*}

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]

[1/192*(48*(c^3*cos(2*x*e + 2*d)^2 - 2*c^3*cos(2*x*e + 2*d) + c^3)*sqrt(a - b + c)*log(2*(a^2 - 2*a*b + b^2 +
2*(a - b)*c + c^2)*cos(2*x*e + 2*d)^2 + 2*a^2 - b^2 + 2*c^2 + 2*((a - b + c)*cos(2*x*e + 2*d)^2 - (2*a - b)*co
s(2*x*e + 2*d) + a - c)*sqrt(a - b + c)*sqrt(((a - b + c)*cos(2*x*e + 2*d)^2 - 2*(a - c)*cos(2*x*e + 2*d) + a
+ b + c)/(cos(2*x*e + 2*d)^2 - 2*cos(2*x*e + 2*d) + 1)) - 4*(a^2 - a*b + b*c - c^2)*cos(2*x*e + 2*d)) - 3*(b^3
- 8*(a - b)*c^2 - 16*c^3 + (b^3 - 8*(a - b)*c^2 - 16*c^3 - 2*(2*a*b - b^2)*c)*cos(2*x*e + 2*d)^2 - 2*(2*a*b -
b^2)*c - 2*(b^3 - 8*(a - b)*c^2 - 16*c^3 - 2*(2*a*b - b^2)*c)*cos(2*x*e + 2*d))*sqrt(c)*log(((b^2 + 4*(a - 2*
b)*c + 8*c^2)*cos(2*x*e + 2*d)^2 + b^2 + 4*(a + 2*b)*c + 8*c^2 + 4*((b - 2*c)*cos(2*x*e + 2*d)^2 - 2*b*cos(2*x
*e + 2*d) + b + 2*c)*sqrt(c)*sqrt(((a - b + c)*cos(2*x*e + 2*d)^2 - 2*(a - c)*cos(2*x*e + 2*d) + a + b + c)/(c
os(2*x*e + 2*d)^2 - 2*cos(2*x*e + 2*d) + 1)) - 2*(b^2 + 4*a*c - 8*c^2)*cos(2*x*e + 2*d))/(cos(2*x*e + 2*d)^2 -
2*cos(2*x*e + 2*d) + 1)) + 4*(3*b^2*c - 4*(2*a - b)*c^2 - 20*c^3 + (3*b^2*c - 8*(a - b)*c^2 - 44*c^3)*cos(2*x
*e + 2*d)^2 - 2*(3*b^2*c - 2*(4*a - 3*b)*c^2 - 16*c^3)*cos(2*x*e + 2*d))*sqrt(((a - b + c)*cos(2*x*e + 2*d)^2
- 2*(a - c)*cos(2*x*e + 2*d) + a + b + c)/(cos(2*x*e + 2*d)^2 - 2*cos(2*x*e + 2*d) + 1)))/(c^3*cos(2*x*e + 2*d
)^2*e - 2*c^3*cos(2*x*e + 2*d)*e + c^3*e), -1/96*(3*(b^3 - 8*(a - b)*c^2 - 16*c^3 + (b^3 - 8*(a - b)*c^2 - 16*
c^3 - 2*(2*a*b - b^2)*c)*cos(2*x*e + 2*d)^2 - 2*(2*a*b - b^2)*c - 2*(b^3 - 8*(a - b)*c^2 - 16*c^3 - 2*(2*a*b -
b^2)*c)*cos(2*x*e + 2*d))*sqrt(-c)*arctan(-1/2*((b - 2*c)*cos(2*x*e + 2*d)^2 - 2*b*cos(2*x*e + 2*d) + b + 2*c
)*sqrt(-c)*sqrt(((a - b + c)*cos(2*x*e + 2*d)^2 - 2*(a - c)*cos(2*x*e + 2*d) + a + b + c)/(cos(2*x*e + 2*d)^2
- 2*cos(2*x*e + 2*d) + 1))/(((a - b)*c + c^2)*cos(2*x*e + 2*d)^2 + (a + b)*c + c^2 - 2*(a*c - c^2)*cos(2*x*e +
2*d))) - 24*(c^3*cos(2*x*e + 2*d)^2 - 2*c^3*cos(2*x*e + 2*d) + c^3)*sqrt(a - b + c)*log(2*(a^2 - 2*a*b + b^2
+ 2*(a - b)*c + c^2)*cos(2*x*e + 2*d)^2 + 2*a^2 - b^2 + 2*c^2 + 2*((a - b + c)*cos(2*x*e + 2*d)^2 - (2*a - b)*
cos(2*x*e + 2*d) + a - c)*sqrt(a - b + c)*sqrt(((a - b + c)*cos(2*x*e + 2*d)^2 - 2*(a - c)*cos(2*x*e + 2*d) +
a + b + c)/(cos(2*x*e + 2*d)^2 - 2*cos(2*x*e + 2*d) + 1)) - 4*(a^2 - a*b + b*c - c^2)*cos(2*x*e + 2*d)) - 2*(3
*b^2*c - 4*(2*a - b)*c^2 - 20*c^3 + (3*b^2*c - 8*(a - b)*c^2 - 44*c^3)*cos(2*x*e + 2*d)^2 - 2*(3*b^2*c - 2*(4*
a - 3*b)*c^2 - 16*c^3)*cos(2*x*e + 2*d))*sqrt(((a - b + c)*cos(2*x*e + 2*d)^2 - 2*(a - c)*cos(2*x*e + 2*d) + a
+ b + c)/(cos(2*x*e + 2*d)^2 - 2*cos(2*x*e + 2*d) + 1)))/(c^3*cos(2*x*e + 2*d)^2*e - 2*c^3*cos(2*x*e + 2*d)*e
+ c^3*e), -1/192*(96*(c^3*cos(2*x*e + 2*d)^2 - 2*c^3*cos(2*x*e + 2*d) + c^3)*sqrt(-a + b - c)*arctan(((a - b
+ c)*cos(2*x*e + 2*d)^2 - (2*a - b)*cos(2*x*e + 2*d) + a - c)*sqrt(-a + b - c)*sqrt(((a - b + c)*cos(2*x*e + 2
*d)^2 - 2*(a - c)*cos(2*x*e + 2*d) + a + b + c)/(cos(2*x*e + 2*d)^2 - 2*cos(2*x*e + 2*d) + 1))/((a^2 - 2*a*b +
b^2 + 2*(a - b)*c + c^2)*cos(2*x*e + 2*d)^2 + a^2 - b^2 + 2*a*c + c^2 - 2*(a^2 - a*b + b*c - c^2)*cos(2*x*e +
2*d))) + 3*(b^3 - 8*(a - b)*c^2 - 16*c^3 + (b^3 - 8*(a - b)*c^2 - 16*c^3 - 2*(2*a*b - b^2)*c)*cos(2*x*e + 2*d
)^2 - 2*(2*a*b - b^2)*c - 2*(b^3 - 8*(a - b)*c^2 - 16*c^3 - 2*(2*a*b - b^2)*c)*cos(2*x*e + 2*d))*sqrt(c)*log((
(b^2 + 4*(a - 2*b)*c + 8*c^2)*cos(2*x*e + 2*d)^2 + b^2 + 4*(a + 2*b)*c + 8*c^2 + 4*((b - 2*c)*cos(2*x*e + 2*d)
^2 - 2*b*cos(2*x*e + 2*d) + b + 2*c)*sqrt(c)*sqrt(((a - b + c)*cos(2*x*e + 2*d)^2 - 2*(a - c)*cos(2*x*e + 2*d)
+ a + b + c)/(cos(2*x*e + 2*d)^2 - 2*cos(2*x*e + 2*d) + 1)) - 2*(b^2 + 4*a*c - 8*c^2)*cos(2*x*e + 2*d))/(cos(
2*x*e + 2*d)^2 - 2*cos(2*x*e + 2*d) + 1)) - 4*(3*b^2*c - 4*(2*a - b)*c^2 - 20*c^3 + (3*b^2*c - 8*(a - b)*c^2 -
44*c^3)*cos(2*x*e + 2*d)^2 - 2*(3*b^2*c - 2*(4*a - 3*b)*c^2 - 16*c^3)*cos(2*x*e + 2*d))*sqrt(((a - b + c)*cos
(2*x*e + 2*d)^2 - 2*(a - c)*cos(2*x*e + 2*d) + a + b + c)/(cos(2*x*e + 2*d)^2 - 2*cos(2*x*e + 2*d) + 1)))/(c^3
*cos(2*x*e + 2*d)^2*e - 2*c^3*cos(2*x*e + 2*d)*e + c^3*e), -1/96*(48*(c^3*cos(2*x*e + 2*d)^2 - 2*c^3*cos(2*x*e
+ 2*d) + c^3)*sqrt(-a + b - c)*arctan(((a - b + c)*cos(2*x*e + 2*d)^2 - (2*a - b)*cos(2*x*e + 2*d) + a - c)*s
qrt(-a + b - c)*sqrt(((a - b + c)*cos(2*x*e + 2*d)^2 - 2*(a - c)*cos(2*x*e + 2*d) + a + b + c)/(cos(2*x*e + 2*
d)^2 - 2*cos(2*x*e + 2*d) + 1))/((a^2 - 2*a*b + b^2 + 2*(a - b)*c + c^2)*cos(2*x*e + 2*d)^2 + a^2 - b^2 + 2*a*
c + c^2 - 2*(a^2 - a*b + b*c - c^2)*cos(2*x*e + 2*d))) + 3*(b^3 - 8*(a - b)*c^2 - 16*c^3 + (b^3 - 8*(a - b)*c^
2 - 16*c^3 - 2*(2*a*b - b^2)*c)*cos(2*x*e + 2*d)^2 - 2*(2*a*b - b^2)*c - 2*(b^3 - 8*(a - b)*c^2 - 16*c^3 - 2*(
2*a*b - b^2)*c)*cos(2*x*e + 2*d))*sqrt(-c)*arctan(-1/2*((b - 2*c)*cos(2*x*e + 2*d)^2 - 2*b*cos(2*x*e + 2*d) +
b + 2*c)*sqrt(-c)*sqrt(((a - b + c)*cos(2*x*e + 2*d)^2 - 2*(a - c)*cos(2*x*e + 2*d) + a + b + c)/(cos(2*x*e +
2*d)^2 - 2*cos(2*x*e + 2*d) + 1))/(((a - b)*c + c^2)*cos(2*x*e + 2*d)^2 + (a + b)*c + c^2 - 2*(a*c - c^2)*cos(
2*x*e + 2*d))) - 2*(3*b^2*c - 4*(2*a - b)*c^2 -...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {a + b \cot ^{2}{\left (d + e x \right )} + c \cot ^{4}{\left (d + e x \right )}} \cot ^{5}{\left (d + e x \right )}\, dx \end {gather*}

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]

Integral(sqrt(a + b*cot(d + e*x)**2 + c*cot(d + e*x)**4)*cot(d + e*x)**5, x)

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

\text{Hanged}

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