3.1.23 \(\int \cot ^3(d+e x) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx\) [23]

Optimal. Leaf size=209 \[ -\frac {\sqrt {a-b+c} \tanh ^{-1}\left (\frac {2 a-b+(b-2 c) \cot ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 e}+\frac {\left (b^2+4 b c-4 c (a+2 c)\right ) \tanh ^{-1}\left (\frac {b+2 c \cot ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{16 c^{3/2} e}-\frac {\left (b-4 c+2 c \cot ^2(d+e x)\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{8 c e} \]

[Out]

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

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Rubi [A]
time = 0.24, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3782, 1265, 828, 857, 635, 212, 738} \begin {gather*} \frac {\left (-4 c (a+2 c)+b^2+4 b 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 )}{16 c^{3/2} e}-\frac {\left (b+2 c \cot ^2(d+e x)-4 c\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{8 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]^3*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4],x]

[Out]

-1/2*(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])])/e + ((b^2 + 4*b*c - 4*c*(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])])/(16*c^(3/2)*e) - ((b - 4*c + 2*c*Cot[d + e*x]^2)*Sqrt[a + b*Cot[d +
e*x]^2 + c*Cot[d + e*x]^4])/(8*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 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 ^3(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^3 \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 \sqrt {a+b x+c x^2}}{1+x} \, dx,x,\cot ^2(d+e x)\right )}{2 e}\\ &=-\frac {\left (b-4 c+2 c \cot ^2(d+e x)\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{8 c e}+\frac {\text {Subst}\left (\int \frac {\frac {1}{2} \left (b^2+4 a c-4 b c\right )+\frac {1}{2} \left (b^2+4 b c-4 c (a+2 c)\right ) x}{(1+x) \sqrt {a+b x+c x^2}} \, dx,x,\cot ^2(d+e x)\right )}{8 c e}\\ &=-\frac {\left (b-4 c+2 c \cot ^2(d+e x)\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{8 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^2+4 b c-4 c (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 )}{16 c e}\\ &=-\frac {\left (b-4 c+2 c \cot ^2(d+e x)\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{8 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^2+4 b c-4 c (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 )}{8 c 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^2+4 b c-4 c (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 )}{16 c^{3/2} e}-\frac {\left (b-4 c+2 c \cot ^2(d+e x)\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{8 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 = 36.40, size = 412434, normalized size = 1973.37 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[Cot[d + e*x]^3*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. \(424\) vs. \(2(185)=370\).
time = 0.23, size = 425, normalized size = 2.03

method	result	size
derivativedivides	\(\frac {-\frac {\left (b +2 c \left (\cot ^{2}\left (e x +d \right )\right )\right ) \sqrt {a +b \left (\cot ^{2}\left (e x +d \right )\right )+c \left (\cot ^{4}\left (e x +d \right )\right )}}{8 c}-\frac {\ln \left (\frac {\frac {b}{2}+c \left (\cot ^{2}\left (e x +d \right )\right )}{\sqrt {c}}+\sqrt {a +b \left (\cot ^{2}\left (e x +d \right )\right )+c \left (\cot ^{4}\left (e x +d \right )\right )}\right ) a}{4 \sqrt {c}}+\frac {\ln \left (\frac {\frac {b}{2}+c \left (\cot ^{2}\left (e x +d \right )\right )}{\sqrt {c}}+\sqrt {a +b \left (\cot ^{2}\left (e x +d \right )\right )+c \left (\cot ^{4}\left (e x +d \right )\right )}\right ) b^{2}}{16 c^{\frac {3}{2}}}+\frac {\sqrt {c \left (\cot ^{2}\left (e x +d \right )+1\right )^{2}+\left (b -2 c \right ) \left (\cot ^{2}\left (e x +d \right )+1\right )+a -b +c}}{2}+\frac {\ln \left (\frac {\frac {b}{2}-c +\left (\cot ^{2}\left (e x +d \right )+1\right ) c}{\sqrt {c}}+\sqrt {c \left (\cot ^{2}\left (e x +d \right )+1\right )^{2}+\left (b -2 c \right ) \left (\cot ^{2}\left (e x +d \right )+1\right )+a -b +c}\right ) b}{4 \sqrt {c}}-\frac {\ln \left (\frac {\frac {b}{2}-c +\left (\cot ^{2}\left (e x +d \right )+1\right ) c}{\sqrt {c}}+\sqrt {c \left (\cot ^{2}\left (e x +d \right )+1\right )^{2}+\left (b -2 c \right ) \left (\cot ^{2}\left (e x +d \right )+1\right )+a -b +c}\right ) \sqrt {c}}{2}-\frac {\sqrt {a -b +c}\, \ln \left (\frac {2 a -2 b +2 c +\left (b -2 c \right ) \left (\cot ^{2}\left (e x +d \right )+1\right )+2 \sqrt {a -b +c}\, \sqrt {c \left (\cot ^{2}\left (e x +d \right )+1\right )^{2}+\left (b -2 c \right ) \left (\cot ^{2}\left (e x +d \right )+1\right )+a -b +c}}{\cot ^{2}\left (e x +d \right )+1}\right )}{2}}{e}\)	\(425\)
default	\(\frac {-\frac {\left (b +2 c \left (\cot ^{2}\left (e x +d \right )\right )\right ) \sqrt {a +b \left (\cot ^{2}\left (e x +d \right )\right )+c \left (\cot ^{4}\left (e x +d \right )\right )}}{8 c}-\frac {\ln \left (\frac {\frac {b}{2}+c \left (\cot ^{2}\left (e x +d \right )\right )}{\sqrt {c}}+\sqrt {a +b \left (\cot ^{2}\left (e x +d \right )\right )+c \left (\cot ^{4}\left (e x +d \right )\right )}\right ) a}{4 \sqrt {c}}+\frac {\ln \left (\frac {\frac {b}{2}+c \left (\cot ^{2}\left (e x +d \right )\right )}{\sqrt {c}}+\sqrt {a +b \left (\cot ^{2}\left (e x +d \right )\right )+c \left (\cot ^{4}\left (e x +d \right )\right )}\right ) b^{2}}{16 c^{\frac {3}{2}}}+\frac {\sqrt {c \left (\cot ^{2}\left (e x +d \right )+1\right )^{2}+\left (b -2 c \right ) \left (\cot ^{2}\left (e x +d \right )+1\right )+a -b +c}}{2}+\frac {\ln \left (\frac {\frac {b}{2}-c +\left (\cot ^{2}\left (e x +d \right )+1\right ) c}{\sqrt {c}}+\sqrt {c \left (\cot ^{2}\left (e x +d \right )+1\right )^{2}+\left (b -2 c \right ) \left (\cot ^{2}\left (e x +d \right )+1\right )+a -b +c}\right ) b}{4 \sqrt {c}}-\frac {\ln \left (\frac {\frac {b}{2}-c +\left (\cot ^{2}\left (e x +d \right )+1\right ) c}{\sqrt {c}}+\sqrt {c \left (\cot ^{2}\left (e x +d \right )+1\right )^{2}+\left (b -2 c \right ) \left (\cot ^{2}\left (e x +d \right )+1\right )+a -b +c}\right ) \sqrt {c}}{2}-\frac {\sqrt {a -b +c}\, \ln \left (\frac {2 a -2 b +2 c +\left (b -2 c \right ) \left (\cot ^{2}\left (e x +d \right )+1\right )+2 \sqrt {a -b +c}\, \sqrt {c \left (\cot ^{2}\left (e x +d \right )+1\right )^{2}+\left (b -2 c \right ) \left (\cot ^{2}\left (e x +d \right )+1\right )+a -b +c}}{\cot ^{2}\left (e x +d \right )+1}\right )}{2}}{e}\)	\(425\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(e*x+d)^3*(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/e*(-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)^3*(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)^3, x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 595 vs. \(2 (191) = 382\).
time = 7.16, size = 2452, normalized size = 11.73 \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)^3*(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2),x, algorithm="fricas")

[Out]

[1/32*(8*(c^2*cos(2*x*e + 2*d) - c^2)*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)) + (b^2 - 4*(a - b)*c - 8*c^2 - (b^2 -
4*(a - b)*c - 8*c^2)*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*(b*c - 2*c^
2 - (b*c - 6*c^2)*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^2*cos(2*x*e + 2*d)*e - c^2*e), -1/16*((b^2 - 4*(a - b)
*c - 8*c^2 - (b^2 - 4*(a - b)*c - 8*c^2)*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))) - 4*(c^2*cos(2*x*e + 2*d) - c^2)*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*(b*c - 2*c^2 - (b*c - 6*c^2)*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^2*cos(2*x*e + 2*d)*e - c^2*e), 1/32*(16*(
c^2*cos(2*x*e + 2*d) - c^2)*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)/(c
os(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))) + (b^2 - 4*(a - b)*c - 8*c^2 - (b^2 - 4*(a
- b)*c - 8*c^2)*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*(b*c - 2*c^2 - (
b*c - 6*c^2)*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^2*cos(2*x*e + 2*d)*e - c^2*e), 1/16*(8*(c^2*cos(2*x*e + 2*d
) - c^2)*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))) - (b^2 - 4*(a - b)*c - 8*c^2 - (b^2 - 4*(a - b)*c - 8*c^2)*co
s(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
*(b*c - 2*c^2 - (b*c - 6*c^2)*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^2*cos(2*x*e + 2*d)*e - c^2*e)]

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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 ^{3}{\left (d + e x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(e*x+d)**3*(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)**3, 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)^3*(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]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\mathrm {cot}\left (d+e\,x\right )}^3\,\sqrt {c\,{\mathrm {cot}\left (d+e\,x\right )}^4+b\,{\mathrm {cot}\left (d+e\,x\right )}^2+a} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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