3.1.27 \(\int \frac {\cot ^7(d+e x)}{(a+b \cot ^2(d+e x)+c \cot ^4(d+e x))^{3/2}} \, dx\) [27]

Optimal. Leaf size=236 \[ -\frac {\tanh ^{-1}\left (\frac {2 a-b+(b-2 c) \cot ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 (a-b+c)^{3/2} e}-\frac {\tanh ^{-1}\left (\frac {b+2 c \cot ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 c^{3/2} e}-\frac {a \left (b^2-a (b+2 c)\right )+\left (b^3+2 a^2 c-a b (b+3 c)\right ) \cot ^2(d+e x)}{c (a-b+c) \left (b^2-4 a c\right ) e \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \]

[Out]

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

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Rubi [A]
time = 0.37, antiderivative size = 236, 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, 1660, 857, 635, 212, 738} \begin {gather*} -\frac {\left (2 a^2 c-a b (b+3 c)+b^3\right ) \cot ^2(d+e x)+a \left (b^2-a (b+2 c)\right )}{c e (a-b+c) \left (b^2-4 a c\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}-\frac {\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 )}{2 c^{3/2} 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 (a-b+c)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cot[d + e*x]^7/(a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4)^(3/2),x]

[Out]

-1/2*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])]/((a - b + c)^(3/2)*e) - 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])]/(2*c^(3/2)*e) - (a*(b^2 - a*(b + 2*c)) + (b^3 + 2*a^2*c - a*b*(b + 3*c))*Cot[d + e*x]^2)/(c*(a -
b + c)*(b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])

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 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 1660

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomi
alQuotient[(d + e*x)^m*Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2,
x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2, x], x, 1]}, Simp[(b*f - 2*a*g + (2*
c*f - b*g)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(d
+ e*x)^m*(a + b*x + c*x^2)^(p + 1)*ExpandToSum[((p + 1)*(b^2 - 4*a*c)*Q)/(d + e*x)^m - ((2*p + 3)*(2*c*f - b*
g))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && LtQ[p, -1] && ILtQ[m, 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 \frac {\cot ^7(d+e x)}{\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}} \, dx &=-\frac {\text {Subst}\left (\int \frac {x^7}{\left (1+x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}} \, dx,x,\cot (d+e x)\right )}{e}\\ &=-\frac {\text {Subst}\left (\int \frac {x^3}{(1+x) \left (a+b x+c x^2\right )^{3/2}} \, dx,x,\cot ^2(d+e x)\right )}{2 e}\\ &=-\frac {a \left (b^2-a (b+2 c)\right )+\left (b^3+2 a^2 c-a b (b+3 c)\right ) \cot ^2(d+e x)}{c (a-b+c) \left (b^2-4 a c\right ) e \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}+\frac {\text {Subst}\left (\int \frac {-\frac {(a-b) \left (b^2-4 a c\right )}{2 c (a-b+c)}-\frac {\left (b^2-4 a c\right ) x}{2 c}}{(1+x) \sqrt {a+b x+c x^2}} \, dx,x,\cot ^2(d+e x)\right )}{\left (b^2-4 a c\right ) e}\\ &=-\frac {a \left (b^2-a (b+2 c)\right )+\left (b^3+2 a^2 c-a b (b+3 c)\right ) \cot ^2(d+e x)}{c (a-b+c) \left (b^2-4 a c\right ) e \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,\cot ^2(d+e x)\right )}{2 c e}+\frac {\text {Subst}\left (\int \frac {1}{(1+x) \sqrt {a+b x+c x^2}} \, dx,x,\cot ^2(d+e x)\right )}{2 (a-b+c) e}\\ &=-\frac {a \left (b^2-a (b+2 c)\right )+\left (b^3+2 a^2 c-a b (b+3 c)\right ) \cot ^2(d+e x)}{c (a-b+c) \left (b^2-4 a c\right ) e \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}-\frac {\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 )}{c e}-\frac {\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 )}{(a-b+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 (a-b+c)^{3/2} e}-\frac {\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 )}{2 c^{3/2} e}-\frac {a \left (b^2-a (b+2 c)\right )+\left (b^3+2 a^2 c-a b (b+3 c)\right ) \cot ^2(d+e x)}{c (a-b+c) \left (b^2-4 a c\right ) e \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 36.14, size = 243520, normalized size = 1031.86 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[Cot[d + e*x]^7/(a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4)^(3/2),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. \(720\) vs. \(2(215)=430\).
time = 2.03, size = 721, normalized size = 3.06

method	result	size
derivativedivides	\(\frac {\frac {\cot ^{2}\left (e x +d \right )}{2 c \sqrt {a +b \left (\cot ^{2}\left (e x +d \right )\right )+c \left (\cot ^{4}\left (e x +d \right )\right )}}-\frac {b}{4 c^{2} \sqrt {a +b \left (\cot ^{2}\left (e x +d \right )\right )+c \left (\cot ^{4}\left (e x +d \right )\right )}}-\frac {b^{2} \left (\cot ^{2}\left (e x +d \right )\right )}{2 c \left (4 a c -b^{2}\right ) \sqrt {a +b \left (\cot ^{2}\left (e x +d \right )\right )+c \left (\cot ^{4}\left (e x +d \right )\right )}}-\frac {b^{3}}{4 c^{2} \left (4 a c -b^{2}\right ) \sqrt {a +b \left (\cot ^{2}\left (e x +d \right )\right )+c \left (\cot ^{4}\left (e x +d \right )\right )}}-\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 )}{2 c^{\frac {3}{2}}}-\frac {2 a +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 )}\, \left (4 a c -b^{2}\right )}-\frac {b +2 c \left (\cot ^{2}\left (e x +d \right )\right )}{\left (4 a c -b^{2}\right ) \sqrt {a +b \left (\cot ^{2}\left (e x +d \right )\right )+c \left (\cot ^{4}\left (e x +d \right )\right )}}-\frac {2 c \sqrt {\left (\cot ^{2}\left (e x +d \right )-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )^{2} c +\sqrt {-4 a c +b^{2}}\, \left (\cot ^{2}\left (e x +d \right )-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{\left (\sqrt {-4 a c +b^{2}}-b +2 c \right ) \left (-4 a c +b^{2}\right ) \left (\cot ^{2}\left (e x +d \right )-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}+\frac {2 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 )}{\left (\sqrt {-4 a c +b^{2}}-b +2 c \right ) \left (\sqrt {-4 a c +b^{2}}+b -2 c \right ) \sqrt {a -b +c}}+\frac {2 c \sqrt {\left (\cot ^{2}\left (e x +d \right )+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )^{2} c -\sqrt {-4 a c +b^{2}}\, \left (\cot ^{2}\left (e x +d \right )+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{\left (\sqrt {-4 a c +b^{2}}+b -2 c \right ) \left (-4 a c +b^{2}\right ) \left (\cot ^{2}\left (e x +d \right )+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{e}\)	\(721\)
default	\(\frac {\frac {\cot ^{2}\left (e x +d \right )}{2 c \sqrt {a +b \left (\cot ^{2}\left (e x +d \right )\right )+c \left (\cot ^{4}\left (e x +d \right )\right )}}-\frac {b}{4 c^{2} \sqrt {a +b \left (\cot ^{2}\left (e x +d \right )\right )+c \left (\cot ^{4}\left (e x +d \right )\right )}}-\frac {b^{2} \left (\cot ^{2}\left (e x +d \right )\right )}{2 c \left (4 a c -b^{2}\right ) \sqrt {a +b \left (\cot ^{2}\left (e x +d \right )\right )+c \left (\cot ^{4}\left (e x +d \right )\right )}}-\frac {b^{3}}{4 c^{2} \left (4 a c -b^{2}\right ) \sqrt {a +b \left (\cot ^{2}\left (e x +d \right )\right )+c \left (\cot ^{4}\left (e x +d \right )\right )}}-\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 )}{2 c^{\frac {3}{2}}}-\frac {2 a +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 )}\, \left (4 a c -b^{2}\right )}-\frac {b +2 c \left (\cot ^{2}\left (e x +d \right )\right )}{\left (4 a c -b^{2}\right ) \sqrt {a +b \left (\cot ^{2}\left (e x +d \right )\right )+c \left (\cot ^{4}\left (e x +d \right )\right )}}-\frac {2 c \sqrt {\left (\cot ^{2}\left (e x +d \right )-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )^{2} c +\sqrt {-4 a c +b^{2}}\, \left (\cot ^{2}\left (e x +d \right )-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{\left (\sqrt {-4 a c +b^{2}}-b +2 c \right ) \left (-4 a c +b^{2}\right ) \left (\cot ^{2}\left (e x +d \right )-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}+\frac {2 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 )}{\left (\sqrt {-4 a c +b^{2}}-b +2 c \right ) \left (\sqrt {-4 a c +b^{2}}+b -2 c \right ) \sqrt {a -b +c}}+\frac {2 c \sqrt {\left (\cot ^{2}\left (e x +d \right )+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )^{2} c -\sqrt {-4 a c +b^{2}}\, \left (\cot ^{2}\left (e x +d \right )+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{\left (\sqrt {-4 a c +b^{2}}+b -2 c \right ) \left (-4 a c +b^{2}\right ) \left (\cot ^{2}\left (e x +d \right )+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{e}\)	\(721\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(e*x+d)^7/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(e*x+d)^7/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1515 vs. \(2 (221) = 442\).
time = 8.57, size = 6139, normalized size = 26.01 \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)^7/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(3/2),x, algorithm="fricas")

[Out]

[1/4*((4*a*c^4 + (4*a^2 + 4*a*b - b^2)*c^3 - (a*b^2 + b^3)*c^2 + (4*a*c^4 + (4*a^2 - 4*a*b - b^2)*c^3 - (a*b^2
- b^3)*c^2)*cos(2*x*e + 2*d)^2 + 2*(a*b^2*c^2 + 4*a*c^4 - (4*a^2 + b^2)*c^3)*cos(2*x*e + 2*d))*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)) - (a^3*b^2 - a^2*b^3 - a*b^4 + b^5 - 4*a*c^4 - (12*a^2 - 4*a*b - b^2)*c^3 - (12*a^3 - 8*a
^2*b - 7*a*b^2 + b^3)*c^2 + (a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5 - 4*a*c^4 - (12*a^2 - 12*a*b - b^2)*c^3 - 3*(
4*a^3 - 8*a^2*b + 3*a*b^2 + b^3)*c^2 - (4*a^4 - 12*a^3*b + 9*a^2*b^2 + 2*a*b^3 - 3*b^4)*c)*cos(2*x*e + 2*d)^2
- (4*a^4 - 4*a^3*b - 7*a^2*b^2 + 6*a*b^3 + b^4)*c - 2*(a^3*b^2 - 2*a^2*b^3 + a*b^4 + 4*a*c^4 + (4*a^2 - 8*a*b
- b^2)*c^3 - (4*a^3 - 3*a*b^2 - 2*b^3)*c^2 - (4*a^4 - 8*a^3*b + 3*a^2*b^2 + b^4)*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*a*b*c^3 + (4*a^2*b - 3*a*b^2 - b^3)*c^2 + ((4*a^2 - 3*a*
b)*c^3 + (4*a^3 - 6*a^2*b + a*b^2 + b^3)*c^2 + (a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4)*c)*cos(2*x*e + 2*d)^2 + (a^
3*b - a^2*b^2 - a*b^3 + b^4)*c - 2*(2*a^2*c^3 + (2*a^3 - a^2*b - a*b^2)*c^2 + (a^3*b - 2*a^2*b^2 + a*b^3)*c)*c
os(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)))/((4*a*c^6 + (12*a^2 - 12*a*b - b^2)*c^5 + 3*(4*a^3 - 8*a^2*b + 3*a*b^2 + b^3
)*c^4 + (4*a^4 - 12*a^3*b + 9*a^2*b^2 + 2*a*b^3 - 3*b^4)*c^3 - (a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*c^2)*cos(
2*x*e + 2*d)^2*e + 2*(4*a*c^6 + (4*a^2 - 8*a*b - b^2)*c^5 - (4*a^3 - 3*a*b^2 - 2*b^3)*c^4 - (4*a^4 - 8*a^3*b +
3*a^2*b^2 + b^4)*c^3 + (a^3*b^2 - 2*a^2*b^3 + a*b^4)*c^2)*cos(2*x*e + 2*d)*e + (4*a*c^6 + (12*a^2 - 4*a*b - b
^2)*c^5 + (12*a^3 - 8*a^2*b - 7*a*b^2 + b^3)*c^4 + (4*a^4 - 4*a^3*b - 7*a^2*b^2 + 6*a*b^3 + b^4)*c^3 - (a^3*b^
2 - a^2*b^3 - a*b^4 + b^5)*c^2)*e), 1/4*(2*(a^3*b^2 - a^2*b^3 - a*b^4 + b^5 - 4*a*c^4 - (12*a^2 - 4*a*b - b^2)
*c^3 - (12*a^3 - 8*a^2*b - 7*a*b^2 + b^3)*c^2 + (a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5 - 4*a*c^4 - (12*a^2 - 12*
a*b - b^2)*c^3 - 3*(4*a^3 - 8*a^2*b + 3*a*b^2 + b^3)*c^2 - (4*a^4 - 12*a^3*b + 9*a^2*b^2 + 2*a*b^3 - 3*b^4)*c)
*cos(2*x*e + 2*d)^2 - (4*a^4 - 4*a^3*b - 7*a^2*b^2 + 6*a*b^3 + b^4)*c - 2*(a^3*b^2 - 2*a^2*b^3 + a*b^4 + 4*a*c
^4 + (4*a^2 - 8*a*b - b^2)*c^3 - (4*a^3 - 3*a*b^2 - 2*b^3)*c^2 - (4*a^4 - 8*a^3*b + 3*a^2*b^2 + b^4)*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))) + (4*a*
c^4 + (4*a^2 + 4*a*b - b^2)*c^3 - (a*b^2 + b^3)*c^2 + (4*a*c^4 + (4*a^2 - 4*a*b - b^2)*c^3 - (a*b^2 - b^3)*c^2
)*cos(2*x*e + 2*d)^2 + 2*(a*b^2*c^2 + 4*a*c^4 - (4*a^2 + b^2)*c^3)*cos(2*x*e + 2*d))*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)) - 4*(3*a*b*c^3 + (4*a^2*b - 3*a*b^2 - b^3)*c^2 + ((4*a^2 - 3*a*b)*c^3 + (4*a^3 - 6*a^2*b + a*b^2 + b
^3)*c^2 + (a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4)*c)*cos(2*x*e + 2*d)^2 + (a^3*b - a^2*b^2 - a*b^3 + b^4)*c - 2*(2
*a^2*c^3 + (2*a^3 - a^2*b - a*b^2)*c^2 + (a^3*b - 2*a^2*b^2 + a*b^3)*c)*cos(2*x*e + 2*d))*sqrt(((a - b + c)*co
s(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*c^6 + (12*a^2 - 12*a*b - b^2)*c^5 + 3*(4*a^3 - 8*a^2*b + 3*a*b^2 + b^3)*c^4 + (4*a^4 - 12*a^3*b + 9*a^2*b^2
+ 2*a*b^3 - 3*b^4)*c^3 - (a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*c^2)*cos(2*x*e + 2*d)^2*e + 2*(4*a*c^6 + (4*a^
2 - 8*a*b - b^2)*c^5 - (4*a^3 - 3*a*b^2 - 2*b^3)*c^4 - (4*a^4 - 8*a^3*b + 3*a^2*b^2 + b^4)*c^3 + (a^3*b^2 - 2*
a^2*b^3 + a*b^4)*c^2)*cos(2*x*e + 2*d)*e + (4*a*c^6 + (12*a^2 - 4*a*b - b^2)*c^5 + (12*a^3 - 8*a^2*b - 7*a*b^2
+ b^3)*c^4 + (4*a^4 - 4*a^3*b - 7*a^2*b^2 + 6*a*b^3 + b^4)*c^3 - (a^3*b^2 - a^2*b^3 - a*b^4 + b^5)*c^2)*e), 1
/4*(2*(4*a*c^4 + (4*a^2 + 4*a*b - b^2)*c^3 - (a*b^2 + b^3)*c^2 + (4*a*c^4 + (4*a^2 - 4*a*b - b^2)*c^3 - (a*b^2
- b^3)*c^2)*cos(2*x*e + 2*d)^2 + 2*(a*b^2*c^2 ...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cot ^{7}{\left (d + e x \right )}}{\left (a + b \cot ^{2}{\left (d + e x \right )} + c \cot ^{4}{\left (d + e x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(e*x+d)**7/(a+b*cot(e*x+d)**2+c*cot(e*x+d)**4)**(3/2),x)

[Out]

Integral(cot(d + e*x)**7/(a + b*cot(d + e*x)**2 + c*cot(d + e*x)**4)**(3/2), 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)^7/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(3/2),x, algorithm="giac")

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {cot}\left (d+e\,x\right )}^7}{{\left (c\,{\mathrm {cot}\left (d+e\,x\right )}^4+b\,{\mathrm {cot}\left (d+e\,x\right )}^2+a\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d + e*x)^7/(a + b*cot(d + e*x)^2 + c*cot(d + e*x)^4)^(3/2),x)

[Out]

int(cot(d + e*x)^7/(a + b*cot(d + e*x)^2 + c*cot(d + e*x)^4)^(3/2), x)

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