3.1.68 \(\int \frac {(a+b \cot (c+d x))^3}{(e \cot (c+d x))^{9/2}} \, dx\) [68]

Optimal. Leaf size=377 \[ \frac {(a-b) \left (a^2+4 a b+b^2\right ) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{9/2}}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{9/2}}+\frac {32 a^2 b}{35 d e^2 (e \cot (c+d x))^{5/2}}-\frac {2 a \left (a^2-3 b^2\right )}{3 d e^3 (e \cot (c+d x))^{3/2}}-\frac {2 b \left (3 a^2-b^2\right )}{d e^4 \sqrt {e \cot (c+d x)}}+\frac {2 a^2 (a+b \cot (c+d x))}{7 d e (e \cot (c+d x))^{7/2}}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d e^{9/2}}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d e^{9/2}} \]

[Out]

32/35*a^2*b/d/e^2/(e*cot(d*x+c))^(5/2)-2/3*a*(a^2-3*b^2)/d/e^3/(e*cot(d*x+c))^(3/2)+2/7*a^2*(a+b*cot(d*x+c))/d
/e/(e*cot(d*x+c))^(7/2)+1/2*(a-b)*(a^2+4*a*b+b^2)*arctan(1-2^(1/2)*(e*cot(d*x+c))^(1/2)/e^(1/2))/d/e^(9/2)*2^(
1/2)-1/2*(a-b)*(a^2+4*a*b+b^2)*arctan(1+2^(1/2)*(e*cot(d*x+c))^(1/2)/e^(1/2))/d/e^(9/2)*2^(1/2)+1/4*(a+b)*(a^2
-4*a*b+b^2)*ln(e^(1/2)+cot(d*x+c)*e^(1/2)-2^(1/2)*(e*cot(d*x+c))^(1/2))/d/e^(9/2)*2^(1/2)-1/4*(a+b)*(a^2-4*a*b
+b^2)*ln(e^(1/2)+cot(d*x+c)*e^(1/2)+2^(1/2)*(e*cot(d*x+c))^(1/2))/d/e^(9/2)*2^(1/2)-2*b*(3*a^2-b^2)/d/e^4/(e*c
ot(d*x+c))^(1/2)

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Rubi [A]
time = 0.47, antiderivative size = 377, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3646, 3709, 3610, 3615, 1182, 1176, 631, 210, 1179, 642} \begin {gather*} \frac {(a-b) \left (a^2+4 a b+b^2\right ) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{9/2}}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} d e^{9/2}}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d e^{9/2}}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d e^{9/2}}-\frac {2 b \left (3 a^2-b^2\right )}{d e^4 \sqrt {e \cot (c+d x)}}-\frac {2 a \left (a^2-3 b^2\right )}{3 d e^3 (e \cot (c+d x))^{3/2}}+\frac {32 a^2 b}{35 d e^2 (e \cot (c+d x))^{5/2}}+\frac {2 a^2 (a+b \cot (c+d x))}{7 d e (e \cot (c+d x))^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Cot[c + d*x])^3/(e*Cot[c + d*x])^(9/2),x]

[Out]

((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(9/2)) - ((a - b
)*(a^2 + 4*a*b + b^2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(9/2)) + (32*a^2*b)/(35
*d*e^2*(e*Cot[c + d*x])^(5/2)) - (2*a*(a^2 - 3*b^2))/(3*d*e^3*(e*Cot[c + d*x])^(3/2)) - (2*b*(3*a^2 - b^2))/(d
*e^4*Sqrt[e*Cot[c + d*x]]) + (2*a^2*(a + b*Cot[c + d*x]))/(7*d*e*(e*Cot[c + d*x])^(7/2)) + ((a + b)*(a^2 - 4*a
*b + b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*e^(9/2)) - ((a + b)
*(a^2 - 4*a*b + b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*e^(9/2))

Rule 210

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

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 1182

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a*c, 2]}, Dist[(d*q + a*e)/(2*a*c),
Int[(q + c*x^2)/(a + c*x^4), x], x] + Dist[(d*q - a*e)/(2*a*c), Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ
[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(-a)*c]

Rule 3610

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b
*c - a*d)*((a + b*Tan[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 + b^2))), x] + Dist[1/(a^2 + b^2), Int[(a + b*Tan[e +
f*x])^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c
- a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1]

Rule 3615

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

Rule 3646

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(b*c - a*d)^2*(a + b*Tan[e + f*x])^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - D
ist[1/(d*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a^2*d*(b*d*(
m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c*(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3
*a*b^2*d)*Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*(n + 1)))*Tan[e + f*x]^2, x
], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && Gt
Q[m, 2] && LtQ[n, -1] && IntegerQ[2*m]

Rule 3709

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f
_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*b*B + a^2*C)*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1)*(a^2 + b^2)
)), x] + Dist[1/(a^2 + b^2), Int[(a + b*Tan[e + f*x])^(m + 1)*Simp[b*B + a*(A - C) - (A*b - a*B - b*C)*Tan[e +
f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && LtQ[m, -1] && NeQ[a^2
+ b^2, 0]

Rubi steps

\begin {align*} \int \frac {(a+b \cot (c+d x))^3}{(e \cot (c+d x))^{9/2}} \, dx &=\frac {2 a^2 (a+b \cot (c+d x))}{7 d e (e \cot (c+d x))^{7/2}}-\frac {2 \int \frac {-8 a^2 b e^2+\frac {7}{2} a \left (a^2-3 b^2\right ) e^2 \cot (c+d x)+\frac {1}{2} b \left (5 a^2-7 b^2\right ) e^2 \cot ^2(c+d x)}{(e \cot (c+d x))^{7/2}} \, dx}{7 e^3}\\ &=\frac {32 a^2 b}{35 d e^2 (e \cot (c+d x))^{5/2}}+\frac {2 a^2 (a+b \cot (c+d x))}{7 d e (e \cot (c+d x))^{7/2}}-\frac {2 \int \frac {\frac {7}{2} a \left (a^2-3 b^2\right ) e^3+\frac {7}{2} b \left (3 a^2-b^2\right ) e^3 \cot (c+d x)}{(e \cot (c+d x))^{5/2}} \, dx}{7 e^5}\\ &=\frac {32 a^2 b}{35 d e^2 (e \cot (c+d x))^{5/2}}-\frac {2 a \left (a^2-3 b^2\right )}{3 d e^3 (e \cot (c+d x))^{3/2}}+\frac {2 a^2 (a+b \cot (c+d x))}{7 d e (e \cot (c+d x))^{7/2}}-\frac {2 \int \frac {\frac {7}{2} b \left (3 a^2-b^2\right ) e^4-\frac {7}{2} a \left (a^2-3 b^2\right ) e^4 \cot (c+d x)}{(e \cot (c+d x))^{3/2}} \, dx}{7 e^7}\\ &=\frac {32 a^2 b}{35 d e^2 (e \cot (c+d x))^{5/2}}-\frac {2 a \left (a^2-3 b^2\right )}{3 d e^3 (e \cot (c+d x))^{3/2}}-\frac {2 b \left (3 a^2-b^2\right )}{d e^4 \sqrt {e \cot (c+d x)}}+\frac {2 a^2 (a+b \cot (c+d x))}{7 d e (e \cot (c+d x))^{7/2}}-\frac {2 \int \frac {-\frac {7}{2} a \left (a^2-3 b^2\right ) e^5-\frac {7}{2} b \left (3 a^2-b^2\right ) e^5 \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx}{7 e^9}\\ &=\frac {32 a^2 b}{35 d e^2 (e \cot (c+d x))^{5/2}}-\frac {2 a \left (a^2-3 b^2\right )}{3 d e^3 (e \cot (c+d x))^{3/2}}-\frac {2 b \left (3 a^2-b^2\right )}{d e^4 \sqrt {e \cot (c+d x)}}+\frac {2 a^2 (a+b \cot (c+d x))}{7 d e (e \cot (c+d x))^{7/2}}-\frac {4 \text {Subst}\left (\int \frac {\frac {7}{2} a \left (a^2-3 b^2\right ) e^6+\frac {7}{2} b \left (3 a^2-b^2\right ) e^5 x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{7 d e^9}\\ &=\frac {32 a^2 b}{35 d e^2 (e \cot (c+d x))^{5/2}}-\frac {2 a \left (a^2-3 b^2\right )}{3 d e^3 (e \cot (c+d x))^{3/2}}-\frac {2 b \left (3 a^2-b^2\right )}{d e^4 \sqrt {e \cot (c+d x)}}+\frac {2 a^2 (a+b \cot (c+d x))}{7 d e (e \cot (c+d x))^{7/2}}-\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {e-x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{d e^4}-\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {e+x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{d e^4}\\ &=\frac {32 a^2 b}{35 d e^2 (e \cot (c+d x))^{5/2}}-\frac {2 a \left (a^2-3 b^2\right )}{3 d e^3 (e \cot (c+d x))^{3/2}}-\frac {2 b \left (3 a^2-b^2\right )}{d e^4 \sqrt {e \cot (c+d x)}}+\frac {2 a^2 (a+b \cot (c+d x))}{7 d e (e \cot (c+d x))^{7/2}}+\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {e}+2 x}{-e-\sqrt {2} \sqrt {e} x-x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d e^{9/2}}+\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {e}-2 x}{-e+\sqrt {2} \sqrt {e} x-x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d e^{9/2}}-\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{e-\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{2 d e^4}-\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{e+\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{2 d e^4}\\ &=\frac {32 a^2 b}{35 d e^2 (e \cot (c+d x))^{5/2}}-\frac {2 a \left (a^2-3 b^2\right )}{3 d e^3 (e \cot (c+d x))^{3/2}}-\frac {2 b \left (3 a^2-b^2\right )}{d e^4 \sqrt {e \cot (c+d x)}}+\frac {2 a^2 (a+b \cot (c+d x))}{7 d e (e \cot (c+d x))^{7/2}}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d e^{9/2}}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d e^{9/2}}-\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{9/2}}+\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{9/2}}\\ &=\frac {(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{9/2}}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{9/2}}+\frac {32 a^2 b}{35 d e^2 (e \cot (c+d x))^{5/2}}-\frac {2 a \left (a^2-3 b^2\right )}{3 d e^3 (e \cot (c+d x))^{3/2}}-\frac {2 b \left (3 a^2-b^2\right )}{d e^4 \sqrt {e \cot (c+d x)}}+\frac {2 a^2 (a+b \cot (c+d x))}{7 d e (e \cot (c+d x))^{7/2}}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d e^{9/2}}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d e^{9/2}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.74, size = 116, normalized size = 0.31 \begin {gather*} \frac {2 \sqrt {e \cot (c+d x)} \left (5 a \left (a^2-3 b^2\right ) \, _2F_1\left (-\frac {7}{4},1;-\frac {3}{4};-\cot ^2(c+d x)\right )+b \left (b (15 a+7 b \cot (c+d x))+7 \left (3 a^2-b^2\right ) \cot (c+d x) \, _2F_1\left (-\frac {5}{4},1;-\frac {1}{4};-\cot ^2(c+d x)\right )\right )\right ) \tan ^4(c+d x)}{35 d e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Cot[c + d*x])^3/(e*Cot[c + d*x])^(9/2),x]

[Out]

(2*Sqrt[e*Cot[c + d*x]]*(5*a*(a^2 - 3*b^2)*Hypergeometric2F1[-7/4, 1, -3/4, -Cot[c + d*x]^2] + b*(b*(15*a + 7*
b*Cot[c + d*x]) + 7*(3*a^2 - b^2)*Cot[c + d*x]*Hypergeometric2F1[-5/4, 1, -1/4, -Cot[c + d*x]^2]))*Tan[c + d*x
]^4)/(35*d*e^5)

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Maple [A]
time = 0.46, size = 388, normalized size = 1.03

method	result	size
derivativedivides	\(-\frac {2 \left (-\frac {a^{3} e}{7 \left (e \cot \left (d x +c \right )\right )^{\frac {7}{2}}}-\frac {3 a^{2} b}{5 \left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}+\frac {a \left (a^{2}-3 b^{2}\right )}{3 e \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {b \left (3 a^{2}-b^{2}\right )}{e^{2} \sqrt {e \cot \left (d x +c \right )}}+\frac {\frac {\left (a^{3} e -3 a \,b^{2} e \right ) \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2}}+\frac {\left (3 a^{2} b -b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}}{e^{2}}\right )}{d \,e^{2}}\)	\(388\)
default	\(-\frac {2 \left (-\frac {a^{3} e}{7 \left (e \cot \left (d x +c \right )\right )^{\frac {7}{2}}}-\frac {3 a^{2} b}{5 \left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}+\frac {a \left (a^{2}-3 b^{2}\right )}{3 e \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {b \left (3 a^{2}-b^{2}\right )}{e^{2} \sqrt {e \cot \left (d x +c \right )}}+\frac {\frac {\left (a^{3} e -3 a \,b^{2} e \right ) \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2}}+\frac {\left (3 a^{2} b -b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}}{e^{2}}\right )}{d \,e^{2}}\)	\(388\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*cot(d*x+c))^3/(e*cot(d*x+c))^(9/2),x,method=_RETURNVERBOSE)

[Out]

-2/d/e^2*(-1/7*a^3*e/(e*cot(d*x+c))^(7/2)-3/5*a^2*b/(e*cot(d*x+c))^(5/2)+1/3*a/e*(a^2-3*b^2)/(e*cot(d*x+c))^(3
/2)+b*(3*a^2-b^2)/e^2/(e*cot(d*x+c))^(1/2)+1/e^2*(1/8*(a^3*e-3*a*b^2*e)*(e^2)^(1/4)/e^2*2^(1/2)*(ln((e*cot(d*x
+c)+(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2))/(e*cot(d*x+c)-(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/
2)+(e^2)^(1/2)))+2*arctan(2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)-2*arctan(-2^(1/2)/(e^2)^(1/4)*(e*cot(d*x
+c))^(1/2)+1))+1/8*(3*a^2*b-b^3)/(e^2)^(1/4)*2^(1/2)*(ln((e*cot(d*x+c)-(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/2
)+(e^2)^(1/2))/(e*cot(d*x+c)+(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2)))+2*arctan(2^(1/2)/(e^2)^(1/
4)*(e*cot(d*x+c))^(1/2)+1)-2*arctan(-2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1))))

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Maxima [A]
time = 0.52, size = 267, normalized size = 0.71 \begin {gather*} \frac {{\left (8 \, {\left (15 \, a^{3} + \frac {63 \, a^{2} b}{\tan \left (d x + c\right )} - \frac {35 \, {\left (a^{3} - 3 \, a b^{2}\right )}}{\tan \left (d x + c\right )^{2}} - \frac {105 \, {\left (3 \, a^{2} b - b^{3}\right )}}{\tan \left (d x + c\right )^{3}}\right )} \tan \left (d x + c\right )^{\frac {7}{2}} - 210 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - 210 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - 105 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + 105 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )\right )} e^{\left (-\frac {9}{2}\right )}}{420 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cot(d*x+c))^3/(e*cot(d*x+c))^(9/2),x, algorithm="maxima")

[Out]

1/420*(8*(15*a^3 + 63*a^2*b/tan(d*x + c) - 35*(a^3 - 3*a*b^2)/tan(d*x + c)^2 - 105*(3*a^2*b - b^3)/tan(d*x + c
)^3)*tan(d*x + c)^(7/2) - 210*sqrt(2)*(a^3 + 3*a^2*b - 3*a*b^2 - b^3)*arctan(1/2*sqrt(2)*(sqrt(2) + 2/sqrt(tan
(d*x + c)))) - 210*sqrt(2)*(a^3 + 3*a^2*b - 3*a*b^2 - b^3)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2/sqrt(tan(d*x + c))
)) - 105*sqrt(2)*(a^3 - 3*a^2*b - 3*a*b^2 + b^3)*log(sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1) + 105*sq
rt(2)*(a^3 - 3*a^2*b - 3*a*b^2 + b^3)*log(-sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1))*e^(-9/2)/d

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Fricas [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((a+b*cot(d*x+c))^3/(e*cot(d*x+c))^(9/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \cot {\left (c + d x \right )}\right )^{3}}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {9}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cot(d*x+c))**3/(e*cot(d*x+c))**(9/2),x)

[Out]

Integral((a + b*cot(c + d*x))**3/(e*cot(c + d*x))**(9/2), 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((a+b*cot(d*x+c))^3/(e*cot(d*x+c))^(9/2),x, algorithm="giac")

[Out]

Timed out

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Mupad [B]
time = 5.26, size = 1992, normalized size = 5.28 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*cot(c + d*x))^3/(e*cot(c + d*x))^(9/2),x)

[Out]

atan((((e*cot(c + d*x))^(1/2)*(16*a^6*d^3*e^14 - 16*b^6*d^3*e^14 + 240*a^2*b^4*d^3*e^14 - 240*a^4*b^2*d^3*e^14
) + (32*a^3*d^4*e^19 - 96*a*b^2*d^4*e^19)*(((a*b^5*6i + a^5*b*6i - a^6 + b^6 - 15*a^2*b^4 - a^3*b^3*20i + 15*a
^4*b^2)*1i)/(4*d^2*e^9))^(1/2))*(((a*b^5*6i + a^5*b*6i - a^6 + b^6 - 15*a^2*b^4 - a^3*b^3*20i + 15*a^4*b^2)*1i
)/(4*d^2*e^9))^(1/2)*1i + ((e*cot(c + d*x))^(1/2)*(16*a^6*d^3*e^14 - 16*b^6*d^3*e^14 + 240*a^2*b^4*d^3*e^14 -
240*a^4*b^2*d^3*e^14) - (32*a^3*d^4*e^19 - 96*a*b^2*d^4*e^19)*(((a*b^5*6i + a^5*b*6i - a^6 + b^6 - 15*a^2*b^4
- a^3*b^3*20i + 15*a^4*b^2)*1i)/(4*d^2*e^9))^(1/2))*(((a*b^5*6i + a^5*b*6i - a^6 + b^6 - 15*a^2*b^4 - a^3*b^3*
20i + 15*a^4*b^2)*1i)/(4*d^2*e^9))^(1/2)*1i)/(((e*cot(c + d*x))^(1/2)*(16*a^6*d^3*e^14 - 16*b^6*d^3*e^14 + 240
*a^2*b^4*d^3*e^14 - 240*a^4*b^2*d^3*e^14) + (32*a^3*d^4*e^19 - 96*a*b^2*d^4*e^19)*(((a*b^5*6i + a^5*b*6i - a^6
+ b^6 - 15*a^2*b^4 - a^3*b^3*20i + 15*a^4*b^2)*1i)/(4*d^2*e^9))^(1/2))*(((a*b^5*6i + a^5*b*6i - a^6 + b^6 - 1
5*a^2*b^4 - a^3*b^3*20i + 15*a^4*b^2)*1i)/(4*d^2*e^9))^(1/2) - ((e*cot(c + d*x))^(1/2)*(16*a^6*d^3*e^14 - 16*b
^6*d^3*e^14 + 240*a^2*b^4*d^3*e^14 - 240*a^4*b^2*d^3*e^14) - (32*a^3*d^4*e^19 - 96*a*b^2*d^4*e^19)*(((a*b^5*6i
+ a^5*b*6i - a^6 + b^6 - 15*a^2*b^4 - a^3*b^3*20i + 15*a^4*b^2)*1i)/(4*d^2*e^9))^(1/2))*(((a*b^5*6i + a^5*b*6
i - a^6 + b^6 - 15*a^2*b^4 - a^3*b^3*20i + 15*a^4*b^2)*1i)/(4*d^2*e^9))^(1/2) - 16*b^9*d^2*e^10 + 48*a^8*b*d^2
*e^10 + 96*a^4*b^5*d^2*e^10 + 128*a^6*b^3*d^2*e^10))*(((a*b^5*6i + a^5*b*6i - a^6 + b^6 - 15*a^2*b^4 - a^3*b^3
*20i + 15*a^4*b^2)*1i)/(4*d^2*e^9))^(1/2)*2i + atan((((e*cot(c + d*x))^(1/2)*(16*a^6*d^3*e^14 - 16*b^6*d^3*e^1
4 + 240*a^2*b^4*d^3*e^14 - 240*a^4*b^2*d^3*e^14) + (32*a^3*d^4*e^19 - 96*a*b^2*d^4*e^19)*(((a*b^5*6i + a^5*b*6
i + a^6 - b^6 + 15*a^2*b^4 - a^3*b^3*20i - 15*a^4*b^2)*1i)/(4*d^2*e^9))^(1/2))*(((a*b^5*6i + a^5*b*6i + a^6 -
b^6 + 15*a^2*b^4 - a^3*b^3*20i - 15*a^4*b^2)*1i)/(4*d^2*e^9))^(1/2)*1i + ((e*cot(c + d*x))^(1/2)*(16*a^6*d^3*e
^14 - 16*b^6*d^3*e^14 + 240*a^2*b^4*d^3*e^14 - 240*a^4*b^2*d^3*e^14) - (32*a^3*d^4*e^19 - 96*a*b^2*d^4*e^19)*(
((a*b^5*6i + a^5*b*6i + a^6 - b^6 + 15*a^2*b^4 - a^3*b^3*20i - 15*a^4*b^2)*1i)/(4*d^2*e^9))^(1/2))*(((a*b^5*6i
+ a^5*b*6i + a^6 - b^6 + 15*a^2*b^4 - a^3*b^3*20i - 15*a^4*b^2)*1i)/(4*d^2*e^9))^(1/2)*1i)/(((e*cot(c + d*x))
^(1/2)*(16*a^6*d^3*e^14 - 16*b^6*d^3*e^14 + 240*a^2*b^4*d^3*e^14 - 240*a^4*b^2*d^3*e^14) + (32*a^3*d^4*e^19 -
96*a*b^2*d^4*e^19)*(((a*b^5*6i + a^5*b*6i + a^6 - b^6 + 15*a^2*b^4 - a^3*b^3*20i - 15*a^4*b^2)*1i)/(4*d^2*e^9)
)^(1/2))*(((a*b^5*6i + a^5*b*6i + a^6 - b^6 + 15*a^2*b^4 - a^3*b^3*20i - 15*a^4*b^2)*1i)/(4*d^2*e^9))^(1/2) -
((e*cot(c + d*x))^(1/2)*(16*a^6*d^3*e^14 - 16*b^6*d^3*e^14 + 240*a^2*b^4*d^3*e^14 - 240*a^4*b^2*d^3*e^14) - (3
2*a^3*d^4*e^19 - 96*a*b^2*d^4*e^19)*(((a*b^5*6i + a^5*b*6i + a^6 - b^6 + 15*a^2*b^4 - a^3*b^3*20i - 15*a^4*b^2
)*1i)/(4*d^2*e^9))^(1/2))*(((a*b^5*6i + a^5*b*6i + a^6 - b^6 + 15*a^2*b^4 - a^3*b^3*20i - 15*a^4*b^2)*1i)/(4*d
^2*e^9))^(1/2) - 16*b^9*d^2*e^10 + 48*a^8*b*d^2*e^10 + 96*a^4*b^5*d^2*e^10 + 128*a^6*b^3*d^2*e^10))*(((a*b^5*6
i + a^5*b*6i + a^6 - b^6 + 15*a^2*b^4 - a^3*b^3*20i - 15*a^4*b^2)*1i)/(4*d^2*e^9))^(1/2)*2i + ((2*a^3*e)/7 + (
2*e*cot(c + d*x)^2*(3*a*b^2 - a^3))/3 - 2*e*cot(c + d*x)^3*(3*a^2*b - b^3) + (6*a^2*b*e*cot(c + d*x))/5)/(d*e^
2*(e*cot(c + d*x))^(7/2))

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