3.1.65 \(\int \frac {(a+b \cot (c+d x))^3}{(e \cot (c+d x))^{3/2}} \, dx\) [65]

Optimal. Leaf size=313 \[ -\frac {(a+b) \left (a^2-4 a b+b^2\right ) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{3/2}}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{3/2}}-\frac {2 b \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)}}{d e^2}+\frac {2 a^2 (a+b \cot (c+d x))}{d e \sqrt {e \cot (c+d x)}}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d e^{3/2}}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d e^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.29, antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3646, 3711, 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^{3/2}}+\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^{3/2}}+\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^{3/2}}-\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^{3/2}}-\frac {2 b \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)}}{d e^2}+\frac {2 a^2 (a+b \cot (c+d x))}{d e \sqrt {e \cot (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

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 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 3711

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

Rubi steps

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

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 3.55, size = 193, normalized size = 0.62 \begin {gather*} \frac {2 \left (3 a b^2-b^3 \cot (c+d x)+a \left (a^2-3 b^2\right ) \, _2F_1\left (-\frac {1}{4},1;\frac {3}{4};-\cot ^2(c+d x)\right )-\frac {b \left (-3 a^2+b^2\right ) \sqrt {\cot (c+d x)} \left (2 \text {ArcTan}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-2 \text {ArcTan}\left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )+\log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-\log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )}{4 \sqrt {2}}\right )}{d e \sqrt {e \cot (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(3*a*b^2 - b^3*Cot[c + d*x] + a*(a^2 - 3*b^2)*Hypergeometric2F1[-1/4, 1, 3/4, -Cot[c + d*x]^2] - (b*(-3*a^2
+ b^2)*Sqrt[Cot[c + d*x]]*(2*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]
] + Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/
(4*Sqrt[2])))/(d*e*Sqrt[e*Cot[c + d*x]])

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Maple [A]
time = 0.49, size = 332, normalized size = 1.06

method	result	size
derivativedivides	\(-\frac {2 \left (b^{3} \sqrt {e \cot \left (d x +c \right )}-\frac {a^{3} e}{\sqrt {e \cot \left (d x +c \right )}}-e \left (\frac {\left (-3 a^{2} b e +b^{3} e \right ) \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2}}+\frac {\left (a^{3}-3 a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )\right )}{d \,e^{2}}\)	\(332\)
default	\(-\frac {2 \left (b^{3} \sqrt {e \cot \left (d x +c \right )}-\frac {a^{3} e}{\sqrt {e \cot \left (d x +c \right )}}-e \left (\frac {\left (-3 a^{2} b e +b^{3} e \right ) \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2}}+\frac {\left (a^{3}-3 a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )\right )}{d \,e^{2}}\)	\(332\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/d/e^2*(b^3*(e*cot(d*x+c))^(1/2)-a^3*e/(e*cot(d*x+c))^(1/2)-e*(1/8*(-3*a^2*b*e+b^3*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*(a^3-3*a*b^2)/(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 = 0.51, size = 220, normalized size = 0.70 \begin {gather*} \frac {{\left (8 \, a^{3} \sqrt {\tan \left (d x + c\right )} + 2 \, \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 ) + 2 \, \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 ) - \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 ) + \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 ) - \frac {8 \, b^{3}}{\sqrt {\tan \left (d x + c\right )}}\right )} e^{\left (-\frac {3}{2}\right )}}{4 \, 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))^(3/2),x, algorithm="maxima")

[Out]

1/4*(8*a^3*sqrt(tan(d*x + c)) + 2*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)))) + 2*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
)))) - 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) + 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) - 8*b^3/sqrt(tan(d*x + c
)))*e^(-3/2)/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))^(3/2),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 {3}{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))**(3/2),x)

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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))^(3/2),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 1.20, size = 1951, normalized size = 6.23 \begin {gather*} \frac {2\,a^3}{d\,e\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}-\frac {2\,b^3\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{d\,e^2}-\mathrm {atan}\left (\frac {\left (\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\left (16\,a^6\,d^3\,e^5-240\,a^4\,b^2\,d^3\,e^5+240\,a^2\,b^4\,d^3\,e^5-16\,b^6\,d^3\,e^5\right )+\left (32\,b^3\,d^4\,e^7-96\,a^2\,b\,d^4\,e^7\right )\,\sqrt {-\frac {\left (-a^6+a^5\,b\,6{}\mathrm {i}+15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}-15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}+b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e^3}}\right )\,\sqrt {-\frac {\left (-a^6+a^5\,b\,6{}\mathrm {i}+15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}-15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}+b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e^3}}\,1{}\mathrm {i}+\left (\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\left (16\,a^6\,d^3\,e^5-240\,a^4\,b^2\,d^3\,e^5+240\,a^2\,b^4\,d^3\,e^5-16\,b^6\,d^3\,e^5\right )-\left (32\,b^3\,d^4\,e^7-96\,a^2\,b\,d^4\,e^7\right )\,\sqrt {-\frac {\left (-a^6+a^5\,b\,6{}\mathrm {i}+15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}-15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}+b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e^3}}\right )\,\sqrt {-\frac {\left (-a^6+a^5\,b\,6{}\mathrm {i}+15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}-15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}+b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e^3}}\,1{}\mathrm {i}}{48\,a\,b^8\,d^2\,e^4+\left (\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\left (16\,a^6\,d^3\,e^5-240\,a^4\,b^2\,d^3\,e^5+240\,a^2\,b^4\,d^3\,e^5-16\,b^6\,d^3\,e^5\right )-\left (32\,b^3\,d^4\,e^7-96\,a^2\,b\,d^4\,e^7\right )\,\sqrt {-\frac {\left (-a^6+a^5\,b\,6{}\mathrm {i}+15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}-15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}+b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e^3}}\right )\,\sqrt {-\frac {\left (-a^6+a^5\,b\,6{}\mathrm {i}+15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}-15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}+b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e^3}}-16\,a^9\,d^2\,e^4-\left (\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\left (16\,a^6\,d^3\,e^5-240\,a^4\,b^2\,d^3\,e^5+240\,a^2\,b^4\,d^3\,e^5-16\,b^6\,d^3\,e^5\right )+\left (32\,b^3\,d^4\,e^7-96\,a^2\,b\,d^4\,e^7\right )\,\sqrt {-\frac {\left (-a^6+a^5\,b\,6{}\mathrm {i}+15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}-15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}+b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e^3}}\right )\,\sqrt {-\frac {\left (-a^6+a^5\,b\,6{}\mathrm {i}+15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}-15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}+b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e^3}}+128\,a^3\,b^6\,d^2\,e^4+96\,a^5\,b^4\,d^2\,e^4}\right )\,\sqrt {-\frac {\left (-a^6+a^5\,b\,6{}\mathrm {i}+15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}-15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}+b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e^3}}\,2{}\mathrm {i}-\mathrm {atan}\left (\frac {\left (\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\left (16\,a^6\,d^3\,e^5-240\,a^4\,b^2\,d^3\,e^5+240\,a^2\,b^4\,d^3\,e^5-16\,b^6\,d^3\,e^5\right )+\left (32\,b^3\,d^4\,e^7-96\,a^2\,b\,d^4\,e^7\right )\,\sqrt {-\frac {\left (a^6+a^5\,b\,6{}\mathrm {i}-15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}+15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}-b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e^3}}\right )\,\sqrt {-\frac {\left (a^6+a^5\,b\,6{}\mathrm {i}-15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}+15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}-b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e^3}}\,1{}\mathrm {i}+\left (\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\left (16\,a^6\,d^3\,e^5-240\,a^4\,b^2\,d^3\,e^5+240\,a^2\,b^4\,d^3\,e^5-16\,b^6\,d^3\,e^5\right )-\left (32\,b^3\,d^4\,e^7-96\,a^2\,b\,d^4\,e^7\right )\,\sqrt {-\frac {\left (a^6+a^5\,b\,6{}\mathrm {i}-15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}+15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}-b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e^3}}\right )\,\sqrt {-\frac {\left (a^6+a^5\,b\,6{}\mathrm {i}-15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}+15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}-b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e^3}}\,1{}\mathrm {i}}{48\,a\,b^8\,d^2\,e^4+\left (\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\left (16\,a^6\,d^3\,e^5-240\,a^4\,b^2\,d^3\,e^5+240\,a^2\,b^4\,d^3\,e^5-16\,b^6\,d^3\,e^5\right )-\left (32\,b^3\,d^4\,e^7-96\,a^2\,b\,d^4\,e^7\right )\,\sqrt {-\frac {\left (a^6+a^5\,b\,6{}\mathrm {i}-15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}+15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}-b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e^3}}\right )\,\sqrt {-\frac {\left (a^6+a^5\,b\,6{}\mathrm {i}-15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}+15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}-b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e^3}}-16\,a^9\,d^2\,e^4-\left (\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\left (16\,a^6\,d^3\,e^5-240\,a^4\,b^2\,d^3\,e^5+240\,a^2\,b^4\,d^3\,e^5-16\,b^6\,d^3\,e^5\right )+\left (32\,b^3\,d^4\,e^7-96\,a^2\,b\,d^4\,e^7\right )\,\sqrt {-\frac {\left (a^6+a^5\,b\,6{}\mathrm {i}-15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}+15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}-b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e^3}}\right )\,\sqrt {-\frac {\left (a^6+a^5\,b\,6{}\mathrm {i}-15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}+15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}-b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e^3}}+128\,a^3\,b^6\,d^2\,e^4+96\,a^5\,b^4\,d^2\,e^4}\right )\,\sqrt {-\frac {\left (a^6+a^5\,b\,6{}\mathrm {i}-15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}+15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}-b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e^3}}\,2{}\mathrm {i} \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))^(3/2),x)

[Out]

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

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