3.1.57 \(\int \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2 \, dx\) [57]

Optimal. Leaf size=288 \[ \frac {\left (a^2-2 a b-b^2\right ) \sqrt {e} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d}-\frac {\left (a^2-2 a b-b^2\right ) \sqrt {e} \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d}-\frac {4 a b \sqrt {e \cot (c+d x)}}{d}-\frac {2 b^2 (e \cot (c+d x))^{3/2}}{3 d e}-\frac {\left (a^2+2 a b-b^2\right ) \sqrt {e} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d}+\frac {\left (a^2+2 a b-b^2\right ) \sqrt {e} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d} \]

[Out]

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

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Rubi [A]
time = 0.19, antiderivative size = 288, 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 = {3624, 3609, 3615, 1182, 1176, 631, 210, 1179, 642} \begin {gather*} \frac {\sqrt {e} \left (a^2-2 a b-b^2\right ) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d}-\frac {\sqrt {e} \left (a^2-2 a b-b^2\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} d}-\frac {\sqrt {e} \left (a^2+2 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}+\frac {\sqrt {e} \left (a^2+2 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}-\frac {4 a b \sqrt {e \cot (c+d x)}}{d}-\frac {2 b^2 (e \cot (c+d x))^{3/2}}{3 d e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 3609

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

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 3624

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

Rubi steps

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

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.57, size = 220, normalized size = 0.76 \begin {gather*} -\frac {\sqrt {e \cot (c+d x)} \left (4 \left (a^2-b^2\right ) \cot ^{\frac {3}{2}}(c+d x) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\cot ^2(c+d x)\right )+b \left (6 \sqrt {2} a \text {ArcTan}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-6 \sqrt {2} a \text {ArcTan}\left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )+24 a \sqrt {\cot (c+d x)}+4 b \cot ^{\frac {3}{2}}(c+d x)+3 \sqrt {2} a \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-3 \sqrt {2} a \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )\right )}{6 d \sqrt {\cot (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6*(Sqrt[e*Cot[c + d*x]]*(4*(a^2 - b^2)*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2] +
b*(6*Sqrt[2]*a*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 6*Sqrt[2]*a*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] +
24*a*Sqrt[Cot[c + d*x]] + 4*b*Cot[c + d*x]^(3/2) + 3*Sqrt[2]*a*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*
x]] - 3*Sqrt[2]*a*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])))/(d*Sqrt[Cot[c + d*x]])

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Maple [A]
time = 0.57, size = 321, normalized size = 1.11

method	result	size
derivativedivides	\(-\frac {2 \left (\frac {b^{2} \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+2 a b e \sqrt {e \cot \left (d x +c \right )}+e^{2} \left (-\frac {a b \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 e}+\frac {\left (a^{2}-b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )\right )}{d e}\)	\(321\)
default	\(-\frac {2 \left (\frac {b^{2} \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+2 a b e \sqrt {e \cot \left (d x +c \right )}+e^{2} \left (-\frac {a b \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 e}+\frac {\left (a^{2}-b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )\right )}{d e}\)	\(321\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/d/e*(1/3*b^2*(e*cot(d*x+c))^(3/2)+2*a*b*e*(e*cot(d*x+c))^(1/2)+e^2*(-1/4*a/e*b*(e^2)^(1/4)*2^(1/2)*(ln((e*c
ot(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*c
ot(d*x+c))^(1/2)+1))+1/8*(a^2-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 = 0.52, size = 192, normalized size = 0.67 \begin {gather*} -\frac {{\left (6 \, \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 6 \, \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - 3 \, \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + 3 \, \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + \frac {48 \, a b}{\sqrt {\tan \left (d x + c\right )}} + \frac {8 \, b^{2}}{\tan \left (d x + c\right )^{\frac {3}{2}}}\right )} e^{\frac {1}{2}}}{12 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*cot(d*x+c))^(1/2)*(a+b*cot(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/12*(6*sqrt(2)*(a^2 - 2*a*b - b^2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2/sqrt(tan(d*x + c)))) + 6*sqrt(2)*(a^2 - 2
*a*b - b^2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2/sqrt(tan(d*x + c)))) - 3*sqrt(2)*(a^2 + 2*a*b - b^2)*log(sqrt(2)/
sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1) + 3*sqrt(2)*(a^2 + 2*a*b - b^2)*log(-sqrt(2)/sqrt(tan(d*x + c)) + 1/t
an(d*x + c) + 1) + 48*a*b/sqrt(tan(d*x + c)) + 8*b^2/tan(d*x + c)^(3/2))*e^(1/2)/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((e*cot(d*x+c))^(1/2)*(a+b*cot(d*x+c))^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 \sqrt {e \cot {\left (c + d x \right )}} \left (a + b \cot {\left (c + d x \right )}\right )^{2}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*cot(d*x+c))**(1/2)*(a+b*cot(d*x+c))**2,x)

[Out]

Integral(sqrt(e*cot(c + d*x))*(a + b*cot(c + d*x))**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((e*cot(d*x+c))^(1/2)*(a+b*cot(d*x+c))^2,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 1.21, size = 1157, normalized size = 4.02 \begin {gather*} -\frac {2\,b^2\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}{3\,d\,e}-\frac {4\,a\,b\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{d}-\mathrm {atan}\left (\frac {a^4\,e^4\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\sqrt {\frac {a^3\,b\,e}{d^2}-\frac {a\,b^3\,e}{d^2}-\frac {a^4\,e\,1{}\mathrm {i}}{4\,d^2}-\frac {b^4\,e\,1{}\mathrm {i}}{4\,d^2}+\frac {a^2\,b^2\,e\,3{}\mathrm {i}}{2\,d^2}}\,32{}\mathrm {i}}{\frac {16\,a^6\,e^5}{d}-\frac {16\,b^6\,e^5}{d}+\frac {112\,a^2\,b^4\,e^5}{d}-\frac {112\,a^4\,b^2\,e^5}{d}+\frac {a\,b^5\,e^5\,32{}\mathrm {i}}{d}+\frac {a^5\,b\,e^5\,32{}\mathrm {i}}{d}-\frac {a^3\,b^3\,e^5\,192{}\mathrm {i}}{d}}+\frac {b^4\,e^4\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\sqrt {\frac {a^3\,b\,e}{d^2}-\frac {a\,b^3\,e}{d^2}-\frac {a^4\,e\,1{}\mathrm {i}}{4\,d^2}-\frac {b^4\,e\,1{}\mathrm {i}}{4\,d^2}+\frac {a^2\,b^2\,e\,3{}\mathrm {i}}{2\,d^2}}\,32{}\mathrm {i}}{\frac {16\,a^6\,e^5}{d}-\frac {16\,b^6\,e^5}{d}+\frac {112\,a^2\,b^4\,e^5}{d}-\frac {112\,a^4\,b^2\,e^5}{d}+\frac {a\,b^5\,e^5\,32{}\mathrm {i}}{d}+\frac {a^5\,b\,e^5\,32{}\mathrm {i}}{d}-\frac {a^3\,b^3\,e^5\,192{}\mathrm {i}}{d}}-\frac {a^2\,b^2\,e^4\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\sqrt {\frac {a^3\,b\,e}{d^2}-\frac {a\,b^3\,e}{d^2}-\frac {a^4\,e\,1{}\mathrm {i}}{4\,d^2}-\frac {b^4\,e\,1{}\mathrm {i}}{4\,d^2}+\frac {a^2\,b^2\,e\,3{}\mathrm {i}}{2\,d^2}}\,192{}\mathrm {i}}{\frac {16\,a^6\,e^5}{d}-\frac {16\,b^6\,e^5}{d}+\frac {112\,a^2\,b^4\,e^5}{d}-\frac {112\,a^4\,b^2\,e^5}{d}+\frac {a\,b^5\,e^5\,32{}\mathrm {i}}{d}+\frac {a^5\,b\,e^5\,32{}\mathrm {i}}{d}-\frac {a^3\,b^3\,e^5\,192{}\mathrm {i}}{d}}\right )\,\sqrt {-\frac {1{}\mathrm {i}\,e\,a^4-4\,e\,a^3\,b-6{}\mathrm {i}\,e\,a^2\,b^2+4\,e\,a\,b^3+1{}\mathrm {i}\,e\,b^4}{4\,d^2}}\,2{}\mathrm {i}+\mathrm {atan}\left (\frac {a^4\,e^4\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\sqrt {\frac {a^3\,b\,e}{d^2}-\frac {a\,b^3\,e}{d^2}+\frac {a^4\,e\,1{}\mathrm {i}}{4\,d^2}+\frac {b^4\,e\,1{}\mathrm {i}}{4\,d^2}-\frac {a^2\,b^2\,e\,3{}\mathrm {i}}{2\,d^2}}\,32{}\mathrm {i}}{\frac {16\,b^6\,e^5}{d}-\frac {16\,a^6\,e^5}{d}-\frac {112\,a^2\,b^4\,e^5}{d}+\frac {112\,a^4\,b^2\,e^5}{d}+\frac {a\,b^5\,e^5\,32{}\mathrm {i}}{d}+\frac {a^5\,b\,e^5\,32{}\mathrm {i}}{d}-\frac {a^3\,b^3\,e^5\,192{}\mathrm {i}}{d}}+\frac {b^4\,e^4\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\sqrt {\frac {a^3\,b\,e}{d^2}-\frac {a\,b^3\,e}{d^2}+\frac {a^4\,e\,1{}\mathrm {i}}{4\,d^2}+\frac {b^4\,e\,1{}\mathrm {i}}{4\,d^2}-\frac {a^2\,b^2\,e\,3{}\mathrm {i}}{2\,d^2}}\,32{}\mathrm {i}}{\frac {16\,b^6\,e^5}{d}-\frac {16\,a^6\,e^5}{d}-\frac {112\,a^2\,b^4\,e^5}{d}+\frac {112\,a^4\,b^2\,e^5}{d}+\frac {a\,b^5\,e^5\,32{}\mathrm {i}}{d}+\frac {a^5\,b\,e^5\,32{}\mathrm {i}}{d}-\frac {a^3\,b^3\,e^5\,192{}\mathrm {i}}{d}}-\frac {a^2\,b^2\,e^4\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\sqrt {\frac {a^3\,b\,e}{d^2}-\frac {a\,b^3\,e}{d^2}+\frac {a^4\,e\,1{}\mathrm {i}}{4\,d^2}+\frac {b^4\,e\,1{}\mathrm {i}}{4\,d^2}-\frac {a^2\,b^2\,e\,3{}\mathrm {i}}{2\,d^2}}\,192{}\mathrm {i}}{\frac {16\,b^6\,e^5}{d}-\frac {16\,a^6\,e^5}{d}-\frac {112\,a^2\,b^4\,e^5}{d}+\frac {112\,a^4\,b^2\,e^5}{d}+\frac {a\,b^5\,e^5\,32{}\mathrm {i}}{d}+\frac {a^5\,b\,e^5\,32{}\mathrm {i}}{d}-\frac {a^3\,b^3\,e^5\,192{}\mathrm {i}}{d}}\right )\,\sqrt {\frac {1{}\mathrm {i}\,e\,a^4+4\,e\,a^3\,b-6{}\mathrm {i}\,e\,a^2\,b^2-4\,e\,a\,b^3+1{}\mathrm {i}\,e\,b^4}{4\,d^2}}\,2{}\mathrm {i} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atan((a^4*e^4*(e*cot(c + d*x))^(1/2)*((a^4*e*1i)/(4*d^2) + (b^4*e*1i)/(4*d^2) - (a^2*b^2*e*3i)/(2*d^2) - (a*b^
3*e)/d^2 + (a^3*b*e)/d^2)^(1/2)*32i)/((16*b^6*e^5)/d - (16*a^6*e^5)/d + (a*b^5*e^5*32i)/d + (a^5*b*e^5*32i)/d
- (112*a^2*b^4*e^5)/d - (a^3*b^3*e^5*192i)/d + (112*a^4*b^2*e^5)/d) + (b^4*e^4*(e*cot(c + d*x))^(1/2)*((a^4*e*
1i)/(4*d^2) + (b^4*e*1i)/(4*d^2) - (a^2*b^2*e*3i)/(2*d^2) - (a*b^3*e)/d^2 + (a^3*b*e)/d^2)^(1/2)*32i)/((16*b^6
*e^5)/d - (16*a^6*e^5)/d + (a*b^5*e^5*32i)/d + (a^5*b*e^5*32i)/d - (112*a^2*b^4*e^5)/d - (a^3*b^3*e^5*192i)/d
+ (112*a^4*b^2*e^5)/d) - (a^2*b^2*e^4*(e*cot(c + d*x))^(1/2)*((a^4*e*1i)/(4*d^2) + (b^4*e*1i)/(4*d^2) - (a^2*b
^2*e*3i)/(2*d^2) - (a*b^3*e)/d^2 + (a^3*b*e)/d^2)^(1/2)*192i)/((16*b^6*e^5)/d - (16*a^6*e^5)/d + (a*b^5*e^5*32
i)/d + (a^5*b*e^5*32i)/d - (112*a^2*b^4*e^5)/d - (a^3*b^3*e^5*192i)/d + (112*a^4*b^2*e^5)/d))*((a^4*e*1i + b^4
*e*1i - a^2*b^2*e*6i - 4*a*b^3*e + 4*a^3*b*e)/(4*d^2))^(1/2)*2i - atan((a^4*e^4*(e*cot(c + d*x))^(1/2)*((a^2*b
^2*e*3i)/(2*d^2) - (b^4*e*1i)/(4*d^2) - (a^4*e*1i)/(4*d^2) - (a*b^3*e)/d^2 + (a^3*b*e)/d^2)^(1/2)*32i)/((16*a^
6*e^5)/d - (16*b^6*e^5)/d + (a*b^5*e^5*32i)/d + (a^5*b*e^5*32i)/d + (112*a^2*b^4*e^5)/d - (a^3*b^3*e^5*192i)/d
- (112*a^4*b^2*e^5)/d) + (b^4*e^4*(e*cot(c + d*x))^(1/2)*((a^2*b^2*e*3i)/(2*d^2) - (b^4*e*1i)/(4*d^2) - (a^4*
e*1i)/(4*d^2) - (a*b^3*e)/d^2 + (a^3*b*e)/d^2)^(1/2)*32i)/((16*a^6*e^5)/d - (16*b^6*e^5)/d + (a*b^5*e^5*32i)/d
+ (a^5*b*e^5*32i)/d + (112*a^2*b^4*e^5)/d - (a^3*b^3*e^5*192i)/d - (112*a^4*b^2*e^5)/d) - (a^2*b^2*e^4*(e*cot
(c + d*x))^(1/2)*((a^2*b^2*e*3i)/(2*d^2) - (b^4*e*1i)/(4*d^2) - (a^4*e*1i)/(4*d^2) - (a*b^3*e)/d^2 + (a^3*b*e)
/d^2)^(1/2)*192i)/((16*a^6*e^5)/d - (16*b^6*e^5)/d + (a*b^5*e^5*32i)/d + (a^5*b*e^5*32i)/d + (112*a^2*b^4*e^5)
/d - (a^3*b^3*e^5*192i)/d - (112*a^4*b^2*e^5)/d))*(-(a^4*e*1i + b^4*e*1i - a^2*b^2*e*6i + 4*a*b^3*e - 4*a^3*b*
e)/(4*d^2))^(1/2)*2i - (2*b^2*(e*cot(c + d*x))^(3/2))/(3*d*e) - (4*a*b*(e*cot(c + d*x))^(1/2))/d

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