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

Optimal. Leaf size=267 \[ \frac {\left (a^2+2 a b-b^2\right ) \log \left (\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d e^{3/2}}-\frac {\left (a^2+2 a b-b^2\right ) \log \left (\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d e^{3/2}}-\frac {\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{3/2}}+\frac {\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} d e^{3/2}}+\frac {2 a^2}{d e \sqrt {e \cot (c+d x)}} \]

[Out]

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

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Rubi [A]  time = 0.26, antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3542, 3534, 1168, 1162, 617, 204, 1165, 628} \[ \frac {\left (a^2+2 a b-b^2\right ) \log \left (\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d e^{3/2}}-\frac {\left (a^2+2 a b-b^2\right ) \log \left (\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d e^{3/2}}-\frac {\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{3/2}}+\frac {\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} d e^{3/2}}+\frac {2 a^2}{d e \sqrt {e \cot (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((a^2 - 2*a*b - b^2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(3/2))) + ((a^2 - 2*a*
b - b^2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(3/2)) + (2*a^2)/(d*e*Sqrt[e*Cot[c +
d*x]]) + ((a^2 + 2*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^2 + 2*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 204

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

Rule 617

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 628

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 1162

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 1165

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 1168

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 3534

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 3542

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

Rubi steps

\begin {align*} \int \frac {(a+b \cot (c+d x))^2}{(e \cot (c+d x))^{3/2}} \, dx &=\frac {2 a^2}{d e \sqrt {e \cot (c+d x)}}+\frac {\int \frac {2 a b e-\left (a^2-b^2\right ) e \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx}{e^2}\\ &=\frac {2 a^2}{d e \sqrt {e \cot (c+d x)}}+\frac {2 \operatorname {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 e^2}\\ &=\frac {2 a^2}{d e \sqrt {e \cot (c+d x)}}+\frac {\left (a^2-2 a b-b^2\right ) \operatorname {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^2+2 a b-b^2\right ) \operatorname {Subst}\left (\int \frac {e-x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{d e}\\ &=\frac {2 a^2}{d e \sqrt {e \cot (c+d x)}}+\frac {\left (a^2+2 a b-b^2\right ) \operatorname {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^2+2 a b-b^2\right ) \operatorname {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^2-2 a b-b^2\right ) \operatorname {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^2-2 a b-b^2\right ) \operatorname {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 a^2}{d e \sqrt {e \cot (c+d x)}}+\frac {\left (a^2+2 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^2+2 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^2-2 a b-b^2\right ) \operatorname {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^2-2 a b-b^2\right ) \operatorname {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^2-2 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 {\left (a^2-2 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 a^2}{d e \sqrt {e \cot (c+d x)}}+\frac {\left (a^2+2 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^2+2 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]  time = 0.33, size = 218, normalized size = 0.82 \[ -\frac {\cot ^{\frac {3}{2}}(c+d x) \left (-\frac {2 \left (a^2-b^2\right ) \, _2F_1\left (-\frac {1}{4},1;\frac {3}{4};-\cot ^2(c+d x)\right )}{\sqrt {\cot (c+d x)}}+4 a b \left (\frac {1}{2} \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\tan ^{-1}\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )-\frac {2 b^2}{\sqrt {\cot (c+d x)}}\right )}{d (e \cot (c+d x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \cot \left (d x + c\right ) + a\right )}^{2}}{\left (e \cot \left (d x + c\right )\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.46, size = 538, normalized size = 2.01 \[ -\frac {a b \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \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 e^{2} d}-\frac {a b \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{e^{2} d}+\frac {a b \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{e^{2} d}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right ) a^{2}}{2 e d \left (e^{2}\right )^{\frac {1}{4}}}-\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right ) b^{2}}{2 e d \left (e^{2}\right )^{\frac {1}{4}}}-\frac {\sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right ) a^{2}}{2 e d \left (e^{2}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right ) b^{2}}{2 e d \left (e^{2}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \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 ) a^{2}}{4 e d \left (e^{2}\right )^{\frac {1}{4}}}-\frac {\sqrt {2}\, \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 ) b^{2}}{4 e d \left (e^{2}\right )^{\frac {1}{4}}}+\frac {2 a^{2}}{d e \sqrt {e \cot \left (d x +c \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.90, size = 242, normalized size = 0.91 \[ \frac {e {\left (\frac {8 \, a^{2}}{e^{2} \sqrt {\frac {e}{\tan \left (d x + c\right )}}} + \frac {\frac {2 \, \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} + 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right )}{\sqrt {e}} + \frac {2 \, \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} - 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right )}{\sqrt {e}} - \frac {\sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \log \left (\sqrt {2} \sqrt {e} \sqrt {\frac {e}{\tan \left (d x + c\right )}} + e + \frac {e}{\tan \left (d x + c\right )}\right )}{\sqrt {e}} + \frac {\sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \log \left (-\sqrt {2} \sqrt {e} \sqrt {\frac {e}{\tan \left (d x + c\right )}} + e + \frac {e}{\tan \left (d x + c\right )}\right )}{\sqrt {e}}}{e^{2}}\right )}}{4 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*e*(8*a^2/(e^2*sqrt(e/tan(d*x + c))) + (2*sqrt(2)*(a^2 - 2*a*b - b^2)*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(e) +
2*sqrt(e/tan(d*x + c)))/sqrt(e))/sqrt(e) + 2*sqrt(2)*(a^2 - 2*a*b - b^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*sqrt(e)
- 2*sqrt(e/tan(d*x + c)))/sqrt(e))/sqrt(e) - sqrt(2)*(a^2 + 2*a*b - b^2)*log(sqrt(2)*sqrt(e)*sqrt(e/tan(d*x +
c)) + e + e/tan(d*x + c))/sqrt(e) + sqrt(2)*(a^2 + 2*a*b - b^2)*log(-sqrt(2)*sqrt(e)*sqrt(e/tan(d*x + c)) + e
+ e/tan(d*x + c))/sqrt(e))/e^2)/d

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mupad [B]  time = 0.94, size = 1196, normalized size = 4.48 \[ \frac {2\,a^2}{d\,e\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}+2\,\mathrm {atanh}\left (\frac {32\,a^4\,d^3\,e^5\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\sqrt {\frac {a^4\,1{}\mathrm {i}}{4\,d^2\,e^3}+\frac {b^4\,1{}\mathrm {i}}{4\,d^2\,e^3}-\frac {a\,b^3}{d^2\,e^3}+\frac {a^3\,b}{d^2\,e^3}-\frac {a^2\,b^2\,3{}\mathrm {i}}{2\,d^2\,e^3}}}{-16\,a^6\,d^2\,e^4+a^5\,b\,d^2\,e^4\,32{}\mathrm {i}+112\,a^4\,b^2\,d^2\,e^4-a^3\,b^3\,d^2\,e^4\,192{}\mathrm {i}-112\,a^2\,b^4\,d^2\,e^4+a\,b^5\,d^2\,e^4\,32{}\mathrm {i}+16\,b^6\,d^2\,e^4}+\frac {32\,b^4\,d^3\,e^5\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\sqrt {\frac {a^4\,1{}\mathrm {i}}{4\,d^2\,e^3}+\frac {b^4\,1{}\mathrm {i}}{4\,d^2\,e^3}-\frac {a\,b^3}{d^2\,e^3}+\frac {a^3\,b}{d^2\,e^3}-\frac {a^2\,b^2\,3{}\mathrm {i}}{2\,d^2\,e^3}}}{-16\,a^6\,d^2\,e^4+a^5\,b\,d^2\,e^4\,32{}\mathrm {i}+112\,a^4\,b^2\,d^2\,e^4-a^3\,b^3\,d^2\,e^4\,192{}\mathrm {i}-112\,a^2\,b^4\,d^2\,e^4+a\,b^5\,d^2\,e^4\,32{}\mathrm {i}+16\,b^6\,d^2\,e^4}-\frac {192\,a^2\,b^2\,d^3\,e^5\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\sqrt {\frac {a^4\,1{}\mathrm {i}}{4\,d^2\,e^3}+\frac {b^4\,1{}\mathrm {i}}{4\,d^2\,e^3}-\frac {a\,b^3}{d^2\,e^3}+\frac {a^3\,b}{d^2\,e^3}-\frac {a^2\,b^2\,3{}\mathrm {i}}{2\,d^2\,e^3}}}{-16\,a^6\,d^2\,e^4+a^5\,b\,d^2\,e^4\,32{}\mathrm {i}+112\,a^4\,b^2\,d^2\,e^4-a^3\,b^3\,d^2\,e^4\,192{}\mathrm {i}-112\,a^2\,b^4\,d^2\,e^4+a\,b^5\,d^2\,e^4\,32{}\mathrm {i}+16\,b^6\,d^2\,e^4}\right )\,\sqrt {\frac {\left (a^4-a^3\,b\,4{}\mathrm {i}-6\,a^2\,b^2+a\,b^3\,4{}\mathrm {i}+b^4\right )\,1{}\mathrm {i}}{4\,d^2\,e^3}}-2\,\mathrm {atanh}\left (\frac {32\,a^4\,d^3\,e^5\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\sqrt {\frac {a^3\,b}{d^2\,e^3}-\frac {b^4\,1{}\mathrm {i}}{4\,d^2\,e^3}-\frac {a\,b^3}{d^2\,e^3}-\frac {a^4\,1{}\mathrm {i}}{4\,d^2\,e^3}+\frac {a^2\,b^2\,3{}\mathrm {i}}{2\,d^2\,e^3}}}{16\,a^6\,d^2\,e^4+a^5\,b\,d^2\,e^4\,32{}\mathrm {i}-112\,a^4\,b^2\,d^2\,e^4-a^3\,b^3\,d^2\,e^4\,192{}\mathrm {i}+112\,a^2\,b^4\,d^2\,e^4+a\,b^5\,d^2\,e^4\,32{}\mathrm {i}-16\,b^6\,d^2\,e^4}+\frac {32\,b^4\,d^3\,e^5\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\sqrt {\frac {a^3\,b}{d^2\,e^3}-\frac {b^4\,1{}\mathrm {i}}{4\,d^2\,e^3}-\frac {a\,b^3}{d^2\,e^3}-\frac {a^4\,1{}\mathrm {i}}{4\,d^2\,e^3}+\frac {a^2\,b^2\,3{}\mathrm {i}}{2\,d^2\,e^3}}}{16\,a^6\,d^2\,e^4+a^5\,b\,d^2\,e^4\,32{}\mathrm {i}-112\,a^4\,b^2\,d^2\,e^4-a^3\,b^3\,d^2\,e^4\,192{}\mathrm {i}+112\,a^2\,b^4\,d^2\,e^4+a\,b^5\,d^2\,e^4\,32{}\mathrm {i}-16\,b^6\,d^2\,e^4}-\frac {192\,a^2\,b^2\,d^3\,e^5\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\sqrt {\frac {a^3\,b}{d^2\,e^3}-\frac {b^4\,1{}\mathrm {i}}{4\,d^2\,e^3}-\frac {a\,b^3}{d^2\,e^3}-\frac {a^4\,1{}\mathrm {i}}{4\,d^2\,e^3}+\frac {a^2\,b^2\,3{}\mathrm {i}}{2\,d^2\,e^3}}}{16\,a^6\,d^2\,e^4+a^5\,b\,d^2\,e^4\,32{}\mathrm {i}-112\,a^4\,b^2\,d^2\,e^4-a^3\,b^3\,d^2\,e^4\,192{}\mathrm {i}+112\,a^2\,b^4\,d^2\,e^4+a\,b^5\,d^2\,e^4\,32{}\mathrm {i}-16\,b^6\,d^2\,e^4}\right )\,\sqrt {-\frac {\left (a^4+a^3\,b\,4{}\mathrm {i}-6\,a^2\,b^2-a\,b^3\,4{}\mathrm {i}+b^4\right )\,1{}\mathrm {i}}{4\,d^2\,e^3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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