3.1.58 \(\int \frac {(a+b \cot (c+d x))^2}{\sqrt {e \cot (c+d x)}} \, dx\) [58]

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

[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*2^(1/2)/e^(1/2)-1/2*(a^2+2*a*b-b^2)*arcta
n(1+2^(1/2)*(e*cot(d*x+c))^(1/2)/e^(1/2))/d*2^(1/2)/e^(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*2^(1/2)/e^(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*c
ot(d*x+c))^(1/2))/d*2^(1/2)/e^(1/2)-2*b^2*(e*cot(d*x+c))^(1/2)/d/e

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Rubi [A]
time = 0.17, 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 = {3624, 3615, 1182, 1176, 631, 210, 1179, 642} \begin {gather*} \frac {\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 \sqrt {e}}-\frac {\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 \sqrt {e}}+\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 \sqrt {e}}-\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 \sqrt {e}}-\frac {2 b^2 \sqrt {e \cot (c+d x)}}{d e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.91, size = 192, normalized size = 0.72 \begin {gather*} -\frac {\sqrt {\cot (c+d x)} \left (2 b^2 \sqrt {\cot (c+d x)}+\frac {4}{3} a b \cot ^{\frac {3}{2}}(c+d x) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\cot ^2(c+d x)\right )-\frac {\left (a^2-b^2\right ) \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 )}{2 \sqrt {2}}\right )}{d \sqrt {e \cot (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.47, size = 306, normalized size = 1.15

method	result	size
derivativedivides	\(-\frac {2 \left (b^{2} \sqrt {e \cot \left (d x +c \right )}+e \left (\frac {\left (a^{2} e -b^{2} e \right ) \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2}}+\frac {a b \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (e^{2}\right )^{\frac {1}{4}}}\right )\right )}{d e}\)	\(306\)
default	\(-\frac {2 \left (b^{2} \sqrt {e \cot \left (d x +c \right )}+e \left (\frac {\left (a^{2} e -b^{2} e \right ) \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2}}+\frac {a b \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (e^{2}\right )^{\frac {1}{4}}}\right )\right )}{d e}\)	\(306\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/d/e*(b^2*(e*cot(d*x+c))^(1/2)+e*(1/8*(a^2*e-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))+1/4
*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*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 = 0.50, size = 179, normalized size = 0.67 \begin {gather*} -\frac {{\left (2 \, \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 ) + 2 \, \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 ) + \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 ) - \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 {8 \, b^{2}}{\sqrt {\tan \left (d x + c\right )}}\right )} e^{\left (-\frac {1}{2}\right )}}{4 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*sqrt(2)*(a^2 + 2*a*b - b^2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2/sqrt(tan(d*x + c)))) + 2*sqrt(2)*(a^2 + 2*
a*b - b^2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2/sqrt(tan(d*x + c)))) + sqrt(2)*(a^2 - 2*a*b - b^2)*log(sqrt(2)/sqr
t(tan(d*x + c)) + 1/tan(d*x + c) + 1) - sqrt(2)*(a^2 - 2*a*b - b^2)*log(-sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*
x + c) + 1) + 8*b^2/sqrt(tan(d*x + c)))*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((a+b*cot(d*x+c))^2/(e*cot(d*x+c))^(1/2),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 )^{2}}{\sqrt {e \cot {\left (c + d x \right )}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*cot(c + d*x))**2/sqrt(e*cot(c + d*x)), 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))^2/(e*cot(d*x+c))^(1/2),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 = 1.01, size = 1234, normalized size = 4.62 \begin {gather*} 2\,\mathrm {atanh}\left (\frac {32\,a^4\,e^2\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\sqrt {\frac {a\,b^3}{d^2\,e}-\frac {b^4\,1{}\mathrm {i}}{4\,d^2\,e}-\frac {a^4\,1{}\mathrm {i}}{4\,d^2\,e}-\frac {a^3\,b}{d^2\,e}+\frac {a^2\,b^2\,3{}\mathrm {i}}{2\,d^2\,e}}}{\frac {a^6\,e^2\,16{}\mathrm {i}}{d}-\frac {b^6\,e^2\,16{}\mathrm {i}}{d}+\frac {32\,a\,b^5\,e^2}{d}+\frac {32\,a^5\,b\,e^2}{d}+\frac {a^2\,b^4\,e^2\,112{}\mathrm {i}}{d}-\frac {192\,a^3\,b^3\,e^2}{d}-\frac {a^4\,b^2\,e^2\,112{}\mathrm {i}}{d}}+\frac {32\,b^4\,e^2\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\sqrt {\frac {a\,b^3}{d^2\,e}-\frac {b^4\,1{}\mathrm {i}}{4\,d^2\,e}-\frac {a^4\,1{}\mathrm {i}}{4\,d^2\,e}-\frac {a^3\,b}{d^2\,e}+\frac {a^2\,b^2\,3{}\mathrm {i}}{2\,d^2\,e}}}{\frac {a^6\,e^2\,16{}\mathrm {i}}{d}-\frac {b^6\,e^2\,16{}\mathrm {i}}{d}+\frac {32\,a\,b^5\,e^2}{d}+\frac {32\,a^5\,b\,e^2}{d}+\frac {a^2\,b^4\,e^2\,112{}\mathrm {i}}{d}-\frac {192\,a^3\,b^3\,e^2}{d}-\frac {a^4\,b^2\,e^2\,112{}\mathrm {i}}{d}}-\frac {192\,a^2\,b^2\,e^2\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\sqrt {\frac {a\,b^3}{d^2\,e}-\frac {b^4\,1{}\mathrm {i}}{4\,d^2\,e}-\frac {a^4\,1{}\mathrm {i}}{4\,d^2\,e}-\frac {a^3\,b}{d^2\,e}+\frac {a^2\,b^2\,3{}\mathrm {i}}{2\,d^2\,e}}}{\frac {a^6\,e^2\,16{}\mathrm {i}}{d}-\frac {b^6\,e^2\,16{}\mathrm {i}}{d}+\frac {32\,a\,b^5\,e^2}{d}+\frac {32\,a^5\,b\,e^2}{d}+\frac {a^2\,b^4\,e^2\,112{}\mathrm {i}}{d}-\frac {192\,a^3\,b^3\,e^2}{d}-\frac {a^4\,b^2\,e^2\,112{}\mathrm {i}}{d}}\right )\,\sqrt {\frac {a\,b^3}{d^2\,e}-\frac {b^4\,1{}\mathrm {i}}{4\,d^2\,e}-\frac {a^4\,1{}\mathrm {i}}{4\,d^2\,e}-\frac {a^3\,b}{d^2\,e}+\frac {a^2\,b^2\,3{}\mathrm {i}}{2\,d^2\,e}}+2\,\mathrm {atanh}\left (\frac {32\,a^4\,e^2\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\sqrt {\frac {a^4\,1{}\mathrm {i}}{4\,d^2\,e}+\frac {b^4\,1{}\mathrm {i}}{4\,d^2\,e}+\frac {a\,b^3}{d^2\,e}-\frac {a^3\,b}{d^2\,e}-\frac {a^2\,b^2\,3{}\mathrm {i}}{2\,d^2\,e}}}{\frac {32\,a\,b^5\,e^2}{d}+\frac {b^6\,e^2\,16{}\mathrm {i}}{d}-\frac {a^6\,e^2\,16{}\mathrm {i}}{d}+\frac {32\,a^5\,b\,e^2}{d}-\frac {a^2\,b^4\,e^2\,112{}\mathrm {i}}{d}-\frac {192\,a^3\,b^3\,e^2}{d}+\frac {a^4\,b^2\,e^2\,112{}\mathrm {i}}{d}}+\frac {32\,b^4\,e^2\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\sqrt {\frac {a^4\,1{}\mathrm {i}}{4\,d^2\,e}+\frac {b^4\,1{}\mathrm {i}}{4\,d^2\,e}+\frac {a\,b^3}{d^2\,e}-\frac {a^3\,b}{d^2\,e}-\frac {a^2\,b^2\,3{}\mathrm {i}}{2\,d^2\,e}}}{\frac {32\,a\,b^5\,e^2}{d}+\frac {b^6\,e^2\,16{}\mathrm {i}}{d}-\frac {a^6\,e^2\,16{}\mathrm {i}}{d}+\frac {32\,a^5\,b\,e^2}{d}-\frac {a^2\,b^4\,e^2\,112{}\mathrm {i}}{d}-\frac {192\,a^3\,b^3\,e^2}{d}+\frac {a^4\,b^2\,e^2\,112{}\mathrm {i}}{d}}-\frac {192\,a^2\,b^2\,e^2\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\sqrt {\frac {a^4\,1{}\mathrm {i}}{4\,d^2\,e}+\frac {b^4\,1{}\mathrm {i}}{4\,d^2\,e}+\frac {a\,b^3}{d^2\,e}-\frac {a^3\,b}{d^2\,e}-\frac {a^2\,b^2\,3{}\mathrm {i}}{2\,d^2\,e}}}{\frac {32\,a\,b^5\,e^2}{d}+\frac {b^6\,e^2\,16{}\mathrm {i}}{d}-\frac {a^6\,e^2\,16{}\mathrm {i}}{d}+\frac {32\,a^5\,b\,e^2}{d}-\frac {a^2\,b^4\,e^2\,112{}\mathrm {i}}{d}-\frac {192\,a^3\,b^3\,e^2}{d}+\frac {a^4\,b^2\,e^2\,112{}\mathrm {i}}{d}}\right )\,\sqrt {\frac {a^4\,1{}\mathrm {i}}{4\,d^2\,e}+\frac {b^4\,1{}\mathrm {i}}{4\,d^2\,e}+\frac {a\,b^3}{d^2\,e}-\frac {a^3\,b}{d^2\,e}-\frac {a^2\,b^2\,3{}\mathrm {i}}{2\,d^2\,e}}-\frac {2\,b^2\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{d\,e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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