3.72 \(\int \frac {1}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))} \, dx\)

Optimal. Leaf size=302 \[ \frac {(a+b) \log \left (\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d \sqrt {e} \left (a^2+b^2\right )}-\frac {(a+b) \log \left (\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d \sqrt {e} \left (a^2+b^2\right )}+\frac {(a-b) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d \sqrt {e} \left (a^2+b^2\right )}-\frac {(a-b) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} d \sqrt {e} \left (a^2+b^2\right )}-\frac {2 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{\sqrt {a} d \sqrt {e} \left (a^2+b^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.37, antiderivative size = 302, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {3574, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205} \[ \frac {(a+b) \log \left (\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d \sqrt {e} \left (a^2+b^2\right )}-\frac {(a+b) \log \left (\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d \sqrt {e} \left (a^2+b^2\right )}-\frac {2 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{\sqrt {a} d \sqrt {e} \left (a^2+b^2\right )}+\frac {(a-b) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d \sqrt {e} \left (a^2+b^2\right )}-\frac {(a-b) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} d \sqrt {e} \left (a^2+b^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[e*Cot[c + d*x]]*(a + b*Cot[c + d*x])),x]

[Out]

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

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

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 205

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

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 3574

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

Rule 3634

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*
tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[A/f, Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x]
/; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]

Rubi steps

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

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Mathematica [C]  time = 0.25, size = 248, normalized size = 0.82 \[ -\frac {\sqrt {\cot (c+d x)} \left (-\frac {2 b \cot ^{\frac {3}{2}}(c+d x) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\cot ^2(c+d x)\right )}{3 \left (a^2+b^2\right )}-\frac {a \left (2 \sqrt {2} \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )-2 \sqrt {2} \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )+4 \left (\sqrt {2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-\sqrt {2} \tan ^{-1}\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )\right )\right )}{8 \left (a^2+b^2\right )}+\frac {2 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\cot (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} \left (a^2+b^2\right )}\right )}{d \sqrt {e \cot (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[e*Cot[c + d*x]]*(a + b*Cot[c + d*x])),x]

[Out]

-((Sqrt[Cot[c + d*x]]*((2*b^(3/2)*ArcTan[(Sqrt[b]*Sqrt[Cot[c + d*x]])/Sqrt[a]])/(Sqrt[a]*(a^2 + b^2)) - (2*b*C
ot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2])/(3*(a^2 + b^2)) - (a*(4*(Sqrt[2]*ArcTan[1 -
Sqrt[2]*Sqrt[Cot[c + d*x]]] - Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]) + 2*Sqrt[2]*Log[1 - Sqrt[2]*Sqr
t[Cot[c + d*x]] + Cot[c + d*x]] - 2*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/(8*(a^2 + b^2
))))/(d*Sqrt[e*Cot[c + d*x]]))

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

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b \cot \left (d x + c\right ) + a\right )} \sqrt {e \cot \left (d x + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.71, size = 423, normalized size = 1.40 \[ -\frac {2 b^{2} \arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}\, b}{\sqrt {a e b}}\right )}{d \left (a^{2}+b^{2}\right ) \sqrt {a e b}}-\frac {a \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )}{4 d e \left (a^{2}+b^{2}\right )}-\frac {a \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{2 d e \left (a^{2}+b^{2}\right )}+\frac {a \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{2 d e \left (a^{2}+b^{2}\right )}+\frac {b \sqrt {2}\, \ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )}{4 d \left (a^{2}+b^{2}\right ) \left (e^{2}\right )^{\frac {1}{4}}}+\frac {b \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{2 d \left (a^{2}+b^{2}\right ) \left (e^{2}\right )^{\frac {1}{4}}}-\frac {b \sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{2 d \left (a^{2}+b^{2}\right ) \left (e^{2}\right )^{\frac {1}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/d*b^2/(a^2+b^2)/(a*e*b)^(1/2)*arctan((e*cot(d*x+c))^(1/2)*b/(a*e*b)^(1/2))-1/4/d/e/(a^2+b^2)*a*(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*co
t(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2)))-1/2/d/e/(a^2+b^2)*a*(e^2)^(1/4)*2^(1/2)*arctan(2^(1/2)/(e^2)^(1/4)*(e*co
t(d*x+c))^(1/2)+1)+1/2/d/e/(a^2+b^2)*a*(e^2)^(1/4)*2^(1/2)*arctan(-2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)
+1/4/d/(a^2+b^2)*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)))+1/2/d/(a^2+b^2)*b/(e^2)^(1/4)*2^(1/2)*arc
tan(2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)-1/2/d/(a^2+b^2)*b/(e^2)^(1/4)*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 = 0.53, size = 239, normalized size = 0.79 \[ -\frac {e {\left (\frac {8 \, b^{2} \arctan \left (\frac {b \sqrt {\frac {e}{\tan \left (d x + c\right )}}}{\sqrt {a b e}}\right )}{\sqrt {a b e} {\left (a^{2} + b^{2}\right )} e} + \frac {\frac {2 \, \sqrt {2} {\left (a - b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} + 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right )}{\sqrt {e}} + \frac {2 \, \sqrt {2} {\left (a - b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} - 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right )}{\sqrt {e}} + \frac {\sqrt {2} {\left (a + b\right )} \log \left (\sqrt {2} \sqrt {e} \sqrt {\frac {e}{\tan \left (d x + c\right )}} + e + \frac {e}{\tan \left (d x + c\right )}\right )}{\sqrt {e}} - \frac {\sqrt {2} {\left (a + b\right )} \log \left (-\sqrt {2} \sqrt {e} \sqrt {\frac {e}{\tan \left (d x + c\right )}} + e + \frac {e}{\tan \left (d x + c\right )}\right )}{\sqrt {e}}}{{\left (a^{2} + b^{2}\right )} e}\right )}}{4 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.77, size = 4871, normalized size = 16.13 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atan(((((((32*(5*a*b^5*e^9 + a^3*b^3*e^9))/d^3 - ((((1/(b^2*d^2*e*1i - a^2*d^2*e*1i + 2*a*b*d^2*e))^(1/2)*((32
*(16*b^8*d^2*e^10 + 28*a^2*b^6*d^2*e^10 + 8*a^4*b^4*d^2*e^10 - 4*a^6*b^2*d^2*e^10))/d^3 - (16*(e*cot(c + d*x))
^(1/2)*(1/(b^2*d^2*e*1i - a^2*d^2*e*1i + 2*a*b*d^2*e))^(1/2)*(16*b^9*d^4*e^10 + 16*a^2*b^7*d^4*e^10 - 16*a^4*b
^5*d^4*e^10 - 16*a^6*b^3*d^4*e^10))/d^4))/2 - (32*(e*cot(c + d*x))^(1/2)*(4*a^3*b^4*d^2*e^9 - 30*a*b^6*d^2*e^9
+ 2*a^5*b^2*d^2*e^9))/d^4)*(1/(b^2*d^2*e*1i - a^2*d^2*e*1i + 2*a*b*d^2*e))^(1/2))/2)*(1/(b^2*d^2*e*1i - a^2*d
^2*e*1i + 2*a*b*d^2*e))^(1/2))/2 + (96*b^5*e^8*(e*cot(c + d*x))^(1/2))/d^4)*(1/(b^2*d^2*e*1i - a^2*d^2*e*1i +
2*a*b*d^2*e))^(1/2)*1i)/2 - (((((32*(5*a*b^5*e^9 + a^3*b^3*e^9))/d^3 - ((((1/(b^2*d^2*e*1i - a^2*d^2*e*1i + 2*
a*b*d^2*e))^(1/2)*((32*(16*b^8*d^2*e^10 + 28*a^2*b^6*d^2*e^10 + 8*a^4*b^4*d^2*e^10 - 4*a^6*b^2*d^2*e^10))/d^3
+ (16*(e*cot(c + d*x))^(1/2)*(1/(b^2*d^2*e*1i - a^2*d^2*e*1i + 2*a*b*d^2*e))^(1/2)*(16*b^9*d^4*e^10 + 16*a^2*b
^7*d^4*e^10 - 16*a^4*b^5*d^4*e^10 - 16*a^6*b^3*d^4*e^10))/d^4))/2 + (32*(e*cot(c + d*x))^(1/2)*(4*a^3*b^4*d^2*
e^9 - 30*a*b^6*d^2*e^9 + 2*a^5*b^2*d^2*e^9))/d^4)*(1/(b^2*d^2*e*1i - a^2*d^2*e*1i + 2*a*b*d^2*e))^(1/2))/2)*(1
/(b^2*d^2*e*1i - a^2*d^2*e*1i + 2*a*b*d^2*e))^(1/2))/2 - (96*b^5*e^8*(e*cot(c + d*x))^(1/2))/d^4)*(1/(b^2*d^2*
e*1i - a^2*d^2*e*1i + 2*a*b*d^2*e))^(1/2)*1i)/2)/((((((32*(5*a*b^5*e^9 + a^3*b^3*e^9))/d^3 - ((((1/(b^2*d^2*e*
1i - a^2*d^2*e*1i + 2*a*b*d^2*e))^(1/2)*((32*(16*b^8*d^2*e^10 + 28*a^2*b^6*d^2*e^10 + 8*a^4*b^4*d^2*e^10 - 4*a
^6*b^2*d^2*e^10))/d^3 - (16*(e*cot(c + d*x))^(1/2)*(1/(b^2*d^2*e*1i - a^2*d^2*e*1i + 2*a*b*d^2*e))^(1/2)*(16*b
^9*d^4*e^10 + 16*a^2*b^7*d^4*e^10 - 16*a^4*b^5*d^4*e^10 - 16*a^6*b^3*d^4*e^10))/d^4))/2 - (32*(e*cot(c + d*x))
^(1/2)*(4*a^3*b^4*d^2*e^9 - 30*a*b^6*d^2*e^9 + 2*a^5*b^2*d^2*e^9))/d^4)*(1/(b^2*d^2*e*1i - a^2*d^2*e*1i + 2*a*
b*d^2*e))^(1/2))/2)*(1/(b^2*d^2*e*1i - a^2*d^2*e*1i + 2*a*b*d^2*e))^(1/2))/2 + (96*b^5*e^8*(e*cot(c + d*x))^(1
/2))/d^4)*(1/(b^2*d^2*e*1i - a^2*d^2*e*1i + 2*a*b*d^2*e))^(1/2))/2 + (((((32*(5*a*b^5*e^9 + a^3*b^3*e^9))/d^3
- ((((1/(b^2*d^2*e*1i - a^2*d^2*e*1i + 2*a*b*d^2*e))^(1/2)*((32*(16*b^8*d^2*e^10 + 28*a^2*b^6*d^2*e^10 + 8*a^4
*b^4*d^2*e^10 - 4*a^6*b^2*d^2*e^10))/d^3 + (16*(e*cot(c + d*x))^(1/2)*(1/(b^2*d^2*e*1i - a^2*d^2*e*1i + 2*a*b*
d^2*e))^(1/2)*(16*b^9*d^4*e^10 + 16*a^2*b^7*d^4*e^10 - 16*a^4*b^5*d^4*e^10 - 16*a^6*b^3*d^4*e^10))/d^4))/2 + (
32*(e*cot(c + d*x))^(1/2)*(4*a^3*b^4*d^2*e^9 - 30*a*b^6*d^2*e^9 + 2*a^5*b^2*d^2*e^9))/d^4)*(1/(b^2*d^2*e*1i -
a^2*d^2*e*1i + 2*a*b*d^2*e))^(1/2))/2)*(1/(b^2*d^2*e*1i - a^2*d^2*e*1i + 2*a*b*d^2*e))^(1/2))/2 - (96*b^5*e^8*
(e*cot(c + d*x))^(1/2))/d^4)*(1/(b^2*d^2*e*1i - a^2*d^2*e*1i + 2*a*b*d^2*e))^(1/2))/2))*(1/(b^2*d^2*e*1i - a^2
*d^2*e*1i + 2*a*b*d^2*e))^(1/2)*1i + atan(((((32*(5*a*b^5*e^9 + a^3*b^3*e^9))/d^3 - (((32*(16*b^8*d^2*e^10 + 2
8*a^2*b^6*d^2*e^10 + 8*a^4*b^4*d^2*e^10 - 4*a^6*b^2*d^2*e^10))/d^3 - (32*(e*cot(c + d*x))^(1/2)*(1i/(4*(b^2*d^
2*e - a^2*d^2*e + a*b*d^2*e*2i)))^(1/2)*(16*b^9*d^4*e^10 + 16*a^2*b^7*d^4*e^10 - 16*a^4*b^5*d^4*e^10 - 16*a^6*
b^3*d^4*e^10))/d^4)*(1i/(4*(b^2*d^2*e - a^2*d^2*e + a*b*d^2*e*2i)))^(1/2) - (32*(e*cot(c + d*x))^(1/2)*(4*a^3*
b^4*d^2*e^9 - 30*a*b^6*d^2*e^9 + 2*a^5*b^2*d^2*e^9))/d^4)*(1i/(4*(b^2*d^2*e - a^2*d^2*e + a*b*d^2*e*2i)))^(1/2
))*(1i/(4*(b^2*d^2*e - a^2*d^2*e + a*b*d^2*e*2i)))^(1/2) + (96*b^5*e^8*(e*cot(c + d*x))^(1/2))/d^4)*(1i/(4*(b^
2*d^2*e - a^2*d^2*e + a*b*d^2*e*2i)))^(1/2)*1i - (((32*(5*a*b^5*e^9 + a^3*b^3*e^9))/d^3 - (((32*(16*b^8*d^2*e^
10 + 28*a^2*b^6*d^2*e^10 + 8*a^4*b^4*d^2*e^10 - 4*a^6*b^2*d^2*e^10))/d^3 + (32*(e*cot(c + d*x))^(1/2)*(1i/(4*(
b^2*d^2*e - a^2*d^2*e + a*b*d^2*e*2i)))^(1/2)*(16*b^9*d^4*e^10 + 16*a^2*b^7*d^4*e^10 - 16*a^4*b^5*d^4*e^10 - 1
6*a^6*b^3*d^4*e^10))/d^4)*(1i/(4*(b^2*d^2*e - a^2*d^2*e + a*b*d^2*e*2i)))^(1/2) + (32*(e*cot(c + d*x))^(1/2)*(
4*a^3*b^4*d^2*e^9 - 30*a*b^6*d^2*e^9 + 2*a^5*b^2*d^2*e^9))/d^4)*(1i/(4*(b^2*d^2*e - a^2*d^2*e + a*b*d^2*e*2i))
)^(1/2))*(1i/(4*(b^2*d^2*e - a^2*d^2*e + a*b*d^2*e*2i)))^(1/2) - (96*b^5*e^8*(e*cot(c + d*x))^(1/2))/d^4)*(1i/
(4*(b^2*d^2*e - a^2*d^2*e + a*b*d^2*e*2i)))^(1/2)*1i)/((((32*(5*a*b^5*e^9 + a^3*b^3*e^9))/d^3 - (((32*(16*b^8*
d^2*e^10 + 28*a^2*b^6*d^2*e^10 + 8*a^4*b^4*d^2*e^10 - 4*a^6*b^2*d^2*e^10))/d^3 - (32*(e*cot(c + d*x))^(1/2)*(1
i/(4*(b^2*d^2*e - a^2*d^2*e + a*b*d^2*e*2i)))^(1/2)*(16*b^9*d^4*e^10 + 16*a^2*b^7*d^4*e^10 - 16*a^4*b^5*d^4*e^
10 - 16*a^6*b^3*d^4*e^10))/d^4)*(1i/(4*(b^2*d^2*e - a^2*d^2*e + a*b*d^2*e*2i)))^(1/2) - (32*(e*cot(c + d*x))^(
1/2)*(4*a^3*b^4*d^2*e^9 - 30*a*b^6*d^2*e^9 + 2*a^5*b^2*d^2*e^9))/d^4)*(1i/(4*(b^2*d^2*e - a^2*d^2*e + a*b*d^2*
e*2i)))^(1/2))*(1i/(4*(b^2*d^2*e - a^2*d^2*e + a*b*d^2*e*2i)))^(1/2) + (96*b^5*e^8*(e*cot(c + d*x))^(1/2))/d^4
)*(1i/(4*(b^2*d^2*e - a^2*d^2*e + a*b*d^2*e*2i)))^(1/2) + (((32*(5*a*b^5*e^9 + a^3*b^3*e^9))/d^3 - (((32*(16*b
^8*d^2*e^10 + 28*a^2*b^6*d^2*e^10 + 8*a^4*b^4*d^2*e^10 - 4*a^6*b^2*d^2*e^10))/d^3 + (32*(e*cot(c + d*x))^(1/2)
*(1i/(4*(b^2*d^2*e - a^2*d^2*e + a*b*d^2*e*2i)))^(1/2)*(16*b^9*d^4*e^10 + 16*a^2*b^7*d^4*e^10 - 16*a^4*b^5*d^4
*e^10 - 16*a^6*b^3*d^4*e^10))/d^4)*(1i/(4*(b^2*d^2*e - a^2*d^2*e + a*b*d^2*e*2i)))^(1/2) + (32*(e*cot(c + d*x)
)^(1/2)*(4*a^3*b^4*d^2*e^9 - 30*a*b^6*d^2*e^9 + 2*a^5*b^2*d^2*e^9))/d^4)*(1i/(4*(b^2*d^2*e - a^2*d^2*e + a*b*d
^2*e*2i)))^(1/2))*(1i/(4*(b^2*d^2*e - a^2*d^2*e + a*b*d^2*e*2i)))^(1/2) - (96*b^5*e^8*(e*cot(c + d*x))^(1/2))/
d^4)*(1i/(4*(b^2*d^2*e - a^2*d^2*e + a*b*d^2*e*2i)))^(1/2)))*(1i/(4*(b^2*d^2*e - a^2*d^2*e + a*b*d^2*e*2i)))^(
1/2)*2i + (atan(((((((32*(5*a*b^5*e^9 + a^3*b^3*e^9))/d^3 - (((((32*(16*b^8*d^2*e^10 + 28*a^2*b^6*d^2*e^10 + 8
*a^4*b^4*d^2*e^10 - 4*a^6*b^2*d^2*e^10))/d^3 - (32*(e*cot(c + d*x))^(1/2)*(-a*b^3*e)^(1/2)*(16*b^9*d^4*e^10 +
16*a^2*b^7*d^4*e^10 - 16*a^4*b^5*d^4*e^10 - 16*a^6*b^3*d^4*e^10))/(d^4*(a^3*d*e + a*b^2*d*e)))*(-a*b^3*e)^(1/2
))/(a^3*d*e + a*b^2*d*e) - (32*(e*cot(c + d*x))^(1/2)*(4*a^3*b^4*d^2*e^9 - 30*a*b^6*d^2*e^9 + 2*a^5*b^2*d^2*e^
9))/d^4)*(-a*b^3*e)^(1/2))/(a^3*d*e + a*b^2*d*e))*(-a*b^3*e)^(1/2))/(a^3*d*e + a*b^2*d*e) + (96*b^5*e^8*(e*cot
(c + d*x))^(1/2))/d^4)*(-a*b^3*e)^(1/2)*1i)/(a^3*d*e + a*b^2*d*e) - (((((32*(5*a*b^5*e^9 + a^3*b^3*e^9))/d^3 -
(((((32*(16*b^8*d^2*e^10 + 28*a^2*b^6*d^2*e^10 + 8*a^4*b^4*d^2*e^10 - 4*a^6*b^2*d^2*e^10))/d^3 + (32*(e*cot(c
+ d*x))^(1/2)*(-a*b^3*e)^(1/2)*(16*b^9*d^4*e^10 + 16*a^2*b^7*d^4*e^10 - 16*a^4*b^5*d^4*e^10 - 16*a^6*b^3*d^4*
e^10))/(d^4*(a^3*d*e + a*b^2*d*e)))*(-a*b^3*e)^(1/2))/(a^3*d*e + a*b^2*d*e) + (32*(e*cot(c + d*x))^(1/2)*(4*a^
3*b^4*d^2*e^9 - 30*a*b^6*d^2*e^9 + 2*a^5*b^2*d^2*e^9))/d^4)*(-a*b^3*e)^(1/2))/(a^3*d*e + a*b^2*d*e))*(-a*b^3*e
)^(1/2))/(a^3*d*e + a*b^2*d*e) - (96*b^5*e^8*(e*cot(c + d*x))^(1/2))/d^4)*(-a*b^3*e)^(1/2)*1i)/(a^3*d*e + a*b^
2*d*e))/((((((32*(5*a*b^5*e^9 + a^3*b^3*e^9))/d^3 - (((((32*(16*b^8*d^2*e^10 + 28*a^2*b^6*d^2*e^10 + 8*a^4*b^4
*d^2*e^10 - 4*a^6*b^2*d^2*e^10))/d^3 - (32*(e*cot(c + d*x))^(1/2)*(-a*b^3*e)^(1/2)*(16*b^9*d^4*e^10 + 16*a^2*b
^7*d^4*e^10 - 16*a^4*b^5*d^4*e^10 - 16*a^6*b^3*d^4*e^10))/(d^4*(a^3*d*e + a*b^2*d*e)))*(-a*b^3*e)^(1/2))/(a^3*
d*e + a*b^2*d*e) - (32*(e*cot(c + d*x))^(1/2)*(4*a^3*b^4*d^2*e^9 - 30*a*b^6*d^2*e^9 + 2*a^5*b^2*d^2*e^9))/d^4)
*(-a*b^3*e)^(1/2))/(a^3*d*e + a*b^2*d*e))*(-a*b^3*e)^(1/2))/(a^3*d*e + a*b^2*d*e) + (96*b^5*e^8*(e*cot(c + d*x
))^(1/2))/d^4)*(-a*b^3*e)^(1/2))/(a^3*d*e + a*b^2*d*e) + (((((32*(5*a*b^5*e^9 + a^3*b^3*e^9))/d^3 - (((((32*(1
6*b^8*d^2*e^10 + 28*a^2*b^6*d^2*e^10 + 8*a^4*b^4*d^2*e^10 - 4*a^6*b^2*d^2*e^10))/d^3 + (32*(e*cot(c + d*x))^(1
/2)*(-a*b^3*e)^(1/2)*(16*b^9*d^4*e^10 + 16*a^2*b^7*d^4*e^10 - 16*a^4*b^5*d^4*e^10 - 16*a^6*b^3*d^4*e^10))/(d^4
*(a^3*d*e + a*b^2*d*e)))*(-a*b^3*e)^(1/2))/(a^3*d*e + a*b^2*d*e) + (32*(e*cot(c + d*x))^(1/2)*(4*a^3*b^4*d^2*e
^9 - 30*a*b^6*d^2*e^9 + 2*a^5*b^2*d^2*e^9))/d^4)*(-a*b^3*e)^(1/2))/(a^3*d*e + a*b^2*d*e))*(-a*b^3*e)^(1/2))/(a
^3*d*e + a*b^2*d*e) - (96*b^5*e^8*(e*cot(c + d*x))^(1/2))/d^4)*(-a*b^3*e)^(1/2))/(a^3*d*e + a*b^2*d*e)))*(-a*b
^3*e)^(1/2)*2i)/(a^3*d*e + a*b^2*d*e)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {e \cot {\left (c + d x \right )}} \left (a + b \cot {\left (c + d x \right )}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(sqrt(e*cot(c + d*x))*(a + b*cot(c + d*x))), x)

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