Integrand size = 25, antiderivative size = 83 \[ \int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \cot (c+d x))} \, dx=-\frac {\arctan \left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{a d \sqrt {e}}-\frac {\text {arctanh}\left (\frac {\sqrt {e} (1+\cot (c+d x))}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{\sqrt {2} a d \sqrt {e}} \]
-arctan((e*cot(d*x+c))^(1/2)/e^(1/2))/a/d/e^(1/2)-1/2*arctanh(1/2*(1+cot(d *x+c))*e^(1/2)*2^(1/2)/(e*cot(d*x+c))^(1/2))/a/d*2^(1/2)/e^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(283\) vs. \(2(83)=166\).
Time = 0.52 (sec) , antiderivative size = 283, normalized size of antiderivative = 3.41 \[ \int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \cot (c+d x))} \, dx=-\frac {8 e^{3/2} \arctan \left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )+4 \left (-e^2\right )^{3/4} \arctan \left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt [4]{-e^2}}\right )-2 \sqrt {2} e^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )+2 \sqrt {2} e^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )-4 \left (-e^2\right )^{3/4} \text {arctanh}\left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt [4]{-e^2}}\right )-\sqrt {2} e^{3/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )+\sqrt {2} e^{3/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{8 a d e^2} \]
-1/8*(8*e^(3/2)*ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]] + 4*(-e^2)^(3/4)*ArcT an[Sqrt[e*Cot[c + d*x]]/(-e^2)^(1/4)] - 2*Sqrt[2]*e^(3/2)*ArcTan[1 - (Sqrt [2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]] + 2*Sqrt[2]*e^(3/2)*ArcTan[1 + (Sqrt[2] *Sqrt[e*Cot[c + d*x]])/Sqrt[e]] - 4*(-e^2)^(3/4)*ArcTanh[Sqrt[e*Cot[c + d* x]]/(-e^2)^(1/4)] - Sqrt[2]*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - S qrt[2]*Sqrt[e*Cot[c + d*x]]] + Sqrt[2]*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(a*d*e^2)
Time = 0.63 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.98, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3042, 4057, 3042, 4015, 221, 4117, 27, 73, 216}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{(a \cot (c+d x)+a) \sqrt {e \cot (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right ) \sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 4057 |
| \(\displaystyle \frac {\int \frac {a-a \cot (c+d x)}{\sqrt {e \cot (c+d x)}}dx}{2 a^2}+\frac {1}{2} \int \frac {\cot ^2(c+d x)+1}{\sqrt {e \cot (c+d x)} (\cot (c+d x) a+a)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\tan \left (c+d x+\frac {\pi }{2}\right ) a+a}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a^2}+\frac {1}{2} \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx\) |
| \(\Big \downarrow \) 4015 |
| \(\displaystyle \frac {1}{2} \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx-\frac {\int \frac {1}{2 a^2-(\cot (c+d x) a+a)^2 \tan (c+d x)}d\frac {\cot (c+d x) a+a}{\sqrt {e \cot (c+d x)}}}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{2} \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx-\frac {\text {arctanh}\left (\frac {\sqrt {e} (a \cot (c+d x)+a)}{\sqrt {2} a \sqrt {e \cot (c+d x)}}\right )}{\sqrt {2} a d \sqrt {e}}\) |
| \(\Big \downarrow \) 4117 |
| \(\displaystyle \frac {\int \frac {1}{a \sqrt {e \cot (c+d x)} (\cot (c+d x)+1)}d(-\cot (c+d x))}{2 d}-\frac {\text {arctanh}\left (\frac {\sqrt {e} (a \cot (c+d x)+a)}{\sqrt {2} a \sqrt {e \cot (c+d x)}}\right )}{\sqrt {2} a d \sqrt {e}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {1}{\sqrt {e \cot (c+d x)} (\cot (c+d x)+1)}d(-\cot (c+d x))}{2 a d}-\frac {\text {arctanh}\left (\frac {\sqrt {e} (a \cot (c+d x)+a)}{\sqrt {2} a \sqrt {e \cot (c+d x)}}\right )}{\sqrt {2} a d \sqrt {e}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {\int \frac {1}{\frac {\cot ^2(c+d x)}{e}+1}d\sqrt {e \cot (c+d x)}}{a d e}-\frac {\text {arctanh}\left (\frac {\sqrt {e} (a \cot (c+d x)+a)}{\sqrt {2} a \sqrt {e \cot (c+d x)}}\right )}{\sqrt {2} a d \sqrt {e}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\arctan \left (\frac {\cot (c+d x)}{\sqrt {e}}\right )}{a d \sqrt {e}}-\frac {\text {arctanh}\left (\frac {\sqrt {e} (a \cot (c+d x)+a)}{\sqrt {2} a \sqrt {e \cot (c+d x)}}\right )}{\sqrt {2} a d \sqrt {e}}\) |
ArcTan[Cot[c + d*x]/Sqrt[e]]/(a*d*Sqrt[e]) - ArcTanh[(Sqrt[e]*(a + a*Cot[c + d*x]))/(Sqrt[2]*a*Sqrt[e*Cot[c + d*x]])]/(Sqrt[2]*a*d*Sqrt[e])
3.1.26.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[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] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_ )]], x_Symbol] :> Simp[-2*(d^2/f) Subst[Int[1/(2*c*d + b*x^2), x], x, (c - d*Tan[e + f*x])/Sqrt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && EqQ[c^2 - d^2, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)/((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[1/(c^2 + d^2) Int[(a + b*Tan[e + f*x])^m *(c - d*Tan[e + f*x]), x], x] + Simp[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]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[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]
Leaf count of result is larger than twice the leaf count of optimal. \(303\) vs. \(2(68)=136\).
Time = 0.04 (sec) , antiderivative size = 304, normalized size of antiderivative = 3.66
| method | result | size |
| derivativedivides | \(-\frac {2 e^{2} \left (\frac {\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}-\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}}{2 e^{2}}+\frac {\arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}}{\sqrt {e}}\right )}{2 e^{\frac {5}{2}}}\right )}{d a}\) | \(304\) |
| default | \(-\frac {2 e^{2} \left (\frac {\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}-\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}}{2 e^{2}}+\frac {\arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}}{\sqrt {e}}\right )}{2 e^{\frac {5}{2}}}\right )}{d a}\) | \(304\) |
-2/d/a*e^2*(1/2/e^2*(1/8/e*(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*co t(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/(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*ar ctan(-2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)))+1/2/e^(5/2)*arctan((e* cot(d*x+c))^(1/2)/e^(1/2)))
Time = 0.28 (sec) , antiderivative size = 321, normalized size of antiderivative = 3.87 \[ \int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \cot (c+d x))} \, dx=\left [\frac {\sqrt {2} \sqrt {-e} \arctan \left (\frac {\sqrt {2} \sqrt {-e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} {\left (\cos \left (2 \, d x + 2 \, c\right ) + \sin \left (2 \, d x + 2 \, c\right ) + 1\right )}}{2 \, {\left (e \cos \left (2 \, d x + 2 \, c\right ) + e\right )}}\right ) - \sqrt {-e} \log \left (\frac {e \cos \left (2 \, d x + 2 \, c\right ) - e \sin \left (2 \, d x + 2 \, c\right ) + 2 \, \sqrt {-e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} \sin \left (2 \, d x + 2 \, c\right ) + e}{\cos \left (2 \, d x + 2 \, c\right ) + \sin \left (2 \, d x + 2 \, c\right ) + 1}\right )}{2 \, a d e}, \frac {\sqrt {2} \sqrt {e} \log \left (\sqrt {2} \sqrt {e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} {\left (\cos \left (2 \, d x + 2 \, c\right ) - \sin \left (2 \, d x + 2 \, c\right ) - 1\right )} + 2 \, e \sin \left (2 \, d x + 2 \, c\right ) + e\right ) - 4 \, \sqrt {e} \arctan \left (\frac {\sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{\sqrt {e}}\right )}{4 \, a d e}\right ] \]
[1/2*(sqrt(2)*sqrt(-e)*arctan(1/2*sqrt(2)*sqrt(-e)*sqrt((e*cos(2*d*x + 2*c ) + e)/sin(2*d*x + 2*c))*(cos(2*d*x + 2*c) + sin(2*d*x + 2*c) + 1)/(e*cos( 2*d*x + 2*c) + e)) - sqrt(-e)*log((e*cos(2*d*x + 2*c) - e*sin(2*d*x + 2*c) + 2*sqrt(-e)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c))*sin(2*d*x + 2*c) + e)/(cos(2*d*x + 2*c) + sin(2*d*x + 2*c) + 1)))/(a*d*e), 1/4*(sqrt(2 )*sqrt(e)*log(sqrt(2)*sqrt(e)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2* c))*(cos(2*d*x + 2*c) - sin(2*d*x + 2*c) - 1) + 2*e*sin(2*d*x + 2*c) + e) - 4*sqrt(e)*arctan(sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c))/sqrt(e) ))/(a*d*e)]
\[ \int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \cot (c+d x))} \, dx=\frac {\int \frac {1}{\sqrt {e \cot {\left (c + d x \right )}} \cot {\left (c + d x \right )} + \sqrt {e \cot {\left (c + d x \right )}}}\, dx}{a} \]
Exception generated. \[ \int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \cot (c+d x))} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \cot (c+d x))} \, dx=\int { \frac {1}{{\left (a \cot \left (d x + c\right ) + a\right )} \sqrt {e \cot \left (d x + c\right )}} \,d x } \]
Time = 12.78 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.95 \[ \int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \cot (c+d x))} \, dx=-\frac {\mathrm {atan}\left (\frac {\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )}{a\,d\,\sqrt {e}}-\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {12\,\sqrt {2}\,e^{9/2}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{12\,e^5\,\mathrm {cot}\left (c+d\,x\right )+12\,e^5}\right )}{2\,a\,d\,\sqrt {e}} \]