Integrand size = 33, antiderivative size = 179 \[ \int \cot (d+e x) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx=\frac {\sqrt {a-b+c} \text {arctanh}\left (\frac {2 a-b+(b-2 c) \cot ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 e}-\frac {(b-2 c) \text {arctanh}\left (\frac {b+2 c \cot ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{4 \sqrt {c} e}-\frac {\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{2 e} \]
-1/4*(b-2*c)*arctanh(1/2*(b+2*c*cot(e*x+d)^2)/c^(1/2)/(a+b*cot(e*x+d)^2+c* cot(e*x+d)^4)^(1/2))/e/c^(1/2)+1/2*arctanh(1/2*(2*a-b+(b-2*c)*cot(e*x+d)^2 )/(a-b+c)^(1/2)/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2))*(a-b+c)^(1/2)/e-1 /2*(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2)/e
Time = 2.46 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.37 \[ \int \cot (d+e x) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx=-\frac {\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \tan ^2(d+e x) \left ((b-2 c) \text {arctanh}\left (\frac {2 c+b \tan ^2(d+e x)}{2 \sqrt {c} \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}\right )+2 \sqrt {c} \left (-\sqrt {a-b+c} \text {arctanh}\left (\frac {b-2 c+(2 a-b) \tan ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}\right )+\cot ^2(d+e x) \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}\right )\right )}{4 \sqrt {c} e \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}} \]
-1/4*(Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4]*Tan[d + e*x]^2*((b - 2 *c)*ArcTanh[(2*c + b*Tan[d + e*x]^2)/(2*Sqrt[c]*Sqrt[c + b*Tan[d + e*x]^2 + a*Tan[d + e*x]^4])] + 2*Sqrt[c]*(-(Sqrt[a - b + c]*ArcTanh[(b - 2*c + (2 *a - b)*Tan[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[c + b*Tan[d + e*x]^2 + a*T an[d + e*x]^4])]) + Cot[d + e*x]^2*Sqrt[c + b*Tan[d + e*x]^2 + a*Tan[d + e *x]^4])))/(Sqrt[c]*e*Sqrt[c + b*Tan[d + e*x]^2 + a*Tan[d + e*x]^4])
Time = 0.42 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.97, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.303, Rules used = {3042, 4184, 1576, 1162, 25, 1269, 1092, 219, 1154, 219}
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.
\(\displaystyle \int \cot (d+e x) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \cot (d+e x) \sqrt {a+b \cot (d+e x)^2+c \cot (d+e x)^4}dx\) |
\(\Big \downarrow \) 4184 |
\(\displaystyle -\frac {\int \frac {\cot (d+e x) \sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}{\cot ^2(d+e x)+1}d\cot (d+e x)}{e}\) |
\(\Big \downarrow \) 1576 |
\(\displaystyle -\frac {\int \frac {\sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}{\cot ^2(d+e x)+1}d\cot ^2(d+e x)}{2 e}\) |
\(\Big \downarrow \) 1162 |
\(\displaystyle -\frac {\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}-\frac {1}{2} \int -\frac {(b-2 c) \cot ^2(d+e x)+2 a-b}{\left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}d\cot ^2(d+e x)}{2 e}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\frac {1}{2} \int \frac {(b-2 c) \cot ^2(d+e x)+2 a-b}{\left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}d\cot ^2(d+e x)+\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{2 e}\) |
\(\Big \downarrow \) 1269 |
\(\displaystyle -\frac {\frac {1}{2} \left ((b-2 c) \int \frac {1}{\sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}d\cot ^2(d+e x)+2 (a-b+c) \int \frac {1}{\left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}d\cot ^2(d+e x)\right )+\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{2 e}\) |
\(\Big \downarrow \) 1092 |
\(\displaystyle -\frac {\frac {1}{2} \left (2 (b-2 c) \int \frac {1}{4 c-\cot ^4(d+e x)}d\frac {2 c \cot ^2(d+e x)+b}{\sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}+2 (a-b+c) \int \frac {1}{\left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}d\cot ^2(d+e x)\right )+\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{2 e}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {\frac {1}{2} \left (2 (a-b+c) \int \frac {1}{\left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}d\cot ^2(d+e x)+\frac {(b-2 c) \text {arctanh}\left (\frac {b+2 c \cot ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{\sqrt {c}}\right )+\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{2 e}\) |
\(\Big \downarrow \) 1154 |
\(\displaystyle -\frac {\frac {1}{2} \left (\frac {(b-2 c) \text {arctanh}\left (\frac {b+2 c \cot ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{\sqrt {c}}-4 (a-b+c) \int \frac {1}{4 (a-b+c)-\cot ^4(d+e x)}d\frac {(b-2 c) \cot ^2(d+e x)+2 a-b}{\sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}\right )+\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{2 e}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {\frac {1}{2} \left (\frac {(b-2 c) \text {arctanh}\left (\frac {b+2 c \cot ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{\sqrt {c}}-2 \sqrt {a-b+c} \text {arctanh}\left (\frac {2 a+(b-2 c) \cot ^2(d+e x)-b}{2 \sqrt {a-b+c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )\right )+\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{2 e}\) |
-1/2*((-2*Sqrt[a - b + c]*ArcTanh[(2*a - b + (b - 2*c)*Cot[d + e*x]^2)/(2* Sqrt[a - b + c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])] + ((b - 2* c)*ArcTanh[(b + 2*c*Cot[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])])/Sqrt[c])/2 + Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e *x]^4])/e
3.1.24.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 2*p + 1))), x ] - Simp[p/(e*(m + 2*p + 1)) Int[(d + e*x)^m*Simp[b*d - 2*a*e + (2*c*d - b*e)*x, x]*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x ] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && ( !RationalQ[m] || LtQ[m, 1]) && !ILtQ[m + 2*p, 0] && IntQuadraticQ[a, b, c, d, e, m, p, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^( p_.), x_Symbol] :> Simp[1/2 Subst[Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x] , x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]
Int[cot[(d_.) + (e_.)*(x_)]^(m_.)*((a_.) + (b_.)*(cot[(d_.) + (e_.)*(x_)]*( f_.))^(n_.) + (c_.)*(cot[(d_.) + (e_.)*(x_)]*(f_.))^(n2_.))^(p_), x_Symbol] :> Simp[-f/e Subst[Int[(x/f)^m*((a + b*x^n + c*x^(2*n))^p/(f^2 + x^2)), x], x, f*Cot[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[ n2, 2*n] && NeQ[b^2 - 4*a*c, 0]
Time = 0.09 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.21
method | result | size |
derivativedivides | \(\frac {-\frac {\sqrt {\left (\cot \left (e x +d \right )^{2}+1\right )^{2} c +\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+a -b +c}}{2}-\frac {\left (b -2 c \right ) \ln \left (\frac {\frac {b}{2}-c +\left (\cot \left (e x +d \right )^{2}+1\right ) c}{\sqrt {c}}+\sqrt {\left (\cot \left (e x +d \right )^{2}+1\right )^{2} c +\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+a -b +c}\right )}{4 \sqrt {c}}+\frac {\sqrt {a -b +c}\, \ln \left (\frac {2 a -2 b +2 c +\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+2 \sqrt {a -b +c}\, \sqrt {\left (\cot \left (e x +d \right )^{2}+1\right )^{2} c +\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+a -b +c}}{\cot \left (e x +d \right )^{2}+1}\right )}{2}}{e}\) | \(217\) |
default | \(\frac {-\frac {\sqrt {\left (\cot \left (e x +d \right )^{2}+1\right )^{2} c +\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+a -b +c}}{2}-\frac {\left (b -2 c \right ) \ln \left (\frac {\frac {b}{2}-c +\left (\cot \left (e x +d \right )^{2}+1\right ) c}{\sqrt {c}}+\sqrt {\left (\cot \left (e x +d \right )^{2}+1\right )^{2} c +\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+a -b +c}\right )}{4 \sqrt {c}}+\frac {\sqrt {a -b +c}\, \ln \left (\frac {2 a -2 b +2 c +\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+2 \sqrt {a -b +c}\, \sqrt {\left (\cot \left (e x +d \right )^{2}+1\right )^{2} c +\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+a -b +c}}{\cot \left (e x +d \right )^{2}+1}\right )}{2}}{e}\) | \(217\) |
1/e*(-1/2*((cot(e*x+d)^2+1)^2*c+(b-2*c)*(cot(e*x+d)^2+1)+a-b+c)^(1/2)-1/4* (b-2*c)*ln((1/2*b-c+(cot(e*x+d)^2+1)*c)/c^(1/2)+((cot(e*x+d)^2+1)^2*c+(b-2 *c)*(cot(e*x+d)^2+1)+a-b+c)^(1/2))/c^(1/2)+1/2*(a-b+c)^(1/2)*ln((2*a-2*b+2 *c+(b-2*c)*(cot(e*x+d)^2+1)+2*(a-b+c)^(1/2)*((cot(e*x+d)^2+1)^2*c+(b-2*c)* (cot(e*x+d)^2+1)+a-b+c)^(1/2))/(cot(e*x+d)^2+1)))
Leaf count of result is larger than twice the leaf count of optimal. 465 vs. \(2 (155) = 310\).
Time = 1.75 (sec) , antiderivative size = 1932, normalized size of antiderivative = 10.79 \[ \int \cot (d+e x) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx=\text {Too large to display} \]
[1/8*(2*sqrt(a - b + c)*c*log(2*(a^2 - 2*a*b + b^2 + 2*(a - b)*c + c^2)*co s(2*e*x + 2*d)^2 + 2*a^2 - b^2 + 2*c^2 + 2*((a - b + c)*cos(2*e*x + 2*d)^2 - (2*a - b)*cos(2*e*x + 2*d) + a - c)*sqrt(a - b + c)*sqrt(((a - b + c)*c os(2*e*x + 2*d)^2 - 2*(a - c)*cos(2*e*x + 2*d) + a + b + c)/(cos(2*e*x + 2 *d)^2 - 2*cos(2*e*x + 2*d) + 1)) - 4*(a^2 - a*b + b*c - c^2)*cos(2*e*x + 2 *d)) - (b - 2*c)*sqrt(c)*log(((b^2 + 4*(a - 2*b)*c + 8*c^2)*cos(2*e*x + 2* d)^2 + b^2 + 4*(a + 2*b)*c + 8*c^2 + 4*((b - 2*c)*cos(2*e*x + 2*d)^2 - 2*b *cos(2*e*x + 2*d) + b + 2*c)*sqrt(c)*sqrt(((a - b + c)*cos(2*e*x + 2*d)^2 - 2*(a - c)*cos(2*e*x + 2*d) + a + b + c)/(cos(2*e*x + 2*d)^2 - 2*cos(2*e* x + 2*d) + 1)) - 2*(b^2 + 4*a*c - 8*c^2)*cos(2*e*x + 2*d))/(cos(2*e*x + 2* d)^2 - 2*cos(2*e*x + 2*d) + 1)) - 4*c*sqrt(((a - b + c)*cos(2*e*x + 2*d)^2 - 2*(a - c)*cos(2*e*x + 2*d) + a + b + c)/(cos(2*e*x + 2*d)^2 - 2*cos(2*e *x + 2*d) + 1)))/(c*e), -1/4*((b - 2*c)*sqrt(-c)*arctan(-1/2*((b - 2*c)*co s(2*e*x + 2*d)^2 - 2*b*cos(2*e*x + 2*d) + b + 2*c)*sqrt(-c)*sqrt(((a - b + c)*cos(2*e*x + 2*d)^2 - 2*(a - c)*cos(2*e*x + 2*d) + a + b + c)/(cos(2*e* x + 2*d)^2 - 2*cos(2*e*x + 2*d) + 1))/(((a - b)*c + c^2)*cos(2*e*x + 2*d)^ 2 + (a + b)*c + c^2 - 2*(a*c - c^2)*cos(2*e*x + 2*d))) - sqrt(a - b + c)*c *log(2*(a^2 - 2*a*b + b^2 + 2*(a - b)*c + c^2)*cos(2*e*x + 2*d)^2 + 2*a^2 - b^2 + 2*c^2 + 2*((a - b + c)*cos(2*e*x + 2*d)^2 - (2*a - b)*cos(2*e*x + 2*d) + a - c)*sqrt(a - b + c)*sqrt(((a - b + c)*cos(2*e*x + 2*d)^2 - 2*...
\[ \int \cot (d+e x) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx=\int \sqrt {a + b \cot ^{2}{\left (d + e x \right )} + c \cot ^{4}{\left (d + e x \right )}} \cot {\left (d + e x \right )}\, dx \]
\[ \int \cot (d+e x) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx=\int { \sqrt {c \cot \left (e x + d\right )^{4} + b \cot \left (e x + d\right )^{2} + a} \cot \left (e x + d\right ) \,d x } \]
Timed out. \[ \int \cot (d+e x) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx=\text {Timed out} \]
Timed out. \[ \int \cot (d+e x) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx=\int \mathrm {cot}\left (d+e\,x\right )\,\sqrt {c\,{\mathrm {cot}\left (d+e\,x\right )}^4+b\,{\mathrm {cot}\left (d+e\,x\right )}^2+a} \,d x \]