Integrand size = 14, antiderivative size = 74 \[ \int \sqrt {a+b \coth (c+d x)} \, dx=-\frac {\sqrt {a-b} \text {arctanh}\left (\frac {\sqrt {a+b \coth (c+d x)}}{\sqrt {a-b}}\right )}{d}+\frac {\sqrt {a+b} \text {arctanh}\left (\frac {\sqrt {a+b \coth (c+d x)}}{\sqrt {a+b}}\right )}{d} \] Output:
-(a-b)^(1/2)*arctanh((a+b*coth(d*x+c))^(1/2)/(a-b)^(1/2))/d+(a+b)^(1/2)*ar ctanh((a+b*coth(d*x+c))^(1/2)/(a+b)^(1/2))/d
Result contains complex when optimal does not.
Time = 4.91 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.73 \[ \int \sqrt {a+b \coth (c+d x)} \, dx=\frac {\left (-\sqrt {i (a-b)} \text {arctanh}\left (\frac {\sqrt {i (a+b \coth (c+d x))}}{\sqrt {i (a-b)}}\right )+\sqrt {i (a+b)} \text {arctanh}\left (\frac {\sqrt {i (a+b \coth (c+d x))}}{\sqrt {i (a+b)}}\right )\right ) \sqrt {a+b \coth (c+d x)}}{d \sqrt {i (a+b \coth (c+d x))}} \] Input:
Integrate[Sqrt[a + b*Coth[c + d*x]],x]
Output:
((-(Sqrt[I*(a - b)]*ArcTanh[Sqrt[I*(a + b*Coth[c + d*x])]/Sqrt[I*(a - b)]] ) + Sqrt[I*(a + b)]*ArcTanh[Sqrt[I*(a + b*Coth[c + d*x])]/Sqrt[I*(a + b)]] )*Sqrt[a + b*Coth[c + d*x]])/(d*Sqrt[I*(a + b*Coth[c + d*x])])
Time = 0.44 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.22, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 3966, 25, 483, 25, 1450, 220}
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.
\(\displaystyle \int \sqrt {a+b \coth (c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sqrt {a-i b \tan \left (i c+i d x+\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 3966 |
\(\displaystyle -\frac {b \int -\frac {\sqrt {a+b \coth (c+d x)}}{b^2-b^2 \coth ^2(c+d x)}d(b \coth (c+d x))}{d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {b \int \frac {\sqrt {a+b \coth (c+d x)}}{b^2-b^2 \coth ^2(c+d x)}d(b \coth (c+d x))}{d}\) |
\(\Big \downarrow \) 483 |
\(\displaystyle \frac {2 b \int -\frac {b^2 \coth ^2(c+d x)}{b^4 \coth ^4(c+d x)-2 a b^2 \coth ^2(c+d x)+a^2-b^2}d\sqrt {a+b \coth (c+d x)}}{d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {2 b \int \frac {b^2 \coth ^2(c+d x)}{b^4 \coth ^4(c+d x)-2 a b^2 \coth ^2(c+d x)+a^2-b^2}d\sqrt {a+b \coth (c+d x)}}{d}\) |
\(\Big \downarrow \) 1450 |
\(\displaystyle \frac {2 b \left (-\frac {(a+b) \int \frac {1}{b^2 \coth ^2(c+d x)-a-b}d\sqrt {a+b \coth (c+d x)}}{2 b}-\frac {1}{2} \left (1-\frac {a}{b}\right ) \int \frac {1}{b^2 \coth ^2(c+d x)-a+b}d\sqrt {a+b \coth (c+d x)}\right )}{d}\) |
\(\Big \downarrow \) 220 |
\(\displaystyle \frac {2 b \left (\frac {\left (1-\frac {a}{b}\right ) \text {arctanh}\left (\frac {\sqrt {a+b \coth (c+d x)}}{\sqrt {a-b}}\right )}{2 \sqrt {a-b}}+\frac {\sqrt {a+b} \text {arctanh}\left (\frac {\sqrt {a+b \coth (c+d x)}}{\sqrt {a+b}}\right )}{2 b}\right )}{d}\) |
Input:
Int[Sqrt[a + b*Coth[c + d*x]],x]
Output:
(2*b*(((1 - a/b)*ArcTanh[Sqrt[a + b*Coth[c + d*x]]/Sqrt[a - b]])/(2*Sqrt[a - b]) + (Sqrt[a + b]*ArcTanh[Sqrt[a + b*Coth[c + d*x]]/Sqrt[a + b]])/(2*b )))/d
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(- 1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Int[Sqrt[(c_) + (d_.)*(x_)]/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[2*d Subst[Int[x^2/(b*c^2 + a*d^2 - 2*b*c*x^2 + b*x^4), x], x, Sqrt[c + d*x]], x ] /; FreeQ[{a, b, c, d}, x]
Int[((d_.)*(x_))^(m_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Wi th[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(d^2/2)*(b/q + 1) Int[(d*x)^(m - 2)/(b/ 2 + q/2 + c*x^2), x], x] - Simp[(d^2/2)*(b/q - 1) Int[(d*x)^(m - 2)/(b/2 - q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - 4*a*c, 0] && GeQ[m, 2]
Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b/d Su bst[Int[(a + x)^n/(b^2 + x^2), x], x, b*Tan[c + d*x]], x] /; FreeQ[{a, b, c , d, n}, x] && NeQ[a^2 + b^2, 0]
Time = 0.23 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.85
method | result | size |
derivativedivides | \(-\frac {\sqrt {-a +b}\, \arctan \left (\frac {\sqrt {a +b \coth \left (d x +c \right )}}{\sqrt {-a +b}}\right )}{d}+\frac {\sqrt {a +b}\, \operatorname {arctanh}\left (\frac {\sqrt {a +b \coth \left (d x +c \right )}}{\sqrt {a +b}}\right )}{d}\) | \(63\) |
default | \(-\frac {\sqrt {-a +b}\, \arctan \left (\frac {\sqrt {a +b \coth \left (d x +c \right )}}{\sqrt {-a +b}}\right )}{d}+\frac {\sqrt {a +b}\, \operatorname {arctanh}\left (\frac {\sqrt {a +b \coth \left (d x +c \right )}}{\sqrt {a +b}}\right )}{d}\) | \(63\) |
Input:
int((a+b*coth(d*x+c))^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/d*(-a+b)^(1/2)*arctan((a+b*coth(d*x+c))^(1/2)/(-a+b)^(1/2))+(a+b)^(1/2) *arctanh((a+b*coth(d*x+c))^(1/2)/(a+b)^(1/2))/d
Leaf count of result is larger than twice the leaf count of optimal. 338 vs. \(2 (62) = 124\).
Time = 0.14 (sec) , antiderivative size = 2231, normalized size of antiderivative = 30.15 \[ \int \sqrt {a+b \coth (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((a+b*coth(d*x+c))^(1/2),x, algorithm="fricas")
Output:
[1/4*(sqrt(a + b)*log(2*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^4 + 8*(a^2 + 2*a *b + b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + 2*(a^2 + 2*a*b + b^2)*sinh(d*x + c)^4 - 4*(a^2 + a*b)*cosh(d*x + c)^2 + 4*(3*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 - a^2 - a*b)*sinh(d*x + c)^2 + 2*a^2 - b^2 + 2*((a + b)*cosh(d*x + c)^4 + 4*(a + b)*cosh(d*x + c)*sinh(d*x + c)^3 + (a + b)*sinh(d*x + c)^4 - (2*a + b)*cosh(d*x + c)^2 + (6*(a + b)*cosh(d*x + c)^2 - 2*a - b)*sinh(d* x + c)^2 + 2*(2*(a + b)*cosh(d*x + c)^3 - (2*a + b)*cosh(d*x + c))*sinh(d* x + c) + a)*sqrt(a + b)*sqrt((b*cosh(d*x + c) + a*sinh(d*x + c))/sinh(d*x + c)) + 8*((a^2 + 2*a*b + b^2)*cosh(d*x + c)^3 - (a^2 + a*b)*cosh(d*x + c) )*sinh(d*x + c)) + sqrt(a - b)*log(((2*a^2 - b^2)*cosh(d*x + c)^4 + 4*(2*a ^2 - b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (2*a^2 - b^2)*sinh(d*x + c)^4 - 4*(a^2 - a*b)*cosh(d*x + c)^2 + 2*(3*(2*a^2 - b^2)*cosh(d*x + c)^2 - 2*a^2 + 2*a*b)*sinh(d*x + c)^2 + 2*a^2 - 4*a*b + 2*b^2 - 2*(a*cosh(d*x + c)^4 + 4*a*cosh(d*x + c)*sinh(d*x + c)^3 + a*sinh(d*x + c)^4 - (2*a - b)*cosh(d* x + c)^2 + (6*a*cosh(d*x + c)^2 - 2*a + b)*sinh(d*x + c)^2 + 2*(2*a*cosh(d *x + c)^3 - (2*a - b)*cosh(d*x + c))*sinh(d*x + c) + a - b)*sqrt(a - b)*sq rt((b*cosh(d*x + c) + a*sinh(d*x + c))/sinh(d*x + c)) + 4*((2*a^2 - b^2)*c osh(d*x + c)^3 - 2*(a^2 - a*b)*cosh(d*x + c))*sinh(d*x + c))/(cosh(d*x + c )^4 + 4*cosh(d*x + c)^3*sinh(d*x + c) + 6*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*cosh(d*x + c)*sinh(d*x + c)^3 + sinh(d*x + c)^4)))/d, -1/4*(2*sqrt(...
\[ \int \sqrt {a+b \coth (c+d x)} \, dx=\int \sqrt {a + b \coth {\left (c + d x \right )}}\, dx \] Input:
integrate((a+b*coth(d*x+c))**(1/2),x)
Output:
Integral(sqrt(a + b*coth(c + d*x)), x)
\[ \int \sqrt {a+b \coth (c+d x)} \, dx=\int { \sqrt {b \coth \left (d x + c\right ) + a} \,d x } \] Input:
integrate((a+b*coth(d*x+c))^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(b*coth(d*x + c) + a), x)
Exception generated. \[ \int \sqrt {a+b \coth (c+d x)} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+b*coth(d*x+c))^(1/2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Time = 2.56 (sec) , antiderivative size = 151, normalized size of antiderivative = 2.04 \[ \int \sqrt {a+b \coth (c+d x)} \, dx=\frac {\mathrm {atan}\left (\frac {b^2\,\sqrt {a-b}\,\sqrt {a+b\,\mathrm {coth}\left (c+d\,x\right )}\,1{}\mathrm {i}+a\,b\,\sqrt {a-b}\,\sqrt {a+b\,\mathrm {coth}\left (c+d\,x\right )}\,1{}\mathrm {i}}{a^2\,b-b^3}\right )\,\sqrt {a-b}\,1{}\mathrm {i}}{d}+\frac {\mathrm {atan}\left (\frac {b^2\,\sqrt {a+b}\,\sqrt {a+b\,\mathrm {coth}\left (c+d\,x\right )}\,1{}\mathrm {i}-a\,b\,\sqrt {a+b}\,\sqrt {a+b\,\mathrm {coth}\left (c+d\,x\right )}\,1{}\mathrm {i}}{a^2\,b-b^3}\right )\,\sqrt {a+b}\,1{}\mathrm {i}}{d} \] Input:
int((a + b*coth(c + d*x))^(1/2),x)
Output:
(atan((b^2*(a - b)^(1/2)*(a + b*coth(c + d*x))^(1/2)*1i + a*b*(a - b)^(1/2 )*(a + b*coth(c + d*x))^(1/2)*1i)/(a^2*b - b^3))*(a - b)^(1/2)*1i)/d + (at an((b^2*(a + b)^(1/2)*(a + b*coth(c + d*x))^(1/2)*1i - a*b*(a + b)^(1/2)*( a + b*coth(c + d*x))^(1/2)*1i)/(a^2*b - b^3))*(a + b)^(1/2)*1i)/d
\[ \int \sqrt {a+b \coth (c+d x)} \, dx=\int \sqrt {\coth \left (d x +c \right ) b +a}d x \] Input:
int((a+b*coth(d*x+c))^(1/2),x)
Output:
int(sqrt(coth(c + d*x)*b + a),x)