Integrand size = 21, antiderivative size = 106 \[ \int \frac {\coth (c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b} d}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{\sqrt {a+b} d} \] Output:
2*arctanh((a+b*sech(d*x+c))^(1/2)/a^(1/2))/a^(1/2)/d-arctanh((a+b*sech(d*x +c))^(1/2)/(a-b)^(1/2))/(a-b)^(1/2)/d-arctanh((a+b*sech(d*x+c))^(1/2)/(a+b )^(1/2))/(a+b)^(1/2)/d
Time = 0.28 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.94 \[ \int \frac {\coth (c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=-\frac {-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{\sqrt {a}}+\frac {\text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b}}+\frac {\text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{\sqrt {a+b}}}{d} \] Input:
Integrate[Coth[c + d*x]/Sqrt[a + b*Sech[c + d*x]],x]
Output:
-(((-2*ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a]])/Sqrt[a] + ArcTanh[Sqrt[ a + b*Sech[c + d*x]]/Sqrt[a - b]]/Sqrt[a - b] + ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]]/Sqrt[a + b])/d)
Time = 0.40 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 26, 4373, 561, 1484, 2009}
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.
\(\displaystyle \int \frac {\coth (c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {i}{\cot \left (i c+i d x+\frac {\pi }{2}\right ) \sqrt {a+b \csc \left (i c+i d x+\frac {\pi }{2}\right )}}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \frac {1}{\cot \left (\frac {1}{2} (2 i c+\pi )+i d x\right ) \sqrt {a+b \csc \left (\frac {1}{2} (2 i c+\pi )+i d x\right )}}dx\) |
\(\Big \downarrow \) 4373 |
\(\displaystyle -\frac {b^2 \int \frac {\cosh (c+d x)}{b \sqrt {a+b \text {sech}(c+d x)} \left (b^2-b^2 \text {sech}^2(c+d x)\right )}d(b \text {sech}(c+d x))}{d}\) |
\(\Big \downarrow \) 561 |
\(\displaystyle -\frac {2 b^2 \int \frac {1}{\left (a-b^2 \text {sech}^2(c+d x)\right ) \left (b^4 \text {sech}^4(c+d x)-2 a b^2 \text {sech}^2(c+d x)+a^2-b^2\right )}d\sqrt {a+b \text {sech}(c+d x)}}{d}\) |
\(\Big \downarrow \) 1484 |
\(\displaystyle -\frac {2 b^2 \int \left (\frac {1}{2 b^2 \left (-b^2 \text {sech}^2(c+d x)+a+b\right )}-\frac {1}{2 b^2 \left (b^2 \text {sech}^2(c+d x)-a+b\right )}-\frac {1}{b^2 \left (a-b^2 \text {sech}^2(c+d x)\right )}\right )d\sqrt {a+b \text {sech}(c+d x)}}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 b^2 \left (-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} b^2}+\frac {\text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{2 b^2 \sqrt {a-b}}+\frac {\text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{2 b^2 \sqrt {a+b}}\right )}{d}\) |
Input:
Int[Coth[c + d*x]/Sqrt[a + b*Sech[c + d*x]],x]
Output:
(-2*b^2*(-(ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a]]/(Sqrt[a]*b^2)) + Arc Tanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a - b]]/(2*Sqrt[a - b]*b^2) + ArcTanh[ Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]]/(2*b^2*Sqrt[a + b])))/d
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{k = Denominator[n]}, Simp[k/d Subst[Int[x^(k*(n + 1) - 1)*(-c /d + x^k/d)^m*Simp[(b*c^2 + a*d^2)/d^2 - 2*b*c*(x^k/d^2) + b*(x^(2*k)/d^2), x]^p, x], x, (c + d*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, m, p}, x] && Frac tionQ[n] && IntegerQ[p] && IntegerQ[m]
Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symb ol] :> Int[ExpandIntegrand[(d + e*x^2)^q/(a + b*x^2 + c*x^4), x], x] /; Fre eQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[q]
Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n _), x_Symbol] :> Simp[-(-1)^((m - 1)/2)/(d*b^(m - 1)) Subst[Int[(b^2 - x^ 2)^((m - 1)/2)*((a + x)^n/x), x], x, b*Csc[c + d*x]], x] /; FreeQ[{a, b, c, d, n}, x] && IntegerQ[(m - 1)/2] && NeQ[a^2 - b^2, 0]
\[\int \frac {\coth \left (d x +c \right )}{\sqrt {a +b \,\operatorname {sech}\left (d x +c \right )}}d x\]
Input:
int(coth(d*x+c)/(a+b*sech(d*x+c))^(1/2),x)
Output:
int(coth(d*x+c)/(a+b*sech(d*x+c))^(1/2),x)
Leaf count of result is larger than twice the leaf count of optimal. 585 vs. \(2 (88) = 176\).
Time = 0.83 (sec) , antiderivative size = 9458, normalized size of antiderivative = 89.23 \[ \int \frac {\coth (c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\text {Too large to display} \] Input:
integrate(coth(d*x+c)/(a+b*sech(d*x+c))^(1/2),x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {\coth (c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\int \frac {\coth {\left (c + d x \right )}}{\sqrt {a + b \operatorname {sech}{\left (c + d x \right )}}}\, dx \] Input:
integrate(coth(d*x+c)/(a+b*sech(d*x+c))**(1/2),x)
Output:
Integral(coth(c + d*x)/sqrt(a + b*sech(c + d*x)), x)
\[ \int \frac {\coth (c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\int { \frac {\coth \left (d x + c\right )}{\sqrt {b \operatorname {sech}\left (d x + c\right ) + a}} \,d x } \] Input:
integrate(coth(d*x+c)/(a+b*sech(d*x+c))^(1/2),x, algorithm="maxima")
Output:
integrate(coth(d*x + c)/sqrt(b*sech(d*x + c) + a), x)
\[ \int \frac {\coth (c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\int { \frac {\coth \left (d x + c\right )}{\sqrt {b \operatorname {sech}\left (d x + c\right ) + a}} \,d x } \] Input:
integrate(coth(d*x+c)/(a+b*sech(d*x+c))^(1/2),x, algorithm="giac")
Output:
integrate(coth(d*x + c)/sqrt(b*sech(d*x + c) + a), x)
Timed out. \[ \int \frac {\coth (c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\int \frac {\mathrm {coth}\left (c+d\,x\right )}{\sqrt {a+\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}}} \,d x \] Input:
int(coth(c + d*x)/(a + b/cosh(c + d*x))^(1/2),x)
Output:
int(coth(c + d*x)/(a + b/cosh(c + d*x))^(1/2), x)
\[ \int \frac {\coth (c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\int \frac {\sqrt {\mathrm {sech}\left (d x +c \right ) b +a}\, \coth \left (d x +c \right )}{\mathrm {sech}\left (d x +c \right ) b +a}d x \] Input:
int(coth(d*x+c)/(a+b*sech(d*x+c))^(1/2),x)
Output:
int((sqrt(sech(c + d*x)*b + a)*coth(c + d*x))/(sech(c + d*x)*b + a),x)