Integrand size = 21, antiderivative size = 142 \[ \int \frac {\coth (c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{(a-b)^{3/2} d}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{(a+b)^{3/2} d}+\frac {2 b^2}{a \left (a^2-b^2\right ) d \sqrt {a+b \text {sech}(c+d x)}} \] Output:
2*arctanh((a+b*sech(d*x+c))^(1/2)/a^(1/2))/a^(3/2)/d-arctanh((a+b*sech(d*x +c))^(1/2)/(a-b)^(1/2))/(a-b)^(3/2)/d-arctanh((a+b*sech(d*x+c))^(1/2)/(a+b )^(1/2))/(a+b)^(3/2)/d+2*b^2/a/(a^2-b^2)/d/(a+b*sech(d*x+c))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.70 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.49 \[ \int \frac {\coth (c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=-\frac {-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b}}+\frac {\text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{\sqrt {a+b}}-\frac {a \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {a+b \text {sech}(c+d x)}{a-b}\right )}{(a-b) \sqrt {a+b \text {sech}(c+d x)}}+\frac {a \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {a+b \text {sech}(c+d x)}{a+b}\right )}{(a+b) \sqrt {a+b \text {sech}(c+d x)}}+\frac {2 b \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},1+\frac {b \text {sech}(c+d x)}{a}\right )}{a \sqrt {a+b \text {sech}(c+d x)}}}{b d} \] Input:
Integrate[Coth[c + d*x]/(a + b*Sech[c + d*x])^(3/2),x]
Output:
-((-(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] - (a*Hypergeometric2F1 [-1/2, 1, 1/2, (a + b*Sech[c + d*x])/(a - b)])/((a - b)*Sqrt[a + b*Sech[c + d*x]]) + (a*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Sech[c + d*x])/(a + b )])/((a + b)*Sqrt[a + b*Sech[c + d*x]]) + (2*b*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (b*Sech[c + d*x])/a])/(a*Sqrt[a + b*Sech[c + d*x]]))/(b*d))
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.
Failed to integrate
Input:
Int[Coth[c + d*x]/(a + b*Sech[c + d*x])^(3/2),x]
Output:
$Aborted
\[\int \frac {\coth \left (d x +c \right )}{\left (a +b \,\operatorname {sech}\left (d x +c \right )\right )^{\frac {3}{2}}}d x\]
Input:
int(coth(d*x+c)/(a+b*sech(d*x+c))^(3/2),x)
Output:
int(coth(d*x+c)/(a+b*sech(d*x+c))^(3/2),x)
Leaf count of result is larger than twice the leaf count of optimal. 1273 vs. \(2 (122) = 244\).
Time = 3.58 (sec) , antiderivative size = 14962, normalized size of antiderivative = 105.37 \[ \int \frac {\coth (c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=\text {Too large to display} \] Input:
integrate(coth(d*x+c)/(a+b*sech(d*x+c))^(3/2),x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {\coth (c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=\int \frac {\coth {\left (c + d x \right )}}{\left (a + b \operatorname {sech}{\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(coth(d*x+c)/(a+b*sech(d*x+c))**(3/2),x)
Output:
Integral(coth(c + d*x)/(a + b*sech(c + d*x))**(3/2), x)
\[ \int \frac {\coth (c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=\int { \frac {\coth \left (d x + c\right )}{{\left (b \operatorname {sech}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(coth(d*x+c)/(a+b*sech(d*x+c))^(3/2),x, algorithm="maxima")
Output:
integrate(coth(d*x + c)/(b*sech(d*x + c) + a)^(3/2), x)
\[ \int \frac {\coth (c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=\int { \frac {\coth \left (d x + c\right )}{{\left (b \operatorname {sech}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(coth(d*x+c)/(a+b*sech(d*x+c))^(3/2),x, algorithm="giac")
Output:
integrate(coth(d*x + c)/(b*sech(d*x + c) + a)^(3/2), x)
Timed out. \[ \int \frac {\coth (c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=\int \frac {\mathrm {coth}\left (c+d\,x\right )}{{\left (a+\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}\right )}^{3/2}} \,d x \] Input:
int(coth(c + d*x)/(a + b/cosh(c + d*x))^(3/2),x)
Output:
int(coth(c + d*x)/(a + b/cosh(c + d*x))^(3/2), x)
\[ \int \frac {\coth (c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=\int \frac {\sqrt {\mathrm {sech}\left (d x +c \right ) b +a}\, \coth \left (d x +c \right )}{\mathrm {sech}\left (d x +c \right )^{2} b^{2}+2 \,\mathrm {sech}\left (d x +c \right ) a b +a^{2}}d x \] Input:
int(coth(d*x+c)/(a+b*sech(d*x+c))^(3/2),x)
Output:
int((sqrt(sech(c + d*x)*b + a)*coth(c + d*x))/(sech(c + d*x)**2*b**2 + 2*s ech(c + d*x)*a*b + a**2),x)