Integrand size = 22, antiderivative size = 959 \[ \int \frac {\sqrt {x}}{\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx=-\frac {2 b^2 x}{a^2 \left (a^2+b^2\right ) d}+\frac {2 x^{3/2}}{3 a^2}+\frac {4 b^2 \sqrt {x} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {2 b^3 x \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {4 b x \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}+\frac {4 b^2 \sqrt {x} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {2 b^3 x \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}+\frac {4 b x \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}+\frac {4 b^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {4 b^3 \sqrt {x} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {8 b \sqrt {x} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}+\frac {4 b^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {4 b^3 \sqrt {x} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}+\frac {8 b \sqrt {x} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}-\frac {4 b^3 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}+\frac {8 b \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^3}+\frac {4 b^3 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}-\frac {8 b \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^3}-\frac {2 b^2 x \cosh \left (c+d \sqrt {x}\right )}{a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )} \]
-2*b^2*x/a^2/(a^2+b^2)/d+2/3*x^(3/2)/a^2+2*b^3*x*ln(1+a*exp(c+d*x^(1/2))/( b-(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)^(3/2)/d-2*b^3*x*ln(1+a*exp(c+d*x^(1/2))/ (b+(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)^(3/2)/d+4*b^2*polylog(2,-a*exp(c+d*x^(1 /2))/(b-(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)/d^3+4*b^2*polylog(2,-a*exp(c+d*x^( 1/2))/(b+(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)/d^3-4*b^3*polylog(3,-a*exp(c+d*x^ (1/2))/(b-(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)^(3/2)/d^3+4*b^3*polylog(3,-a*exp (c+d*x^(1/2))/(b+(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)^(3/2)/d^3-2*b^2*x*cosh(c+ d*x^(1/2))/a/(a^2+b^2)/d/(b+a*sinh(c+d*x^(1/2)))-4*b*x*ln(1+a*exp(c+d*x^(1 /2))/(b-(a^2+b^2)^(1/2)))/a^2/d/(a^2+b^2)^(1/2)+4*b*x*ln(1+a*exp(c+d*x^(1/ 2))/(b+(a^2+b^2)^(1/2)))/a^2/d/(a^2+b^2)^(1/2)+8*b*polylog(3,-a*exp(c+d*x^ (1/2))/(b-(a^2+b^2)^(1/2)))/a^2/d^3/(a^2+b^2)^(1/2)-8*b*polylog(3,-a*exp(c +d*x^(1/2))/(b+(a^2+b^2)^(1/2)))/a^2/d^3/(a^2+b^2)^(1/2)+4*b^2*ln(1+a*exp( c+d*x^(1/2))/(b-(a^2+b^2)^(1/2)))*x^(1/2)/a^2/(a^2+b^2)/d^2+4*b^2*ln(1+a*e xp(c+d*x^(1/2))/(b+(a^2+b^2)^(1/2)))*x^(1/2)/a^2/(a^2+b^2)/d^2+4*b^3*polyl og(2,-a*exp(c+d*x^(1/2))/(b-(a^2+b^2)^(1/2)))*x^(1/2)/a^2/(a^2+b^2)^(3/2)/ d^2-4*b^3*polylog(2,-a*exp(c+d*x^(1/2))/(b+(a^2+b^2)^(1/2)))*x^(1/2)/a^2/( a^2+b^2)^(3/2)/d^2-8*b*polylog(2,-a*exp(c+d*x^(1/2))/(b-(a^2+b^2)^(1/2)))* x^(1/2)/a^2/d^2/(a^2+b^2)^(1/2)+8*b*polylog(2,-a*exp(c+d*x^(1/2))/(b+(a^2+ b^2)^(1/2)))*x^(1/2)/a^2/d^2/(a^2+b^2)^(1/2)
Time = 3.17 (sec) , antiderivative size = 948, normalized size of antiderivative = 0.99 \[ \int \frac {\sqrt {x}}{\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\frac {\text {csch}^2\left (c+d \sqrt {x}\right ) \left (b+a \sinh \left (c+d \sqrt {x}\right )\right ) \left (2 x^{3/2} \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )-\frac {6 b e^c \left (2 b e^c x-\frac {e^{-c} \left (-1+e^{2 c}\right ) \left (2 b d \sqrt {\left (a^2+b^2\right ) e^{2 c}} \sqrt {x} \log \left (1+\frac {a e^{2 c+d \sqrt {x}}}{b e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-2 a^2 d^2 e^c x \log \left (1+\frac {a e^{2 c+d \sqrt {x}}}{b e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-b^2 d^2 e^c x \log \left (1+\frac {a e^{2 c+d \sqrt {x}}}{b e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+2 b d \sqrt {\left (a^2+b^2\right ) e^{2 c}} \sqrt {x} \log \left (1+\frac {a e^{2 c+d \sqrt {x}}}{b e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+2 a^2 d^2 e^c x \log \left (1+\frac {a e^{2 c+d \sqrt {x}}}{b e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+b^2 d^2 e^c x \log \left (1+\frac {a e^{2 c+d \sqrt {x}}}{b e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+2 \left (b \sqrt {\left (a^2+b^2\right ) e^{2 c}}-2 a^2 d e^c \sqrt {x}-b^2 d e^c \sqrt {x}\right ) \operatorname {PolyLog}\left (2,-\frac {a e^{2 c+d \sqrt {x}}}{b e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+2 \left (b \sqrt {\left (a^2+b^2\right ) e^{2 c}}+2 a^2 d e^c \sqrt {x}+b^2 d e^c \sqrt {x}\right ) \operatorname {PolyLog}\left (2,-\frac {a e^{2 c+d \sqrt {x}}}{b e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+4 a^2 e^c \operatorname {PolyLog}\left (3,-\frac {a e^{2 c+d \sqrt {x}}}{b e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+2 b^2 e^c \operatorname {PolyLog}\left (3,-\frac {a e^{2 c+d \sqrt {x}}}{b e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-4 a^2 e^c \operatorname {PolyLog}\left (3,-\frac {a e^{2 c+d \sqrt {x}}}{b e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-2 b^2 e^c \operatorname {PolyLog}\left (3,-\frac {a e^{2 c+d \sqrt {x}}}{b e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )\right )}{d^2 \sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right ) \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )}{\left (a^2+b^2\right ) d \left (-1+e^{2 c}\right )}+\frac {6 b^2 x \text {csch}(c) \left (b \cosh (c)+a \sinh \left (d \sqrt {x}\right )\right )}{\left (a^2+b^2\right ) d}\right )}{3 a^2 \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \]
(Csch[c + d*Sqrt[x]]^2*(b + a*Sinh[c + d*Sqrt[x]])*(2*x^(3/2)*(b + a*Sinh[ c + d*Sqrt[x]]) - (6*b*E^c*(2*b*E^c*x - ((-1 + E^(2*c))*(2*b*d*Sqrt[(a^2 + b^2)*E^(2*c)]*Sqrt[x]*Log[1 + (a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])] - 2*a^2*d^2*E^c*x*Log[1 + (a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])] - b^2*d^2*E^c*x*Log[1 + (a*E^(2*c + d*Sqrt[ x]))/(b*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])] + 2*b*d*Sqrt[(a^2 + b^2)*E^(2*c) ]*Sqrt[x]*Log[1 + (a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(a^2 + b^2)*E^(2*c )])] + 2*a^2*d^2*E^c*x*Log[1 + (a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])] + b^2*d^2*E^c*x*Log[1 + (a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])] + 2*(b*Sqrt[(a^2 + b^2)*E^(2*c)] - 2*a^2*d*E^ c*Sqrt[x] - b^2*d*E^c*Sqrt[x])*PolyLog[2, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] + 2*(b*Sqrt[(a^2 + b^2)*E^(2*c)] + 2*a^2*d *E^c*Sqrt[x] + b^2*d*E^c*Sqrt[x])*PolyLog[2, -((a*E^(2*c + d*Sqrt[x]))/(b* E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))] + 4*a^2*E^c*PolyLog[3, -((a*E^(2*c + d* Sqrt[x]))/(b*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] + 2*b^2*E^c*PolyLog[3, -(( a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] - 4*a^2*E^c*P olyLog[3, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))] - 2*b^2*E^c*PolyLog[3, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(a^2 + b^2) *E^(2*c)]))]))/(d^2*E^c*Sqrt[(a^2 + b^2)*E^(2*c)]))*(b + a*Sinh[c + d*Sqrt [x]]))/((a^2 + b^2)*d*(-1 + E^(2*c))) + (6*b^2*x*Csch[c]*(b*Cosh[c] + a...
Time = 2.08 (sec) , antiderivative size = 960, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {5960, 3042, 4679, 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.
\(\displaystyle \int \frac {\sqrt {x}}{\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx\) |
\(\Big \downarrow \) 5960 |
\(\displaystyle 2 \int \frac {x}{\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2}d\sqrt {x}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 2 \int \frac {x}{\left (a+i b \csc \left (i c+i d \sqrt {x}\right )\right )^2}d\sqrt {x}\) |
\(\Big \downarrow \) 4679 |
\(\displaystyle 2 \int \left (\frac {x b^2}{a^2 \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )^2}-\frac {2 x b}{a^2 \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )}+\frac {x}{a^2}\right )d\sqrt {x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (\frac {x \log \left (\frac {e^{c+d \sqrt {x}} a}{b-\sqrt {a^2+b^2}}+1\right ) b^3}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {x \log \left (\frac {e^{c+d \sqrt {x}} a}{b+\sqrt {a^2+b^2}}+1\right ) b^3}{a^2 \left (a^2+b^2\right )^{3/2} d}+\frac {2 \sqrt {x} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) b^3}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {2 \sqrt {x} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) b^3}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {2 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) b^3}{a^2 \left (a^2+b^2\right )^{3/2} d^3}+\frac {2 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) b^3}{a^2 \left (a^2+b^2\right )^{3/2} d^3}-\frac {x b^2}{a^2 \left (a^2+b^2\right ) d}+\frac {2 \sqrt {x} \log \left (\frac {e^{c+d \sqrt {x}} a}{b-\sqrt {a^2+b^2}}+1\right ) b^2}{a^2 \left (a^2+b^2\right ) d^2}+\frac {2 \sqrt {x} \log \left (\frac {e^{c+d \sqrt {x}} a}{b+\sqrt {a^2+b^2}}+1\right ) b^2}{a^2 \left (a^2+b^2\right ) d^2}+\frac {2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) b^2}{a^2 \left (a^2+b^2\right ) d^3}+\frac {2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) b^2}{a^2 \left (a^2+b^2\right ) d^3}-\frac {x \cosh \left (c+d \sqrt {x}\right ) b^2}{a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )}-\frac {2 x \log \left (\frac {e^{c+d \sqrt {x}} a}{b-\sqrt {a^2+b^2}}+1\right ) b}{a^2 \sqrt {a^2+b^2} d}+\frac {2 x \log \left (\frac {e^{c+d \sqrt {x}} a}{b+\sqrt {a^2+b^2}}+1\right ) b}{a^2 \sqrt {a^2+b^2} d}-\frac {4 \sqrt {x} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) b}{a^2 \sqrt {a^2+b^2} d^2}+\frac {4 \sqrt {x} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) b}{a^2 \sqrt {a^2+b^2} d^2}+\frac {4 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) b}{a^2 \sqrt {a^2+b^2} d^3}-\frac {4 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) b}{a^2 \sqrt {a^2+b^2} d^3}+\frac {x^{3/2}}{3 a^2}\right )\) |
2*(-((b^2*x)/(a^2*(a^2 + b^2)*d)) + x^(3/2)/(3*a^2) + (2*b^2*Sqrt[x]*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)*d^2) + ( b^3*x*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^ 2)^(3/2)*d) - (2*b*x*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2])]) /(a^2*Sqrt[a^2 + b^2]*d) + (2*b^2*Sqrt[x]*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)*d^2) - (b^3*x*Log[1 + (a*E^(c + d*S qrt[x]))/(b + Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)^(3/2)*d) + (2*b*x*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2])])/(a^2*Sqrt[a^2 + b^2]*d) + (2*b^2*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a ^2 + b^2)*d^3) + (2*b^3*Sqrt[x]*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sq rt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^2) - (4*b*Sqrt[x]*PolyLog[2, -( (a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^2) + (2*b^2*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*( a^2 + b^2)*d^3) - (2*b^3*Sqrt[x]*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + S qrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^2) + (4*b*Sqrt[x]*PolyLog[2, - ((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^2) - (2*b^3*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2* (a^2 + b^2)^(3/2)*d^3) + (4*b*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt [a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^3) + (2*b^3*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^3) - (4*b...
3.1.67.3.1 Defintions of rubi rules used
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(n_.)*((c_.) + (d_.)*(x_))^(m_.) , x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, 1/(Sin[e + f*x]^n/(b + a*Si n[e + f*x])^n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[n, 0] && IGt Q[m, 0]
Int[((a_.) + Csch[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbo l] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Csch[c + d*x] )^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[(m + 1)/n], 0] && IntegerQ[p]
\[\int \frac {\sqrt {x}}{\left (a +b \,\operatorname {csch}\left (c +d \sqrt {x}\right )\right )^{2}}d x\]
\[ \int \frac {\sqrt {x}}{\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\int { \frac {\sqrt {x}}{{\left (b \operatorname {csch}\left (d \sqrt {x} + c\right ) + a\right )}^{2}} \,d x } \]
\[ \int \frac {\sqrt {x}}{\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\int \frac {\sqrt {x}}{\left (a + b \operatorname {csch}{\left (c + d \sqrt {x} \right )}\right )^{2}}\, dx \]
\[ \int \frac {\sqrt {x}}{\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\int { \frac {\sqrt {x}}{{\left (b \operatorname {csch}\left (d \sqrt {x} + c\right ) + a\right )}^{2}} \,d x } \]
2/3*(6*a*b^2*x - (a^3*d*e^(2*c) + a*b^2*d*e^(2*c))*x^(3/2)*e^(2*d*sqrt(x)) + (a^3*d + a*b^2*d)*x^(3/2) - 2*(3*b^3*x*e^c + (a^2*b*d*e^c + b^3*d*e^c)* x^(3/2))*e^(d*sqrt(x)))/(a^5*d + a^3*b^2*d - (a^5*d*e^(2*c) + a^3*b^2*d*e^ (2*c))*e^(2*d*sqrt(x)) - 2*(a^4*b*d*e^c + a^2*b^3*d*e^c)*e^(d*sqrt(x))) - integrate(-2*(2*a*b^2*x - (2*b^3*x*e^c + (2*a^2*b*d*e^c + b^3*d*e^c)*x^(3/ 2))*e^(d*sqrt(x)))/((a^5*d*e^(2*c) + a^3*b^2*d*e^(2*c))*x*e^(2*d*sqrt(x)) + 2*(a^4*b*d*e^c + a^2*b^3*d*e^c)*x*e^(d*sqrt(x)) - (a^5*d + a^3*b^2*d)*x) , x)
\[ \int \frac {\sqrt {x}}{\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\int { \frac {\sqrt {x}}{{\left (b \operatorname {csch}\left (d \sqrt {x} + c\right ) + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {x}}{\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\int \frac {\sqrt {x}}{{\left (a+\frac {b}{\mathrm {sinh}\left (c+d\,\sqrt {x}\right )}\right )}^2} \,d x \]