3.1.66 \(\int \frac {x^{3/2}}{(a+b \text {csch}(c+d \sqrt {x}))^2} \, dx\) [66]

3.1.66.1 Optimal result

Integrand size = 22, antiderivative size = 1639 \[ \int \frac {x^{3/2}}{\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx =\text {Too large to display} \]

output

-2*b^2*x^2/a^2/(a^2+b^2)/d-48*b^3*polylog(5,-a*exp(c+d*x^(1/2))/(b-(a^2+b^ 
2)^(1/2)))/a^2/(a^2+b^2)^(3/2)/d^5+48*b^3*polylog(5,-a*exp(c+d*x^(1/2))/(b 
+(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)^(3/2)/d^5+96*b*polylog(5,-a*exp(c+d*x^(1/ 
2))/(b-(a^2+b^2)^(1/2)))/a^2/d^5/(a^2+b^2)^(1/2)-96*b*polylog(5,-a*exp(c+d 
*x^(1/2))/(b+(a^2+b^2)^(1/2)))/a^2/d^5/(a^2+b^2)^(1/2)+48*b^2*polylog(4,-a 
*exp(c+d*x^(1/2))/(b-(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)/d^5+48*b^2*polylog(4, 
-a*exp(c+d*x^(1/2))/(b+(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)/d^5+2/5*x^(5/2)/a^2 
-48*b^2*polylog(3,-a*exp(c+d*x^(1/2))/(b+(a^2+b^2)^(1/2)))*x^(1/2)/a^2/(a^ 
2+b^2)/d^4+48*b^3*polylog(4,-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^4-48*b^3*polylog(4,-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^4-96*b*polylog(4,-a*exp(c+d*x^(1/ 
2))/(b-(a^2+b^2)^(1/2)))*x^(1/2)/a^2/d^4/(a^2+b^2)^(1/2)+96*b*polylog(4,-a 
*exp(c+d*x^(1/2))/(b+(a^2+b^2)^(1/2)))*x^(1/2)/a^2/d^4/(a^2+b^2)^(1/2)+8*b 
^2*x^(3/2)*ln(1+a*exp(c+d*x^(1/2))/(b-(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)/d^2+ 
2*b^3*x^2*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+8*b^2*x^(3/2)*ln(1+a*exp(c+d*x^(1/2))/(b+(a^2+b^2)^(1/2)))/a^2/(a^2+b^2 
)/d^2-2*b^3*x^2*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+24*b^2*x*polylog(2,-a*exp(c+d*x^(1/2))/(b-(a^2+b^2)^(1/2)))/a^2/( 
a^2+b^2)/d^3+8*b^3*x^(3/2)*polylog(2,-a*exp(c+d*x^(1/2))/(b-(a^2+b^2)^(1/2 
)))/a^2/(a^2+b^2)^(3/2)/d^2+24*b^2*x*polylog(2,-a*exp(c+d*x^(1/2))/(b+(...
 

3.1.66.2 Mathematica [A] (verified)

Time = 6.35 (sec) , antiderivative size = 1696, normalized size of antiderivative = 1.03 \[ \int \frac {x^{3/2}}{\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx =\text {Too large to display} \]

input

Integrate[x^(3/2)/(a + b*Csch[c + d*Sqrt[x]])^2,x]
 

output

(2*Csch[c + d*Sqrt[x]]^2*(b + a*Sinh[c + d*Sqrt[x]])*(x^(5/2)*(b + a*Sinh[ 
c + d*Sqrt[x]]) - (5*b*E^c*(2*b*E^c*x^2 - ((-1 + E^(2*c))*(4*b*d^3*Sqrt[(a 
^2 + b^2)*E^(2*c)]*x^(3/2)*Log[1 + (a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[( 
a^2 + b^2)*E^(2*c)])] - 2*a^2*d^4*E^c*x^2*Log[1 + (a*E^(2*c + d*Sqrt[x]))/ 
(b*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])] - b^2*d^4*E^c*x^2*Log[1 + (a*E^(2*c + 
 d*Sqrt[x]))/(b*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])] + 4*b*d^3*Sqrt[(a^2 + b^ 
2)*E^(2*c)]*x^(3/2)*Log[1 + (a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(a^2 + b 
^2)*E^(2*c)])] + 2*a^2*d^4*E^c*x^2*Log[1 + (a*E^(2*c + d*Sqrt[x]))/(b*E^c 
+ Sqrt[(a^2 + b^2)*E^(2*c)])] + b^2*d^4*E^c*x^2*Log[1 + (a*E^(2*c + d*Sqrt 
[x]))/(b*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])] - 4*d^2*(-3*b*Sqrt[(a^2 + b^2)* 
E^(2*c)] + 2*a^2*d*E^c*Sqrt[x] + b^2*d*E^c*Sqrt[x])*x*PolyLog[2, -((a*E^(2 
*c + d*Sqrt[x]))/(b*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] + 4*d^2*(3*b*Sqrt[( 
a^2 + b^2)*E^(2*c)] + 2*a^2*d*E^c*Sqrt[x] + b^2*d*E^c*Sqrt[x])*x*PolyLog[2 
, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))] - 24*b*d 
*Sqrt[(a^2 + b^2)*E^(2*c)]*Sqrt[x]*PolyLog[3, -((a*E^(2*c + d*Sqrt[x]))/(b 
*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] + 24*a^2*d^2*E^c*x*PolyLog[3, -((a*E^( 
2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] + 12*b^2*d^2*E^c*x 
*PolyLog[3, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])) 
] - 24*b*d*Sqrt[(a^2 + b^2)*E^(2*c)]*Sqrt[x]*PolyLog[3, -((a*E^(2*c + d*Sq 
rt[x]))/(b*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))] - 24*a^2*d^2*E^c*x*PolyLo...
 

3.1.66.3 Rubi [A] (verified)

Time = 3.07 (sec) , antiderivative size = 1640, 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.

\(\displaystyle \int \frac {x^{3/2}}{\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx\)

\(\Big \downarrow \) 5960

\(\displaystyle 2 \int \frac {x^2}{\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2}d\sqrt {x}\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 \int \frac {x^2}{\left (a+i b \csc \left (i c+i d \sqrt {x}\right )\right )^2}d\sqrt {x}\)

\(\Big \downarrow \) 4679

\(\displaystyle 2 \int \left (-\frac {2 b x^2}{a^2 \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )}+\frac {x^2}{a^2}+\frac {b^2 x^2}{a^2 \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )^2}\right )d\sqrt {x}\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (\frac {x^2 \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^2 \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 {4 x^{3/2} \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 {4 x^{3/2} \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 {12 x \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 {12 x \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 {24 \sqrt {x} \operatorname {PolyLog}\left (4,-\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^4}-\frac {24 \sqrt {x} \operatorname {PolyLog}\left (4,-\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^4}-\frac {24 \operatorname {PolyLog}\left (5,-\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^5}+\frac {24 \operatorname {PolyLog}\left (5,-\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^5}-\frac {x^2 b^2}{a^2 \left (a^2+b^2\right ) d}+\frac {4 x^{3/2} \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 {4 x^{3/2} \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 {12 x \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 {12 x \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 {24 \sqrt {x} \operatorname {PolyLog}\left (3,-\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^4}-\frac {24 \sqrt {x} \operatorname {PolyLog}\left (3,-\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^4}+\frac {24 \operatorname {PolyLog}\left (4,-\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^5}+\frac {24 \operatorname {PolyLog}\left (4,-\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^5}-\frac {x^2 \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^2 \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^2 \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 {8 x^{3/2} \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 {8 x^{3/2} \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 {24 x \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 {24 x \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 {48 \sqrt {x} \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) b}{a^2 \sqrt {a^2+b^2} d^4}+\frac {48 \sqrt {x} \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) b}{a^2 \sqrt {a^2+b^2} d^4}+\frac {48 \operatorname {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) b}{a^2 \sqrt {a^2+b^2} d^5}-\frac {48 \operatorname {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) b}{a^2 \sqrt {a^2+b^2} d^5}+\frac {x^{5/2}}{5 a^2}\right )\)

input

Int[x^(3/2)/(a + b*Csch[c + d*Sqrt[x]])^2,x]
 

output

2*(-((b^2*x^2)/(a^2*(a^2 + b^2)*d)) + x^(5/2)/(5*a^2) + (4*b^2*x^(3/2)*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^2*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^2*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b 
^2])])/(a^2*Sqrt[a^2 + b^2]*d) + (4*b^2*x^(3/2)*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^2*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^2*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2])])/(a^2*Sqrt[a^2 + 
b^2]*d) + (12*b^2*x*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2 
]))])/(a^2*(a^2 + b^2)*d^3) + (4*b^3*x^(3/2)*PolyLog[2, -((a*E^(c + d*Sqrt 
[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^2) - (8*b*x^(3/2)* 
PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 
+ b^2]*d^2) + (12*b^2*x*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + 
 b^2]))])/(a^2*(a^2 + b^2)*d^3) - (4*b^3*x^(3/2)*PolyLog[2, -((a*E^(c + d* 
Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^2) + (8*b*x^(3 
/2)*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*Sqrt[ 
a^2 + b^2]*d^2) - (24*b^2*Sqrt[x]*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - 
Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^4) - (12*b^3*x*PolyLog[3, -((a*E^(c 
 + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^3) + (24* 
b*x*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*Sq...
 

3.1.66.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4679

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]
 

rule 5960

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]
 

3.1.66.4 Maple [F]

\[\int \frac {x^{\frac {3}{2}}}{\left (a +b \,\operatorname {csch}\left (c +d \sqrt {x}\right )\right )^{2}}d x\]

input

int(x^(3/2)/(a+b*csch(c+d*x^(1/2)))^2,x)
 

output

int(x^(3/2)/(a+b*csch(c+d*x^(1/2)))^2,x)
 

3.1.66.5 Fricas [F]

\[ \int \frac {x^{3/2}}{\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\int { \frac {x^{\frac {3}{2}}}{{\left (b \operatorname {csch}\left (d \sqrt {x} + c\right ) + a\right )}^{2}} \,d x } \]

input

integrate(x^(3/2)/(a+b*csch(c+d*x^(1/2)))^2,x, algorithm="fricas")
 

output

integral(x^(3/2)/(b^2*csch(d*sqrt(x) + c)^2 + 2*a*b*csch(d*sqrt(x) + c) + 
a^2), x)
 

3.1.66.6 Sympy [F]

\[ \int \frac {x^{3/2}}{\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\int \frac {x^{\frac {3}{2}}}{\left (a + b \operatorname {csch}{\left (c + d \sqrt {x} \right )}\right )^{2}}\, dx \]

input

integrate(x**(3/2)/(a+b*csch(c+d*x**(1/2)))**2,x)
 

output

Integral(x**(3/2)/(a + b*csch(c + d*sqrt(x)))**2, x)
 

3.1.66.7 Maxima [F]

\[ \int \frac {x^{3/2}}{\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\int { \frac {x^{\frac {3}{2}}}{{\left (b \operatorname {csch}\left (d \sqrt {x} + c\right ) + a\right )}^{2}} \,d x } \]

input

integrate(x^(3/2)/(a+b*csch(c+d*x^(1/2)))^2,x, algorithm="maxima")
 

output

2/5*(10*a*b^2*x^2 - (a^3*d*e^(2*c) + a*b^2*d*e^(2*c))*x^(5/2)*e^(2*d*sqrt( 
x)) + (a^3*d + a*b^2*d)*x^(5/2) - 2*(5*b^3*x^2*e^c + (a^2*b*d*e^c + b^3*d* 
e^c)*x^(5/2))*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*(4*a*b^2*x^2 - (4*b^3*x^2*e^c + (2*a^2*b*d*e^c + b^3*d*e 
^c)*x^(5/2))*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)
 

3.1.66.8 Giac [F]

\[ \int \frac {x^{3/2}}{\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\int { \frac {x^{\frac {3}{2}}}{{\left (b \operatorname {csch}\left (d \sqrt {x} + c\right ) + a\right )}^{2}} \,d x } \]

input

integrate(x^(3/2)/(a+b*csch(c+d*x^(1/2)))^2,x, algorithm="giac")
 

output

integrate(x^(3/2)/(b*csch(d*sqrt(x) + c) + a)^2, x)
 

3.1.66.9 Mupad [F(-1)]

Timed out. \[ \int \frac {x^{3/2}}{\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\int \frac {x^{3/2}}{{\left (a+\frac {b}{\mathrm {sinh}\left (c+d\,\sqrt {x}\right )}\right )}^2} \,d x \]

input

int(x^(3/2)/(a + b/sinh(c + d*x^(1/2)))^2,x)
 

output

int(x^(3/2)/(a + b/sinh(c + d*x^(1/2)))^2, x)