\(\int \frac {\sqrt {x}}{(a+b \text {sech}(c+d \sqrt {x}))^2} \, dx\) [75]

Optimal result

Integrand size = 22, antiderivative size = 1027 \[ \int \frac {\sqrt {x}}{\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2} \, dx =\text {Too large to display} \] Output:

2*b^2*x/a^2/(a^2-b^2)/d+2/3*x^(3/2)/a^2-4*b^2*x^(1/2)*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*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*x*ln(1+a*exp(c+d*x^(1/2 
))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(1/2)/d-4*b^2*x^(1/2)*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*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*x*ln(1+a*exp(c+ 
d*x^(1/2))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(1/2)/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^3*x^(1/2)*po 
lylog(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 
-8*b*x^(1/2)*polylog(2,-a*exp(c+d*x^(1/2))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2 
+b^2)^(1/2)/d^2-4*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*x^(1/2)*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+8*b*x^(1/2)*polylog(2,-a*exp(c+d*x^(1/2 
))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(1/2)/d^2-4*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+8*b*polylog(3,- 
a*exp(c+d*x^(1/2))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(1/2)/d^3+4*b^3*po 
lylog(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 
-8*b*polylog(3,-a*exp(c+d*x^(1/2))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(1 
/2)/d^3+2*b^2*x*sinh(c+d*x^(1/2))/a/(a^2-b^2)/d/(b+a*cosh(c+d*x^(1/2)))
 

Mathematica [A] (verified)

Time = 3.21 (sec) , antiderivative size = 986, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt {x}}{\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(2*(b + a*Cosh[c + d*Sqrt[x]])*Sech[c + d*Sqrt[x]]^2*(x^(3/2)*(b + a*Cosh[ 
c + d*Sqrt[x]]) + (3*b*E^c*(b + a*Cosh[c + d*Sqrt[x]])*(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*S 
qrt[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*Sqr 
t[(-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])*Po 
lyLog[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 - Sq 
rt[(-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)]))]))/(d^2*E^c*Sqrt[(- 
a^2 + b^2)*E^(2*c)])))/((a^2 - b^2)*d*(1 + E^(2*c))) + (3*b^2*x*Sech[c]...
 

Rubi [A] (verified)

Time = 2.18 (sec) , antiderivative size = 1026, 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 = {5959, 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.

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

\(\Big \downarrow \) 5959

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4679

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

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (\frac {x \log \left (\frac {e^{c+d \sqrt {x}} a}{b-\sqrt {b^2-a^2}}+1\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d}-\frac {x \log \left (\frac {e^{c+d \sqrt {x}} a}{b+\sqrt {b^2-a^2}}+1\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d}+\frac {2 \sqrt {x} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^2}-\frac {2 \sqrt {x} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^2}-\frac {2 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^3}+\frac {2 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^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 {b^2-a^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 {b^2-a^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 {b^2-a^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 {b^2-a^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^3}+\frac {x \sinh \left (c+d \sqrt {x}\right ) b^2}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt {x}\right )\right )}-\frac {2 x \log \left (\frac {e^{c+d \sqrt {x}} a}{b-\sqrt {b^2-a^2}}+1\right ) b}{a^2 \sqrt {b^2-a^2} d}+\frac {2 x \log \left (\frac {e^{c+d \sqrt {x}} a}{b+\sqrt {b^2-a^2}}+1\right ) b}{a^2 \sqrt {b^2-a^2} d}-\frac {4 \sqrt {x} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^2}+\frac {4 \sqrt {x} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^2}+\frac {4 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^3}-\frac {4 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^3}+\frac {x^{3/2}}{3 a^2}\right )\)

Input:

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

Output:

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*Sqrt[x]))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) + (2*b*x*L 
og[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 - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2) - (4*b*Sqrt[x]*Pol 
yLog[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*Sqr 
t[x]))/(b + Sqrt[-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*Po 
lyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 +...
 

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 5959

Int[(x_)^(m_.)*((a_.) + (b_.)*Sech[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbo 
l] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Sech[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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {x}}{\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(a-b>0)', see `assume?` for more 
details)Is
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {x}}{\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\int \frac {\sqrt {x}}{{\left (a+\frac {b}{\mathrm {cosh}\left (c+d\,\sqrt {x}\right )}\right )}^2} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {x}}{\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\text {too large to display} \] Input:

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

Output:

(2*( - 24*e**(2*sqrt(x)*d + 2*c)*sqrt(a**2 - b**2)*atan((e**(sqrt(x)*d + c 
)*a + b)/sqrt(a**2 - b**2))*a**3*b + 30*e**(2*sqrt(x)*d + 2*c)*sqrt(a**2 - 
 b**2)*atan((e**(sqrt(x)*d + c)*a + b)/sqrt(a**2 - b**2))*a*b**3 - 48*e**( 
sqrt(x)*d + c)*sqrt(a**2 - b**2)*atan((e**(sqrt(x)*d + c)*a + b)/sqrt(a**2 
 - b**2))*a**2*b**2 + 60*e**(sqrt(x)*d + c)*sqrt(a**2 - b**2)*atan((e**(sq 
rt(x)*d + c)*a + b)/sqrt(a**2 - b**2))*b**4 - 24*sqrt(a**2 - b**2)*atan((e 
**(sqrt(x)*d + c)*a + b)/sqrt(a**2 - b**2))*a**3*b + 30*sqrt(a**2 - b**2)* 
atan((e**(sqrt(x)*d + c)*a + b)/sqrt(a**2 - b**2))*a*b**3 - 24*e**(2*sqrt( 
x)*d + 3*c)*int(e**(sqrt(x)*d)/(e**(4*sqrt(x)*d + 4*c)*a**2 + 4*e**(3*sqrt 
(x)*d + 3*c)*a*b + 2*e**(2*sqrt(x)*d + 2*c)*a**2 + 4*e**(2*sqrt(x)*d + 2*c 
)*b**2 + 4*e**(sqrt(x)*d + c)*a*b + a**2),x)*a**6*b*d**2 + 42*e**(2*sqrt(x 
)*d + 3*c)*int(e**(sqrt(x)*d)/(e**(4*sqrt(x)*d + 4*c)*a**2 + 4*e**(3*sqrt( 
x)*d + 3*c)*a*b + 2*e**(2*sqrt(x)*d + 2*c)*a**2 + 4*e**(2*sqrt(x)*d + 2*c) 
*b**2 + 4*e**(sqrt(x)*d + c)*a*b + a**2),x)*a**4*b**3*d**2 - 18*e**(2*sqrt 
(x)*d + 3*c)*int(e**(sqrt(x)*d)/(e**(4*sqrt(x)*d + 4*c)*a**2 + 4*e**(3*sqr 
t(x)*d + 3*c)*a*b + 2*e**(2*sqrt(x)*d + 2*c)*a**2 + 4*e**(2*sqrt(x)*d + 2* 
c)*b**2 + 4*e**(sqrt(x)*d + c)*a*b + a**2),x)*a**2*b**5*d**2 - 12*e**(2*sq 
rt(x)*d + 3*c)*int((sqrt(x)*e**(sqrt(x)*d))/(e**(4*sqrt(x)*d + 4*c)*a**2 + 
 4*e**(3*sqrt(x)*d + 3*c)*a*b + 2*e**(2*sqrt(x)*d + 2*c)*a**2 + 4*e**(2*sq 
rt(x)*d + 2*c)*b**2 + 4*e**(sqrt(x)*d + c)*a*b + a**2),x)*a**6*b*d**3 +...