3.1.44 \(\int \frac {x}{a+b \text {sech}(c+d \sqrt {x})} \, dx\) [44]

3.1.44.1 Optimal result

Integrand size = 18, antiderivative size = 481 \[ \int \frac {x}{a+b \text {sech}\left (c+d \sqrt {x}\right )} \, dx=\frac {x^2}{2 a}-\frac {2 b x^{3/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}+\frac {2 b x^{3/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {6 b x \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {6 b x \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {12 b \sqrt {x} \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {12 b \sqrt {x} \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {12 b \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}+\frac {12 b \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4} \]

output

1/2*x^2/a-2*b*x^(3/2)*ln(1+a*exp(c+d*x^(1/2))/(b-(-a^2+b^2)^(1/2)))/a/d/(- 
a^2+b^2)^(1/2)+2*b*x^(3/2)*ln(1+a*exp(c+d*x^(1/2))/(b+(-a^2+b^2)^(1/2)))/a 
/d/(-a^2+b^2)^(1/2)-6*b*x*polylog(2,-a*exp(c+d*x^(1/2))/(b-(-a^2+b^2)^(1/2 
)))/a/d^2/(-a^2+b^2)^(1/2)+6*b*x*polylog(2,-a*exp(c+d*x^(1/2))/(b+(-a^2+b^ 
2)^(1/2)))/a/d^2/(-a^2+b^2)^(1/2)-12*b*polylog(4,-a*exp(c+d*x^(1/2))/(b-(- 
a^2+b^2)^(1/2)))/a/d^4/(-a^2+b^2)^(1/2)+12*b*polylog(4,-a*exp(c+d*x^(1/2)) 
/(b+(-a^2+b^2)^(1/2)))/a/d^4/(-a^2+b^2)^(1/2)+12*b*polylog(3,-a*exp(c+d*x^ 
(1/2))/(b-(-a^2+b^2)^(1/2)))*x^(1/2)/a/d^3/(-a^2+b^2)^(1/2)-12*b*polylog(3 
,-a*exp(c+d*x^(1/2))/(b+(-a^2+b^2)^(1/2)))*x^(1/2)/a/d^3/(-a^2+b^2)^(1/2)
 

3.1.44.2 Mathematica [A] (verified)

Time = 0.67 (sec) , antiderivative size = 373, normalized size of antiderivative = 0.78 \[ \int \frac {x}{a+b \text {sech}\left (c+d \sqrt {x}\right )} \, dx=\frac {\sqrt {-a^2+b^2} d^4 x^2-4 b d^3 x^{3/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )+4 b d^3 x^{3/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )-12 b d^2 x \operatorname {PolyLog}\left (2,\frac {a e^{c+d \sqrt {x}}}{-b+\sqrt {-a^2+b^2}}\right )+12 b d^2 x \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )+24 b d \sqrt {x} \operatorname {PolyLog}\left (3,\frac {a e^{c+d \sqrt {x}}}{-b+\sqrt {-a^2+b^2}}\right )-24 b d \sqrt {x} \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )-24 b \operatorname {PolyLog}\left (4,\frac {a e^{c+d \sqrt {x}}}{-b+\sqrt {-a^2+b^2}}\right )+24 b \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{2 a \sqrt {-a^2+b^2} d^4} \]

input

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

output

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

3.1.44.3 Rubi [A] (verified)

Time = 1.10 (sec) , antiderivative size = 482, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, 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.

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

\(\Big \downarrow \) 5959

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

\(\Big \downarrow \) 3042

\(\displaystyle 2 \int \frac {x^{3/2}}{a+b \csc \left (i c+i d \sqrt {x}+\frac {\pi }{2}\right )}d\sqrt {x}\)

\(\Big \downarrow \) 4679

\(\displaystyle 2 \int \left (\frac {x^{3/2}}{a}-\frac {b x^{3/2}}{a \left (b+a \cosh \left (c+d \sqrt {x}\right )\right )}\right )d\sqrt {x}\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (-\frac {6 b \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right )}{a d^4 \sqrt {b^2-a^2}}+\frac {6 b \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right )}{a d^4 \sqrt {b^2-a^2}}+\frac {6 b \sqrt {x} \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right )}{a d^3 \sqrt {b^2-a^2}}-\frac {6 b \sqrt {x} \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right )}{a d^3 \sqrt {b^2-a^2}}-\frac {3 b x \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right )}{a d^2 \sqrt {b^2-a^2}}+\frac {3 b x \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right )}{a d^2 \sqrt {b^2-a^2}}-\frac {b x^{3/2} \log \left (\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}+1\right )}{a d \sqrt {b^2-a^2}}+\frac {b x^{3/2} \log \left (\frac {a e^{c+d \sqrt {x}}}{\sqrt {b^2-a^2}+b}+1\right )}{a d \sqrt {b^2-a^2}}+\frac {x^2}{4 a}\right )\)

input

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

output

2*(x^2/(4*a) - (b*x^(3/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b 
^2])])/(a*Sqrt[-a^2 + b^2]*d) + (b*x^(3/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/( 
b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) - (3*b*x*PolyLog[2, -((a*E^ 
(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^2) + (3*b 
*x*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a 
^2 + b^2]*d^2) + (6*b*Sqrt[x]*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt 
[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^3) - (6*b*Sqrt[x]*PolyLog[3, -((a*E 
^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^3) - (6* 
b*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^ 
2 + b^2]*d^4) + (6*b*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b 
^2]))])/(a*Sqrt[-a^2 + b^2]*d^4))
 

3.1.44.3.1 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 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]
 

3.1.44.4 Maple [F]

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

input

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

output

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

3.1.44.5 Fricas [F]

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

input

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

output

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

3.1.44.6 Sympy [F]

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

input

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

output

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

3.1.44.7 Maxima [F(-2)]

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

input

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

3.1.44.8 Giac [F]

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

input

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

output

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

3.1.44.9 Mupad [F(-1)]

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

input

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

output

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