\(\int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^3} \, dx\) [79]

Optimal result

Integrand size = 16, antiderivative size = 234 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^3} \, dx=\frac {b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{2 d \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {a+b \text {sech}^{-1}(c x)}{2 e (d+e x)^2}+\frac {b \left (2 c^2 d^2-e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{2 d^2 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{2 d^2 e} \] Output:

1/2*b*e*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*(-c^2*x^2+1)^(1/2)/d/(c^2*d^2-e^2) 
/(e*x+d)-1/2*(a+b*arcsech(c*x))/e/(e*x+d)^2+1/2*b*(2*c^2*d^2-e^2)*(1/(c*x+ 
1))^(1/2)*(c*x+1)^(1/2)*arctan((c^2*d*x+e)/(c^2*d^2-e^2)^(1/2)/(-c^2*x^2+1 
)^(1/2))/d^2/(c^2*d^2-e^2)^(3/2)+1/2*b*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*arc 
tanh((-c^2*x^2+1)^(1/2))/d^2/e
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.40 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.46 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^3} \, dx=\frac {1}{2} \left (-\frac {a}{e (d+e x)^2}+\frac {b \sqrt {\frac {1-c x}{1+c x}} (e+c e x)}{d (c d-e) (c d+e) (d+e x)}-\frac {b \text {sech}^{-1}(c x)}{e (d+e x)^2}-\frac {b \log (x)}{d^2 e}+\frac {b \log \left (1+\sqrt {\frac {1-c x}{1+c x}}+c x \sqrt {\frac {1-c x}{1+c x}}\right )}{d^2 e}-\frac {i b \left (2 c^2 d^2-e^2\right ) \log \left (\frac {4 d^2 e \sqrt {c^2 d^2-e^2} \left (i e+i c^2 d x+\sqrt {c^2 d^2-e^2} \sqrt {\frac {1-c x}{1+c x}}+c \sqrt {c^2 d^2-e^2} x \sqrt {\frac {1-c x}{1+c x}}\right )}{b \left (2 c^2 d^2-e^2\right ) (d+e x)}\right )}{d^2 (c d-e) (c d+e) \sqrt {c^2 d^2-e^2}}\right ) \] Input:

Integrate[(a + b*ArcSech[c*x])/(d + e*x)^3,x]
 

Output:

(-(a/(e*(d + e*x)^2)) + (b*Sqrt[(1 - c*x)/(1 + c*x)]*(e + c*e*x))/(d*(c*d 
- e)*(c*d + e)*(d + e*x)) - (b*ArcSech[c*x])/(e*(d + e*x)^2) - (b*Log[x])/ 
(d^2*e) + (b*Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c 
*x)]])/(d^2*e) - (I*b*(2*c^2*d^2 - e^2)*Log[(4*d^2*e*Sqrt[c^2*d^2 - e^2]*( 
I*e + I*c^2*d*x + Sqrt[c^2*d^2 - e^2]*Sqrt[(1 - c*x)/(1 + c*x)] + c*Sqrt[c 
^2*d^2 - e^2]*x*Sqrt[(1 - c*x)/(1 + c*x)]))/(b*(2*c^2*d^2 - e^2)*(d + e*x) 
)])/(d^2*(c*d - e)*(c*d + e)*Sqrt[c^2*d^2 - e^2]))/2
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.04, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {6842, 617, 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.

\(\displaystyle \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^3} \, dx\)

\(\Big \downarrow \) 6842

\(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {1}{x (d+e x)^2 \sqrt {1-c^2 x^2}}dx}{2 e}-\frac {a+b \text {sech}^{-1}(c x)}{2 e (d+e x)^2}\)

\(\Big \downarrow \) 617

\(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \left (-\frac {e}{d^2 (d+e x) \sqrt {1-c^2 x^2}}-\frac {e}{d (d+e x)^2 \sqrt {1-c^2 x^2}}+\frac {1}{d^2 x \sqrt {1-c^2 x^2}}\right )dx}{2 e}-\frac {a+b \text {sech}^{-1}(c x)}{2 e (d+e x)^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {a+b \text {sech}^{-1}(c x)}{2 e (d+e x)^2}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (-\frac {c^2 e \arctan \left (\frac {c^2 d x+e}{\sqrt {1-c^2 x^2} \sqrt {c^2 d^2-e^2}}\right )}{\left (c^2 d^2-e^2\right )^{3/2}}-\frac {e \arctan \left (\frac {c^2 d x+e}{\sqrt {1-c^2 x^2} \sqrt {c^2 d^2-e^2}}\right )}{d^2 \sqrt {c^2 d^2-e^2}}-\frac {\text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{d^2}-\frac {e^2 \sqrt {1-c^2 x^2}}{d \left (c^2 d^2-e^2\right ) (d+e x)}\right )}{2 e}\)

Input:

Int[(a + b*ArcSech[c*x])/(d + e*x)^3,x]
 

Output:

-1/2*(a + b*ArcSech[c*x])/(e*(d + e*x)^2) - (b*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 
 + c*x]*(-((e^2*Sqrt[1 - c^2*x^2])/(d*(c^2*d^2 - e^2)*(d + e*x))) - (c^2*e 
*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(c^2*d^2 - 
 e^2)^(3/2) - (e*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^ 
2])])/(d^2*Sqrt[c^2*d^2 - e^2]) - ArcTanh[Sqrt[1 - c^2*x^2]]/d^2))/(2*e)
 

Defintions of rubi rules used

rule 617

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> Int[ExpandIntegrand[(a + b*x^2)^p, x^m*(c + d*x)^n, x], x] /; FreeQ[{ 
a, b, c, d, p}, x] && ILtQ[n, 0] && IntegerQ[m] && IntegerQ[2*p]
 

rule 2009

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

rule 6842

Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbo 
l] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcSech[c*x])/(e*(m + 1))), x] + Simp[ 
b*(Sqrt[1 + c*x]/(e*(m + 1)))*Sqrt[1/(1 + c*x)]   Int[(d + e*x)^(m + 1)/(x* 
Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[m, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(592\) vs. \(2(204)=408\).

Time = 1.97 (sec) , antiderivative size = 593, normalized size of antiderivative = 2.53

method	result	size
parts	\(-\frac {a}{2 \left (e x +d \right )^{2} e}+\frac {b \left (-\frac {c^{3} \operatorname {arcsech}\left (c x \right )}{2 \left (c e x +c d \right )^{2} e}+\frac {c^{2} \sqrt {-\frac {c x -1}{c x}}\, x \sqrt {\frac {c x +1}{c x}}\, \left (\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) c^{3} d^{3} \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}+\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) c^{3} d^{2} e \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, x -2 \ln \left (\frac {2 \sqrt {-c^{2} x^{2}+1}\, \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +2 d \,c^{2} x +2 e}{c e x +c d}\right ) c^{3} d^{3}-2 \ln \left (\frac {2 \sqrt {-c^{2} x^{2}+1}\, \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +2 d \,c^{2} x +2 e}{c e x +c d}\right ) c^{3} d^{2} e x -\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) c d \,e^{2} \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) e^{3} \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, c x +c d \,e^{2} \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-c^{2} x^{2}+1}+\ln \left (\frac {2 \sqrt {-c^{2} x^{2}+1}\, \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +2 d \,c^{2} x +2 e}{c e x +c d}\right ) c d \,e^{2}+\ln \left (\frac {2 \sqrt {-c^{2} x^{2}+1}\, \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +2 d \,c^{2} x +2 e}{c e x +c d}\right ) e^{3} c x \right )}{2 e \sqrt {-c^{2} x^{2}+1}\, \left (c d -e \right ) \left (c d +e \right ) d^{2} \left (c e x +c d \right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{c}\)	\(593\)
derivativedivides	\(\frac {-\frac {a \,c^{3}}{2 \left (c e x +c d \right )^{2} e}+b \,c^{3} \left (-\frac {\operatorname {arcsech}\left (c x \right )}{2 \left (c e x +c d \right )^{2} e}+\frac {\sqrt {-\frac {c x -1}{c x}}\, x \sqrt {\frac {c x +1}{c x}}\, \left (\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) c^{3} d^{3} \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}+\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) c^{3} d^{2} e \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, x -2 \ln \left (\frac {2 \sqrt {-c^{2} x^{2}+1}\, \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +2 d \,c^{2} x +2 e}{c e x +c d}\right ) c^{3} d^{3}-2 \ln \left (\frac {2 \sqrt {-c^{2} x^{2}+1}\, \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +2 d \,c^{2} x +2 e}{c e x +c d}\right ) c^{3} d^{2} e x -\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) c d \,e^{2} \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) e^{3} \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, c x +c d \,e^{2} \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-c^{2} x^{2}+1}+\ln \left (\frac {2 \sqrt {-c^{2} x^{2}+1}\, \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +2 d \,c^{2} x +2 e}{c e x +c d}\right ) c d \,e^{2}+\ln \left (\frac {2 \sqrt {-c^{2} x^{2}+1}\, \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +2 d \,c^{2} x +2 e}{c e x +c d}\right ) e^{3} c x \right )}{2 e c \sqrt {-c^{2} x^{2}+1}\, \left (c d -e \right ) \left (c d +e \right ) d^{2} \left (c e x +c d \right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{c}\)	\(600\)
default	\(\frac {-\frac {a \,c^{3}}{2 \left (c e x +c d \right )^{2} e}+b \,c^{3} \left (-\frac {\operatorname {arcsech}\left (c x \right )}{2 \left (c e x +c d \right )^{2} e}+\frac {\sqrt {-\frac {c x -1}{c x}}\, x \sqrt {\frac {c x +1}{c x}}\, \left (\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) c^{3} d^{3} \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}+\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) c^{3} d^{2} e \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, x -2 \ln \left (\frac {2 \sqrt {-c^{2} x^{2}+1}\, \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +2 d \,c^{2} x +2 e}{c e x +c d}\right ) c^{3} d^{3}-2 \ln \left (\frac {2 \sqrt {-c^{2} x^{2}+1}\, \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +2 d \,c^{2} x +2 e}{c e x +c d}\right ) c^{3} d^{2} e x -\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) c d \,e^{2} \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) e^{3} \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, c x +c d \,e^{2} \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-c^{2} x^{2}+1}+\ln \left (\frac {2 \sqrt {-c^{2} x^{2}+1}\, \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +2 d \,c^{2} x +2 e}{c e x +c d}\right ) c d \,e^{2}+\ln \left (\frac {2 \sqrt {-c^{2} x^{2}+1}\, \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +2 d \,c^{2} x +2 e}{c e x +c d}\right ) e^{3} c x \right )}{2 e c \sqrt {-c^{2} x^{2}+1}\, \left (c d -e \right ) \left (c d +e \right ) d^{2} \left (c e x +c d \right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{c}\)	\(600\)

Input:

int((a+b*arcsech(c*x))/(e*x+d)^3,x,method=_RETURNVERBOSE)
 

Output:

-1/2*a/(e*x+d)^2/e+b/c*(-1/2*c^3/(c*e*x+c*d)^2/e*arcsech(c*x)+1/2*c^2/e*(- 
(c*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)*(arctanh(1/(-c^2*x^2+1)^(1/2))*c^ 
3*d^3*(-(c^2*d^2-e^2)/e^2)^(1/2)+arctanh(1/(-c^2*x^2+1)^(1/2))*c^3*d^2*e*( 
-(c^2*d^2-e^2)/e^2)^(1/2)*x-2*ln(2*((-c^2*x^2+1)^(1/2)*(-(c^2*d^2-e^2)/e^2 
)^(1/2)*e+d*c^2*x+e)/(c*e*x+c*d))*c^3*d^3-2*ln(2*((-c^2*x^2+1)^(1/2)*(-(c^ 
2*d^2-e^2)/e^2)^(1/2)*e+d*c^2*x+e)/(c*e*x+c*d))*c^3*d^2*e*x-arctanh(1/(-c^ 
2*x^2+1)^(1/2))*c*d*e^2*(-(c^2*d^2-e^2)/e^2)^(1/2)-arctanh(1/(-c^2*x^2+1)^ 
(1/2))*e^3*(-(c^2*d^2-e^2)/e^2)^(1/2)*c*x+c*d*e^2*(-(c^2*d^2-e^2)/e^2)^(1/ 
2)*(-c^2*x^2+1)^(1/2)+ln(2*((-c^2*x^2+1)^(1/2)*(-(c^2*d^2-e^2)/e^2)^(1/2)* 
e+d*c^2*x+e)/(c*e*x+c*d))*c*d*e^2+ln(2*((-c^2*x^2+1)^(1/2)*(-(c^2*d^2-e^2) 
/e^2)^(1/2)*e+d*c^2*x+e)/(c*e*x+c*d))*e^3*c*x)/(-c^2*x^2+1)^(1/2)/(c*d-e)/ 
(c*d+e)/d^2/(c*e*x+c*d)/(-(c^2*d^2-e^2)/e^2)^(1/2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 596 vs. \(2 (156) = 312\).

Time = 0.22 (sec) , antiderivative size = 1212, normalized size of antiderivative = 5.18 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^3} \, dx=\text {Too large to display} \] Input:

integrate((a+b*arcsech(c*x))/(e*x+d)^3,x, algorithm="fricas")
 

Output:

[-1/2*(a*c^4*d^6 - (2*a + b)*c^2*d^4*e^2 + (a + b)*d^2*e^4 - (b*c^2*d^2*e^ 
4 - b*e^6)*x^2 + (2*b*c^2*d^4*e - b*d^2*e^3 + (2*b*c^2*d^2*e^3 - b*e^5)*x^ 
2 + 2*(2*b*c^2*d^3*e^2 - b*d*e^4)*x)*sqrt(-c^2*d^2 + e^2)*log((c^2*d*e*x - 
 (c^3*d^2 - c*e^2)*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + e^2 - sqrt(-c^2*d^2 
+ e^2)*(c^2*d*x + c*e*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + e))/(e*x + d)) - 
2*(b*c^2*d^3*e^3 - b*d*e^5)*x + (b*c^4*d^6 - 2*b*c^2*d^4*e^2 + b*d^2*e^4 + 
 (b*c^4*d^4*e^2 - 2*b*c^2*d^2*e^4 + b*e^6)*x^2 + 2*(b*c^4*d^5*e - 2*b*c^2* 
d^3*e^3 + b*d*e^5)*x)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - 1)/x) + (b 
*c^4*d^6 - 2*b*c^2*d^4*e^2 + b*d^2*e^4)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2* 
x^2)) + 1)/(c*x)) - ((b*c^3*d^3*e^3 - b*c*d*e^5)*x^2 + (b*c^3*d^4*e^2 - b* 
c*d^2*e^4)*x)*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)))/(c^4*d^8*e - 2*c^2*d^6*e^3 + 
 d^4*e^5 + (c^4*d^6*e^3 - 2*c^2*d^4*e^5 + d^2*e^7)*x^2 + 2*(c^4*d^7*e^2 - 
2*c^2*d^5*e^4 + d^3*e^6)*x), -1/2*(a*c^4*d^6 - (2*a + b)*c^2*d^4*e^2 + (a 
+ b)*d^2*e^4 - (b*c^2*d^2*e^4 - b*e^6)*x^2 - 2*(2*b*c^2*d^4*e - b*d^2*e^3 
+ (2*b*c^2*d^2*e^3 - b*e^5)*x^2 + 2*(2*b*c^2*d^3*e^2 - b*d*e^4)*x)*sqrt(c^ 
2*d^2 - e^2)*arctan(-(sqrt(c^2*d^2 - e^2)*c*d*x*sqrt(-(c^2*x^2 - 1)/(c^2*x 
^2)) - sqrt(c^2*d^2 - e^2)*(e*x + d))/((c^2*d^2 - e^2)*x)) - 2*(b*c^2*d^3* 
e^3 - b*d*e^5)*x + (b*c^4*d^6 - 2*b*c^2*d^4*e^2 + b*d^2*e^4 + (b*c^4*d^4*e 
^2 - 2*b*c^2*d^2*e^4 + b*e^6)*x^2 + 2*(b*c^4*d^5*e - 2*b*c^2*d^3*e^3 + b*d 
*e^5)*x)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - 1)/x) + (b*c^4*d^6 -...
 

Sympy [F]

\[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^3} \, dx=\int \frac {a + b \operatorname {asech}{\left (c x \right )}}{\left (d + e x\right )^{3}}\, dx \] Input:

integrate((a+b*asech(c*x))/(e*x+d)**3,x)
 

Output:

Integral((a + b*asech(c*x))/(d + e*x)**3, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^3} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((a+b*arcsech(c*x))/(e*x+d)^3,x, algorithm="maxima")
 

Output:

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un 
defined.
 

Giac [F]

\[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^3} \, dx=\int { \frac {b \operatorname {arsech}\left (c x\right ) + a}{{\left (e x + d\right )}^{3}} \,d x } \] Input:

integrate((a+b*arcsech(c*x))/(e*x+d)^3,x, algorithm="giac")
 

Output:

integrate((b*arcsech(c*x) + a)/(e*x + d)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^3} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )}{{\left (d+e\,x\right )}^3} \,d x \] Input:

int((a + b*acosh(1/(c*x)))/(d + e*x)^3,x)
 

Output:

int((a + b*acosh(1/(c*x)))/(d + e*x)^3, x)
 

Reduce [F]

\[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^3} \, dx=\frac {2 \left (\int \frac {\mathit {asech} \left (c x \right )}{e^{3} x^{3}+3 d \,e^{2} x^{2}+3 d^{2} e x +d^{3}}d x \right ) b \,d^{2} e +4 \left (\int \frac {\mathit {asech} \left (c x \right )}{e^{3} x^{3}+3 d \,e^{2} x^{2}+3 d^{2} e x +d^{3}}d x \right ) b d \,e^{2} x +2 \left (\int \frac {\mathit {asech} \left (c x \right )}{e^{3} x^{3}+3 d \,e^{2} x^{2}+3 d^{2} e x +d^{3}}d x \right ) b \,e^{3} x^{2}-a}{2 e \left (e^{2} x^{2}+2 d e x +d^{2}\right )} \] Input:

int((a+b*asech(c*x))/(e*x+d)^3,x)
 

Output:

(2*int(asech(c*x)/(d**3 + 3*d**2*e*x + 3*d*e**2*x**2 + e**3*x**3),x)*b*d** 
2*e + 4*int(asech(c*x)/(d**3 + 3*d**2*e*x + 3*d*e**2*x**2 + e**3*x**3),x)* 
b*d*e**2*x + 2*int(asech(c*x)/(d**3 + 3*d**2*e*x + 3*d*e**2*x**2 + e**3*x* 
*3),x)*b*e**3*x**2 - a)/(2*e*(d**2 + 2*d*e*x + e**2*x**2))