\(\int \frac {a+b \text {csch}^{-1}(c x)}{(d+e x)^2} \, dx\) [49]

Optimal result

Integrand size = 16, antiderivative size = 98 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{(d+e x)^2} \, dx=\frac {b \text {csch}^{-1}(c x)}{d e}-\frac {a+b \text {csch}^{-1}(c x)}{e (d+e x)}+\frac {b \text {arctanh}\left (\frac {c^2 d-\frac {e}{x}}{c \sqrt {c^2 d^2+e^2} \sqrt {1+\frac {1}{c^2 x^2}}}\right )}{d \sqrt {c^2 d^2+e^2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.37 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{(d+e x)^2} \, dx=-\frac {a}{e (d+e x)}-\frac {b \text {csch}^{-1}(c x)}{e (d+e x)}+\frac {b \text {arcsinh}\left (\frac {1}{c x}\right )}{d e}+\frac {b \log (d+e x)}{d \sqrt {c^2 d^2+e^2}}-\frac {b \log \left (e+c \left (-c d+\sqrt {c^2 d^2+e^2} \sqrt {1+\frac {1}{c^2 x^2}}\right ) x\right )}{d \sqrt {c^2 d^2+e^2}} \] Input:

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

Output:

-(a/(e*(d + e*x))) - (b*ArcCsch[c*x])/(e*(d + e*x)) + (b*ArcSinh[1/(c*x)]) 
/(d*e) + (b*Log[d + e*x])/(d*Sqrt[c^2*d^2 + e^2]) - (b*Log[e + c*(-(c*d) + 
 Sqrt[c^2*d^2 + e^2]*Sqrt[1 + 1/(c^2*x^2)])*x])/(d*Sqrt[c^2*d^2 + e^2])
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.08, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {6844, 1892, 1803, 605, 222, 488, 219}

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.

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

\(\Big \downarrow \) 6844

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

\(\Big \downarrow \) 1892

\(\displaystyle -\frac {b \int \frac {1}{\sqrt {1+\frac {1}{c^2 x^2}} \left (\frac {d}{x}+e\right ) x^3}dx}{c e}-\frac {a+b \text {csch}^{-1}(c x)}{e (d+e x)}\)

\(\Big \downarrow \) 1803

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

\(\Big \downarrow \) 605

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

\(\Big \downarrow \) 222

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

\(\displaystyle \frac {b \left (\frac {c \text {arcsinh}\left (\frac {1}{c x}\right )}{d}+\frac {c e \text {arctanh}\left (\frac {c \left (d-\frac {e}{c^2 x}\right )}{\sqrt {\frac {1}{c^2 x^2}+1} \sqrt {c^2 d^2+e^2}}\right )}{d \sqrt {c^2 d^2+e^2}}\right )}{c e}-\frac {a+b \text {csch}^{-1}(c x)}{e (d+e x)}\)

Input:

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

Output:

-((a + b*ArcCsch[c*x])/(e*(d + e*x))) + (b*((c*ArcSinh[1/(c*x)])/d + (c*e* 
ArcTanh[(c*(d - e/(c^2*x)))/(Sqrt[c^2*d^2 + e^2]*Sqrt[1 + 1/(c^2*x^2)])])/ 
(d*Sqrt[c^2*d^2 + e^2])))/(c*e)
 

Defintions of rubi rules used

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt 
[a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
 

rule 488

Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ 
Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ 
[{a, b, c, d}, x]
 

rule 605

Int[((x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_))/((c_) + (d_.)*(x_)), x_Symbol] 
:> Simp[1/d   Int[x^(m - 1)*(a + b*x^2)^p, x], x] - Simp[c/d   Int[x^(m - 1 
)*((a + b*x^2)^p/(c + d*x)), x], x] /; FreeQ[{a, b, c, d, p}, x] && IGtQ[m, 
 0] && LtQ[-1, p, 0]
 

rule 1803

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q 
_.), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(d + e*x 
)^q*(a + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x] && 
 EqQ[n2, 2*n] && IntegerQ[Simplify[(m + 1)/n]]
 

rule 1892

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^(mn_.))^(q_.)*((a_) + (c_.)*(x_)^(n2_.))^ 
(p_.), x_Symbol] :> Int[x^(m + mn*q)*(e + d/x^mn)^q*(a + c*x^n2)^p, x] /; F 
reeQ[{a, c, d, e, m, mn, p}, x] && EqQ[n2, -2*mn] && IntegerQ[q] && (PosQ[n 
2] ||  !IntegerQ[p])
 

rule 6844

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

Maple [A] (verified)

Time = 1.98 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.88

method	result	size
parts	\(-\frac {a}{\left (e x +d \right ) e}+\frac {b \left (-\frac {c^{2} \operatorname {arccsch}\left (c x \right )}{\left (c e x +c d \right ) e}+\frac {\sqrt {c^{2} x^{2}+1}\, \left (\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right ) \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}-\ln \left (\frac {2 \sqrt {c^{2} x^{2}+1}\, \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, e -2 d \,c^{2} x +2 e}{c e x +c d}\right )\right )}{e \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, x d \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}\right )}{c}\)	\(184\)
derivativedivides	\(\frac {-\frac {a \,c^{2}}{\left (c e x +c d \right ) e}+b \,c^{2} \left (-\frac {\operatorname {arccsch}\left (c x \right )}{\left (c e x +c d \right ) e}+\frac {\sqrt {c^{2} x^{2}+1}\, \left (\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right ) \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}-\ln \left (\frac {2 \sqrt {c^{2} x^{2}+1}\, \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, e -2 d \,c^{2} x +2 e}{c e x +c d}\right )\right )}{e \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c^{2} x d \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}\right )}{c}\)	\(194\)
default	\(\frac {-\frac {a \,c^{2}}{\left (c e x +c d \right ) e}+b \,c^{2} \left (-\frac {\operatorname {arccsch}\left (c x \right )}{\left (c e x +c d \right ) e}+\frac {\sqrt {c^{2} x^{2}+1}\, \left (\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right ) \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}-\ln \left (\frac {2 \sqrt {c^{2} x^{2}+1}\, \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, e -2 d \,c^{2} x +2 e}{c e x +c d}\right )\right )}{e \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c^{2} x d \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}\right )}{c}\)	\(194\)

Input:

int((a+b*arccsch(c*x))/(e*x+d)^2,x,method=_RETURNVERBOSE)
 

Output:

-a/(e*x+d)/e+b/c*(-c^2/(c*e*x+c*d)/e*arccsch(c*x)+1/e*(c^2*x^2+1)^(1/2)*(a 
rctanh(1/(c^2*x^2+1)^(1/2))*((c^2*d^2+e^2)/e^2)^(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^2*x^2+1)/c^2/ 
x^2)^(1/2)/x/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. 354 vs. \(2 (92) = 184\).

Time = 0.14 (sec) , antiderivative size = 354, normalized size of antiderivative = 3.61 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{(d+e x)^2} \, dx=-\frac {a c^{2} d^{3} + a d e^{2} - \sqrt {c^{2} d^{2} + e^{2}} {\left (b e^{2} x + b d e\right )} \log \left (-\frac {c^{3} d^{2} x - c d e + {\left (c^{3} d^{2} + c e^{2}\right )} x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + {\left (c^{2} d x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + c^{2} d x - e\right )} \sqrt {c^{2} d^{2} + e^{2}}}{e x + d}\right ) - {\left (b c^{2} d^{3} + b d e^{2} + {\left (b c^{2} d^{2} e + b e^{3}\right )} x\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) + {\left (b c^{2} d^{3} + b d e^{2} + {\left (b c^{2} d^{2} e + b e^{3}\right )} x\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) + {\left (b c^{2} d^{3} + b d e^{2}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )}{c^{2} d^{4} e + d^{2} e^{3} + {\left (c^{2} d^{3} e^{2} + d e^{4}\right )} x} \] Input:

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

Output:

-(a*c^2*d^3 + a*d*e^2 - sqrt(c^2*d^2 + e^2)*(b*e^2*x + b*d*e)*log(-(c^3*d^ 
2*x - c*d*e + (c^3*d^2 + c*e^2)*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + (c^2*d*x 
*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + c^2*d*x - e)*sqrt(c^2*d^2 + e^2))/(e*x + 
d)) - (b*c^2*d^3 + b*d*e^2 + (b*c^2*d^2*e + b*e^3)*x)*log(c*x*sqrt((c^2*x^ 
2 + 1)/(c^2*x^2)) - c*x + 1) + (b*c^2*d^3 + b*d*e^2 + (b*c^2*d^2*e + b*e^3 
)*x)*log(c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) - c*x - 1) + (b*c^2*d^3 + b*d*e 
^2)*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x)))/(c^2*d^4*e + d^2*e 
^3 + (c^2*d^3*e^2 + d*e^4)*x)
 

Sympy [F]

\[ \int \frac {a+b \text {csch}^{-1}(c x)}{(d+e x)^2} \, dx=\int \frac {a + b \operatorname {acsch}{\left (c x \right )}}{\left (d + e x\right )^{2}}\, dx \] Input:

integrate((a+b*acsch(c*x))/(e*x+d)**2,x)
 

Output:

Integral((a + b*acsch(c*x))/(d + e*x)**2, x)
 

Maxima [F]

\[ \int \frac {a+b \text {csch}^{-1}(c x)}{(d+e x)^2} \, dx=\int { \frac {b \operatorname {arcsch}\left (c x\right ) + a}{{\left (e x + d\right )}^{2}} \,d x } \] Input:

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

Output:

-1/2*(2*c^2*integrate(x/(c^2*e^2*x^3 + c^2*d*e*x^2 + e^2*x + d*e + (c^2*e^ 
2*x^3 + c^2*d*e*x^2 + e^2*x + d*e)*sqrt(c^2*x^2 + 1)), x) + I*c*(log(I*c*x 
 + 1) - log(-I*c*x + 1))/(c^2*d^2 + e^2) - 2*e*log(e*x + d)/(c^2*d^3 + d*e 
^2) - (2*c^2*d^3*log(c) + 2*d*e^2*log(c) - 2*(c^2*d^2*e + e^3)*x*log(x) + 
(c^2*d^2*e*x + c^2*d^3)*log(c^2*x^2 + 1) - 2*(c^2*d^3 + d*e^2)*log(sqrt(c^ 
2*x^2 + 1) + 1))/(c^2*d^4*e + d^2*e^3 + (c^2*d^3*e^2 + d*e^4)*x))*b - a/(e 
^2*x + d*e)
 

Giac [F]

\[ \int \frac {a+b \text {csch}^{-1}(c x)}{(d+e x)^2} \, dx=\int { \frac {b \operatorname {arcsch}\left (c x\right ) + a}{{\left (e x + d\right )}^{2}} \,d x } \] Input:

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

Output:

integrate((b*arccsch(c*x) + a)/(e*x + d)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \text {csch}^{-1}(c x)}{(d+e x)^2} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )}{{\left (d+e\,x\right )}^2} \,d x \] Input:

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

Output:

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

Reduce [F]

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

int((a+b*acsch(c*x))/(e*x+d)^2,x)
 

Output:

(int(acsch(c*x)/(d**2 + 2*d*e*x + e**2*x**2),x)*b*d**2 + int(acsch(c*x)/(d 
**2 + 2*d*e*x + e**2*x**2),x)*b*d*e*x + a*x)/(d*(d + e*x))