\(\int (d+e x) (a+b \text {csch}^{-1}(c x)) \, dx\) [46]

Optimal result

Integrand size = 14, antiderivative size = 81 \[ \int (d+e x) \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {b e \sqrt {1+\frac {1}{c^2 x^2}} x}{2 c}-\frac {b d^2 \text {csch}^{-1}(c x)}{2 e}+\frac {(d+e x)^2 \left (a+b \text {csch}^{-1}(c x)\right )}{2 e}+\frac {b d \text {arctanh}\left (\sqrt {1+\frac {1}{c^2 x^2}}\right )}{c} \] Output:

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

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.47 \[ \int (d+e x) \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=a d x+\frac {1}{2} a e x^2+\frac {b e x \sqrt {\frac {1+c^2 x^2}{c^2 x^2}}}{2 c}+b d x \text {csch}^{-1}(c x)+\frac {1}{2} b e x^2 \text {csch}^{-1}(c x)+\frac {2 b d \sqrt {1+\frac {1}{c^2 x^2}} x \text {arctanh}\left (\frac {-1+\sqrt {1+c^2 x^2}}{c x}\right )}{\sqrt {1+c^2 x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.05, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {6844, 1892, 1730, 540, 25, 27, 538, 222, 243, 73, 221}

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.

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

\(\Big \downarrow \) 6844

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

\(\Big \downarrow \) 1892

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

\(\Big \downarrow \) 1730

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

\(\Big \downarrow \) 540

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 538

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

\(\Big \downarrow \) 222

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

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

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 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 538

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

rule 540

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol 
] :> With[{Qx = PolynomialQuotient[(c + d*x)^n, x, x], R = PolynomialRemain 
der[(c + d*x)^n, x, x]}, Simp[R*x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))) 
, x] + Simp[1/(a*(m + 1))   Int[x^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*(m + 
1)*Qx - b*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, p}, x] && IG 
tQ[n, 1] && ILtQ[m, -1] && GtQ[p, -1] && IntegerQ[2*p]
 

rule 1730

Int[((d_) + (e_.)*(x_)^(n_))^(q_.)*((a_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol 
] :> -Subst[Int[(d + e/x^n)^q*((a + c/x^(2*n))^p/x^2), x], x, 1/x] /; FreeQ 
[{a, c, d, e, p, q}, x] && EqQ[n2, 2*n] && ILtQ[n, 0]
 

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 = 0.21 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.21

method	result	size
parts	\(a \left (\frac {1}{2} x^{2} e +d x \right )+\frac {b \left (\frac {c \,\operatorname {arccsch}\left (c x \right ) x^{2} e}{2}+\operatorname {arccsch}\left (c x \right ) d c x +\frac {\sqrt {c^{2} x^{2}+1}\, \left (e \sqrt {c^{2} x^{2}+1}+2 d c \,\operatorname {arcsinh}\left (c x \right )\right )}{2 c^{2} \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, x}\right )}{c}\)	\(98\)
derivativedivides	\(\frac {\frac {a \left (d \,c^{2} x +\frac {1}{2} e \,c^{2} x^{2}\right )}{c}+\frac {b \left (\operatorname {arccsch}\left (c x \right ) d \,c^{2} x +\frac {\operatorname {arccsch}\left (c x \right ) e \,c^{2} x^{2}}{2}+\frac {\sqrt {c^{2} x^{2}+1}\, \left (e \sqrt {c^{2} x^{2}+1}+2 d c \,\operatorname {arcsinh}\left (c x \right )\right )}{2 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c x}\right )}{c}}{c}\)	\(115\)
default	\(\frac {\frac {a \left (d \,c^{2} x +\frac {1}{2} e \,c^{2} x^{2}\right )}{c}+\frac {b \left (\operatorname {arccsch}\left (c x \right ) d \,c^{2} x +\frac {\operatorname {arccsch}\left (c x \right ) e \,c^{2} x^{2}}{2}+\frac {\sqrt {c^{2} x^{2}+1}\, \left (e \sqrt {c^{2} x^{2}+1}+2 d c \,\operatorname {arcsinh}\left (c x \right )\right )}{2 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c x}\right )}{c}}{c}\)	\(115\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 207 vs. \(2 (71) = 142\).

Time = 0.12 (sec) , antiderivative size = 207, normalized size of antiderivative = 2.56 \[ \int (d+e x) \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {a c e x^{2} + 2 \, a c d x + b e x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - 2 \, b d \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x\right ) + {\left (2 \, b c d + b c e\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) - {\left (2 \, b c d + b c e\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) + {\left (b c e x^{2} + 2 \, b c d x - 2 \, b c d - b c e\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )}{2 \, c} \] Input:

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

Output:

1/2*(a*c*e*x^2 + 2*a*c*d*x + b*e*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) - 2*b*d*l 
og(c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) - c*x) + (2*b*c*d + b*c*e)*log(c*x*sq 
rt((c^2*x^2 + 1)/(c^2*x^2)) - c*x + 1) - (2*b*c*d + b*c*e)*log(c*x*sqrt((c 
^2*x^2 + 1)/(c^2*x^2)) - c*x - 1) + (b*c*e*x^2 + 2*b*c*d*x - 2*b*c*d - b*c 
*e)*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x)))/c
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.07 \[ \int (d+e x) \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {1}{2} \, a e x^{2} + \frac {1}{2} \, {\left (x^{2} \operatorname {arcsch}\left (c x\right ) + \frac {x \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c}\right )} b e + a d x + \frac {{\left (2 \, c x \operatorname {arcsch}\left (c x\right ) + \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + 1\right ) - \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} - 1\right )\right )} b d}{2 \, c} \] Input:

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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