\(\int \frac {a+b \text {arcsinh}(c x)}{x^3 (d+c^2 d x^2)^{3/2}} \, dx\) [171]

Optimal result

Integrand size = 26, antiderivative size = 287 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (d+c^2 d x^2\right )^{3/2}} \, dx=-\frac {b c \sqrt {1+c^2 x^2}}{2 d x \sqrt {d+c^2 d x^2}}-\frac {3 c^2 (a+b \text {arcsinh}(c x))}{2 d \sqrt {d+c^2 d x^2}}-\frac {a+b \text {arcsinh}(c x)}{2 d x^2 \sqrt {d+c^2 d x^2}}+\frac {b c^2 \sqrt {1+c^2 x^2} \arctan (c x)}{d \sqrt {d+c^2 d x^2}}+\frac {3 c^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}+\frac {3 b c^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{2 d \sqrt {d+c^2 d x^2}}-\frac {3 b c^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{2 d \sqrt {d+c^2 d x^2}} \] Output:

-1/2*b*c*(c^2*x^2+1)^(1/2)/d/x/(c^2*d*x^2+d)^(1/2)-3/2*c^2*(a+b*arcsinh(c* 
x))/d/(c^2*d*x^2+d)^(1/2)-1/2*(a+b*arcsinh(c*x))/d/x^2/(c^2*d*x^2+d)^(1/2) 
+b*c^2*(c^2*x^2+1)^(1/2)*arctan(c*x)/d/(c^2*d*x^2+d)^(1/2)+3*c^2*(c^2*x^2+ 
1)^(1/2)*(a+b*arcsinh(c*x))*arctanh(c*x+(c^2*x^2+1)^(1/2))/d/(c^2*d*x^2+d) 
^(1/2)+3/2*b*c^2*(c^2*x^2+1)^(1/2)*polylog(2,-c*x-(c^2*x^2+1)^(1/2))/d/(c^ 
2*d*x^2+d)^(1/2)-3/2*b*c^2*(c^2*x^2+1)^(1/2)*polylog(2,c*x+(c^2*x^2+1)^(1/ 
2))/d/(c^2*d*x^2+d)^(1/2)
 

Mathematica [A] (verified)

Time = 5.62 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.29 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (d+c^2 d x^2\right )^{3/2}} \, dx=\frac {-\frac {4 a \left (1+3 c^2 x^2\right ) \sqrt {d+c^2 d x^2}}{x^2+c^2 x^4}-12 a c^2 \sqrt {d} \log (x)+12 a c^2 \sqrt {d} \log \left (d+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+\frac {b c^2 d \left (-8 \text {arcsinh}(c x)+16 \sqrt {1+c^2 x^2} \arctan \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )-2 \sqrt {1+c^2 x^2} \coth \left (\frac {1}{2} \text {arcsinh}(c x)\right )-\sqrt {1+c^2 x^2} \text {arcsinh}(c x) \text {csch}^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )-12 \sqrt {1+c^2 x^2} \text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )+12 \sqrt {1+c^2 x^2} \text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )-12 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )+12 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )-\sqrt {1+c^2 x^2} \text {arcsinh}(c x) \text {sech}^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )+2 \sqrt {1+c^2 x^2} \tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )}{\sqrt {d+c^2 d x^2}}}{8 d^2} \] Input:

Integrate[(a + b*ArcSinh[c*x])/(x^3*(d + c^2*d*x^2)^(3/2)),x]
 

Output:

((-4*a*(1 + 3*c^2*x^2)*Sqrt[d + c^2*d*x^2])/(x^2 + c^2*x^4) - 12*a*c^2*Sqr 
t[d]*Log[x] + 12*a*c^2*Sqrt[d]*Log[d + Sqrt[d]*Sqrt[d + c^2*d*x^2]] + (b*c 
^2*d*(-8*ArcSinh[c*x] + 16*Sqrt[1 + c^2*x^2]*ArcTan[Tanh[ArcSinh[c*x]/2]] 
- 2*Sqrt[1 + c^2*x^2]*Coth[ArcSinh[c*x]/2] - Sqrt[1 + c^2*x^2]*ArcSinh[c*x 
]*Csch[ArcSinh[c*x]/2]^2 - 12*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 - E^(-A 
rcSinh[c*x])] + 12*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x] 
)] - 12*Sqrt[1 + c^2*x^2]*PolyLog[2, -E^(-ArcSinh[c*x])] + 12*Sqrt[1 + c^2 
*x^2]*PolyLog[2, E^(-ArcSinh[c*x])] - Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Sech[ 
ArcSinh[c*x]/2]^2 + 2*Sqrt[1 + c^2*x^2]*Tanh[ArcSinh[c*x]/2]))/Sqrt[d + c^ 
2*d*x^2])/(8*d^2)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.84 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.82, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {6224, 264, 216, 6226, 216, 6231, 3042, 26, 4670, 2715, 2838}

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.

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

\(\Big \downarrow \) 6224

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

\(\Big \downarrow \) 264

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

\(\Big \downarrow \) 216

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

\(\Big \downarrow \) 6226

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

\(\Big \downarrow \) 216

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

\(\Big \downarrow \) 6231

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

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {3}{2} c^2 \left (\frac {\sqrt {c^2 x^2+1} \int i (a+b \text {arcsinh}(c x)) \csc (i \text {arcsinh}(c x))d\text {arcsinh}(c x)}{d \sqrt {c^2 d x^2+d}}+\frac {a+b \text {arcsinh}(c x)}{d \sqrt {c^2 d x^2+d}}-\frac {b \sqrt {c^2 x^2+1} \arctan (c x)}{d \sqrt {c^2 d x^2+d}}\right )-\frac {a+b \text {arcsinh}(c x)}{2 d x^2 \sqrt {c^2 d x^2+d}}+\frac {b c \sqrt {c^2 x^2+1} \left (-c \arctan (c x)-\frac {1}{x}\right )}{2 d \sqrt {c^2 d x^2+d}}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {3}{2} c^2 \left (\frac {i \sqrt {c^2 x^2+1} \int (a+b \text {arcsinh}(c x)) \csc (i \text {arcsinh}(c x))d\text {arcsinh}(c x)}{d \sqrt {c^2 d x^2+d}}+\frac {a+b \text {arcsinh}(c x)}{d \sqrt {c^2 d x^2+d}}-\frac {b \sqrt {c^2 x^2+1} \arctan (c x)}{d \sqrt {c^2 d x^2+d}}\right )-\frac {a+b \text {arcsinh}(c x)}{2 d x^2 \sqrt {c^2 d x^2+d}}+\frac {b c \sqrt {c^2 x^2+1} \left (-c \arctan (c x)-\frac {1}{x}\right )}{2 d \sqrt {c^2 d x^2+d}}\)

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 2715

\(\displaystyle -\frac {3}{2} c^2 \left (\frac {i \sqrt {c^2 x^2+1} \left (i b \int e^{-\text {arcsinh}(c x)} \log \left (1-e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-i b \int e^{-\text {arcsinh}(c x)} \log \left (1+e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}+2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )}{d \sqrt {c^2 d x^2+d}}+\frac {a+b \text {arcsinh}(c x)}{d \sqrt {c^2 d x^2+d}}-\frac {b \sqrt {c^2 x^2+1} \arctan (c x)}{d \sqrt {c^2 d x^2+d}}\right )-\frac {a+b \text {arcsinh}(c x)}{2 d x^2 \sqrt {c^2 d x^2+d}}+\frac {b c \sqrt {c^2 x^2+1} \left (-c \arctan (c x)-\frac {1}{x}\right )}{2 d \sqrt {c^2 d x^2+d}}\)

\(\Big \downarrow \) 2838

\(\displaystyle -\frac {3}{2} c^2 \left (\frac {i \sqrt {c^2 x^2+1} \left (2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )\right )}{d \sqrt {c^2 d x^2+d}}+\frac {a+b \text {arcsinh}(c x)}{d \sqrt {c^2 d x^2+d}}-\frac {b \sqrt {c^2 x^2+1} \arctan (c x)}{d \sqrt {c^2 d x^2+d}}\right )-\frac {a+b \text {arcsinh}(c x)}{2 d x^2 \sqrt {c^2 d x^2+d}}+\frac {b c \sqrt {c^2 x^2+1} \left (-c \arctan (c x)-\frac {1}{x}\right )}{2 d \sqrt {c^2 d x^2+d}}\)

Input:

Int[(a + b*ArcSinh[c*x])/(x^3*(d + c^2*d*x^2)^(3/2)),x]
 

Output:

-1/2*(a + b*ArcSinh[c*x])/(d*x^2*Sqrt[d + c^2*d*x^2]) + (b*c*Sqrt[1 + c^2* 
x^2]*(-x^(-1) - c*ArcTan[c*x]))/(2*d*Sqrt[d + c^2*d*x^2]) - (3*c^2*((a + b 
*ArcSinh[c*x])/(d*Sqrt[d + c^2*d*x^2]) - (b*Sqrt[1 + c^2*x^2]*ArcTan[c*x]) 
/(d*Sqrt[d + c^2*d*x^2]) + (I*Sqrt[1 + c^2*x^2]*((2*I)*(a + b*ArcSinh[c*x] 
)*ArcTanh[E^ArcSinh[c*x]] + I*b*PolyLog[2, -E^ArcSinh[c*x]] - I*b*PolyLog[ 
2, E^ArcSinh[c*x]]))/(d*Sqrt[d + c^2*d*x^2])))/2
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 216

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

rule 264

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( 
m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c 
^2*(m + 1)))   Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p 
}, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4670

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x 
_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] 
 + (-Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x 
)], x], x] + Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e 
+ f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
 

rule 6224

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ 
.)*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + 
b*ArcSinh[c*x])^n/(d*f*(m + 1))), x] + (-Simp[c^2*((m + 2*p + 3)/(f^2*(m + 
1)))   Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] - Sim 
p[b*c*(n/(f*(m + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]   Int[(f*x)^(m + 
1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{ 
a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && ILtQ[m, -1]
 

rule 6226

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ 
.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(f*x)^(m + 1))*(d + e*x^2)^(p + 1)*((a 
 + b*ArcSinh[c*x])^n/(2*d*f*(p + 1))), x] + (Simp[(m + 2*p + 3)/(2*d*(p + 1 
))   Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n, x], x] + Simp[ 
b*c*(n/(2*f*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]   Int[(f*x)^(m + 
1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{ 
a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[p, -1] &&  !G 
tQ[m, 1] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 1])
 

rule 6231

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.) 
*(x_)^2], x_Symbol] :> Simp[(1/c^(m + 1))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e 
*x^2]]   Subst[Int[(a + b*x)^n*Sinh[x]^m, x], x, ArcSinh[c*x]], x] /; FreeQ 
[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0] && IntegerQ[m]
 

Maple [A] (verified)

Time = 1.02 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.19

method	result	size
default	\(a \left (-\frac {1}{2 d \,x^{2} \sqrt {c^{2} d \,x^{2}+d}}-\frac {3 c^{2} \left (\frac {1}{d \sqrt {c^{2} d \,x^{2}+d}}-\frac {\ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c^{2} d \,x^{2}+d}}{x}\right )}{d^{\frac {3}{2}}}\right )}{2}\right )+b \left (-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (3 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +\operatorname {arcsinh}\left (x c \right )\right )}{2 d^{2} \left (c^{2} x^{2}+1\right ) x^{2}}+\frac {2 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arctan \left (x c +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{\sqrt {c^{2} x^{2}+1}\, d^{2}}+\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {dilog}\left (1+x c +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}\, d^{2}}+\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}\, d^{2}}+\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {dilog}\left (x c +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}\, d^{2}}\right )\)	\(342\)
parts	\(a \left (-\frac {1}{2 d \,x^{2} \sqrt {c^{2} d \,x^{2}+d}}-\frac {3 c^{2} \left (\frac {1}{d \sqrt {c^{2} d \,x^{2}+d}}-\frac {\ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c^{2} d \,x^{2}+d}}{x}\right )}{d^{\frac {3}{2}}}\right )}{2}\right )+b \left (-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (3 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +\operatorname {arcsinh}\left (x c \right )\right )}{2 d^{2} \left (c^{2} x^{2}+1\right ) x^{2}}+\frac {2 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arctan \left (x c +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{\sqrt {c^{2} x^{2}+1}\, d^{2}}+\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {dilog}\left (1+x c +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}\, d^{2}}+\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}\, d^{2}}+\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {dilog}\left (x c +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}\, d^{2}}\right )\)	\(342\)

Input:

int((a+b*arcsinh(x*c))/x^3/(c^2*d*x^2+d)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

a*(-1/2/d/x^2/(c^2*d*x^2+d)^(1/2)-3/2*c^2*(1/d/(c^2*d*x^2+d)^(1/2)-1/d^(3/ 
2)*ln((2*d+2*d^(1/2)*(c^2*d*x^2+d)^(1/2))/x)))+b*(-1/2*(d*(c^2*x^2+1))^(1/ 
2)*(3*arcsinh(x*c)*c^2*x^2+(c^2*x^2+1)^(1/2)*x*c+arcsinh(x*c))/d^2/(c^2*x^ 
2+1)/x^2+2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/d^2*arctan(x*c+(c^2*x^2 
+1)^(1/2))*c^2+3/2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/d^2*dilog(1+x*c 
+(c^2*x^2+1)^(1/2))*c^2+3/2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/d^2*ar 
csinh(x*c)*ln(1+x*c+(c^2*x^2+1)^(1/2))*c^2+3/2*(d*(c^2*x^2+1))^(1/2)/(c^2* 
x^2+1)^(1/2)/d^2*dilog(x*c+(c^2*x^2+1)^(1/2))*c^2)
 

Fricas [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (d+c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{3}} \,d x } \] Input:

integrate((a+b*arcsinh(c*x))/x^3/(c^2*d*x^2+d)^(3/2),x, algorithm="fricas" 
)
 

Output:

integral(sqrt(c^2*d*x^2 + d)*(b*arcsinh(c*x) + a)/(c^4*d^2*x^7 + 2*c^2*d^2 
*x^5 + d^2*x^3), x)
 

Sympy [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (d+c^2 d x^2\right )^{3/2}} \, dx=\int \frac {a + b \operatorname {asinh}{\left (c x \right )}}{x^{3} \left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:

integrate((a+b*asinh(c*x))/x**3/(c**2*d*x**2+d)**(3/2),x)
 

Output:

Integral((a + b*asinh(c*x))/(x**3*(d*(c**2*x**2 + 1))**(3/2)), x)
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (d+c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{3}} \,d x } \] Input:

integrate((a+b*arcsinh(c*x))/x^3/(c^2*d*x^2+d)^(3/2),x, algorithm="maxima" 
)
 

Output:

1/2*(3*c^2*arcsinh(1/(c*abs(x)))/d^(3/2) - 3*c^2/(sqrt(c^2*d*x^2 + d)*d) - 
 1/(sqrt(c^2*d*x^2 + d)*d*x^2))*a + b*integrate(log(c*x + sqrt(c^2*x^2 + 1 
))/((c^2*d*x^2 + d)^(3/2)*x^3), x)
 

Giac [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (d+c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{3}} \,d x } \] Input:

integrate((a+b*arcsinh(c*x))/x^3/(c^2*d*x^2+d)^(3/2),x, algorithm="giac")
 

Output:

integrate((b*arcsinh(c*x) + a)/((c^2*d*x^2 + d)^(3/2)*x^3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (d+c^2 d x^2\right )^{3/2}} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{x^3\,{\left (d\,c^2\,x^2+d\right )}^{3/2}} \,d x \] Input:

int((a + b*asinh(c*x))/(x^3*(d + c^2*d*x^2)^(3/2)),x)
 

Output:

int((a + b*asinh(c*x))/(x^3*(d + c^2*d*x^2)^(3/2)), x)
 

Reduce [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (d+c^2 d x^2\right )^{3/2}} \, dx=\frac {-3 \sqrt {c^{2} x^{2}+1}\, a \,c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, a +2 \left (\int \frac {\mathit {asinh} \left (c x \right )}{\sqrt {c^{2} x^{2}+1}\, c^{2} x^{5}+\sqrt {c^{2} x^{2}+1}\, x^{3}}d x \right ) b \,c^{2} x^{4}+2 \left (\int \frac {\mathit {asinh} \left (c x \right )}{\sqrt {c^{2} x^{2}+1}\, c^{2} x^{5}+\sqrt {c^{2} x^{2}+1}\, x^{3}}d x \right ) b \,x^{2}-3 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x -1\right ) a \,c^{4} x^{4}-3 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x -1\right ) a \,c^{2} x^{2}+3 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x +1\right ) a \,c^{4} x^{4}+3 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x +1\right ) a \,c^{2} x^{2}}{2 \sqrt {d}\, d \,x^{2} \left (c^{2} x^{2}+1\right )} \] Input:

int((a+b*asinh(c*x))/x^3/(c^2*d*x^2+d)^(3/2),x)
 

Output:

( - 3*sqrt(c**2*x**2 + 1)*a*c**2*x**2 - sqrt(c**2*x**2 + 1)*a + 2*int(asin 
h(c*x)/(sqrt(c**2*x**2 + 1)*c**2*x**5 + sqrt(c**2*x**2 + 1)*x**3),x)*b*c** 
2*x**4 + 2*int(asinh(c*x)/(sqrt(c**2*x**2 + 1)*c**2*x**5 + sqrt(c**2*x**2 
+ 1)*x**3),x)*b*x**2 - 3*log(sqrt(c**2*x**2 + 1) + c*x - 1)*a*c**4*x**4 - 
3*log(sqrt(c**2*x**2 + 1) + c*x - 1)*a*c**2*x**2 + 3*log(sqrt(c**2*x**2 + 
1) + c*x + 1)*a*c**4*x**4 + 3*log(sqrt(c**2*x**2 + 1) + c*x + 1)*a*c**2*x* 
*2)/(2*sqrt(d)*d*x**2*(c**2*x**2 + 1))