\(\int \frac {a+b \text {arcsinh}(c x)}{x^2 (d+c^2 d x^2)^3} \, dx\) [52]

Optimal result

Integrand size = 24, antiderivative size = 211 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (d+c^2 d x^2\right )^3} \, dx=-\frac {b c}{12 d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac {7 b c}{8 d^3 \sqrt {1+c^2 x^2}}-\frac {a+b \text {arcsinh}(c x)}{d^3 x}-\frac {c^2 x (a+b \text {arcsinh}(c x))}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac {7 c^2 x (a+b \text {arcsinh}(c x))}{8 d^3 \left (1+c^2 x^2\right )}-\frac {15 c (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{4 d^3}-\frac {b c \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )}{d^3}+\frac {15 i b c \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{8 d^3}-\frac {15 i b c \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{8 d^3} \] Output:

-1/12*b*c/d^3/(c^2*x^2+1)^(3/2)-7/8*b*c/d^3/(c^2*x^2+1)^(1/2)-(a+b*arcsinh 
(c*x))/d^3/x-1/4*c^2*x*(a+b*arcsinh(c*x))/d^3/(c^2*x^2+1)^2-7/8*c^2*x*(a+b 
*arcsinh(c*x))/d^3/(c^2*x^2+1)-15/4*c*(a+b*arcsinh(c*x))*arctan(c*x+(c^2*x 
^2+1)^(1/2))/d^3-b*c*arctanh((c^2*x^2+1)^(1/2))/d^3+15/8*I*b*c*polylog(2,- 
I*(c*x+(c^2*x^2+1)^(1/2)))/d^3-15/8*I*b*c*polylog(2,I*(c*x+(c^2*x^2+1)^(1/ 
2)))/d^3
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.70 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.41 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (d+c^2 d x^2\right )^3} \, dx=-\frac {\frac {45 (a+b \text {arcsinh}(c x))}{x}-\frac {6 (a+b \text {arcsinh}(c x))}{x \left (1+c^2 x^2\right )^2}-\frac {15 (a+b \text {arcsinh}(c x))}{x+c^2 x^3}+45 a c \arctan (c x)+45 b c \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )+\frac {2 b c \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},1+c^2 x^2\right )}{\left (1+c^2 x^2\right )^{3/2}}+\frac {15 b c \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},1+c^2 x^2\right )}{\sqrt {1+c^2 x^2}}+45 b \sqrt {-c^2} \text {arcsinh}(c x) \log \left (1+\frac {c e^{\text {arcsinh}(c x)}}{\sqrt {-c^2}}\right )-45 b \sqrt {-c^2} \text {arcsinh}(c x) \log \left (1+\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )-45 b \sqrt {-c^2} \operatorname {PolyLog}\left (2,\frac {c e^{\text {arcsinh}(c x)}}{\sqrt {-c^2}}\right )+45 b \sqrt {-c^2} \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )}{24 d^3} \] Input:

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

Output:

-1/24*((45*(a + b*ArcSinh[c*x]))/x - (6*(a + b*ArcSinh[c*x]))/(x*(1 + c^2* 
x^2)^2) - (15*(a + b*ArcSinh[c*x]))/(x + c^2*x^3) + 45*a*c*ArcTan[c*x] + 4 
5*b*c*ArcTanh[Sqrt[1 + c^2*x^2]] + (2*b*c*Hypergeometric2F1[-3/2, 1, -1/2, 
 1 + c^2*x^2])/(1 + c^2*x^2)^(3/2) + (15*b*c*Hypergeometric2F1[-1/2, 1, 1/ 
2, 1 + c^2*x^2])/Sqrt[1 + c^2*x^2] + 45*b*Sqrt[-c^2]*ArcSinh[c*x]*Log[1 + 
(c*E^ArcSinh[c*x])/Sqrt[-c^2]] - 45*b*Sqrt[-c^2]*ArcSinh[c*x]*Log[1 + (Sqr 
t[-c^2]*E^ArcSinh[c*x])/c] - 45*b*Sqrt[-c^2]*PolyLog[2, (c*E^ArcSinh[c*x]) 
/Sqrt[-c^2]] + 45*b*Sqrt[-c^2]*PolyLog[2, (Sqrt[-c^2]*E^ArcSinh[c*x])/c])/ 
d^3
 

Rubi [A] (verified)

Time = 1.27 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.18, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6224, 27, 243, 61, 61, 73, 221, 6203, 241, 6203, 241, 6204, 3042, 4668, 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.

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

\(\Big \downarrow \) 6224

\(\displaystyle -5 c^2 \int \frac {a+b \text {arcsinh}(c x)}{d^3 \left (c^2 x^2+1\right )^3}dx+\frac {b c \int \frac {1}{x \left (c^2 x^2+1\right )^{5/2}}dx}{d^3}-\frac {a+b \text {arcsinh}(c x)}{d^3 x \left (c^2 x^2+1\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {5 c^2 \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^3}dx}{d^3}+\frac {b c \int \frac {1}{x \left (c^2 x^2+1\right )^{5/2}}dx}{d^3}-\frac {a+b \text {arcsinh}(c x)}{d^3 x \left (c^2 x^2+1\right )^2}\)

\(\Big \downarrow \) 243

\(\displaystyle -\frac {5 c^2 \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^3}dx}{d^3}+\frac {b c \int \frac {1}{x^2 \left (c^2 x^2+1\right )^{5/2}}dx^2}{2 d^3}-\frac {a+b \text {arcsinh}(c x)}{d^3 x \left (c^2 x^2+1\right )^2}\)

\(\Big \downarrow \) 61

\(\displaystyle -\frac {5 c^2 \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^3}dx}{d^3}+\frac {b c \left (\int \frac {1}{x^2 \left (c^2 x^2+1\right )^{3/2}}dx^2+\frac {2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )}{2 d^3}-\frac {a+b \text {arcsinh}(c x)}{d^3 x \left (c^2 x^2+1\right )^2}\)

\(\Big \downarrow \) 61

\(\displaystyle -\frac {5 c^2 \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^3}dx}{d^3}+\frac {b c \left (\int \frac {1}{x^2 \sqrt {c^2 x^2+1}}dx^2+\frac {2}{\sqrt {c^2 x^2+1}}+\frac {2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )}{2 d^3}-\frac {a+b \text {arcsinh}(c x)}{d^3 x \left (c^2 x^2+1\right )^2}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {5 c^2 \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^3}dx}{d^3}+\frac {b c \left (\frac {2 \int \frac {1}{\frac {x^4}{c^2}-\frac {1}{c^2}}d\sqrt {c^2 x^2+1}}{c^2}+\frac {2}{\sqrt {c^2 x^2+1}}+\frac {2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )}{2 d^3}-\frac {a+b \text {arcsinh}(c x)}{d^3 x \left (c^2 x^2+1\right )^2}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {5 c^2 \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^3}dx}{d^3}-\frac {a+b \text {arcsinh}(c x)}{d^3 x \left (c^2 x^2+1\right )^2}+\frac {b c \left (-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )+\frac {2}{\sqrt {c^2 x^2+1}}+\frac {2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )}{2 d^3}\)

\(\Big \downarrow \) 6203

\(\displaystyle -\frac {5 c^2 \left (\frac {3}{4} \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^2}dx-\frac {1}{4} b c \int \frac {x}{\left (c^2 x^2+1\right )^{5/2}}dx+\frac {x (a+b \text {arcsinh}(c x))}{4 \left (c^2 x^2+1\right )^2}\right )}{d^3}-\frac {a+b \text {arcsinh}(c x)}{d^3 x \left (c^2 x^2+1\right )^2}+\frac {b c \left (-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )+\frac {2}{\sqrt {c^2 x^2+1}}+\frac {2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )}{2 d^3}\)

\(\Big \downarrow \) 241

\(\displaystyle -\frac {5 c^2 \left (\frac {3}{4} \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^2}dx+\frac {x (a+b \text {arcsinh}(c x))}{4 \left (c^2 x^2+1\right )^2}+\frac {b}{12 c \left (c^2 x^2+1\right )^{3/2}}\right )}{d^3}-\frac {a+b \text {arcsinh}(c x)}{d^3 x \left (c^2 x^2+1\right )^2}+\frac {b c \left (-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )+\frac {2}{\sqrt {c^2 x^2+1}}+\frac {2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )}{2 d^3}\)

\(\Big \downarrow \) 6203

\(\displaystyle -\frac {5 c^2 \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \text {arcsinh}(c x)}{c^2 x^2+1}dx-\frac {1}{2} b c \int \frac {x}{\left (c^2 x^2+1\right )^{3/2}}dx+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}\right )+\frac {x (a+b \text {arcsinh}(c x))}{4 \left (c^2 x^2+1\right )^2}+\frac {b}{12 c \left (c^2 x^2+1\right )^{3/2}}\right )}{d^3}-\frac {a+b \text {arcsinh}(c x)}{d^3 x \left (c^2 x^2+1\right )^2}+\frac {b c \left (-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )+\frac {2}{\sqrt {c^2 x^2+1}}+\frac {2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )}{2 d^3}\)

\(\Big \downarrow \) 241

\(\displaystyle -\frac {5 c^2 \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \text {arcsinh}(c x)}{c^2 x^2+1}dx+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))}{4 \left (c^2 x^2+1\right )^2}+\frac {b}{12 c \left (c^2 x^2+1\right )^{3/2}}\right )}{d^3}-\frac {a+b \text {arcsinh}(c x)}{d^3 x \left (c^2 x^2+1\right )^2}+\frac {b c \left (-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )+\frac {2}{\sqrt {c^2 x^2+1}}+\frac {2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )}{2 d^3}\)

\(\Big \downarrow \) 6204

\(\displaystyle -\frac {5 c^2 \left (\frac {3}{4} \left (\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}d\text {arcsinh}(c x)}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))}{4 \left (c^2 x^2+1\right )^2}+\frac {b}{12 c \left (c^2 x^2+1\right )^{3/2}}\right )}{d^3}-\frac {a+b \text {arcsinh}(c x)}{d^3 x \left (c^2 x^2+1\right )^2}+\frac {b c \left (-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )+\frac {2}{\sqrt {c^2 x^2+1}}+\frac {2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )}{2 d^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {5 c^2 \left (\frac {3}{4} \left (\frac {\int (a+b \text {arcsinh}(c x)) \csc \left (i \text {arcsinh}(c x)+\frac {\pi }{2}\right )d\text {arcsinh}(c x)}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))}{4 \left (c^2 x^2+1\right )^2}+\frac {b}{12 c \left (c^2 x^2+1\right )^{3/2}}\right )}{d^3}-\frac {a+b \text {arcsinh}(c x)}{d^3 x \left (c^2 x^2+1\right )^2}+\frac {b c \left (-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )+\frac {2}{\sqrt {c^2 x^2+1}}+\frac {2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )}{2 d^3}\)

\(\Big \downarrow \) 4668

\(\displaystyle -\frac {5 c^2 \left (\frac {3}{4} \left (\frac {-i b \int \log \left (1-i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+i b \int \log \left (1+i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))}{4 \left (c^2 x^2+1\right )^2}+\frac {b}{12 c \left (c^2 x^2+1\right )^{3/2}}\right )}{d^3}-\frac {a+b \text {arcsinh}(c x)}{d^3 x \left (c^2 x^2+1\right )^2}+\frac {b c \left (-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )+\frac {2}{\sqrt {c^2 x^2+1}}+\frac {2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )}{2 d^3}\)

\(\Big \downarrow \) 2715

\(\displaystyle -\frac {5 c^2 \left (\frac {3}{4} \left (\frac {-i b \int e^{-\text {arcsinh}(c x)} \log \left (1-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}+i b \int e^{-\text {arcsinh}(c x)} \log \left (1+i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))}{4 \left (c^2 x^2+1\right )^2}+\frac {b}{12 c \left (c^2 x^2+1\right )^{3/2}}\right )}{d^3}-\frac {a+b \text {arcsinh}(c x)}{d^3 x \left (c^2 x^2+1\right )^2}+\frac {b c \left (-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )+\frac {2}{\sqrt {c^2 x^2+1}}+\frac {2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )}{2 d^3}\)

\(\Big \downarrow \) 2838

\(\displaystyle -\frac {5 c^2 \left (\frac {3}{4} \left (\frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))}{4 \left (c^2 x^2+1\right )^2}+\frac {b}{12 c \left (c^2 x^2+1\right )^{3/2}}\right )}{d^3}-\frac {a+b \text {arcsinh}(c x)}{d^3 x \left (c^2 x^2+1\right )^2}+\frac {b c \left (-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )+\frac {2}{\sqrt {c^2 x^2+1}}+\frac {2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )}{2 d^3}\)

Input:

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

Output:

-((a + b*ArcSinh[c*x])/(d^3*x*(1 + c^2*x^2)^2)) + (b*c*(2/(3*(1 + c^2*x^2) 
^(3/2)) + 2/Sqrt[1 + c^2*x^2] - 2*ArcTanh[Sqrt[1 + c^2*x^2]]))/(2*d^3) - ( 
5*c^2*(b/(12*c*(1 + c^2*x^2)^(3/2)) + (x*(a + b*ArcSinh[c*x]))/(4*(1 + c^2 
*x^2)^2) + (3*(b/(2*c*Sqrt[1 + c^2*x^2]) + (x*(a + b*ArcSinh[c*x]))/(2*(1 
+ c^2*x^2)) + (2*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]] - I*b*PolyLog 
[2, (-I)*E^ArcSinh[c*x]] + I*b*PolyLog[2, I*E^ArcSinh[c*x]])/(2*c)))/4))/d 
^3
 

Defintions of rubi rules used

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 61

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( 
m + n + 2)/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], 
x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0 
] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d 
, m, n, 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 241

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

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 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 4668

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

rule 6203

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

rule 6204

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symb 
ol] :> Simp[1/(c*d)   Subst[Int[(a + b*x)^n*Sech[x], x], x, ArcSinh[c*x]], 
x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 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]
 

Maple [A] (verified)

Time = 1.24 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.38

method	result	size
derivativedivides	\(c \left (\frac {a \left (-\frac {\frac {7}{8} x^{3} c^{3}+\frac {9}{8} x c}{\left (c^{2} x^{2}+1\right )^{2}}-\frac {15 \arctan \left (x c \right )}{8}-\frac {1}{x c}\right )}{d^{3}}+\frac {b \left (-\frac {7 \,\operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}}{8 \left (c^{2} x^{2}+1\right )^{2}}-\frac {9 \,\operatorname {arcsinh}\left (x c \right ) x c}{8 \left (c^{2} x^{2}+1\right )^{2}}-\frac {15 \,\operatorname {arcsinh}\left (x c \right ) \arctan \left (x c \right )}{8}-\frac {\operatorname {arcsinh}\left (x c \right )}{x c}-\frac {15 \arctan \left (x c \right ) \ln \left (1+\frac {i \left (i x c +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}+\frac {15 \arctan \left (x c \right ) \ln \left (1-\frac {i \left (i x c +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}+\frac {15 i \operatorname {dilog}\left (1+\frac {i \left (i x c +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}-\frac {15 i \operatorname {dilog}\left (1-\frac {i \left (i x c +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}-\frac {47}{24 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {1}{\sqrt {c^{2} x^{2}+1}}-\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )-\frac {15 c^{2} x^{2}}{8 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{d^{3}}\right )\)	\(291\)
default	\(c \left (\frac {a \left (-\frac {\frac {7}{8} x^{3} c^{3}+\frac {9}{8} x c}{\left (c^{2} x^{2}+1\right )^{2}}-\frac {15 \arctan \left (x c \right )}{8}-\frac {1}{x c}\right )}{d^{3}}+\frac {b \left (-\frac {7 \,\operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}}{8 \left (c^{2} x^{2}+1\right )^{2}}-\frac {9 \,\operatorname {arcsinh}\left (x c \right ) x c}{8 \left (c^{2} x^{2}+1\right )^{2}}-\frac {15 \,\operatorname {arcsinh}\left (x c \right ) \arctan \left (x c \right )}{8}-\frac {\operatorname {arcsinh}\left (x c \right )}{x c}-\frac {15 \arctan \left (x c \right ) \ln \left (1+\frac {i \left (i x c +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}+\frac {15 \arctan \left (x c \right ) \ln \left (1-\frac {i \left (i x c +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}+\frac {15 i \operatorname {dilog}\left (1+\frac {i \left (i x c +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}-\frac {15 i \operatorname {dilog}\left (1-\frac {i \left (i x c +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}-\frac {47}{24 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {1}{\sqrt {c^{2} x^{2}+1}}-\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )-\frac {15 c^{2} x^{2}}{8 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{d^{3}}\right )\)	\(291\)
parts	\(\frac {a \left (-\frac {1}{x}-c^{2} \left (\frac {\frac {7}{8} x^{3} c^{2}+\frac {9}{8} x}{\left (c^{2} x^{2}+1\right )^{2}}+\frac {15 \arctan \left (x c \right )}{8 c}\right )\right )}{d^{3}}+\frac {b c \left (-\frac {7 \,\operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}}{8 \left (c^{2} x^{2}+1\right )^{2}}-\frac {9 \,\operatorname {arcsinh}\left (x c \right ) x c}{8 \left (c^{2} x^{2}+1\right )^{2}}-\frac {15 \,\operatorname {arcsinh}\left (x c \right ) \arctan \left (x c \right )}{8}-\frac {\operatorname {arcsinh}\left (x c \right )}{x c}-\frac {15 \arctan \left (x c \right ) \ln \left (1+\frac {i \left (i x c +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}+\frac {15 \arctan \left (x c \right ) \ln \left (1-\frac {i \left (i x c +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}+\frac {15 i \operatorname {dilog}\left (1+\frac {i \left (i x c +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}-\frac {15 i \operatorname {dilog}\left (1-\frac {i \left (i x c +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}-\frac {47}{24 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {1}{\sqrt {c^{2} x^{2}+1}}-\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )-\frac {15 c^{2} x^{2}}{8 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{d^{3}}\)	\(294\)

Input:

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

Output:

c*(a/d^3*(-(7/8*x^3*c^3+9/8*x*c)/(c^2*x^2+1)^2-15/8*arctan(x*c)-1/x/c)+b/d 
^3*(-7/8*arcsinh(x*c)/(c^2*x^2+1)^2*x^3*c^3-9/8*arcsinh(x*c)/(c^2*x^2+1)^2 
*x*c-15/8*arcsinh(x*c)*arctan(x*c)-arcsinh(x*c)/x/c-15/8*arctan(x*c)*ln(1+ 
I*(1+I*x*c)/(c^2*x^2+1)^(1/2))+15/8*arctan(x*c)*ln(1-I*(1+I*x*c)/(c^2*x^2+ 
1)^(1/2))+15/8*I*dilog(1+I*(1+I*x*c)/(c^2*x^2+1)^(1/2))-15/8*I*dilog(1-I*( 
1+I*x*c)/(c^2*x^2+1)^(1/2))-47/24/(c^2*x^2+1)^(3/2)+1/(c^2*x^2+1)^(1/2)-ar 
ctanh(1/(c^2*x^2+1)^(1/2))-15/8*c^2*x^2/(c^2*x^2+1)^(3/2)))
 

Fricas [F]

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

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

Output:

integral((b*arcsinh(c*x) + a)/(c^6*d^3*x^8 + 3*c^4*d^3*x^6 + 3*c^2*d^3*x^4 
 + d^3*x^2), x)
 

Sympy [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (d+c^2 d x^2\right )^3} \, dx=\frac {\int \frac {a}{c^{6} x^{8} + 3 c^{4} x^{6} + 3 c^{2} x^{4} + x^{2}}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{c^{6} x^{8} + 3 c^{4} x^{6} + 3 c^{2} x^{4} + x^{2}}\, dx}{d^{3}} \] Input:

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

Output:

(Integral(a/(c**6*x**8 + 3*c**4*x**6 + 3*c**2*x**4 + x**2), x) + Integral( 
b*asinh(c*x)/(c**6*x**8 + 3*c**4*x**6 + 3*c**2*x**4 + x**2), x))/d**3
 

Maxima [F]

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

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

Output:

-1/8*a*((15*c^4*x^4 + 25*c^2*x^2 + 8)/(c^4*d^3*x^5 + 2*c^2*d^3*x^3 + d^3*x 
) + 15*c*arctan(c*x)/d^3) + b*integrate(log(c*x + sqrt(c^2*x^2 + 1))/(c^6* 
d^3*x^8 + 3*c^4*d^3*x^6 + 3*c^2*d^3*x^4 + d^3*x^2), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (d+c^2 d x^2\right )^3} \, dx=\frac {-15 \mathit {atan} \left (c x \right ) a \,c^{5} x^{5}-30 \mathit {atan} \left (c x \right ) a \,c^{3} x^{3}-15 \mathit {atan} \left (c x \right ) a c x +8 \left (\int \frac {\mathit {asinh} \left (c x \right )}{c^{6} x^{8}+3 c^{4} x^{6}+3 c^{2} x^{4}+x^{2}}d x \right ) b \,c^{4} x^{5}+16 \left (\int \frac {\mathit {asinh} \left (c x \right )}{c^{6} x^{8}+3 c^{4} x^{6}+3 c^{2} x^{4}+x^{2}}d x \right ) b \,c^{2} x^{3}+8 \left (\int \frac {\mathit {asinh} \left (c x \right )}{c^{6} x^{8}+3 c^{4} x^{6}+3 c^{2} x^{4}+x^{2}}d x \right ) b x -15 a \,c^{4} x^{4}-25 a \,c^{2} x^{2}-8 a}{8 d^{3} x \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )} \] Input:

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

Output:

( - 15*atan(c*x)*a*c**5*x**5 - 30*atan(c*x)*a*c**3*x**3 - 15*atan(c*x)*a*c 
*x + 8*int(asinh(c*x)/(c**6*x**8 + 3*c**4*x**6 + 3*c**2*x**4 + x**2),x)*b* 
c**4*x**5 + 16*int(asinh(c*x)/(c**6*x**8 + 3*c**4*x**6 + 3*c**2*x**4 + x** 
2),x)*b*c**2*x**3 + 8*int(asinh(c*x)/(c**6*x**8 + 3*c**4*x**6 + 3*c**2*x** 
4 + x**2),x)*b*x - 15*a*c**4*x**4 - 25*a*c**2*x**2 - 8*a)/(8*d**3*x*(c**4* 
x**4 + 2*c**2*x**2 + 1))