\(\int \frac {a+b \text {arcsinh}(c x)}{(d+e x)^3} \, dx\) [10]

Optimal result

Integrand size = 16, antiderivative size = 128 \[ \int \frac {a+b \text {arcsinh}(c x)}{(d+e x)^3} \, dx=-\frac {b c \sqrt {1+c^2 x^2}}{2 \left (c^2 d^2+e^2\right ) (d+e x)}-\frac {a+b \text {arcsinh}(c x)}{2 e (d+e x)^2}-\frac {b c^3 d \text {arctanh}\left (\frac {e-c^2 d x}{\sqrt {c^2 d^2+e^2} \sqrt {1+c^2 x^2}}\right )}{2 e \left (c^2 d^2+e^2\right )^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.30 \[ \int \frac {a+b \text {arcsinh}(c x)}{(d+e x)^3} \, dx=\frac {1}{2} \left (-\frac {a}{e (d+e x)^2}-\frac {b c \sqrt {1+c^2 x^2}}{\left (c^2 d^2+e^2\right ) (d+e x)}-\frac {b \text {arcsinh}(c x)}{e (d+e x)^2}+\frac {b c^3 d \log (d+e x)}{e \left (c^2 d^2+e^2\right )^{3/2}}-\frac {b c^3 d \log \left (e-c^2 d x+\sqrt {c^2 d^2+e^2} \sqrt {1+c^2 x^2}\right )}{e \left (c^2 d^2+e^2\right )^{3/2}}\right ) \] Input:

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

Output:

(-(a/(e*(d + e*x)^2)) - (b*c*Sqrt[1 + c^2*x^2])/((c^2*d^2 + e^2)*(d + e*x) 
) - (b*ArcSinh[c*x])/(e*(d + e*x)^2) + (b*c^3*d*Log[d + e*x])/(e*(c^2*d^2 
+ e^2)^(3/2)) - (b*c^3*d*Log[e - c^2*d*x + Sqrt[c^2*d^2 + e^2]*Sqrt[1 + c^ 
2*x^2]])/(e*(c^2*d^2 + e^2)^(3/2)))/2
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.01, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6243, 491, 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.

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

\(\Big \downarrow \) 6243

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

\(\Big \downarrow \) 491

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

-1/2*(a + b*ArcSinh[c*x])/(e*(d + e*x)^2) + (b*c*(-((e*Sqrt[1 + c^2*x^2])/ 
((c^2*d^2 + e^2)*(d + e*x))) - (c^2*d*ArcTanh[(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)))/(2*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 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 491

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
d*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/((n + 1)*(b*c^2 + a*d^2))), x] + S 
imp[b*(c/(b*c^2 + a*d^2))   Int[(c + d*x)^(n + 1)*(a + b*x^2)^p, x], x] /; 
FreeQ[{a, b, c, d, n, p}, x] && EqQ[n + 2*p + 3, 0]
 

rule 6243

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(272\) vs. \(2(114)=228\).

Time = 5.03 (sec) , antiderivative size = 273, normalized size of antiderivative = 2.13

method	result	size
parts	\(-\frac {a}{2 \left (e x +d \right )^{2} e}-\frac {b \,c^{2} \operatorname {arcsinh}\left (x c \right )}{2 \left (c e x +c d \right )^{2} e}-\frac {b \,c^{2} \sqrt {\left (x c +\frac {d c}{e}\right )^{2}-\frac {2 d c \left (x c +\frac {d c}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{2 e \left (c^{2} d^{2}+e^{2}\right ) \left (x c +\frac {d c}{e}\right )}-\frac {b \,c^{3} d \ln \left (\frac {\frac {2 c^{2} d^{2}+2 e^{2}}{e^{2}}-\frac {2 d c \left (x c +\frac {d c}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \sqrt {\left (x c +\frac {d c}{e}\right )^{2}-\frac {2 d c \left (x c +\frac {d c}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{x c +\frac {d c}{e}}\right )}{2 e^{2} \left (c^{2} d^{2}+e^{2}\right ) \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}\)	\(273\)
derivativedivides	\(\frac {-\frac {a \,c^{3}}{2 \left (c e x +c d \right )^{2} e}+b \,c^{3} \left (-\frac {\operatorname {arcsinh}\left (x c \right )}{2 \left (c e x +c d \right )^{2} e}+\frac {-\frac {e^{2} \sqrt {\left (x c +\frac {d c}{e}\right )^{2}-\frac {2 d c \left (x c +\frac {d c}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{\left (c^{2} d^{2}+e^{2}\right ) \left (x c +\frac {d c}{e}\right )}-\frac {d c e \ln \left (\frac {\frac {2 c^{2} d^{2}+2 e^{2}}{e^{2}}-\frac {2 d c \left (x c +\frac {d c}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \sqrt {\left (x c +\frac {d c}{e}\right )^{2}-\frac {2 d c \left (x c +\frac {d c}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{x c +\frac {d c}{e}}\right )}{\left (c^{2} d^{2}+e^{2}\right ) \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}}{2 e^{3}}\right )}{c}\)	\(282\)
default	\(\frac {-\frac {a \,c^{3}}{2 \left (c e x +c d \right )^{2} e}+b \,c^{3} \left (-\frac {\operatorname {arcsinh}\left (x c \right )}{2 \left (c e x +c d \right )^{2} e}+\frac {-\frac {e^{2} \sqrt {\left (x c +\frac {d c}{e}\right )^{2}-\frac {2 d c \left (x c +\frac {d c}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{\left (c^{2} d^{2}+e^{2}\right ) \left (x c +\frac {d c}{e}\right )}-\frac {d c e \ln \left (\frac {\frac {2 c^{2} d^{2}+2 e^{2}}{e^{2}}-\frac {2 d c \left (x c +\frac {d c}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \sqrt {\left (x c +\frac {d c}{e}\right )^{2}-\frac {2 d c \left (x c +\frac {d c}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{x c +\frac {d c}{e}}\right )}{\left (c^{2} d^{2}+e^{2}\right ) \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}}{2 e^{3}}\right )}{c}\)	\(282\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 566 vs. \(2 (116) = 232\).

Time = 0.17 (sec) , antiderivative size = 566, normalized size of antiderivative = 4.42 \[ \int \frac {a+b \text {arcsinh}(c x)}{(d+e x)^3} \, dx=-\frac {{\left (a + b\right )} c^{4} d^{6} + {\left (2 \, a + b\right )} c^{2} d^{4} e^{2} + a d^{2} e^{4} + {\left (b c^{4} d^{4} e^{2} + b c^{2} d^{2} e^{4}\right )} x^{2} - {\left (b c^{3} d^{3} e^{2} x^{2} + 2 \, b c^{3} d^{4} e x + b c^{3} d^{5}\right )} \sqrt {c^{2} d^{2} + e^{2}} \log \left (-\frac {c^{3} d^{2} x - c d e + \sqrt {c^{2} d^{2} + e^{2}} {\left (c^{2} d x - e\right )} + {\left (c^{2} d^{2} + \sqrt {c^{2} d^{2} + e^{2}} c d + e^{2}\right )} \sqrt {c^{2} x^{2} + 1}}{e x + d}\right ) + 2 \, {\left (b c^{4} d^{5} e + b c^{2} d^{3} e^{3}\right )} x - {\left ({\left (b c^{4} d^{4} e^{2} + 2 \, b c^{2} d^{2} e^{4} + b e^{6}\right )} x^{2} + 2 \, {\left (b c^{4} d^{5} e + 2 \, b c^{2} d^{3} e^{3} + b d e^{5}\right )} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (b c^{4} d^{6} + 2 \, b c^{2} d^{4} e^{2} + b d^{2} e^{4} + {\left (b c^{4} d^{4} e^{2} + 2 \, b c^{2} d^{2} e^{4} + b e^{6}\right )} x^{2} + 2 \, {\left (b c^{4} d^{5} e + 2 \, b c^{2} d^{3} e^{3} + b d e^{5}\right )} x\right )} \log \left (-c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (b c^{3} d^{5} e + b c d^{3} e^{3} + {\left (b c^{3} d^{4} e^{2} + b c d^{2} e^{4}\right )} x\right )} \sqrt {c^{2} x^{2} + 1}}{2 \, {\left (c^{4} d^{8} e + 2 \, c^{2} d^{6} e^{3} + d^{4} e^{5} + {\left (c^{4} d^{6} e^{3} + 2 \, c^{2} d^{4} e^{5} + d^{2} e^{7}\right )} x^{2} + 2 \, {\left (c^{4} d^{7} e^{2} + 2 \, c^{2} d^{5} e^{4} + d^{3} e^{6}\right )} x\right )}} \] Input:

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

Output:

-1/2*((a + b)*c^4*d^6 + (2*a + b)*c^2*d^4*e^2 + a*d^2*e^4 + (b*c^4*d^4*e^2 
 + b*c^2*d^2*e^4)*x^2 - (b*c^3*d^3*e^2*x^2 + 2*b*c^3*d^4*e*x + b*c^3*d^5)* 
sqrt(c^2*d^2 + e^2)*log(-(c^3*d^2*x - c*d*e + sqrt(c^2*d^2 + e^2)*(c^2*d*x 
 - e) + (c^2*d^2 + sqrt(c^2*d^2 + e^2)*c*d + e^2)*sqrt(c^2*x^2 + 1))/(e*x 
+ d)) + 2*(b*c^4*d^5*e + b*c^2*d^3*e^3)*x - ((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)) - (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)) + (b*c^3*d^5*e + b*c*d^3*e^3 + 
(b*c^3*d^4*e^2 + b*c*d^2*e^4)*x)*sqrt(c^2*x^2 + 1))/(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)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.23 \[ \int \frac {a+b \text {arcsinh}(c x)}{(d+e x)^3} \, dx=-\frac {1}{2} \, {\left (c {\left (\frac {\sqrt {c^{2} x^{2} + 1}}{c^{2} d^{2} e x + c^{2} d^{3} + e^{3} x + d e^{2}} - \frac {c^{2} d \operatorname {arsinh}\left (\frac {c d x}{e {\left | x + \frac {d}{e} \right |}} - \frac {1}{c {\left | x + \frac {d}{e} \right |}}\right )}{{\left (\frac {c^{2} d^{2}}{e^{2}} + 1\right )}^{\frac {3}{2}} e^{4}}\right )} + \frac {\operatorname {arsinh}\left (c x\right )}{e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e}\right )} b - \frac {a}{2 \, {\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} \] Input:

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(2*int(asinh(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(asinh(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(asinh(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))