\(\int \frac {a+b \text {arccosh}(c x)}{(d+e x)^4} \, dx\) [20]

Optimal result

Integrand size = 16, antiderivative size = 202 \[ \int \frac {a+b \text {arccosh}(c x)}{(d+e x)^4} \, dx=-\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {b c^3 d \sqrt {-1+c x} \sqrt {1+c x}}{2 (c d-e)^2 (c d+e)^2 (d+e x)}-\frac {a+b \text {arccosh}(c x)}{3 e (d+e x)^3}+\frac {b c^3 \left (2 c^2 d^2+e^2\right ) \text {arctanh}\left (\frac {\sqrt {c d+e} \sqrt {1+c x}}{\sqrt {c d-e} \sqrt {-1+c x}}\right )}{3 (c d-e)^{5/2} e (c d+e)^{5/2}} \] Output:

-1/6*b*c*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*d^2-e^2)/(e*x+d)^2-1/2*b*c^3*d*( 
c*x-1)^(1/2)*(c*x+1)^(1/2)/(c*d-e)^2/(c*d+e)^2/(e*x+d)-1/3*(a+b*arccosh(c* 
x))/e/(e*x+d)^3+1/3*b*c^3*(2*c^2*d^2+e^2)*arctanh((c*d+e)^(1/2)*(c*x+1)^(1 
/2)/(c*d-e)^(1/2)/(c*x-1)^(1/2))/(c*d-e)^(5/2)/e/(c*d+e)^(5/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.66 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.28 \[ \int \frac {a+b \text {arccosh}(c x)}{(d+e x)^4} \, dx=-\frac {\frac {2 a+\frac {b c e \sqrt {-1+c x} \sqrt {1+c x} (d+e x) \left (-e^2+c^2 d (4 d+3 e x)\right )}{\left (-c^2 d^2+e^2\right )^2}}{(d+e x)^3}+\frac {2 b \text {arccosh}(c x)}{(d+e x)^3}+\frac {i b c^3 \left (2 c^2 d^2+e^2\right ) \log \left (\frac {12 e^2 (-c d+e)^2 (c d+e)^2 \left (-i e-i c^2 d x+\sqrt {-c^2 d^2+e^2} \sqrt {-1+c x} \sqrt {1+c x}\right )}{b c^3 \sqrt {-c^2 d^2+e^2} \left (2 c^2 d^2+e^2\right ) (d+e x)}\right )}{(-c d+e)^2 (c d+e)^2 \sqrt {-c^2 d^2+e^2}}}{6 e} \] Input:

Integrate[(a + b*ArcCosh[c*x])/(d + e*x)^4,x]
 

Output:

-1/6*((2*a + (b*c*e*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x)*(-e^2 + c^2*d*( 
4*d + 3*e*x)))/(-(c^2*d^2) + e^2)^2)/(d + e*x)^3 + (2*b*ArcCosh[c*x])/(d + 
 e*x)^3 + (I*b*c^3*(2*c^2*d^2 + e^2)*Log[(12*e^2*(-(c*d) + e)^2*(c*d + e)^ 
2*((-I)*e - I*c^2*d*x + Sqrt[-(c^2*d^2) + e^2]*Sqrt[-1 + c*x]*Sqrt[1 + c*x 
]))/(b*c^3*Sqrt[-(c^2*d^2) + e^2]*(2*c^2*d^2 + e^2)*(d + e*x))])/((-(c*d) 
+ e)^2*(c*d + e)^2*Sqrt[-(c^2*d^2) + e^2]))/e
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.16, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {6378, 114, 25, 27, 168, 25, 27, 104, 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.

\(\displaystyle \int \frac {a+b \text {arccosh}(c x)}{(d+e x)^4} \, dx\)

\(\Big \downarrow \) 6378

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

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 168

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 104

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

\(\Big \downarrow \) 221

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

Input:

Int[(a + b*ArcCosh[c*x])/(d + e*x)^4,x]
 

Output:

-1/3*(a + b*ArcCosh[c*x])/(e*(d + e*x)^3) + (b*c*(-1/2*(e*Sqrt[-1 + c*x]*S 
qrt[1 + c*x])/((c^2*d^2 - e^2)*(d + e*x)^2) + (c^2*((-3*d*e*Sqrt[-1 + c*x] 
*Sqrt[1 + c*x])/((c^2*d^2 - e^2)*(d + e*x)) + (2*(2*c^2*d^2 + e^2)*ArcTanh 
[(Sqrt[c*d + e]*Sqrt[1 + c*x])/(Sqrt[c*d - e]*Sqrt[-1 + c*x])])/(Sqrt[c*d 
- e]*Sqrt[c*d + e]*(c^2*d^2 - e^2))))/(2*(c^2*d^2 - e^2))))/(3*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 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 114

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

rule 168

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
 

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 6378

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x 
_Symbol] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcCosh[c*x])^n/(e*(m + 1))), x] 
 - Simp[b*c*(n/(e*(m + 1)))   Int[(d + e*x)^(m + 1)*((a + b*ArcCosh[c*x])^( 
n - 1)/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])), 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. \(638\) vs. \(2(174)=348\).

Time = 1.06 (sec) , antiderivative size = 639, normalized size of antiderivative = 3.16

method	result	size
parts	\(-\frac {a}{3 \left (e x +d \right )^{3} e}+\frac {b \left (-\frac {c^{4} \operatorname {arccosh}\left (c x \right )}{3 \left (e c x +c d \right )^{3} e}-\frac {c^{4} \left (2 \ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right ) c^{4} d^{4}+4 \ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right ) c^{4} d^{3} e x +2 \ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right ) c^{4} d^{2} e^{2} x^{2}+4 c^{2} d^{2} e^{2} \sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}+3 c^{2} d \,e^{3} \sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, x +\ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right ) c^{2} d^{2} e^{2}+2 \ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right ) c^{2} d \,e^{3} x +\ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right ) e^{4} c^{2} x^{2}-e^{4} \sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\right ) \sqrt {c x -1}\, \sqrt {c x +1}}{6 e^{2} \sqrt {c^{2} x^{2}-1}\, \left (c d +e \right ) \left (c d -e \right ) \left (c^{2} d^{2}-e^{2}\right ) \left (e c x +c d \right )^{2} \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{c}\)	\(639\)
derivativedivides	\(\frac {-\frac {a \,c^{4}}{3 \left (e c x +c d \right )^{3} e}+b \,c^{4} \left (-\frac {\operatorname {arccosh}\left (c x \right )}{3 \left (e c x +c d \right )^{3} e}-\frac {\left (2 \ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right ) c^{4} d^{4}+4 \ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right ) c^{4} d^{3} e x +2 \ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right ) c^{4} d^{2} e^{2} x^{2}+4 c^{2} d^{2} e^{2} \sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}+3 c^{2} d \,e^{3} \sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, x +\ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right ) c^{2} d^{2} e^{2}+2 \ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right ) c^{2} d \,e^{3} x +\ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right ) e^{4} c^{2} x^{2}-e^{4} \sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\right ) \sqrt {c x -1}\, \sqrt {c x +1}}{6 e^{2} \sqrt {c^{2} x^{2}-1}\, \left (c d +e \right ) \left (c d -e \right ) \left (c^{2} d^{2}-e^{2}\right ) \left (e c x +c d \right )^{2} \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{c}\)	\(643\)
default	\(\frac {-\frac {a \,c^{4}}{3 \left (e c x +c d \right )^{3} e}+b \,c^{4} \left (-\frac {\operatorname {arccosh}\left (c x \right )}{3 \left (e c x +c d \right )^{3} e}-\frac {\left (2 \ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right ) c^{4} d^{4}+4 \ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right ) c^{4} d^{3} e x +2 \ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right ) c^{4} d^{2} e^{2} x^{2}+4 c^{2} d^{2} e^{2} \sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}+3 c^{2} d \,e^{3} \sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, x +\ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right ) c^{2} d^{2} e^{2}+2 \ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right ) c^{2} d \,e^{3} x +\ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right ) e^{4} c^{2} x^{2}-e^{4} \sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\right ) \sqrt {c x -1}\, \sqrt {c x +1}}{6 e^{2} \sqrt {c^{2} x^{2}-1}\, \left (c d +e \right ) \left (c d -e \right ) \left (c^{2} d^{2}-e^{2}\right ) \left (e c x +c d \right )^{2} \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{c}\)	\(643\)

Input:

int((a+b*arccosh(c*x))/(e*x+d)^4,x,method=_RETURNVERBOSE)
 

Output:

-1/3*a/(e*x+d)^3/e+b/c*(-1/3*c^4/(c*e*x+c*d)^3/e*arccosh(c*x)-1/6*c^4/e^2* 
(2*ln(-2*(c^2*d*x-(c^2*x^2-1)^(1/2)*((c^2*d^2-e^2)/e^2)^(1/2)*e+e)/(c*e*x+ 
c*d))*c^4*d^4+4*ln(-2*(c^2*d*x-(c^2*x^2-1)^(1/2)*((c^2*d^2-e^2)/e^2)^(1/2) 
*e+e)/(c*e*x+c*d))*c^4*d^3*e*x+2*ln(-2*(c^2*d*x-(c^2*x^2-1)^(1/2)*((c^2*d^ 
2-e^2)/e^2)^(1/2)*e+e)/(c*e*x+c*d))*c^4*d^2*e^2*x^2+4*c^2*d^2*e^2*(c^2*x^2 
-1)^(1/2)*((c^2*d^2-e^2)/e^2)^(1/2)+3*c^2*d*e^3*(c^2*x^2-1)^(1/2)*((c^2*d^ 
2-e^2)/e^2)^(1/2)*x+ln(-2*(c^2*d*x-(c^2*x^2-1)^(1/2)*((c^2*d^2-e^2)/e^2)^( 
1/2)*e+e)/(c*e*x+c*d))*c^2*d^2*e^2+2*ln(-2*(c^2*d*x-(c^2*x^2-1)^(1/2)*((c^ 
2*d^2-e^2)/e^2)^(1/2)*e+e)/(c*e*x+c*d))*c^2*d*e^3*x+ln(-2*(c^2*d*x-(c^2*x^ 
2-1)^(1/2)*((c^2*d^2-e^2)/e^2)^(1/2)*e+e)/(c*e*x+c*d))*e^4*c^2*x^2-e^4*(c^ 
2*x^2-1)^(1/2)*((c^2*d^2-e^2)/e^2)^(1/2))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2 
*x^2-1)^(1/2)/(c*d+e)/(c*d-e)/(c^2*d^2-e^2)/(c*e*x+c*d)^2/((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. 971 vs. \(2 (174) = 348\).

Time = 0.52 (sec) , antiderivative size = 1963, normalized size of antiderivative = 9.72 \[ \int \frac {a+b \text {arccosh}(c x)}{(d+e x)^4} \, dx=\text {Too large to display} \] Input:

integrate((a+b*arccosh(c*x))/(e*x+d)^4,x, algorithm="fricas")
 

Output:

[-1/6*((2*a + 3*b)*c^6*d^9 - 3*(2*a + b)*c^4*d^7*e^2 + 6*a*c^2*d^5*e^4 - 2 
*a*d^3*e^6 + 3*(b*c^6*d^6*e^3 - b*c^4*d^4*e^5)*x^3 + 9*(b*c^6*d^7*e^2 - b* 
c^4*d^5*e^4)*x^2 - (2*b*c^5*d^8 + b*c^3*d^6*e^2 + (2*b*c^5*d^5*e^3 + b*c^3 
*d^3*e^5)*x^3 + 3*(2*b*c^5*d^6*e^2 + b*c^3*d^4*e^4)*x^2 + 3*(2*b*c^5*d^7*e 
 + b*c^3*d^5*e^3)*x)*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)) + 9*(b*c^6*d^8*e - b*c^4*d^6*e^3)*x - 2*((b*c^6* 
d^6*e^3 - 3*b*c^4*d^4*e^5 + 3*b*c^2*d^2*e^7 - b*e^9)*x^3 + 3*(b*c^6*d^7*e^ 
2 - 3*b*c^4*d^5*e^4 + 3*b*c^2*d^3*e^6 - b*d*e^8)*x^2 + 3*(b*c^6*d^8*e - 3* 
b*c^4*d^6*e^3 + 3*b*c^2*d^4*e^5 - b*d^2*e^7)*x)*log(c*x + sqrt(c^2*x^2 - 1 
)) - 2*(b*c^6*d^9 - 3*b*c^4*d^7*e^2 + 3*b*c^2*d^5*e^4 - b*d^3*e^6 + (b*c^6 
*d^6*e^3 - 3*b*c^4*d^4*e^5 + 3*b*c^2*d^2*e^7 - b*e^9)*x^3 + 3*(b*c^6*d^7*e 
^2 - 3*b*c^4*d^5*e^4 + 3*b*c^2*d^3*e^6 - b*d*e^8)*x^2 + 3*(b*c^6*d^8*e - 3 
*b*c^4*d^6*e^3 + 3*b*c^2*d^4*e^5 - b*d^2*e^7)*x)*log(-c*x + sqrt(c^2*x^2 - 
 1)) + (4*b*c^5*d^8*e - 5*b*c^3*d^6*e^3 + b*c*d^4*e^5 + 3*(b*c^5*d^6*e^3 - 
 b*c^3*d^4*e^5)*x^2 + (7*b*c^5*d^7*e^2 - 8*b*c^3*d^5*e^4 + b*c*d^3*e^6)*x) 
*sqrt(c^2*x^2 - 1))/(c^6*d^12*e - 3*c^4*d^10*e^3 + 3*c^2*d^8*e^5 - d^6*e^7 
 + (c^6*d^9*e^4 - 3*c^4*d^7*e^6 + 3*c^2*d^5*e^8 - d^3*e^10)*x^3 + 3*(c^6*d 
^10*e^3 - 3*c^4*d^8*e^5 + 3*c^2*d^6*e^7 - d^4*e^9)*x^2 + 3*(c^6*d^11*e^2 - 
 3*c^4*d^9*e^4 + 3*c^2*d^7*e^6 - d^5*e^8)*x), -1/6*((2*a + 3*b)*c^6*d^9...
 

Sympy [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{(d+e x)^4} \, dx=\int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{\left (d + e x\right )^{4}}\, dx \] Input:

integrate((a+b*acosh(c*x))/(e*x+d)**4,x)
 

Output:

Integral((a + b*acosh(c*x))/(d + e*x)**4, x)
 

Maxima [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{(d+e x)^4} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (e x + d\right )}^{4}} \,d x } \] Input:

integrate((a+b*arccosh(c*x))/(e*x+d)^4,x, algorithm="maxima")
 

Output:

-1/6*(6*c*integrate(1/3/(c^3*e^4*x^6 + 3*c^3*d*e^3*x^5 - 3*c*d^2*e^2*x^2 - 
 c*d^3*e*x + (3*c^3*d^2*e^2 - c*e^4)*x^4 + (c^3*d^3*e - 3*c*d*e^3)*x^3 + ( 
c^2*e^4*x^5 + 3*c^2*d*e^3*x^4 - 3*d^2*e^2*x - d^3*e + (3*c^2*d^2*e^2 - e^4 
)*x^3 + (c^2*d^3*e - 3*d*e^3)*x^2)*e^(1/2*log(c*x + 1) + 1/2*log(c*x - 1)) 
), x) + 2*(c^6*d^3 + 3*c^4*d*e^2)*log(e*x + d)/(c^6*d^6*e - 3*c^4*d^4*e^3 
+ 3*c^2*d^2*e^5 - e^7) - (3*c^6*d^6 - 2*c^4*d^4*e^2 - c^2*d^2*e^4 + 2*(c^6 
*d^4*e^2 - c^2*e^6)*x^2 + (5*c^6*d^5*e - 2*c^4*d^3*e^3 - 3*c^2*d*e^5)*x - 
2*(c^6*d^6 - 3*c^4*d^4*e^2 + 3*c^2*d^2*e^4 - e^6)*log(c*x + sqrt(c*x + 1)* 
sqrt(c*x - 1)) + (c^6*d^6 + 3*c^5*d^5*e + 3*c^4*d^4*e^2 + c^3*d^3*e^3 + (c 
^6*d^3*e^3 + 3*c^5*d^2*e^4 + 3*c^4*d*e^5 + c^3*e^6)*x^3 + 3*(c^6*d^4*e^2 + 
 3*c^5*d^3*e^3 + 3*c^4*d^2*e^4 + c^3*d*e^5)*x^2 + 3*(c^6*d^5*e + 3*c^5*d^4 
*e^2 + 3*c^4*d^3*e^3 + c^3*d^2*e^4)*x)*log(c*x + 1) + (c^6*d^6 - 3*c^5*d^5 
*e + 3*c^4*d^4*e^2 - c^3*d^3*e^3 + (c^6*d^3*e^3 - 3*c^5*d^2*e^4 + 3*c^4*d* 
e^5 - c^3*e^6)*x^3 + 3*(c^6*d^4*e^2 - 3*c^5*d^3*e^3 + 3*c^4*d^2*e^4 - c^3* 
d*e^5)*x^2 + 3*(c^6*d^5*e - 3*c^5*d^4*e^2 + 3*c^4*d^3*e^3 - c^3*d^2*e^4)*x 
)*log(c*x - 1))/(c^6*d^9*e - 3*c^4*d^7*e^3 + 3*c^2*d^5*e^5 - d^3*e^7 + (c^ 
6*d^6*e^4 - 3*c^4*d^4*e^6 + 3*c^2*d^2*e^8 - e^10)*x^3 + 3*(c^6*d^7*e^3 - 3 
*c^4*d^5*e^5 + 3*c^2*d^3*e^7 - d*e^9)*x^2 + 3*(c^6*d^8*e^2 - 3*c^4*d^6*e^4 
 + 3*c^2*d^4*e^6 - d^2*e^8)*x))*b - 1/3*a/(e^4*x^3 + 3*d*e^3*x^2 + 3*d^2*e 
^2*x + d^3*e)
 

Giac [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{(d+e x)^4} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (e x + d\right )}^{4}} \,d x } \] Input:

integrate((a+b*arccosh(c*x))/(e*x+d)^4,x, algorithm="giac")
 

Output:

integrate((b*arccosh(c*x) + a)/(e*x + d)^4, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{(d+e x)^4} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{{\left (d+e\,x\right )}^4} \,d x \] Input:

int((a + b*acosh(c*x))/(d + e*x)^4,x)
                                                                                    
                                                                                    
 

Output:

int((a + b*acosh(c*x))/(d + e*x)^4, x)
 

Reduce [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{(d+e x)^4} \, dx=\frac {3 \left (\int \frac {\mathit {acosh} \left (c x \right )}{e^{4} x^{4}+4 d \,e^{3} x^{3}+6 d^{2} e^{2} x^{2}+4 d^{3} e x +d^{4}}d x \right ) b \,d^{3} e +9 \left (\int \frac {\mathit {acosh} \left (c x \right )}{e^{4} x^{4}+4 d \,e^{3} x^{3}+6 d^{2} e^{2} x^{2}+4 d^{3} e x +d^{4}}d x \right ) b \,d^{2} e^{2} x +9 \left (\int \frac {\mathit {acosh} \left (c x \right )}{e^{4} x^{4}+4 d \,e^{3} x^{3}+6 d^{2} e^{2} x^{2}+4 d^{3} e x +d^{4}}d x \right ) b d \,e^{3} x^{2}+3 \left (\int \frac {\mathit {acosh} \left (c x \right )}{e^{4} x^{4}+4 d \,e^{3} x^{3}+6 d^{2} e^{2} x^{2}+4 d^{3} e x +d^{4}}d x \right ) b \,e^{4} x^{3}-a}{3 e \left (e^{3} x^{3}+3 d \,e^{2} x^{2}+3 d^{2} e x +d^{3}\right )} \] Input:

int((a+b*acosh(c*x))/(e*x+d)^4,x)
 

Output:

(3*int(acosh(c*x)/(d**4 + 4*d**3*e*x + 6*d**2*e**2*x**2 + 4*d*e**3*x**3 + 
e**4*x**4),x)*b*d**3*e + 9*int(acosh(c*x)/(d**4 + 4*d**3*e*x + 6*d**2*e**2 
*x**2 + 4*d*e**3*x**3 + e**4*x**4),x)*b*d**2*e**2*x + 9*int(acosh(c*x)/(d* 
*4 + 4*d**3*e*x + 6*d**2*e**2*x**2 + 4*d*e**3*x**3 + e**4*x**4),x)*b*d*e** 
3*x**2 + 3*int(acosh(c*x)/(d**4 + 4*d**3*e*x + 6*d**2*e**2*x**2 + 4*d*e**3 
*x**3 + e**4*x**4),x)*b*e**4*x**3 - a)/(3*e*(d**3 + 3*d**2*e*x + 3*d*e**2* 
x**2 + e**3*x**3))