3.1.7 \(\int \frac {\cosh ^{-1}(c x)}{(d+e x)^4} \, dx\) [7]
Optimal. Leaf size=195 \[ -\frac {c \sqrt {-1+c x} \sqrt {1+c x}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {c^3 d \sqrt {-1+c x} \sqrt {1+c x}}{2 (c d-e)^2 (c d+e)^2 (d+e x)}-\frac {\cosh ^{-1}(c x)}{3 e (d+e x)^3}+\frac {c^3 \left (2 c^2 d^2+e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c d+e} \sqrt {1+c x}}{\sqrt {c d-e} \sqrt {-1+c x}}\right )}{3 (c d-e)^{5/2} e (c d+e)^{5/2}} \]
[Out]
-1/3*arccosh(c*x)/e/(e*x+d)^3+1/3*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)-1/6*c*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*d^2-e^2)/(e*x+d)^2-1/2*c^3*d*(c*
x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*d^2-e^2)^2/(e*x+d)
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Rubi [A]
time = 0.20, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5963, 105, 156,
12, 95, 214} \begin {gather*} -\frac {c^3 d \sqrt {c x-1} \sqrt {c x+1}}{2 (c d-e)^2 (c d+e)^2 (d+e x)}-\frac {c \sqrt {c x-1} \sqrt {c x+1}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {c^3 \left (2 c^2 d^2+e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c x+1} \sqrt {c d+e}}{\sqrt {c x-1} \sqrt {c d-e}}\right )}{3 e (c d-e)^{5/2} (c d+e)^{5/2}}-\frac {\cosh ^{-1}(c x)}{3 e (d+e x)^3} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[ArcCosh[c*x]/(d + e*x)^4,x]
[Out]
-1/6*(c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/((c^2*d^2 - e^2)*(d + e*x)^2) - (c^3*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*
(c*d - e)^2*(c*d + e)^2*(d + e*x)) - ArcCosh[c*x]/(3*e*(d + e*x)^3) + (c^3*(2*c^2*d^2 + e^2)*ArcTanh[(Sqrt[c*d
+ e]*Sqrt[1 + c*x])/(Sqrt[c*d - e]*Sqrt[-1 + c*x])])/(3*(c*d - e)^(5/2)*e*(c*d + e)^(5/2))
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 95
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[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] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]
Rule 105
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> 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] + Dist[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 156
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> 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] + Dist[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 214
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 5963
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] - Dist[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]
Rubi steps
\begin {align*} \int \frac {\cosh ^{-1}(c x)}{(d+e x)^4} \, dx &=-\frac {\cosh ^{-1}(c x)}{3 e (d+e x)^3}+\frac {c \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x} (d+e x)^3} \, dx}{3 e}\\ &=-\frac {c \sqrt {-1+c x} \sqrt {1+c x}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {\cosh ^{-1}(c x)}{3 e (d+e x)^3}-\frac {c \int \frac {-2 c^2 d+c^2 e x}{\sqrt {-1+c x} \sqrt {1+c x} (d+e x)^2} \, dx}{6 e \left (c^2 d^2-e^2\right )}\\ &=-\frac {c \sqrt {-1+c x} \sqrt {1+c x}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {c^3 d \sqrt {-1+c x} \sqrt {1+c x}}{2 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {\cosh ^{-1}(c x)}{3 e (d+e x)^3}+\frac {c \int \frac {c^2 \left (2 c^2 d^2+e^2\right )}{\sqrt {-1+c x} \sqrt {1+c x} (d+e x)} \, dx}{6 e \left (c^2 d^2-e^2\right )^2}\\ &=-\frac {c \sqrt {-1+c x} \sqrt {1+c x}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {c^3 d \sqrt {-1+c x} \sqrt {1+c x}}{2 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {\cosh ^{-1}(c x)}{3 e (d+e x)^3}+\frac {\left (c^3 \left (2 c^2 d^2+e^2\right )\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x} (d+e x)} \, dx}{6 e \left (c^2 d^2-e^2\right )^2}\\ &=-\frac {c \sqrt {-1+c x} \sqrt {1+c x}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {c^3 d \sqrt {-1+c x} \sqrt {1+c x}}{2 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {\cosh ^{-1}(c x)}{3 e (d+e x)^3}+\frac {\left (c^3 \left (2 c^2 d^2+e^2\right )\right ) \text {Subst}\left (\int \frac {1}{c d-e-(c d+e) x^2} \, dx,x,\frac {\sqrt {1+c x}}{\sqrt {-1+c x}}\right )}{3 e \left (c^2 d^2-e^2\right )^2}\\ &=-\frac {c \sqrt {-1+c x} \sqrt {1+c x}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {c^3 d \sqrt {-1+c x} \sqrt {1+c x}}{2 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {\cosh ^{-1}(c x)}{3 e (d+e x)^3}+\frac {c^3 \left (2 c^2 d^2+e^2\right ) \tanh ^{-1}\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}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.39, size = 244, normalized size = 1.25 \begin {gather*} \frac {1}{6} \left (\frac {c \sqrt {-1+c x} \sqrt {1+c 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)^2}-\frac {2 \cosh ^{-1}(c x)}{e (d+e x)^3}-\frac {i 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 )}{c^3 \sqrt {-c^2 d^2+e^2} \left (2 c^2 d^2+e^2\right ) (d+e x)}\right )}{e (-c d+e)^2 (c d+e)^2 \sqrt {-c^2 d^2+e^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[ArcCosh[c*x]/(d + e*x)^4,x]
[Out]
((c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(e^2 - c^2*d*(4*d + 3*e*x)))/((-(c^2*d^2) + e^2)^2*(d + e*x)^2) - (2*ArcCosh[
c*x])/(e*(d + e*x)^3) - (I*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]))/(c^3*Sqrt[-(c^2*d^2) + e^2]*(2*c^2*d^2 + e^2)*(d + e*x))
])/(e*(-(c*d) + e)^2*(c*d + e)^2*Sqrt[-(c^2*d^2) + e^2]))/6
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(623\) vs.
\(2(166)=332\).
time = 6.08, size = 624, normalized size = 3.20
| | |
| method |
result |
size |
| | |
| derivativedivides |
\(\frac {-\frac {c^{4} \mathrm {arccosh}\left (c x \right )}{3 \left (e c x +c d \right )^{3} e}-\frac {c^{4} \left (2 \ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right ) c^{4} d^{4}+4 \ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right ) c^{4} d^{3} e x +2 \ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right ) c^{4} d^{2} e^{2} x^{2}+4 c^{2} d^{2} e^{2} \sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}+3 c^{2} d \,e^{3} \sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, x +\ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right ) c^{2} d^{2} e^{2}+2 \ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right ) c^{2} d \,e^{3} x +\ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right ) e^{4} c^{2} x^{2}-e^{4} \sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\right ) \sqrt {c x -1}\, \sqrt {c x +1}}{6 e^{2} \sqrt {c^{2} x^{2}-1}\, \left (c d +e \right ) \left (c d -e \right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \left (c^{2} d^{2}-e^{2}\right ) \left (e c x +c d \right )^{2}}}{c}\) |
\(624\) |
| default |
\(\frac {-\frac {c^{4} \mathrm {arccosh}\left (c x \right )}{3 \left (e c x +c d \right )^{3} e}-\frac {c^{4} \left (2 \ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right ) c^{4} d^{4}+4 \ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right ) c^{4} d^{3} e x +2 \ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right ) c^{4} d^{2} e^{2} x^{2}+4 c^{2} d^{2} e^{2} \sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}+3 c^{2} d \,e^{3} \sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, x +\ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right ) c^{2} d^{2} e^{2}+2 \ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right ) c^{2} d \,e^{3} x +\ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right ) e^{4} c^{2} x^{2}-e^{4} \sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\right ) \sqrt {c x -1}\, \sqrt {c x +1}}{6 e^{2} \sqrt {c^{2} x^{2}-1}\, \left (c d +e \right ) \left (c d -e \right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \left (c^{2} d^{2}-e^{2}\right ) \left (e c x +c d \right )^{2}}}{c}\) |
\(624\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(arccosh(c*x)/(e*x+d)^4,x,method=_RETURNVERBOSE)
[Out]
1/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)/e^2)^(1/2)/(c^2*d^2-e^2)/(c*e*x+c*d)^2)
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(arccosh(c*x)/(e*x+d)^4,x, algorithm="maxima")
[Out]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(c*d-%e>0)', see `assume?` for
more details
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 4546 vs.
\(2 (168) = 336\).
time = 0.77, size = 9111, normalized size = 46.72 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(arccosh(c*x)/(e*x+d)^4,x, algorithm="fricas")
[Out]
[-1/6*(9*c^6*d^8*x*cosh(1) - 3*c^4*d^4*x^3*cosh(1)^5 - 3*c^4*d^4*x^3*sinh(1)^5 + 3*c^6*d^9 - 9*c^4*d^5*x^2*cos
h(1)^4 - 3*(5*c^4*d^4*x^3*cosh(1) + 3*c^4*d^5*x^2)*sinh(1)^4 + 3*(c^6*d^6*x^3 - 3*c^4*d^6*x)*cosh(1)^3 + 3*(c^
6*d^6*x^3 - 10*c^4*d^4*x^3*cosh(1)^2 - 12*c^4*d^5*x^2*cosh(1) - 3*c^4*d^6*x)*sinh(1)^3 + 3*(3*c^6*d^7*x^2 - c^
4*d^7)*cosh(1)^2 + 3*(3*c^6*d^7*x^2 - 10*c^4*d^4*x^3*cosh(1)^3 - 18*c^4*d^5*x^2*cosh(1)^2 - c^4*d^7 + 3*(c^6*d
^6*x^3 - 3*c^4*d^6*x)*cosh(1))*sinh(1)^2 - (6*c^5*d^7*x*cosh(1) + c^3*d^3*x^3*cosh(1)^5 + c^3*d^3*x^3*sinh(1)^
5 + 2*c^5*d^8 + 3*c^3*d^4*x^2*cosh(1)^4 + (5*c^3*d^3*x^3*cosh(1) + 3*c^3*d^4*x^2)*sinh(1)^4 + (2*c^5*d^5*x^3 +
3*c^3*d^5*x)*cosh(1)^3 + (2*c^5*d^5*x^3 + 10*c^3*d^3*x^3*cosh(1)^2 + 12*c^3*d^4*x^2*cosh(1) + 3*c^3*d^5*x)*si
nh(1)^3 + (6*c^5*d^6*x^2 + c^3*d^6)*cosh(1)^2 + (6*c^5*d^6*x^2 + 10*c^3*d^3*x^3*cosh(1)^3 + 18*c^3*d^4*x^2*cos
h(1)^2 + c^3*d^6 + 3*(2*c^5*d^5*x^3 + 3*c^3*d^5*x)*cosh(1))*sinh(1)^2 + (6*c^5*d^7*x + 5*c^3*d^3*x^3*cosh(1)^4
+ 12*c^3*d^4*x^2*cosh(1)^3 + 3*(2*c^5*d^5*x^3 + 3*c^3*d^5*x)*cosh(1)^2 + 2*(6*c^5*d^6*x^2 + c^3*d^6)*cosh(1))
*sinh(1))*sqrt(((c^2*d^2 - 1)*cosh(1) - (c^2*d^2 + 1)*sinh(1))/(cosh(1) - sinh(1)))*log((c^3*d^2*x + c*d*cosh(
1) + c*d*sinh(1) + (c^2*d^2 + c*d*sqrt(((c^2*d^2 - 1)*cosh(1) - (c^2*d^2 + 1)*sinh(1))/(cosh(1) - sinh(1))) -
cosh(1)^2 - 2*cosh(1)*sinh(1) - sinh(1)^2)*sqrt(c^2*x^2 - 1) + (c^2*d*x + cosh(1) + sinh(1))*sqrt(((c^2*d^2 -
1)*cosh(1) - (c^2*d^2 + 1)*sinh(1))/(cosh(1) - sinh(1))))/(x*cosh(1) + x*sinh(1) + d)) - 2*(3*c^6*d^7*x^2*cosh
(1)^2 + 3*c^6*d^8*x*cosh(1) - 9*c^4*d^5*x^2*cosh(1)^4 + 9*c^2*d^3*x^2*cosh(1)^6 - x^3*cosh(1)^9 - x^3*sinh(1)^
9 - 3*d*x^2*cosh(1)^8 - 3*(3*x^3*cosh(1) + d*x^2)*sinh(1)^8 + 3*(c^2*d^2*x^3 - d^2*x)*cosh(1)^7 + 3*(c^2*d^2*x
^3 - 12*x^3*cosh(1)^2 - 8*d*x^2*cosh(1) - d^2*x)*sinh(1)^7 + 3*(3*c^2*d^3*x^2 - 28*x^3*cosh(1)^3 - 28*d*x^2*co
sh(1)^2 + 7*(c^2*d^2*x^3 - d^2*x)*cosh(1))*sinh(1)^6 - 3*(c^4*d^4*x^3 - 3*c^2*d^4*x)*cosh(1)^5 - 3*(c^4*d^4*x^
3 - 18*c^2*d^3*x^2*cosh(1) - 3*c^2*d^4*x + 42*x^3*cosh(1)^4 + 56*d*x^2*cosh(1)^3 - 21*(c^2*d^2*x^3 - d^2*x)*co
sh(1)^2)*sinh(1)^5 - 3*(3*c^4*d^5*x^2 - 45*c^2*d^3*x^2*cosh(1)^2 + 42*x^3*cosh(1)^5 + 70*d*x^2*cosh(1)^4 - 35*
(c^2*d^2*x^3 - d^2*x)*cosh(1)^3 + 5*(c^4*d^4*x^3 - 3*c^2*d^4*x)*cosh(1))*sinh(1)^4 + (c^6*d^6*x^3 - 9*c^4*d^6*
x)*cosh(1)^3 + (c^6*d^6*x^3 - 36*c^4*d^5*x^2*cosh(1) - 9*c^4*d^6*x + 180*c^2*d^3*x^2*cosh(1)^3 - 84*x^3*cosh(1
)^6 - 168*d*x^2*cosh(1)^5 + 105*(c^2*d^2*x^3 - d^2*x)*cosh(1)^4 - 30*(c^4*d^4*x^3 - 3*c^2*d^4*x)*cosh(1)^2)*si
nh(1)^3 + 3*(c^6*d^7*x^2 - 18*c^4*d^5*x^2*cosh(1)^2 + 45*c^2*d^3*x^2*cosh(1)^4 - 12*x^3*cosh(1)^7 - 28*d*x^2*c
osh(1)^6 + 21*(c^2*d^2*x^3 - d^2*x)*cosh(1)^5 - 10*(c^4*d^4*x^3 - 3*c^2*d^4*x)*cosh(1)^3 + (c^6*d^6*x^3 - 9*c^
4*d^6*x)*cosh(1))*sinh(1)^2 + 3*(2*c^6*d^7*x^2*cosh(1) + c^6*d^8*x - 12*c^4*d^5*x^2*cosh(1)^3 + 18*c^2*d^3*x^2
*cosh(1)^5 - 3*x^3*cosh(1)^8 - 8*d*x^2*cosh(1)^7 + 7*(c^2*d^2*x^3 - d^2*x)*cosh(1)^6 - 5*(c^4*d^4*x^3 - 3*c^2*
d^4*x)*cosh(1)^4 + (c^6*d^6*x^3 - 9*c^4*d^6*x)*cosh(1)^2)*sinh(1))*log(c*x + sqrt(c^2*x^2 - 1)) - 2*(3*c^6*d^8
*x*cosh(1) + c^6*d^9 - x^3*cosh(1)^9 - x^3*sinh(1)^9 - 3*d*x^2*cosh(1)^8 - 3*(3*x^3*cosh(1) + d*x^2)*sinh(1)^8
+ 3*(c^2*d^2*x^3 - d^2*x)*cosh(1)^7 + 3*(c^2*d^2*x^3 - 12*x^3*cosh(1)^2 - 8*d*x^2*cosh(1) - d^2*x)*sinh(1)^7
+ (9*c^2*d^3*x^2 - d^3)*cosh(1)^6 + (9*c^2*d^3*x^2 - 84*x^3*cosh(1)^3 - 84*d*x^2*cosh(1)^2 - d^3 + 21*(c^2*d^2
*x^3 - d^2*x)*cosh(1))*sinh(1)^6 - 3*(c^4*d^4*x^3 - 3*c^2*d^4*x)*cosh(1)^5 - 3*(c^4*d^4*x^3 - 3*c^2*d^4*x + 42
*x^3*cosh(1)^4 + 56*d*x^2*cosh(1)^3 - 21*(c^2*d^2*x^3 - d^2*x)*cosh(1)^2 - 2*(9*c^2*d^3*x^2 - d^3)*cosh(1))*si
nh(1)^5 - 3*(3*c^4*d^5*x^2 - c^2*d^5)*cosh(1)^4 - 3*(3*c^4*d^5*x^2 + 42*x^3*cosh(1)^5 - c^2*d^5 + 70*d*x^2*cos
h(1)^4 - 35*(c^2*d^2*x^3 - d^2*x)*cosh(1)^3 - 5*(9*c^2*d^3*x^2 - d^3)*cosh(1)^2 + 5*(c^4*d^4*x^3 - 3*c^2*d^4*x
)*cosh(1))*sinh(1)^4 + (c^6*d^6*x^3 - 9*c^4*d^6*x)*cosh(1)^3 + (c^6*d^6*x^3 - 9*c^4*d^6*x - 84*x^3*cosh(1)^6 -
168*d*x^2*cosh(1)^5 + 105*(c^2*d^2*x^3 - d^2*x)*cosh(1)^4 + 20*(9*c^2*d^3*x^2 - d^3)*cosh(1)^3 - 30*(c^4*d^4*
x^3 - 3*c^2*d^4*x)*cosh(1)^2 - 12*(3*c^4*d^5*x^2 - c^2*d^5)*cosh(1))*sinh(1)^3 + 3*(c^6*d^7*x^2 - c^4*d^7)*cos
h(1)^2 + 3*(c^6*d^7*x^2 - c^4*d^7 - 12*x^3*cosh(1)^7 - 28*d*x^2*cosh(1)^6 + 21*(c^2*d^2*x^3 - d^2*x)*cosh(1)^5
+ 5*(9*c^2*d^3*x^2 - d^3)*cosh(1)^4 - 10*(c^4*d^4*x^3 - 3*c^2*d^4*x)*cosh(1)^3 - 6*(3*c^4*d^5*x^2 - c^2*d^5)*
cosh(1)^2 + (c^6*d^6*x^3 - 9*c^4*d^6*x)*cosh(1))*sinh(1)^2 + 3*(c^6*d^8*x - 3*x^3*cosh(1)^8 - 8*d*x^2*cosh(1)^
7 + 7*(c^2*d^2*x^3 - d^2*x)*cosh(1)^6 + 2*(9*c^2*d^3*x^2 - d^3)*cosh(1)^5 - 5*(c^4*d^4*x^3 - 3*c^2*d^4*x)*cosh
(1)^4 - 4*(3*c^4*d^5*x^2 - c^2*d^5)*cosh(1)^3 + (c^6*d^6*x^3 - 9*c^4*d^6*x)*cosh(1)^2 + 2*(c^6*d^7*x^2 - c^4*d
^7)*cosh(1))*sinh(1))*log(-c*x + sqrt(c^2*x^2 - 1)) + 3*(3*c^6*d^8*x - 5*c^4*d^4*x^3*cosh(1)^4 - 12*c^4*d^5*x^
2*cosh(1)^3 + 3*(c^6*d^6*x^3 - 3*c^4*d^6*x)*cosh(1)^2 + 2*(3*c^6*d^7*x^2 - c^4*d^7)*cosh(1))*sinh(1) + (7*c^5*
d^7*x*cosh(1)^2 + 4*c^5*d^8*cosh(1) - 8*c^3*d^5...
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {acosh}{\left (c x \right )}}{\left (d + e x\right )^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(acosh(c*x)/(e*x+d)**4,x)
[Out]
Integral(acosh(c*x)/(d + e*x)**4, x)
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(arccosh(c*x)/(e*x+d)^4,x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, replacing 0 by ` u`, a substitution variable should perhaps be purged.Warning, integration of abs
or sign ass
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\mathrm {acosh}\left (c\,x\right )}{{\left (d+e\,x\right )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(acosh(c*x)/(d + e*x)^4,x)
[Out]
int(acosh(c*x)/(d + e*x)^4, x)
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