3.1.20 \(\int \frac {a+b \cosh ^{-1}(c x)}{(d+e x)^4} \, dx\) [20]
Optimal. Leaf size=202 \[ -\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {b c^3 d \sqrt {-1+c x} \sqrt {1+c x}}{2 (c d-e)^2 (c d+e)^2 (d+e x)}-\frac {a+b \cosh ^{-1}(c x)}{3 e (d+e x)^3}+\frac {b c^3 \left (2 c^2 d^2+e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c d+e} \sqrt {1+c x}}{\sqrt {c d-e} \sqrt {-1+c x}}\right )}{3 (c d-e)^{5/2} e (c d+e)^{5/2}} \]
[Out]
1/3*(-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)-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^2*d^2-e^2)^2/(e*x+d)
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Rubi [A]
time = 0.16, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5963, 105, 156,
12, 95, 214} \begin {gather*} -\frac {a+b \cosh ^{-1}(c x)}{3 e (d+e x)^3}-\frac {b c^3 d \sqrt {c x-1} \sqrt {c x+1}}{2 (c d-e)^2 (c d+e)^2 (d+e x)}-\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b 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}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(a + b*ArcCosh[c*x])/(d + e*x)^4,x]
[Out]
-1/6*(b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/((c^2*d^2 - e^2)*(d + e*x)^2) - (b*c^3*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
/(2*(c*d - e)^2*(c*d + e)^2*(d + e*x)) - (a + b*ArcCosh[c*x])/(3*e*(d + e*x)^3) + (b*c^3*(2*c^2*d^2 + e^2)*Arc
Tanh[(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 {a+b \cosh ^{-1}(c x)}{(d+e x)^4} \, dx &=-\frac {a+b \cosh ^{-1}(c x)}{3 e (d+e x)^3}+\frac {(b c) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x} (d+e x)^3} \, dx}{3 e}\\ &=-\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 {a+b \cosh ^{-1}(c x)}{3 e (d+e x)^3}-\frac {(b 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 {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 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {a+b \cosh ^{-1}(c x)}{3 e (d+e x)^3}+\frac {(b 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 {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 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {a+b \cosh ^{-1}(c x)}{3 e (d+e x)^3}+\frac {\left (b 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 {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 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {a+b \cosh ^{-1}(c x)}{3 e (d+e x)^3}+\frac {\left (b 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 {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 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {a+b \cosh ^{-1}(c x)}{3 e (d+e x)^3}+\frac {b 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 = 0.70, size = 259, normalized size = 1.28 \begin {gather*} -\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 \cosh ^{-1}(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} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(a + b*ArcCosh[c*x])/(d + e*x)^4,x]
[Out]
-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
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1140\) vs.
\(2(176)=352\).
time = 5.68, size = 1141, normalized size = 5.65 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*arccosh(c*x))/(e*x+d)^4,x,method=_RETURNVERBOSE)
[Out]
1/c*(-1/3*a*c^4/(c*e*x+c*d)^3/e-1/3*b*c^4/(c*e*x+c*d)^3/e*arccosh(c*x)-1/3*b*c^8/e^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)*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))*d^4-2/3*b*c^8/e*(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)*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))*d^3*x-1/3*b*c^8*(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)*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))*d^2*x^2-2/3*b*c^6*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c*d+e)/(c*d-e)/(c^2*d^2-e^2
)/(c*e*x+c*d)^2*d^2-1/2*b*c^6*e*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c*d+e)/(c*d-e)/(c^2*d^2-e^2)/(c*e*x+c*d)^2*d*x-1/
6*b*c^6*(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)*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))*d^2-1/3*b*c^6*e*(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)*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))*d*x-1/6*b*c^6*e^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)*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))*x^2+1/6*b*c^4*e^2*(c*x-1)^(1/2)*(c*x+1)
^(1/2)/(c*d+e)/(c*d-e)/(c^2*d^2-e^2)/(c*e*x+c*d)^2)
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*arccosh(c*x))/(e*x+d)^4,x, algorithm="maxima")
[Out]
-1/6*(6*c*integrate(1/3/(c^3*x^6*e^4 + 3*c^3*d*x^5*e^3 - 3*c*d^2*x^2*e^2 - c*d^3*x*e + (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*x^5*e^4 + 3*c^2*d*x^4*e^3 + (3*c^2*d^2*e^2 - e^4)*x^3 - 3*d^2*x*e^2
- d^3*e + (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(x*e + 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 + (c^6*d^6*e^4 - 3*c^4*d^4*e^6 + 3*c^2*d^
2*e^8 - e^10)*x^3 - d^3*e^7 + 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/(x^3*e^4 + 3*d*x^2*e^3 + 3*d^2*x*e^2 + d^3*e)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 4943 vs.
\(2 (175) = 350\).
time = 0.79, size = 9905, normalized size = 49.03 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*arccosh(c*x))/(e*x+d)^4,x, algorithm="fricas")
[Out]
[-1/6*(9*b*c^6*d^8*x*cosh(1) - 3*b*c^4*d^4*x^3*cosh(1)^5 + (2*a + 3*b)*c^6*d^9 - 2*a*d^3*cosh(1)^6 - 2*a*d^3*s
inh(1)^6 - 3*(b*c^4*d^4*x^3 + 4*a*d^3*cosh(1))*sinh(1)^5 - 3*(3*b*c^4*d^5*x^2 - 2*a*c^2*d^5)*cosh(1)^4 - 3*(5*
b*c^4*d^4*x^3*cosh(1) + 3*b*c^4*d^5*x^2 - 2*a*c^2*d^5 + 10*a*d^3*cosh(1)^2)*sinh(1)^4 + 3*(b*c^6*d^6*x^3 - 3*b
*c^4*d^6*x)*cosh(1)^3 + (3*b*c^6*d^6*x^3 - 30*b*c^4*d^4*x^3*cosh(1)^2 - 9*b*c^4*d^6*x - 40*a*d^3*cosh(1)^3 - 1
2*(3*b*c^4*d^5*x^2 - 2*a*c^2*d^5)*cosh(1))*sinh(1)^3 + 3*(3*b*c^6*d^7*x^2 - (2*a + b)*c^4*d^7)*cosh(1)^2 + 3*(
3*b*c^6*d^7*x^2 - 10*b*c^4*d^4*x^3*cosh(1)^3 - (2*a + b)*c^4*d^7 - 10*a*d^3*cosh(1)^4 - 6*(3*b*c^4*d^5*x^2 - 2
*a*c^2*d^5)*cosh(1)^2 + 3*(b*c^6*d^6*x^3 - 3*b*c^4*d^6*x)*cosh(1))*sinh(1)^2 - (6*b*c^5*d^7*x*cosh(1) + b*c^3*
d^3*x^3*cosh(1)^5 + b*c^3*d^3*x^3*sinh(1)^5 + 2*b*c^5*d^8 + 3*b*c^3*d^4*x^2*cosh(1)^4 + (5*b*c^3*d^3*x^3*cosh(
1) + 3*b*c^3*d^4*x^2)*sinh(1)^4 + (2*b*c^5*d^5*x^3 + 3*b*c^3*d^5*x)*cosh(1)^3 + (2*b*c^5*d^5*x^3 + 10*b*c^3*d^
3*x^3*cosh(1)^2 + 12*b*c^3*d^4*x^2*cosh(1) + 3*b*c^3*d^5*x)*sinh(1)^3 + (6*b*c^5*d^6*x^2 + b*c^3*d^6)*cosh(1)^
2 + (6*b*c^5*d^6*x^2 + 10*b*c^3*d^3*x^3*cosh(1)^3 + 18*b*c^3*d^4*x^2*cosh(1)^2 + b*c^3*d^6 + 3*(2*b*c^5*d^5*x^
3 + 3*b*c^3*d^5*x)*cosh(1))*sinh(1)^2 + (6*b*c^5*d^7*x + 5*b*c^3*d^3*x^3*cosh(1)^4 + 12*b*c^3*d^4*x^2*cosh(1)^
3 + 3*(2*b*c^5*d^5*x^3 + 3*b*c^3*d^5*x)*cosh(1)^2 + 2*(6*b*c^5*d^6*x^2 + b*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*b*c^6*d^7*x^2*cosh(1)^2 + 3*b*c^6*
d^8*x*cosh(1) - 9*b*c^4*d^5*x^2*cosh(1)^4 + 9*b*c^2*d^3*x^2*cosh(1)^6 - b*x^3*cosh(1)^9 - b*x^3*sinh(1)^9 - 3*
b*d*x^2*cosh(1)^8 - 3*(3*b*x^3*cosh(1) + b*d*x^2)*sinh(1)^8 + 3*(b*c^2*d^2*x^3 - b*d^2*x)*cosh(1)^7 + 3*(b*c^2
*d^2*x^3 - 12*b*x^3*cosh(1)^2 - 8*b*d*x^2*cosh(1) - b*d^2*x)*sinh(1)^7 + 3*(3*b*c^2*d^3*x^2 - 28*b*x^3*cosh(1)
^3 - 28*b*d*x^2*cosh(1)^2 + 7*(b*c^2*d^2*x^3 - b*d^2*x)*cosh(1))*sinh(1)^6 - 3*(b*c^4*d^4*x^3 - 3*b*c^2*d^4*x)
*cosh(1)^5 - 3*(b*c^4*d^4*x^3 - 18*b*c^2*d^3*x^2*cosh(1) - 3*b*c^2*d^4*x + 42*b*x^3*cosh(1)^4 + 56*b*d*x^2*cos
h(1)^3 - 21*(b*c^2*d^2*x^3 - b*d^2*x)*cosh(1)^2)*sinh(1)^5 - 3*(3*b*c^4*d^5*x^2 - 45*b*c^2*d^3*x^2*cosh(1)^2 +
42*b*x^3*cosh(1)^5 + 70*b*d*x^2*cosh(1)^4 - 35*(b*c^2*d^2*x^3 - b*d^2*x)*cosh(1)^3 + 5*(b*c^4*d^4*x^3 - 3*b*c
^2*d^4*x)*cosh(1))*sinh(1)^4 + (b*c^6*d^6*x^3 - 9*b*c^4*d^6*x)*cosh(1)^3 + (b*c^6*d^6*x^3 - 36*b*c^4*d^5*x^2*c
osh(1) - 9*b*c^4*d^6*x + 180*b*c^2*d^3*x^2*cosh(1)^3 - 84*b*x^3*cosh(1)^6 - 168*b*d*x^2*cosh(1)^5 + 105*(b*c^2
*d^2*x^3 - b*d^2*x)*cosh(1)^4 - 30*(b*c^4*d^4*x^3 - 3*b*c^2*d^4*x)*cosh(1)^2)*sinh(1)^3 + 3*(b*c^6*d^7*x^2 - 1
8*b*c^4*d^5*x^2*cosh(1)^2 + 45*b*c^2*d^3*x^2*cosh(1)^4 - 12*b*x^3*cosh(1)^7 - 28*b*d*x^2*cosh(1)^6 + 21*(b*c^2
*d^2*x^3 - b*d^2*x)*cosh(1)^5 - 10*(b*c^4*d^4*x^3 - 3*b*c^2*d^4*x)*cosh(1)^3 + (b*c^6*d^6*x^3 - 9*b*c^4*d^6*x)
*cosh(1))*sinh(1)^2 + 3*(2*b*c^6*d^7*x^2*cosh(1) + b*c^6*d^8*x - 12*b*c^4*d^5*x^2*cosh(1)^3 + 18*b*c^2*d^3*x^2
*cosh(1)^5 - 3*b*x^3*cosh(1)^8 - 8*b*d*x^2*cosh(1)^7 + 7*(b*c^2*d^2*x^3 - b*d^2*x)*cosh(1)^6 - 5*(b*c^4*d^4*x^
3 - 3*b*c^2*d^4*x)*cosh(1)^4 + (b*c^6*d^6*x^3 - 9*b*c^4*d^6*x)*cosh(1)^2)*sinh(1))*log(c*x + sqrt(c^2*x^2 - 1)
) - 2*(3*b*c^6*d^8*x*cosh(1) + b*c^6*d^9 - b*x^3*cosh(1)^9 - b*x^3*sinh(1)^9 - 3*b*d*x^2*cosh(1)^8 - 3*(3*b*x^
3*cosh(1) + b*d*x^2)*sinh(1)^8 + 3*(b*c^2*d^2*x^3 - b*d^2*x)*cosh(1)^7 + 3*(b*c^2*d^2*x^3 - 12*b*x^3*cosh(1)^2
- 8*b*d*x^2*cosh(1) - b*d^2*x)*sinh(1)^7 + (9*b*c^2*d^3*x^2 - b*d^3)*cosh(1)^6 + (9*b*c^2*d^3*x^2 - 84*b*x^3*
cosh(1)^3 - 84*b*d*x^2*cosh(1)^2 - b*d^3 + 21*(b*c^2*d^2*x^3 - b*d^2*x)*cosh(1))*sinh(1)^6 - 3*(b*c^4*d^4*x^3
- 3*b*c^2*d^4*x)*cosh(1)^5 - 3*(b*c^4*d^4*x^3 - 3*b*c^2*d^4*x + 42*b*x^3*cosh(1)^4 + 56*b*d*x^2*cosh(1)^3 - 21
*(b*c^2*d^2*x^3 - b*d^2*x)*cosh(1)^2 - 2*(9*b*c^2*d^3*x^2 - b*d^3)*cosh(1))*sinh(1)^5 - 3*(3*b*c^4*d^5*x^2 - b
*c^2*d^5)*cosh(1)^4 - 3*(3*b*c^4*d^5*x^2 + 42*b*x^3*cosh(1)^5 - b*c^2*d^5 + 70*b*d*x^2*cosh(1)^4 - 35*(b*c^2*d
^2*x^3 - b*d^2*x)*cosh(1)^3 - 5*(9*b*c^2*d^3*x^2 - b*d^3)*cosh(1)^2 + 5*(b*c^4*d^4*x^3 - 3*b*c^2*d^4*x)*cosh(1
))*sinh(1)^4 + (b*c^6*d^6*x^3 - 9*b*c^4*d^6*x)*cosh(1)^3 + (b*c^6*d^6*x^3 - 9*b*c^4*d^6*x - 84*b*x^3*cosh(1)^6
- 168*b*d*x^2*cosh(1)^5 + 105*(b*c^2*d^2*x^3 - b*d^2*x)*cosh(1)^4 + 20*(9*b*c^2*d^3*x^2 - b*d^3)*cosh(1)^3 -
30*(b*c^4*d^4*x^3 - 3*b*c^2*d^4*x)*cosh(1)^2 - 12*(3*b*c^4*d^5*x^2 - b*c^2*d^5)*cosh(1))*sinh(1)^3 + 3*(b*c^6*
d^7*x^2 - b*c^4*d^7)*cosh(1)^2 + 3*(b*c^6*d^7*x^2 - b*c^4*d^7 - 12*b*x^3*cosh(1)^7 - 28*b*d*x^2*cosh(1)^6 + 21
*(b*c^2*d^2*x^3 - b*d^2*x)*cosh(1)^5 + 5*(9*b*c^2*d^3*x^2 - b*d^3)*cosh(1)^4 - 10*(b*c^4*d^4*x^3 - 3*b*c^2*d^4
*x)*cosh(1)^3 - 6*(3*b*c^4*d^5*x^2 - b*c^2*d^5)...
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b \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((a+b*acosh(c*x))/(e*x+d)**4,x)
[Out]
Integral((a + b*acosh(c*x))/(d + e*x)**4, x)
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*arccosh(c*x))/(e*x+d)^4,x, algorithm="giac")
[Out]
integrate((b*arccosh(c*x) + a)/(e*x + d)^4, x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {a+b\,\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((a + b*acosh(c*x))/(d + e*x)^4,x)
[Out]
int((a + b*acosh(c*x))/(d + e*x)^4, x)
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