3.1.50 \(\int \frac {a+b \text {csch}^{-1}(c x)}{(d+e x)^3} \, dx\) [50]

Optimal. Leaf size=163 \[ -\frac {b c e \sqrt {1+\frac {1}{c^2 x^2}}}{2 d \left (c^2 d^2+e^2\right ) \left (e+\frac {d}{x}\right )}+\frac {b \text {csch}^{-1}(c x)}{2 d^2 e}-\frac {a+b \text {csch}^{-1}(c x)}{2 e (d+e x)^2}+\frac {b \left (2 c^2 d^2+e^2\right ) \tanh ^{-1}\left (\frac {c^2 d-\frac {e}{x}}{c \sqrt {c^2 d^2+e^2} \sqrt {1+\frac {1}{c^2 x^2}}}\right )}{2 d^2 \left (c^2 d^2+e^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.20, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6425, 1582, 1489, 1665, 858, 221, 739, 212} \begin {gather*} -\frac {a+b \text {csch}^{-1}(c x)}{2 e (d+e x)^2}-\frac {b c e \sqrt {\frac {1}{c^2 x^2}+1}}{2 d \left (c^2 d^2+e^2\right ) \left (\frac {d}{x}+e\right )}+\frac {b \left (2 c^2 d^2+e^2\right ) \tanh ^{-1}\left (\frac {c^2 d-\frac {e}{x}}{c \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {c^2 d^2+e^2}}\right )}{2 d^2 \left (c^2 d^2+e^2\right )^{3/2}}+\frac {b \text {csch}^{-1}(c x)}{2 d^2 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt[a])]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 739

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 858

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rule 1489

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(d + e*x)^q*(a + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x
] && EqQ[n2, 2*n] && IntegerQ[Simplify[(m + 1)/n]]

Rule 1582

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^(mn_.))^(q_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.), x_Symbol] :> Int[x^(m + mn*q
)*(e + d/x^mn)^q*(a + c*x^n2)^p, x] /; FreeQ[{a, c, d, e, m, mn, p}, x] && EqQ[n2, -2*mn] && IntegerQ[q] && (P
osQ[n2] ||  !IntegerQ[p])

Rule 1665

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, d
+ e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/((m + 1
)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p*ExpandToSum[(m
+ 1)*(c*d^2 + a*e^2)*Q + c*d*R*(m + 1) - c*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, c, d, e, p}, x] && Po
lyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1]

Rule 6425

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

Rubi steps

\begin {align*} \int \frac {a+b \text {csch}^{-1}(c x)}{(d+e x)^3} \, dx &=-\frac {a+b \text {csch}^{-1}(c x)}{2 e (d+e x)^2}-\frac {b \int \frac {1}{\sqrt {1+\frac {1}{c^2 x^2}} x^2 (d+e x)^2} \, dx}{2 c e}\\ &=-\frac {a+b \text {csch}^{-1}(c x)}{2 e (d+e x)^2}-\frac {b \int \frac {1}{\sqrt {1+\frac {1}{c^2 x^2}} \left (e+\frac {d}{x}\right )^2 x^4} \, dx}{2 c e}\\ &=-\frac {a+b \text {csch}^{-1}(c x)}{2 e (d+e x)^2}+\frac {b \text {Subst}\left (\int \frac {x^2}{(e+d x)^2 \sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{2 c e}\\ &=-\frac {b c e \sqrt {1+\frac {1}{c^2 x^2}}}{2 d \left (c^2 d^2+e^2\right ) \left (e+\frac {d}{x}\right )}-\frac {a+b \text {csch}^{-1}(c x)}{2 e (d+e x)^2}-\frac {(b c) \text {Subst}\left (\int \frac {e-\left (d+\frac {e^2}{c^2 d}\right ) x}{(e+d x) \sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{2 e \left (c^2 d^2+e^2\right )}\\ &=-\frac {b c e \sqrt {1+\frac {1}{c^2 x^2}}}{2 d \left (c^2 d^2+e^2\right ) \left (e+\frac {d}{x}\right )}-\frac {a+b \text {csch}^{-1}(c x)}{2 e (d+e x)^2}+\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{2 c d^2 e}-\frac {\left (b c \left (2+\frac {e^2}{c^2 d^2}\right )\right ) \text {Subst}\left (\int \frac {1}{(e+d x) \sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{2 \left (c^2 d^2+e^2\right )}\\ &=-\frac {b c e \sqrt {1+\frac {1}{c^2 x^2}}}{2 d \left (c^2 d^2+e^2\right ) \left (e+\frac {d}{x}\right )}+\frac {b \text {csch}^{-1}(c x)}{2 d^2 e}-\frac {a+b \text {csch}^{-1}(c x)}{2 e (d+e x)^2}+\frac {\left (b c \left (2+\frac {e^2}{c^2 d^2}\right )\right ) \text {Subst}\left (\int \frac {1}{d^2+\frac {e^2}{c^2}-x^2} \, dx,x,\frac {d-\frac {e}{c^2 x}}{\sqrt {1+\frac {1}{c^2 x^2}}}\right )}{2 \left (c^2 d^2+e^2\right )}\\ &=-\frac {b c e \sqrt {1+\frac {1}{c^2 x^2}}}{2 d \left (c^2 d^2+e^2\right ) \left (e+\frac {d}{x}\right )}+\frac {b \text {csch}^{-1}(c x)}{2 d^2 e}-\frac {a+b \text {csch}^{-1}(c x)}{2 e (d+e x)^2}+\frac {b \left (2 c^2 d^2+e^2\right ) \tanh ^{-1}\left (\frac {c^2 d-\frac {e}{x}}{c \sqrt {c^2 d^2+e^2} \sqrt {1+\frac {1}{c^2 x^2}}}\right )}{2 d^2 \left (c^2 d^2+e^2\right )^{3/2}}\\ \end {align*}

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Mathematica [A]
time = 0.32, size = 204, normalized size = 1.25 \begin {gather*} \frac {1}{2} \left (-\frac {a}{e (d+e x)^2}-\frac {b c e \sqrt {1+\frac {1}{c^2 x^2}} x}{d \left (c^2 d^2+e^2\right ) (d+e x)}-\frac {b \text {csch}^{-1}(c x)}{e (d+e x)^2}+\frac {b \sinh ^{-1}\left (\frac {1}{c x}\right )}{d^2 e}+\frac {b \left (2 c^2 d^2+e^2\right ) \log (d+e x)}{d^2 \left (c^2 d^2+e^2\right )^{3/2}}-\frac {b \left (2 c^2 d^2+e^2\right ) \log \left (e+c \left (-c d+\sqrt {c^2 d^2+e^2} \sqrt {1+\frac {1}{c^2 x^2}}\right ) x\right )}{d^2 \left (c^2 d^2+e^2\right )^{3/2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(928\) vs. \(2(150)=300\).
time = 2.32, size = 929, normalized size = 5.70

method	result	size
derivativedivides	\(\frac {-\frac {a \,c^{3}}{2 \left (e c x +c d \right )^{2} e}-\frac {b \,c^{3} \mathrm {arccsch}\left (c x \right )}{2 \left (e c x +c d \right )^{2} e}+\frac {b \,c^{3} \sqrt {c^{2} x^{2}+1}\, d \arctanh \left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{2 e \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, x \left (c^{2} d^{2}+e^{2}\right ) \left (e c x +c d \right )}+\frac {b \,c^{3} \sqrt {c^{2} x^{2}+1}\, \arctanh \left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{2 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, \left (c^{2} d^{2}+e^{2}\right ) \left (e c x +c d \right )}-\frac {b \,c^{3} \sqrt {c^{2} x^{2}+1}\, d \ln \left (\frac {2 \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}+1}\, e -2 d \,c^{2} x +2 e}{e c x +c d}\right )}{e \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, x \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 )}-\frac {b \,c^{3} \sqrt {c^{2} x^{2}+1}\, \ln \left (\frac {2 \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}+1}\, e -2 d \,c^{2} x +2 e}{e c x +c d}\right )}{\sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, \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 )}-\frac {b c e \left (c^{2} x^{2}+1\right )}{2 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, x d \left (c^{2} d^{2}+e^{2}\right ) \left (e c x +c d \right )}+\frac {b c e \sqrt {c^{2} x^{2}+1}\, \arctanh \left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{2 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, x d \left (c^{2} d^{2}+e^{2}\right ) \left (e c x +c d \right )}+\frac {b c \,e^{2} \sqrt {c^{2} x^{2}+1}\, \arctanh \left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{2 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, d^{2} \left (c^{2} d^{2}+e^{2}\right ) \left (e c x +c d \right )}-\frac {b c e \sqrt {c^{2} x^{2}+1}\, \ln \left (\frac {2 \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}+1}\, e -2 d \,c^{2} x +2 e}{e c x +c d}\right )}{2 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, x d \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 )}-\frac {b c \,e^{2} \sqrt {c^{2} x^{2}+1}\, \ln \left (\frac {2 \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}+1}\, e -2 d \,c^{2} x +2 e}{e c x +c d}\right )}{2 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, d^{2} \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 )}}{c}\)	\(929\)
default	\(\frac {-\frac {a \,c^{3}}{2 \left (e c x +c d \right )^{2} e}-\frac {b \,c^{3} \mathrm {arccsch}\left (c x \right )}{2 \left (e c x +c d \right )^{2} e}+\frac {b \,c^{3} \sqrt {c^{2} x^{2}+1}\, d \arctanh \left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{2 e \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, x \left (c^{2} d^{2}+e^{2}\right ) \left (e c x +c d \right )}+\frac {b \,c^{3} \sqrt {c^{2} x^{2}+1}\, \arctanh \left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{2 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, \left (c^{2} d^{2}+e^{2}\right ) \left (e c x +c d \right )}-\frac {b \,c^{3} \sqrt {c^{2} x^{2}+1}\, d \ln \left (\frac {2 \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}+1}\, e -2 d \,c^{2} x +2 e}{e c x +c d}\right )}{e \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, x \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 )}-\frac {b \,c^{3} \sqrt {c^{2} x^{2}+1}\, \ln \left (\frac {2 \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}+1}\, e -2 d \,c^{2} x +2 e}{e c x +c d}\right )}{\sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, \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 )}-\frac {b c e \left (c^{2} x^{2}+1\right )}{2 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, x d \left (c^{2} d^{2}+e^{2}\right ) \left (e c x +c d \right )}+\frac {b c e \sqrt {c^{2} x^{2}+1}\, \arctanh \left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{2 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, x d \left (c^{2} d^{2}+e^{2}\right ) \left (e c x +c d \right )}+\frac {b c \,e^{2} \sqrt {c^{2} x^{2}+1}\, \arctanh \left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{2 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, d^{2} \left (c^{2} d^{2}+e^{2}\right ) \left (e c x +c d \right )}-\frac {b c e \sqrt {c^{2} x^{2}+1}\, \ln \left (\frac {2 \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}+1}\, e -2 d \,c^{2} x +2 e}{e c x +c d}\right )}{2 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, x d \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 )}-\frac {b c \,e^{2} \sqrt {c^{2} x^{2}+1}\, \ln \left (\frac {2 \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}+1}\, e -2 d \,c^{2} x +2 e}{e c x +c d}\right )}{2 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, d^{2} \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 )}}{c}\)	\(929\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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*arccsch(c*x))/(e*x+d)^3,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2579 vs. \(2 (145) = 290\).
time = 1.02, size = 2579, normalized size = 15.82 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(a*c^4*d^6 + b*c^3*d^5*cosh(1) + b*c*d*x^2*cosh(1)^5 + b*c*d*x^2*sinh(1)^5 + (2*b*c*d^2*x + a*d^2)*cosh(1
)^4 + (5*b*c*d*x^2*cosh(1) + 2*b*c*d^2*x + a*d^2)*sinh(1)^4 + (b*c^3*d^3*x^2 + b*c*d^3)*cosh(1)^3 + (b*c^3*d^3
*x^2 + 10*b*c*d*x^2*cosh(1)^2 + b*c*d^3 + 4*(2*b*c*d^2*x + a*d^2)*cosh(1))*sinh(1)^3 + 2*(b*c^3*d^4*x + a*c^2*
d^4)*cosh(1)^2 + (2*b*c^3*d^4*x + 10*b*c*d*x^2*cosh(1)^3 + 2*a*c^2*d^4 + 6*(2*b*c*d^2*x + a*d^2)*cosh(1)^2 + 3
*(b*c^3*d^3*x^2 + b*c*d^3)*cosh(1))*sinh(1)^2 - (4*b*c^2*d^3*x*cosh(1)^2 + 2*b*c^2*d^4*cosh(1) + b*x^2*cosh(1)
^5 + b*x^2*sinh(1)^5 + 2*b*d*x*cosh(1)^4 + (5*b*x^2*cosh(1) + 2*b*d*x)*sinh(1)^4 + (2*b*c^2*d^2*x^2 + b*d^2)*c
osh(1)^3 + (2*b*c^2*d^2*x^2 + 10*b*x^2*cosh(1)^2 + 8*b*d*x*cosh(1) + b*d^2)*sinh(1)^3 + (4*b*c^2*d^3*x + 10*b*
x^2*cosh(1)^3 + 12*b*d*x*cosh(1)^2 + 3*(2*b*c^2*d^2*x^2 + b*d^2)*cosh(1))*sinh(1)^2 + (8*b*c^2*d^3*x*cosh(1) +
2*b*c^2*d^4 + 5*b*x^2*cosh(1)^4 + 8*b*d*x*cosh(1)^3 + 3*(2*b*c^2*d^2*x^2 + b*d^2)*cosh(1)^2)*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*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 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))) + (c^3*d^2*x + c*x*cosh(1)^2 + 2*c*x*cosh(1)*sinh(1) + c*x*sinh(1)^2)*sqr
t((c^2*x^2 + 1)/(c^2*x^2)))/(x*cosh(1) + x*sinh(1) + d)) - (2*b*c^4*d^5*x*cosh(1) + b*c^4*d^6 + 4*b*c^2*d^3*x*
cosh(1)^3 + b*x^2*cosh(1)^6 + b*x^2*sinh(1)^6 + 2*b*d*x*cosh(1)^5 + 2*(3*b*x^2*cosh(1) + b*d*x)*sinh(1)^5 + (2
*b*c^2*d^2*x^2 + b*d^2)*cosh(1)^4 + (2*b*c^2*d^2*x^2 + 15*b*x^2*cosh(1)^2 + 10*b*d*x*cosh(1) + b*d^2)*sinh(1)^
4 + 4*(b*c^2*d^3*x + 5*b*x^2*cosh(1)^3 + 5*b*d*x*cosh(1)^2 + (2*b*c^2*d^2*x^2 + b*d^2)*cosh(1))*sinh(1)^3 + (b
*c^4*d^4*x^2 + 2*b*c^2*d^4)*cosh(1)^2 + (b*c^4*d^4*x^2 + 12*b*c^2*d^3*x*cosh(1) + 2*b*c^2*d^4 + 15*b*x^2*cosh(
1)^4 + 20*b*d*x*cosh(1)^3 + 6*(2*b*c^2*d^2*x^2 + b*d^2)*cosh(1)^2)*sinh(1)^2 + 2*(b*c^4*d^5*x + 6*b*c^2*d^3*x*
cosh(1)^2 + 3*b*x^2*cosh(1)^5 + 5*b*d*x*cosh(1)^4 + 2*(2*b*c^2*d^2*x^2 + b*d^2)*cosh(1)^3 + (b*c^4*d^4*x^2 + 2
*b*c^2*d^4)*cosh(1))*sinh(1))*log(c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) - c*x + 1) + (2*b*c^4*d^5*x*cosh(1) + b*c^
4*d^6 + 4*b*c^2*d^3*x*cosh(1)^3 + b*x^2*cosh(1)^6 + b*x^2*sinh(1)^6 + 2*b*d*x*cosh(1)^5 + 2*(3*b*x^2*cosh(1) +
b*d*x)*sinh(1)^5 + (2*b*c^2*d^2*x^2 + b*d^2)*cosh(1)^4 + (2*b*c^2*d^2*x^2 + 15*b*x^2*cosh(1)^2 + 10*b*d*x*cos
h(1) + b*d^2)*sinh(1)^4 + 4*(b*c^2*d^3*x + 5*b*x^2*cosh(1)^3 + 5*b*d*x*cosh(1)^2 + (2*b*c^2*d^2*x^2 + b*d^2)*c
osh(1))*sinh(1)^3 + (b*c^4*d^4*x^2 + 2*b*c^2*d^4)*cosh(1)^2 + (b*c^4*d^4*x^2 + 12*b*c^2*d^3*x*cosh(1) + 2*b*c^
2*d^4 + 15*b*x^2*cosh(1)^4 + 20*b*d*x*cosh(1)^3 + 6*(2*b*c^2*d^2*x^2 + b*d^2)*cosh(1)^2)*sinh(1)^2 + 2*(b*c^4*
d^5*x + 6*b*c^2*d^3*x*cosh(1)^2 + 3*b*x^2*cosh(1)^5 + 5*b*d*x*cosh(1)^4 + 2*(2*b*c^2*d^2*x^2 + b*d^2)*cosh(1)^
3 + (b*c^4*d^4*x^2 + 2*b*c^2*d^4)*cosh(1))*sinh(1))*log(c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) - c*x - 1) + (b*c^4*
d^6 + 2*b*c^2*d^4*cosh(1)^2 + b*d^2*cosh(1)^4 + 4*b*d^2*cosh(1)*sinh(1)^3 + b*d^2*sinh(1)^4 + 2*(b*c^2*d^4 + 3
*b*d^2*cosh(1)^2)*sinh(1)^2 + 4*(b*c^2*d^4*cosh(1) + b*d^2*cosh(1)^3)*sinh(1))*log((c*x*sqrt((c^2*x^2 + 1)/(c^
2*x^2)) + 1)/(c*x)) + (b*c^3*d^5 + 5*b*c*d*x^2*cosh(1)^4 + 4*(2*b*c*d^2*x + a*d^2)*cosh(1)^3 + 3*(b*c^3*d^3*x^
2 + b*c*d^3)*cosh(1)^2 + 4*(b*c^3*d^4*x + a*c^2*d^4)*cosh(1))*sinh(1) + (b*c^3*d^3*x^2*cosh(1)^3 + b*c^3*d^4*x
*cosh(1)^2 + b*c*d*x^2*cosh(1)^5 + b*c*d*x^2*sinh(1)^5 + b*c*d^2*x*cosh(1)^4 + (5*b*c*d*x^2*cosh(1) + b*c*d^2*
x)*sinh(1)^4 + (b*c^3*d^3*x^2 + 10*b*c*d*x^2*cosh(1)^2 + 4*b*c*d^2*x*cosh(1))*sinh(1)^3 + (3*b*c^3*d^3*x^2*cos
h(1) + b*c^3*d^4*x + 10*b*c*d*x^2*cosh(1)^3 + 6*b*c*d^2*x*cosh(1)^2)*sinh(1)^2 + (3*b*c^3*d^3*x^2*cosh(1)^2 +
2*b*c^3*d^4*x*cosh(1) + 5*b*c*d*x^2*cosh(1)^4 + 4*b*c*d^2*x*cosh(1)^3)*sinh(1))*sqrt((c^2*x^2 + 1)/(c^2*x^2)))
/(2*c^4*d^7*x*cosh(1)^2 + c^4*d^8*cosh(1) + 4*c^2*d^5*x*cosh(1)^4 + d^2*x^2*cosh(1)^7 + d^2*x^2*sinh(1)^7 + 2*
d^3*x*cosh(1)^6 + (7*d^2*x^2*cosh(1) + 2*d^3*x)*sinh(1)^6 + (2*c^2*d^4*x^2 + d^4)*cosh(1)^5 + (2*c^2*d^4*x^2 +
21*d^2*x^2*cosh(1)^2 + 12*d^3*x*cosh(1) + d^4)*sinh(1)^5 + (4*c^2*d^5*x + 35*d^2*x^2*cosh(1)^3 + 30*d^3*x*cos
h(1)^2 + 5*(2*c^2*d^4*x^2 + d^4)*cosh(1))*sinh(1)^4 + (c^4*d^6*x^2 + 2*c^2*d^6)*cosh(1)^3 + (c^4*d^6*x^2 + 16*
c^2*d^5*x*cosh(1) + 2*c^2*d^6 + 35*d^2*x^2*cosh(1)^4 + 40*d^3*x*cosh(1)^3 + 10*(2*c^2*d^4*x^2 + d^4)*cosh(1)^2
)*sinh(1)^3 + (2*c^4*d^7*x + 24*c^2*d^5*x*cosh(1)^2 + 21*d^2*x^2*cosh(1)^5 + 30*d^3*x*cosh(1)^4 + 10*(2*c^2*d^
4*x^2 + d^4)*cosh(1)^3 + 3*(c^4*d^6*x^2 + 2*c^2*d^6)*cosh(1))*sinh(1)^2 + (4*c^4*d^7*x*cosh(1) + c^4*d^8 + 16*
c^2*d^5*x*cosh(1)^3 + 7*d^2*x^2*cosh(1)^6 + 12*d^3*x*cosh(1)^5 + 5*(2*c^2*d^4*x^2 + d^4)*cosh(1)^4 + 3*(c^4*d^
6*x^2 + 2*c^2*d^6)*cosh(1)^2)*sinh(1))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b \operatorname {acsch}{\left (c x \right )}}{\left (d + e x\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*acsch(c*x))/(d + e*x)**3, 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*arccsch(c*x))/(e*x+d)^3,x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )}{{\left (d+e\,x\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asinh(1/(c*x)))/(d + e*x)^3,x)

[Out]

int((a + b*asinh(1/(c*x)))/(d + e*x)^3, x)

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