3.2.46 \(\int \frac {a+b \text {csch}^{-1}(c x)}{x^4 \sqrt {d+e x^2}} \, dx\) [146]

Optimal. Leaf size=425 \[ -\frac {b c^3 \left (2 c^2 d+5 e\right ) x^2 \sqrt {d+e x^2}}{9 d^2 \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2}}-\frac {b c \left (2 c^2 d+5 e\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{9 d^2 \sqrt {-c^2 x^2}}+\frac {b c \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{9 d x^2 \sqrt {-c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d^2 x}+\frac {b c^2 \left (2 c^2 d+5 e\right ) x \sqrt {d+e x^2} E\left (\text {ArcTan}(c x)\left |1-\frac {e}{c^2 d}\right .\right )}{9 d^2 \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2} \sqrt {\frac {d+e x^2}{d \left (1+c^2 x^2\right )}}}-\frac {b e \left (c^2 d+6 e\right ) x \sqrt {d+e x^2} F\left (\text {ArcTan}(c x)\left |1-\frac {e}{c^2 d}\right .\right )}{9 d^3 \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2} \sqrt {\frac {d+e x^2}{d \left (1+c^2 x^2\right )}}} \]

[Out]

-1/3*(a+b*arccsch(c*x))*(e*x^2+d)^(1/2)/d/x^3+2/3*e*(a+b*arccsch(c*x))*(e*x^2+d)^(1/2)/d^2/x-1/9*b*c^3*(2*c^2*
d+5*e)*x^2*(e*x^2+d)^(1/2)/d^2/(-c^2*x^2)^(1/2)/(-c^2*x^2-1)^(1/2)-1/9*b*c*(2*c^2*d+5*e)*(-c^2*x^2-1)^(1/2)*(e
*x^2+d)^(1/2)/d^2/(-c^2*x^2)^(1/2)+1/9*b*c*(-c^2*x^2-1)^(1/2)*(e*x^2+d)^(1/2)/d/x^2/(-c^2*x^2)^(1/2)+1/9*b*c^2
*(2*c^2*d+5*e)*x*(1/(c^2*x^2+1))^(1/2)*(c^2*x^2+1)^(1/2)*EllipticE(c*x/(c^2*x^2+1)^(1/2),(1-e/c^2/d)^(1/2))*(e
*x^2+d)^(1/2)/d^2/(-c^2*x^2)^(1/2)/(-c^2*x^2-1)^(1/2)/((e*x^2+d)/d/(c^2*x^2+1))^(1/2)-1/9*b*e*(c^2*d+6*e)*x*(1
/(c^2*x^2+1))^(1/2)*(c^2*x^2+1)^(1/2)*EllipticF(c*x/(c^2*x^2+1)^(1/2),(1-e/c^2/d)^(1/2))*(e*x^2+d)^(1/2)/d^3/(
-c^2*x^2)^(1/2)/(-c^2*x^2-1)^(1/2)/((e*x^2+d)/d/(c^2*x^2+1))^(1/2)

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Rubi [A]
time = 0.32, antiderivative size = 425, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 10, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {277, 270, 6437, 12, 594, 597, 545, 429, 506, 422} \begin {gather*} \frac {2 e \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d^2 x}-\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d x^3}-\frac {b e x \left (c^2 d+6 e\right ) \sqrt {d+e x^2} F\left (\text {ArcTan}(c x)\left |1-\frac {e}{c^2 d}\right .\right )}{9 d^3 \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1} \sqrt {\frac {d+e x^2}{d \left (c^2 x^2+1\right )}}}+\frac {b c^2 x \left (2 c^2 d+5 e\right ) \sqrt {d+e x^2} E\left (\text {ArcTan}(c x)\left |1-\frac {e}{c^2 d}\right .\right )}{9 d^2 \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1} \sqrt {\frac {d+e x^2}{d \left (c^2 x^2+1\right )}}}-\frac {b c \sqrt {-c^2 x^2-1} \left (2 c^2 d+5 e\right ) \sqrt {d+e x^2}}{9 d^2 \sqrt {-c^2 x^2}}+\frac {b c \sqrt {-c^2 x^2-1} \sqrt {d+e x^2}}{9 d x^2 \sqrt {-c^2 x^2}}-\frac {b c^3 x^2 \left (2 c^2 d+5 e\right ) \sqrt {d+e x^2}}{9 d^2 \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCsch[c*x])/(x^4*Sqrt[d + e*x^2]),x]

[Out]

-1/9*(b*c^3*(2*c^2*d + 5*e)*x^2*Sqrt[d + e*x^2])/(d^2*Sqrt[-(c^2*x^2)]*Sqrt[-1 - c^2*x^2]) - (b*c*(2*c^2*d + 5
*e)*Sqrt[-1 - c^2*x^2]*Sqrt[d + e*x^2])/(9*d^2*Sqrt[-(c^2*x^2)]) + (b*c*Sqrt[-1 - c^2*x^2]*Sqrt[d + e*x^2])/(9
*d*x^2*Sqrt[-(c^2*x^2)]) - (Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x]))/(3*d*x^3) + (2*e*Sqrt[d + e*x^2]*(a + b*ArcC
sch[c*x]))/(3*d^2*x) + (b*c^2*(2*c^2*d + 5*e)*x*Sqrt[d + e*x^2]*EllipticE[ArcTan[c*x], 1 - e/(c^2*d)])/(9*d^2*
Sqrt[-(c^2*x^2)]*Sqrt[-1 - c^2*x^2]*Sqrt[(d + e*x^2)/(d*(1 + c^2*x^2))]) - (b*e*(c^2*d + 6*e)*x*Sqrt[d + e*x^2
]*EllipticF[ArcTan[c*x], 1 - e/(c^2*d)])/(9*d^3*Sqrt[-(c^2*x^2)]*Sqrt[-1 - c^2*x^2]*Sqrt[(d + e*x^2)/(d*(1 + c
^2*x^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 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 277

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^n)^(p + 1)/(a*(m + 1))), x]
- Dist[b*((m + n*(p + 1) + 1)/(a*(m + 1))), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 422

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sq
rt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /;
FreeQ[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]

Rule 429

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*
Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /
; FreeQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]

Rule 506

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt
[c + d*x^2])), x] - Dist[c/b, Int[Sqrt[a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b
*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] &&  !SimplerSqrtQ[b/a, d/c]

Rule 545

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Dist[
e, Int[(a + b*x^n)^p*(c + d*x^n)^q, x], x] + Dist[f, Int[x^n*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, n, p, q}, x]

Rule 594

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(a*g*(m + 1))), x] - Dist[1/(a*g^n*(m + 1
)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f)*(m + 1) + e*n*(b*c*(p + 1) + a*d*q)
+ d*((b*e - a*f)*(m + 1) + b*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && IGtQ[n
, 0] && GtQ[q, 0] && LtQ[m, -1] &&  !(EqQ[q, 1] && SimplerQ[e + f*x^n, c + d*x^n])

Rule 597

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

Rule 6437

Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u
= IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCsch[c*x], u, x] - Dist[b*c*(x/Sqrt[(-c^2)*x^2]), Int[Simp
lifyIntegrand[u/(x*Sqrt[-1 - c^2*x^2]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && ((IGtQ[p, 0] &&
!(ILtQ[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 0])) || (IGtQ[(m + 1)/2, 0] &&  !(ILtQ[p, 0] && GtQ[m + 2*p + 3, 0]))
|| (ILtQ[(m + 2*p + 1)/2, 0] &&  !ILtQ[(m - 1)/2, 0]))

Rubi steps

\begin {align*} \int \frac {a+b \text {csch}^{-1}(c x)}{x^4 \sqrt {d+e x^2}} \, dx &=-\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d^2 x}-\frac {(b c x) \int \frac {\sqrt {d+e x^2} \left (-d+2 e x^2\right )}{3 d^2 x^4 \sqrt {-1-c^2 x^2}} \, dx}{\sqrt {-c^2 x^2}}\\ &=-\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d^2 x}-\frac {(b c x) \int \frac {\sqrt {d+e x^2} \left (-d+2 e x^2\right )}{x^4 \sqrt {-1-c^2 x^2}} \, dx}{3 d^2 \sqrt {-c^2 x^2}}\\ &=\frac {b c \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{9 d x^2 \sqrt {-c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d^2 x}+\frac {(b c x) \int \frac {-d \left (2 c^2 d+5 e\right )-e \left (c^2 d+6 e\right ) x^2}{x^2 \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{9 d^2 \sqrt {-c^2 x^2}}\\ &=-\frac {b c \left (2 c^2 d+5 e\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{9 d^2 \sqrt {-c^2 x^2}}+\frac {b c \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{9 d x^2 \sqrt {-c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d^2 x}+\frac {(b c x) \int \frac {-d e \left (c^2 d+6 e\right )-c^2 d e \left (2 c^2 d+5 e\right ) x^2}{\sqrt {-1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{9 d^3 \sqrt {-c^2 x^2}}\\ &=-\frac {b c \left (2 c^2 d+5 e\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{9 d^2 \sqrt {-c^2 x^2}}+\frac {b c \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{9 d x^2 \sqrt {-c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d^2 x}-\frac {\left (b c^3 e \left (2 c^2 d+5 e\right ) x\right ) \int \frac {x^2}{\sqrt {-1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{9 d^2 \sqrt {-c^2 x^2}}-\frac {\left (b c e \left (c^2 d+6 e\right ) x\right ) \int \frac {1}{\sqrt {-1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{9 d^2 \sqrt {-c^2 x^2}}\\ &=-\frac {b c^3 \left (2 c^2 d+5 e\right ) x^2 \sqrt {d+e x^2}}{9 d^2 \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2}}-\frac {b c \left (2 c^2 d+5 e\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{9 d^2 \sqrt {-c^2 x^2}}+\frac {b c \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{9 d x^2 \sqrt {-c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d^2 x}-\frac {b e \left (c^2 d+6 e\right ) x \sqrt {d+e x^2} F\left (\tan ^{-1}(c x)|1-\frac {e}{c^2 d}\right )}{9 d^3 \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2} \sqrt {\frac {d+e x^2}{d \left (1+c^2 x^2\right )}}}-\frac {\left (b c^3 \left (2 c^2 d+5 e\right ) x\right ) \int \frac {\sqrt {d+e x^2}}{\left (-1-c^2 x^2\right )^{3/2}} \, dx}{9 d^2 \sqrt {-c^2 x^2}}\\ &=-\frac {b c^3 \left (2 c^2 d+5 e\right ) x^2 \sqrt {d+e x^2}}{9 d^2 \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2}}-\frac {b c \left (2 c^2 d+5 e\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{9 d^2 \sqrt {-c^2 x^2}}+\frac {b c \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{9 d x^2 \sqrt {-c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d^2 x}+\frac {b c^2 \left (2 c^2 d+5 e\right ) x \sqrt {d+e x^2} E\left (\tan ^{-1}(c x)|1-\frac {e}{c^2 d}\right )}{9 d^2 \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2} \sqrt {\frac {d+e x^2}{d \left (1+c^2 x^2\right )}}}-\frac {b e \left (c^2 d+6 e\right ) x \sqrt {d+e x^2} F\left (\tan ^{-1}(c x)|1-\frac {e}{c^2 d}\right )}{9 d^3 \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2} \sqrt {\frac {d+e x^2}{d \left (1+c^2 x^2\right )}}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 5.51, size = 239, normalized size = 0.56 \begin {gather*} -\frac {\sqrt {d+e x^2} \left (3 a \left (d-2 e x^2\right )+b c \sqrt {1+\frac {1}{c^2 x^2}} x \left (-d+2 c^2 d x^2+5 e x^2\right )+3 b \left (d-2 e x^2\right ) \text {csch}^{-1}(c x)\right )}{9 d^2 x^3}-\frac {i b c \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {1+\frac {e x^2}{d}} \left (c^2 d \left (2 c^2 d+5 e\right ) E\left (i \sinh ^{-1}\left (\sqrt {c^2} x\right )|\frac {e}{c^2 d}\right )-2 \left (c^4 d^2+2 c^2 d e-3 e^2\right ) F\left (i \sinh ^{-1}\left (\sqrt {c^2} x\right )|\frac {e}{c^2 d}\right )\right )}{9 \sqrt {c^2} d^2 \sqrt {1+c^2 x^2} \sqrt {d+e x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcCsch[c*x])/(x^4*Sqrt[d + e*x^2]),x]

[Out]

-1/9*(Sqrt[d + e*x^2]*(3*a*(d - 2*e*x^2) + b*c*Sqrt[1 + 1/(c^2*x^2)]*x*(-d + 2*c^2*d*x^2 + 5*e*x^2) + 3*b*(d -
2*e*x^2)*ArcCsch[c*x]))/(d^2*x^3) - ((I/9)*b*c*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*(c^2*d*(2*c^2*d +
5*e)*EllipticE[I*ArcSinh[Sqrt[c^2]*x], e/(c^2*d)] - 2*(c^4*d^2 + 2*c^2*d*e - 3*e^2)*EllipticF[I*ArcSinh[Sqrt[c
^2]*x], e/(c^2*d)]))/(Sqrt[c^2]*d^2*Sqrt[1 + c^2*x^2]*Sqrt[d + e*x^2])

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Maple [F]
time = 0.14, size = 0, normalized size = 0.00 \[\int \frac {a +b \,\mathrm {arccsch}\left (c x \right )}{x^{4} \sqrt {e \,x^{2}+d}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccsch(c*x))/x^4/(e*x^2+d)^(1/2),x)

[Out]

int((a+b*arccsch(c*x))/x^4/(e*x^2+d)^(1/2),x)

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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))/x^4/(e*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [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((a+b*arccsch(c*x))/x^4/(e*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >> Symbolic function elliptic_ec takes exactly 1 arguments (2 given)

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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 )}}{x^{4} \sqrt {d + e x^{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acsch(c*x))/x**4/(e*x**2+d)**(1/2),x)

[Out]

Integral((a + b*acsch(c*x))/(x**4*sqrt(d + e*x**2)), 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))/x^4/(e*x^2+d)^(1/2),x, algorithm="giac")

[Out]

integrate((b*arccsch(c*x) + a)/(sqrt(e*x^2 + d)*x^4), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )}{x^4\,\sqrt {e\,x^2+d}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asinh(1/(c*x)))/(x^4*(d + e*x^2)^(1/2)),x)

[Out]

int((a + b*asinh(1/(c*x)))/(x^4*(d + e*x^2)^(1/2)), x)

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