3.2.26 \(\int \frac {\sqrt {d+e x^2} (a+b \text {csch}^{-1}(c x))}{x^4} \, dx\) [126]

Optimal. Leaf size=389 \[ -\frac {2 b c^3 \left (c^2 d-2 e\right ) x^2 \sqrt {d+e x^2}}{9 d \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2}}-\frac {2 b c \left (c^2 d-2 e\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{9 d \sqrt {-c^2 x^2}}+\frac {b c \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{9 x^2 \sqrt {-c^2 x^2}}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d x^3}+\frac {2 b c^2 \left (c^2 d-2 e\right ) x \sqrt {d+e x^2} E\left (\text {ArcTan}(c x)\left |1-\frac {e}{c^2 d}\right .\right )}{9 d \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2} \sqrt {\frac {d+e x^2}{d \left (1+c^2 x^2\right )}}}-\frac {b \left (c^2 d-3 e\right ) e x \sqrt {d+e x^2} F\left (\text {ArcTan}(c x)\left |1-\frac {e}{c^2 d}\right .\right )}{9 d^2 \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2} \sqrt {\frac {d+e x^2}{d \left (1+c^2 x^2\right )}}} \]

[Out]

-1/3*(e*x^2+d)^(3/2)*(a+b*arccsch(c*x))/d/x^3-2/9*b*c^3*(c^2*d-2*e)*x^2*(e*x^2+d)^(1/2)/d/(-c^2*x^2)^(1/2)/(-c
^2*x^2-1)^(1/2)-2/9*b*c*(c^2*d-2*e)*(-c^2*x^2-1)^(1/2)*(e*x^2+d)^(1/2)/d/(-c^2*x^2)^(1/2)+1/9*b*c*(-c^2*x^2-1)
^(1/2)*(e*x^2+d)^(1/2)/x^2/(-c^2*x^2)^(1/2)+2/9*b*c^2*(c^2*d-2*e)*x*(1/(c^2*x^2+1))^(1/2)*(c^2*x^2+1)^(1/2)*El
lipticE(c*x/(c^2*x^2+1)^(1/2),(1-e/c^2/d)^(1/2))*(e*x^2+d)^(1/2)/d/(-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*(c^2*d-3*e)*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^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)

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Rubi [A]
time = 0.28, antiderivative size = 389, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {270, 6437, 12, 485, 597, 545, 429, 506, 422} \begin {gather*} -\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d x^3}-\frac {b e x \left (c^2 d-3 e\right ) \sqrt {d+e x^2} F\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 {2 b c^2 x \left (c^2 d-2 e\right ) \sqrt {d+e x^2} E\left (\text {ArcTan}(c x)\left |1-\frac {e}{c^2 d}\right .\right )}{9 d \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 {2 b c \sqrt {-c^2 x^2-1} \left (c^2 d-2 e\right ) \sqrt {d+e x^2}}{9 d \sqrt {-c^2 x^2}}+\frac {b c \sqrt {-c^2 x^2-1} \sqrt {d+e x^2}}{9 x^2 \sqrt {-c^2 x^2}}-\frac {2 b c^3 x^2 \left (c^2 d-2 e\right ) \sqrt {d+e x^2}}{9 d \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*b*c^3*(c^2*d - 2*e)*x^2*Sqrt[d + e*x^2])/(9*d*Sqrt[-(c^2*x^2)]*Sqrt[-1 - c^2*x^2]) - (2*b*c*(c^2*d - 2*e)*
Sqrt[-1 - c^2*x^2]*Sqrt[d + e*x^2])/(9*d*Sqrt[-(c^2*x^2)]) + (b*c*Sqrt[-1 - c^2*x^2]*Sqrt[d + e*x^2])/(9*x^2*S
qrt[-(c^2*x^2)]) - ((d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]))/(3*d*x^3) + (2*b*c^2*(c^2*d - 2*e)*x*Sqrt[d + e*x^
2]*EllipticE[ArcTan[c*x], 1 - e/(c^2*d)])/(9*d*Sqrt[-(c^2*x^2)]*Sqrt[-1 - c^2*x^2]*Sqrt[(d + e*x^2)/(d*(1 + c^
2*x^2))]) - (b*(c^2*d - 3*e)*e*x*Sqrt[d + e*x^2]*EllipticF[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))])

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 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 485

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

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

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Mathematica [C] Result contains complex when optimal does not.
time = 7.46, size = 237, normalized size = 0.61 \begin {gather*} -\frac {\sqrt {d+e x^2} \left (b c \sqrt {1+\frac {1}{c^2 x^2}} x \left (-d+2 c^2 d x^2-4 e x^2\right )+3 a \left (d+e x^2\right )+3 b \left (d+e x^2\right ) \text {csch}^{-1}(c x)\right )}{9 d x^3}-\frac {i b c \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {1+\frac {e x^2}{d}} \left (2 c^2 d \left (c^2 d-2 e\right ) E\left (i \sinh ^{-1}\left (\sqrt {c^2} x\right )|\frac {e}{c^2 d}\right )+\left (-2 c^4 d^2+5 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 \sqrt {1+c^2 x^2} \sqrt {d+e x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/9*(Sqrt[d + e*x^2]*(b*c*Sqrt[1 + 1/(c^2*x^2)]*x*(-d + 2*c^2*d*x^2 - 4*e*x^2) + 3*a*(d + e*x^2) + 3*b*(d + e
*x^2)*ArcCsch[c*x]))/(d*x^3) - ((I/9)*b*c*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*(2*c^2*d*(c^2*d - 2*e)*E
llipticE[I*ArcSinh[Sqrt[c^2]*x], e/(c^2*d)] + (-2*c^4*d^2 + 5*c^2*d*e - 3*e^2)*EllipticF[I*ArcSinh[Sqrt[c^2]*x
], e/(c^2*d)]))/(Sqrt[c^2]*d*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 \[\int \frac {\left (a +b \,\mathrm {arccsch}\left (c x \right )\right ) \sqrt {e \,x^{2}+d}}{x^{4}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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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))*(e*x^2+d)^(1/2)/x^4,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 {\left (a + b \operatorname {acsch}{\left (c x \right )}\right ) \sqrt {d + e x^{2}}}{x^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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