3.2.58 \(\int \frac {x (a+b \text {csch}^{-1}(c x))}{(d+e x^2)^{5/2}} \, dx\) [158]
Optimal. Leaf size=144 \[ \frac {b c x \sqrt {-1-c^2 x^2}}{3 d \left (c^2 d-e\right ) \sqrt {-c^2 x^2} \sqrt {d+e x^2}}-\frac {a+b \text {csch}^{-1}(c x)}{3 e \left (d+e x^2\right )^{3/2}}-\frac {b c x \text {ArcTan}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1-c^2 x^2}}\right )}{3 d^{3/2} e \sqrt {-c^2 x^2}} \]
[Out]
1/3*(-a-b*arccsch(c*x))/e/(e*x^2+d)^(3/2)-1/3*b*c*x*arctan((e*x^2+d)^(1/2)/d^(1/2)/(-c^2*x^2-1)^(1/2))/d^(3/2)
/e/(-c^2*x^2)^(1/2)+1/3*b*c*x*(-c^2*x^2-1)^(1/2)/d/(c^2*d-e)/(-c^2*x^2)^(1/2)/(e*x^2+d)^(1/2)
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Rubi [A]
time = 0.10, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {6435, 457, 98,
95, 210} \begin {gather*} -\frac {a+b \text {csch}^{-1}(c x)}{3 e \left (d+e x^2\right )^{3/2}}-\frac {b c x \text {ArcTan}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-c^2 x^2-1}}\right )}{3 d^{3/2} e \sqrt {-c^2 x^2}}+\frac {b c x \sqrt {-c^2 x^2-1}}{3 d \sqrt {-c^2 x^2} \left (c^2 d-e\right ) \sqrt {d+e x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(x*(a + b*ArcCsch[c*x]))/(d + e*x^2)^(5/2),x]
[Out]
(b*c*x*Sqrt[-1 - c^2*x^2])/(3*d*(c^2*d - e)*Sqrt[-(c^2*x^2)]*Sqrt[d + e*x^2]) - (a + b*ArcCsch[c*x])/(3*e*(d +
e*x^2)^(3/2)) - (b*c*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 - c^2*x^2])])/(3*d^(3/2)*e*Sqrt[-(c^2*x^2)])
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 98
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[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 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, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])
Rule 210
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])
Rule 457
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
Rule 6435
Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))*(x_)*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x^2)^(p +
1)*((a + b*ArcCsch[c*x])/(2*e*(p + 1))), x] - Dist[b*c*(x/(2*e*(p + 1)*Sqrt[(-c^2)*x^2])), Int[(d + e*x^2)^(p
+ 1)/(x*Sqrt[-1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
Rubi steps
\begin {align*} \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx &=-\frac {a+b \text {csch}^{-1}(c x)}{3 e \left (d+e x^2\right )^{3/2}}+\frac {(b c x) \int \frac {1}{x \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{3 e \sqrt {-c^2 x^2}}\\ &=-\frac {a+b \text {csch}^{-1}(c x)}{3 e \left (d+e x^2\right )^{3/2}}+\frac {(b c x) \text {Subst}\left (\int \frac {1}{x \sqrt {-1-c^2 x} (d+e x)^{3/2}} \, dx,x,x^2\right )}{6 e \sqrt {-c^2 x^2}}\\ &=\frac {b c x \sqrt {-1-c^2 x^2}}{3 d \left (c^2 d-e\right ) \sqrt {-c^2 x^2} \sqrt {d+e x^2}}-\frac {a+b \text {csch}^{-1}(c x)}{3 e \left (d+e x^2\right )^{3/2}}+\frac {(b c x) \text {Subst}\left (\int \frac {1}{x \sqrt {-1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{6 d e \sqrt {-c^2 x^2}}\\ &=\frac {b c x \sqrt {-1-c^2 x^2}}{3 d \left (c^2 d-e\right ) \sqrt {-c^2 x^2} \sqrt {d+e x^2}}-\frac {a+b \text {csch}^{-1}(c x)}{3 e \left (d+e x^2\right )^{3/2}}+\frac {(b c x) \text {Subst}\left (\int \frac {1}{-d-x^2} \, dx,x,\frac {\sqrt {d+e x^2}}{\sqrt {-1-c^2 x^2}}\right )}{3 d e \sqrt {-c^2 x^2}}\\ &=\frac {b c x \sqrt {-1-c^2 x^2}}{3 d \left (c^2 d-e\right ) \sqrt {-c^2 x^2} \sqrt {d+e x^2}}-\frac {a+b \text {csch}^{-1}(c x)}{3 e \left (d+e x^2\right )^{3/2}}-\frac {b c x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1-c^2 x^2}}\right )}{3 d^{3/2} e \sqrt {-c^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 157, normalized size = 1.09 \begin {gather*} \frac {a d \left (-c^2 d+e\right )+b c e \sqrt {1+\frac {1}{c^2 x^2}} x \left (d+e x^2\right )+b d \left (-c^2 d+e\right ) \text {csch}^{-1}(c x)}{3 d \left (c^2 d-e\right ) e \left (d+e x^2\right )^{3/2}}+\frac {b c \sqrt {1+\frac {1}{c^2 x^2}} x \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {1+c^2 x^2}}{\sqrt {d+e x^2}}\right )}{3 d^{3/2} e \sqrt {1+c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(x*(a + b*ArcCsch[c*x]))/(d + e*x^2)^(5/2),x]
[Out]
(a*d*(-(c^2*d) + e) + b*c*e*Sqrt[1 + 1/(c^2*x^2)]*x*(d + e*x^2) + b*d*(-(c^2*d) + e)*ArcCsch[c*x])/(3*d*(c^2*d
- e)*e*(d + e*x^2)^(3/2)) + (b*c*Sqrt[1 + 1/(c^2*x^2)]*x*ArcTanh[(Sqrt[d]*Sqrt[1 + c^2*x^2])/Sqrt[d + e*x^2]]
)/(3*d^(3/2)*e*Sqrt[1 + c^2*x^2])
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Maple [F]
time = 0.13, size = 0, normalized size = 0.00 \[\int \frac {x \left (a +b \,\mathrm {arccsch}\left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x*(a+b*arccsch(c*x))/(e*x^2+d)^(5/2),x)
[Out]
int(x*(a+b*arccsch(c*x))/(e*x^2+d)^(5/2),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(x*(a+b*arccsch(c*x))/(e*x^2+d)^(5/2),x, algorithm="maxima")
[Out]
b*integrate(x*log(sqrt(1/(c^2*x^2) + 1) + 1/(c*x))/(x^2*e + d)^(5/2), x) - 1/3*a*e^(-1)/(x^2*e + d)^(3/2)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 724 vs.
\(2 (122) = 244\).
time = 0.50, size = 1488, normalized size = 10.33 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(a+b*arccsch(c*x))/(e*x^2+d)^(5/2),x, algorithm="fricas")
[Out]
[1/12*(4*(b*c^2*d^3 - b*d^2*cosh(1) - b*d^2*sinh(1))*sqrt(x^2*cosh(1) + x^2*sinh(1) + d)*log((c*x*sqrt((c^2*x^
2 + 1)/(c^2*x^2)) + 1)/(c*x)) + (b*x^4*cosh(1)^3 + b*x^4*sinh(1)^3 - b*c^2*d^3 - (b*c^2*d*x^4 - 2*b*d*x^2)*cos
h(1)^2 - (b*c^2*d*x^4 - 3*b*x^4*cosh(1) - 2*b*d*x^2)*sinh(1)^2 - (2*b*c^2*d^2*x^2 - b*d^2)*cosh(1) - (2*b*c^2*
d^2*x^2 - 3*b*x^4*cosh(1)^2 - b*d^2 + 2*(b*c^2*d*x^4 - 2*b*d*x^2)*cosh(1))*sinh(1))*sqrt(d)*log((c^4*d^2*x^4 +
8*c^2*d^2*x^2 + x^4*cosh(1)^2 + x^4*sinh(1)^2 + 4*(c^3*d*x^3 + c*x^3*cosh(1) + c*x^3*sinh(1) + 2*c*d*x)*sqrt(
x^2*cosh(1) + x^2*sinh(1) + d)*sqrt(d)*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 8*d^2 + 2*(3*c^2*d*x^4 + 4*d*x^2)*cosh(
1) + 2*(3*c^2*d*x^4 + x^4*cosh(1) + 4*d*x^2)*sinh(1))/x^4) + 4*(a*c^2*d^3 - a*d^2*cosh(1) - a*d^2*sinh(1) - (b
*c*d*x^3*cosh(1)^2 + b*c*d*x^3*sinh(1)^2 + b*c*d^2*x*cosh(1) + (2*b*c*d*x^3*cosh(1) + b*c*d^2*x)*sinh(1))*sqrt
((c^2*x^2 + 1)/(c^2*x^2)))*sqrt(x^2*cosh(1) + x^2*sinh(1) + d))/(d^2*x^4*cosh(1)^4 + d^2*x^4*sinh(1)^4 - c^2*d
^5*cosh(1) - (c^2*d^3*x^4 - 2*d^3*x^2)*cosh(1)^3 - (c^2*d^3*x^4 - 4*d^2*x^4*cosh(1) - 2*d^3*x^2)*sinh(1)^3 - (
2*c^2*d^4*x^2 - d^4)*cosh(1)^2 - (2*c^2*d^4*x^2 - 6*d^2*x^4*cosh(1)^2 - d^4 + 3*(c^2*d^3*x^4 - 2*d^3*x^2)*cosh
(1))*sinh(1)^2 + (4*d^2*x^4*cosh(1)^3 - c^2*d^5 - 3*(c^2*d^3*x^4 - 2*d^3*x^2)*cosh(1)^2 - 2*(2*c^2*d^4*x^2 - d
^4)*cosh(1))*sinh(1)), -1/6*((b*x^4*cosh(1)^3 + b*x^4*sinh(1)^3 - b*c^2*d^3 - (b*c^2*d*x^4 - 2*b*d*x^2)*cosh(1
)^2 - (b*c^2*d*x^4 - 3*b*x^4*cosh(1) - 2*b*d*x^2)*sinh(1)^2 - (2*b*c^2*d^2*x^2 - b*d^2)*cosh(1) - (2*b*c^2*d^2
*x^2 - 3*b*x^4*cosh(1)^2 - b*d^2 + 2*(b*c^2*d*x^4 - 2*b*d*x^2)*cosh(1))*sinh(1))*sqrt(-d)*arctan(1/2*(c^3*d*x^
3 + c*x^3*cosh(1) + c*x^3*sinh(1) + 2*c*d*x)*sqrt(x^2*cosh(1) + x^2*sinh(1) + d)*sqrt(-d)*sqrt((c^2*x^2 + 1)/(
c^2*x^2))/(c^2*d^2*x^2 + d^2 + (c^2*d*x^4 + d*x^2)*cosh(1) + (c^2*d*x^4 + d*x^2)*sinh(1))) - 2*(b*c^2*d^3 - b*
d^2*cosh(1) - b*d^2*sinh(1))*sqrt(x^2*cosh(1) + x^2*sinh(1) + d)*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(
c*x)) - 2*(a*c^2*d^3 - a*d^2*cosh(1) - a*d^2*sinh(1) - (b*c*d*x^3*cosh(1)^2 + b*c*d*x^3*sinh(1)^2 + b*c*d^2*x*
cosh(1) + (2*b*c*d*x^3*cosh(1) + b*c*d^2*x)*sinh(1))*sqrt((c^2*x^2 + 1)/(c^2*x^2)))*sqrt(x^2*cosh(1) + x^2*sin
h(1) + d))/(d^2*x^4*cosh(1)^4 + d^2*x^4*sinh(1)^4 - c^2*d^5*cosh(1) - (c^2*d^3*x^4 - 2*d^3*x^2)*cosh(1)^3 - (c
^2*d^3*x^4 - 4*d^2*x^4*cosh(1) - 2*d^3*x^2)*sinh(1)^3 - (2*c^2*d^4*x^2 - d^4)*cosh(1)^2 - (2*c^2*d^4*x^2 - 6*d
^2*x^4*cosh(1)^2 - d^4 + 3*(c^2*d^3*x^4 - 2*d^3*x^2)*cosh(1))*sinh(1)^2 + (4*d^2*x^4*cosh(1)^3 - c^2*d^5 - 3*(
c^2*d^3*x^4 - 2*d^3*x^2)*cosh(1)^2 - 2*(2*c^2*d^4*x^2 - d^4)*cosh(1))*sinh(1))]
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(a+b*acsch(c*x))/(e*x**2+d)**(5/2),x)
[Out]
Timed out
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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(x*(a+b*arccsch(c*x))/(e*x^2+d)^(5/2),x, algorithm="giac")
[Out]
integrate((b*arccsch(c*x) + a)*x/(e*x^2 + d)^(5/2), x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x*(a + b*asinh(1/(c*x))))/(d + e*x^2)^(5/2),x)
[Out]
int((x*(a + b*asinh(1/(c*x))))/(d + e*x^2)^(5/2), x)
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