3.140 \(\int \frac {x (a+b \text {csch}^{-1}(c x))}{\sqrt {d+e x^2}} \, dx\)

Optimal. Leaf size=135 \[ \frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e}+\frac {b x \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {-c^2 x^2-1}}{c \sqrt {d+e x^2}}\right )}{\sqrt {e} \sqrt {-c^2 x^2}}+\frac {b c \sqrt {d} x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-c^2 x^2-1}}\right )}{e \sqrt {-c^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.15, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {6300, 446, 105, 63, 217, 203, 93, 204} \[ \frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e}+\frac {b x \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {-c^2 x^2-1}}{c \sqrt {d+e x^2}}\right )}{\sqrt {e} \sqrt {-c^2 x^2}}+\frac {b c \sqrt {d} x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-c^2 x^2-1}}\right )}{e \sqrt {-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x]))/e + (b*x*ArcTan[(Sqrt[e]*Sqrt[-1 - c^2*x^2])/(c*Sqrt[d + e*x^2])])/(Sqr
t[e]*Sqrt[-(c^2*x^2)]) + (b*c*Sqrt[d]*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 - c^2*x^2])])/(e*Sqrt[-(c^2*x^
2)])

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 93

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_)), x_Symbol] :> Dist[b/f, Int[(a
+ b*x)^(m - 1)*(c + d*x)^n, x], x] - Dist[(b*e - a*f)/f, Int[((a + b*x)^(m - 1)*(c + d*x)^n)/(e + f*x), x], x]
/; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[Simplify[m + n + 1], 0] && (GtQ[m, 0] || ( !RationalQ[m] && (Su
mSimplerQ[m, -1] ||  !SumSimplerQ[n, -1])))

Rule 203

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

Rule 204

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

Rule 217

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

Rule 446

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 6300

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

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Mathematica [A]  time = 0.24, size = 223, normalized size = 1.65 \[ \frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e}-\frac {b x \sqrt {\frac {1}{c^2 x^2}+1} \left (c^3 \sqrt {d} \sqrt {-d-e x^2} \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {c^2 x^2+1}}{\sqrt {-d-e x^2}}\right )-\sqrt {c^2} \sqrt {e} \sqrt {c^2 d-e} \sqrt {\frac {c^2 \left (d+e x^2\right )}{c^2 d-e}} \sinh ^{-1}\left (\frac {c \sqrt {e} \sqrt {c^2 x^2+1}}{\sqrt {c^2} \sqrt {c^2 d-e}}\right )\right )}{c^2 e \sqrt {c^2 x^2+1} \sqrt {d+e x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.69, size = 1064, normalized size = 7.88 \[ \left [\frac {4 \, \sqrt {e x^{2} + d} b c \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) + b c \sqrt {d} \log \left (\frac {{\left (c^{4} d^{2} + 6 \, c^{2} d e + e^{2}\right )} x^{4} + 8 \, {\left (c^{2} d^{2} + d e\right )} x^{2} - 4 \, {\left ({\left (c^{3} d + c e\right )} x^{3} + 2 \, c d x\right )} \sqrt {e x^{2} + d} \sqrt {d} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 8 \, d^{2}}{x^{4}}\right ) + 4 \, \sqrt {e x^{2} + d} a c + b \sqrt {e} \log \left (8 \, c^{4} e^{2} x^{4} + c^{4} d^{2} + 6 \, c^{2} d e + 8 \, {\left (c^{4} d e + c^{2} e^{2}\right )} x^{2} + 4 \, {\left (2 \, c^{4} e x^{3} + {\left (c^{4} d + c^{2} e\right )} x\right )} \sqrt {e x^{2} + d} \sqrt {e} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + e^{2}\right )}{4 \, c e}, \frac {4 \, \sqrt {e x^{2} + d} b c \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) + b c \sqrt {d} \log \left (\frac {{\left (c^{4} d^{2} + 6 \, c^{2} d e + e^{2}\right )} x^{4} + 8 \, {\left (c^{2} d^{2} + d e\right )} x^{2} - 4 \, {\left ({\left (c^{3} d + c e\right )} x^{3} + 2 \, c d x\right )} \sqrt {e x^{2} + d} \sqrt {d} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 8 \, d^{2}}{x^{4}}\right ) + 4 \, \sqrt {e x^{2} + d} a c - 2 \, b \sqrt {-e} \arctan \left (\frac {{\left (2 \, c^{2} e x^{3} + {\left (c^{2} d + e\right )} x\right )} \sqrt {e x^{2} + d} \sqrt {-e} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{2 \, {\left (c^{2} e^{2} x^{4} + {\left (c^{2} d e + e^{2}\right )} x^{2} + d e\right )}}\right )}{4 \, c e}, \frac {2 \, b c \sqrt {-d} \arctan \left (\frac {{\left ({\left (c^{3} d + c e\right )} x^{3} + 2 \, c d x\right )} \sqrt {e x^{2} + d} \sqrt {-d} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{2 \, {\left (c^{2} d e x^{4} + {\left (c^{2} d^{2} + d e\right )} x^{2} + d^{2}\right )}}\right ) + 4 \, \sqrt {e x^{2} + d} b c \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) + 4 \, \sqrt {e x^{2} + d} a c + b \sqrt {e} \log \left (8 \, c^{4} e^{2} x^{4} + c^{4} d^{2} + 6 \, c^{2} d e + 8 \, {\left (c^{4} d e + c^{2} e^{2}\right )} x^{2} + 4 \, {\left (2 \, c^{4} e x^{3} + {\left (c^{4} d + c^{2} e\right )} x\right )} \sqrt {e x^{2} + d} \sqrt {e} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + e^{2}\right )}{4 \, c e}, \frac {b c \sqrt {-d} \arctan \left (\frac {{\left ({\left (c^{3} d + c e\right )} x^{3} + 2 \, c d x\right )} \sqrt {e x^{2} + d} \sqrt {-d} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{2 \, {\left (c^{2} d e x^{4} + {\left (c^{2} d^{2} + d e\right )} x^{2} + d^{2}\right )}}\right ) + 2 \, \sqrt {e x^{2} + d} b c \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) + 2 \, \sqrt {e x^{2} + d} a c - b \sqrt {-e} \arctan \left (\frac {{\left (2 \, c^{2} e x^{3} + {\left (c^{2} d + e\right )} x\right )} \sqrt {e x^{2} + d} \sqrt {-e} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{2 \, {\left (c^{2} e^{2} x^{4} + {\left (c^{2} d e + e^{2}\right )} x^{2} + d e\right )}}\right )}{2 \, c e}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(4*sqrt(e*x^2 + d)*b*c*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x)) + b*c*sqrt(d)*log(((c^4*d^2 + 6
*c^2*d*e + e^2)*x^4 + 8*(c^2*d^2 + d*e)*x^2 - 4*((c^3*d + c*e)*x^3 + 2*c*d*x)*sqrt(e*x^2 + d)*sqrt(d)*sqrt((c^
2*x^2 + 1)/(c^2*x^2)) + 8*d^2)/x^4) + 4*sqrt(e*x^2 + d)*a*c + b*sqrt(e)*log(8*c^4*e^2*x^4 + c^4*d^2 + 6*c^2*d*
e + 8*(c^4*d*e + c^2*e^2)*x^2 + 4*(2*c^4*e*x^3 + (c^4*d + c^2*e)*x)*sqrt(e*x^2 + d)*sqrt(e)*sqrt((c^2*x^2 + 1)
/(c^2*x^2)) + e^2))/(c*e), 1/4*(4*sqrt(e*x^2 + d)*b*c*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x)) + b*c
*sqrt(d)*log(((c^4*d^2 + 6*c^2*d*e + e^2)*x^4 + 8*(c^2*d^2 + d*e)*x^2 - 4*((c^3*d + c*e)*x^3 + 2*c*d*x)*sqrt(e
*x^2 + d)*sqrt(d)*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 8*d^2)/x^4) + 4*sqrt(e*x^2 + d)*a*c - 2*b*sqrt(-e)*arctan(1/
2*(2*c^2*e*x^3 + (c^2*d + e)*x)*sqrt(e*x^2 + d)*sqrt(-e)*sqrt((c^2*x^2 + 1)/(c^2*x^2))/(c^2*e^2*x^4 + (c^2*d*e
+ e^2)*x^2 + d*e)))/(c*e), 1/4*(2*b*c*sqrt(-d)*arctan(1/2*((c^3*d + c*e)*x^3 + 2*c*d*x)*sqrt(e*x^2 + d)*sqrt(
-d)*sqrt((c^2*x^2 + 1)/(c^2*x^2))/(c^2*d*e*x^4 + (c^2*d^2 + d*e)*x^2 + d^2)) + 4*sqrt(e*x^2 + d)*b*c*log((c*x*
sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x)) + 4*sqrt(e*x^2 + d)*a*c + b*sqrt(e)*log(8*c^4*e^2*x^4 + c^4*d^2 + 6*
c^2*d*e + 8*(c^4*d*e + c^2*e^2)*x^2 + 4*(2*c^4*e*x^3 + (c^4*d + c^2*e)*x)*sqrt(e*x^2 + d)*sqrt(e)*sqrt((c^2*x^
2 + 1)/(c^2*x^2)) + e^2))/(c*e), 1/2*(b*c*sqrt(-d)*arctan(1/2*((c^3*d + c*e)*x^3 + 2*c*d*x)*sqrt(e*x^2 + d)*sq
rt(-d)*sqrt((c^2*x^2 + 1)/(c^2*x^2))/(c^2*d*e*x^4 + (c^2*d^2 + d*e)*x^2 + d^2)) + 2*sqrt(e*x^2 + d)*b*c*log((c
*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x)) + 2*sqrt(e*x^2 + d)*a*c - b*sqrt(-e)*arctan(1/2*(2*c^2*e*x^3 + (c
^2*d + e)*x)*sqrt(e*x^2 + d)*sqrt(-e)*sqrt((c^2*x^2 + 1)/(c^2*x^2))/(c^2*e^2*x^4 + (c^2*d*e + e^2)*x^2 + d*e))
)/(c*e)]

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x}{\sqrt {e x^{2} + d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.43, size = 0, normalized size = 0.00 \[ \int \frac {x \left (a +b \,\mathrm {arccsch}\left (c x \right )\right )}{\sqrt {e \,x^{2}+d}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ b {\left (\frac {\sqrt {e x^{2} + d} \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right )}{e} + \int \frac {c^{2} e x^{3} + c^{2} d x}{{\left (c^{2} e x^{2} + e\right )} \sqrt {c^{2} x^{2} + 1} \sqrt {e x^{2} + d} + {\left (c^{2} e x^{2} + e\right )} \sqrt {e x^{2} + d}}\,{d x} - \int \frac {{\left (e \log \relax (c) + e\right )} c^{2} x^{3} + {\left (c^{2} d + e \log \relax (c)\right )} x + {\left (c^{2} e x^{3} + e x\right )} \log \relax (x)}{{\left (c^{2} e x^{2} + e\right )} \sqrt {e x^{2} + d}}\,{d x}\right )} + \frac {\sqrt {e x^{2} + d} a}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{\sqrt {e\,x^2+d}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \left (a + b \operatorname {acsch}{\left (c x \right )}\right )}{\sqrt {d + e x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*(a + b*acsch(c*x))/sqrt(d + e*x**2), x)

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