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

Optimal. Leaf size=91 \[ -\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{x}+e x \left (a+b \text {csch}^{-1}(c x)\right )+\frac {b c d \sqrt {-c^2 x^2-1}}{\sqrt {-c^2 x^2}}-\frac {b e x \tan ^{-1}\left (\frac {c x}{\sqrt {-c^2 x^2-1}}\right )}{\sqrt {-c^2 x^2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {14, 6302, 451, 217, 203} \[ -\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{x}+e x \left (a+b \text {csch}^{-1}(c x)\right )+\frac {b c d \sqrt {-c^2 x^2-1}}{\sqrt {-c^2 x^2}}-\frac {b e x \tan ^{-1}\left (\frac {c x}{\sqrt {-c^2 x^2-1}}\right )}{\sqrt {-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 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 451

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

Rule 6302

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

________________________________________________________________________________________

Mathematica [A]  time = 0.13, size = 89, normalized size = 0.98 \[ -\frac {a d}{x}+a e x+b c d \sqrt {\frac {c^2 x^2+1}{c^2 x^2}}+\frac {b e x \sqrt {\frac {1}{c^2 x^2}+1} \sinh ^{-1}(c x)}{\sqrt {c^2 x^2+1}}-\frac {b d \text {csch}^{-1}(c x)}{x}+b e x \text {csch}^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 0.80, size = 222, normalized size = 2.44 \[ \frac {b c^{2} d x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + b c^{2} d x + a c e x^{2} - b e x \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x\right ) - a c d - {\left (b c d - b c e\right )} x \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) + {\left (b c d - b c e\right )} x \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) + {\left (b c e x^{2} - b c d + {\left (b c d - b c e\right )} x\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )}{c x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.07, size = 107, normalized size = 1.18 \[ c \left (\frac {a \left (c x e -\frac {c d}{x}\right )}{c^{2}}+\frac {b \left (\mathrm {arccsch}\left (c x \right ) c x e -\frac {\mathrm {arccsch}\left (c x \right ) c d}{x}+\frac {\sqrt {c^{2} x^{2}+1}\, \left (c^{2} d \sqrt {c^{2} x^{2}+1}+e \arcsinh \left (c x \right ) c x \right )}{c^{2} x^{2} \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}}\right )}{c^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.37, size = 84, normalized size = 0.92 \[ {\left (c \sqrt {\frac {1}{c^{2} x^{2}} + 1} - \frac {\operatorname {arcsch}\left (c x\right )}{x}\right )} b d + a e x + \frac {{\left (2 \, c x \operatorname {arcsch}\left (c x\right ) + \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + 1\right ) - \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} - 1\right )\right )} b e}{2 \, c} - \frac {a d}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________