3.1.81 \(\int \frac {(d+e x^2) (a+b \text {csch}^{-1}(c x))}{x^6} \, dx\) [81]

Optimal. Leaf size=158 \[ \frac {2 b c^3 \left (12 c^2 d-25 e\right ) \sqrt {-1-c^2 x^2}}{225 \sqrt {-c^2 x^2}}+\frac {b c d \sqrt {-1-c^2 x^2}}{25 x^4 \sqrt {-c^2 x^2}}-\frac {b c \left (12 c^2 d-25 e\right ) \sqrt {-1-c^2 x^2}}{225 x^2 \sqrt {-c^2 x^2}}-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}-\frac {e \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3} \]

[Out]

-1/5*d*(a+b*arccsch(c*x))/x^5-1/3*e*(a+b*arccsch(c*x))/x^3+2/225*b*c^3*(12*c^2*d-25*e)*(-c^2*x^2-1)^(1/2)/(-c^
2*x^2)^(1/2)+1/25*b*c*d*(-c^2*x^2-1)^(1/2)/x^4/(-c^2*x^2)^(1/2)-1/225*b*c*(12*c^2*d-25*e)*(-c^2*x^2-1)^(1/2)/x
^2/(-c^2*x^2)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.07, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {14, 6437, 12, 464, 277, 270} \begin {gather*} -\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}-\frac {e \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}-\frac {b c \sqrt {-c^2 x^2-1} \left (12 c^2 d-25 e\right )}{225 x^2 \sqrt {-c^2 x^2}}+\frac {b c d \sqrt {-c^2 x^2-1}}{25 x^4 \sqrt {-c^2 x^2}}+\frac {2 b c^3 \sqrt {-c^2 x^2-1} \left (12 c^2 d-25 e\right )}{225 \sqrt {-c^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((d + e*x^2)*(a + b*ArcCsch[c*x]))/x^6,x]

[Out]

(2*b*c^3*(12*c^2*d - 25*e)*Sqrt[-1 - c^2*x^2])/(225*Sqrt[-(c^2*x^2)]) + (b*c*d*Sqrt[-1 - c^2*x^2])/(25*x^4*Sqr
t[-(c^2*x^2)]) - (b*c*(12*c^2*d - 25*e)*Sqrt[-1 - c^2*x^2])/(225*x^2*Sqrt[-(c^2*x^2)]) - (d*(a + b*ArcCsch[c*x
]))/(5*x^5) - (e*(a + b*ArcCsch[c*x]))/(3*x^3)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 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 464

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[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -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 {\left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{x^6} \, dx &=-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}-\frac {e \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}-\frac {(b c x) \int \frac {-3 d-5 e x^2}{15 x^6 \sqrt {-1-c^2 x^2}} \, dx}{\sqrt {-c^2 x^2}}\\ &=-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}-\frac {e \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}-\frac {(b c x) \int \frac {-3 d-5 e x^2}{x^6 \sqrt {-1-c^2 x^2}} \, dx}{15 \sqrt {-c^2 x^2}}\\ &=\frac {b c d \sqrt {-1-c^2 x^2}}{25 x^4 \sqrt {-c^2 x^2}}-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}-\frac {e \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}-\frac {\left (b c \left (12 c^2 d-25 e\right ) x\right ) \int \frac {1}{x^4 \sqrt {-1-c^2 x^2}} \, dx}{75 \sqrt {-c^2 x^2}}\\ &=\frac {b c d \sqrt {-1-c^2 x^2}}{25 x^4 \sqrt {-c^2 x^2}}-\frac {b c \left (12 c^2 d-25 e\right ) \sqrt {-1-c^2 x^2}}{225 x^2 \sqrt {-c^2 x^2}}-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}-\frac {e \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}+\frac {\left (2 b c^3 \left (12 c^2 d-25 e\right ) x\right ) \int \frac {1}{x^2 \sqrt {-1-c^2 x^2}} \, dx}{225 \sqrt {-c^2 x^2}}\\ &=\frac {2 b c^3 \left (12 c^2 d-25 e\right ) \sqrt {-1-c^2 x^2}}{225 \sqrt {-c^2 x^2}}+\frac {b c d \sqrt {-1-c^2 x^2}}{25 x^4 \sqrt {-c^2 x^2}}-\frac {b c \left (12 c^2 d-25 e\right ) \sqrt {-1-c^2 x^2}}{225 x^2 \sqrt {-c^2 x^2}}-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}-\frac {e \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.09, size = 93, normalized size = 0.59 \begin {gather*} \frac {-15 a \left (3 d+5 e x^2\right )+b c \sqrt {1+\frac {1}{c^2 x^2}} x \left (25 e x^2 \left (1-2 c^2 x^2\right )+3 d \left (3-4 c^2 x^2+8 c^4 x^4\right )\right )-15 b \left (3 d+5 e x^2\right ) \text {csch}^{-1}(c x)}{225 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x^2)*(a + b*ArcCsch[c*x]))/x^6,x]

[Out]

(-15*a*(3*d + 5*e*x^2) + b*c*Sqrt[1 + 1/(c^2*x^2)]*x*(25*e*x^2*(1 - 2*c^2*x^2) + 3*d*(3 - 4*c^2*x^2 + 8*c^4*x^
4)) - 15*b*(3*d + 5*e*x^2)*ArcCsch[c*x])/(225*x^5)

________________________________________________________________________________________

Maple [A]
time = 0.24, size = 140, normalized size = 0.89

method	result	size
derivativedivides	\(c^{5} \left (\frac {a \left (-\frac {e}{3 c^{3} x^{3}}-\frac {d}{5 c^{3} x^{5}}\right )}{c^{2}}+\frac {b \left (-\frac {\mathrm {arccsch}\left (c x \right ) e}{3 c^{3} x^{3}}-\frac {\mathrm {arccsch}\left (c x \right ) d}{5 c^{3} x^{5}}+\frac {\left (c^{2} x^{2}+1\right ) \left (24 c^{6} d \,x^{4}-50 c^{4} e \,x^{4}-12 c^{4} d \,x^{2}+25 c^{2} e \,x^{2}+9 c^{2} d \right )}{225 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c^{6} x^{6}}\right )}{c^{2}}\right )\)	\(140\)
default	\(c^{5} \left (\frac {a \left (-\frac {e}{3 c^{3} x^{3}}-\frac {d}{5 c^{3} x^{5}}\right )}{c^{2}}+\frac {b \left (-\frac {\mathrm {arccsch}\left (c x \right ) e}{3 c^{3} x^{3}}-\frac {\mathrm {arccsch}\left (c x \right ) d}{5 c^{3} x^{5}}+\frac {\left (c^{2} x^{2}+1\right ) \left (24 c^{6} d \,x^{4}-50 c^{4} e \,x^{4}-12 c^{4} d \,x^{2}+25 c^{2} e \,x^{2}+9 c^{2} d \right )}{225 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c^{6} x^{6}}\right )}{c^{2}}\right )\)	\(140\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)*(a+b*arccsch(c*x))/x^6,x,method=_RETURNVERBOSE)

[Out]

c^5*(a/c^2*(-1/3*e/c^3/x^3-1/5*d/c^3/x^5)+b/c^2*(-1/3*arccsch(c*x)*e/c^3/x^3-1/5*arccsch(c*x)*d/c^3/x^5+1/225*
(c^2*x^2+1)*(24*c^6*d*x^4-50*c^4*e*x^4-12*c^4*d*x^2+25*c^2*e*x^2+9*c^2*d)/((c^2*x^2+1)/c^2/x^2)^(1/2)/c^6/x^6)
)

________________________________________________________________________________________

Maxima [A]
time = 0.25, size = 134, normalized size = 0.85 \begin {gather*} \frac {1}{75} \, b d {\left (\frac {3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} - 10 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 15 \, c^{6} \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c} - \frac {15 \, \operatorname {arcsch}\left (c x\right )}{x^{5}}\right )} + \frac {1}{9} \, b {\left (\frac {c^{4} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 3 \, c^{4} \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c} - \frac {3 \, \operatorname {arcsch}\left (c x\right )}{x^{3}}\right )} e - \frac {a e}{3 \, x^{3}} - \frac {a d}{5 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)*(a+b*arccsch(c*x))/x^6,x, algorithm="maxima")

[Out]

1/75*b*d*((3*c^6*(1/(c^2*x^2) + 1)^(5/2) - 10*c^6*(1/(c^2*x^2) + 1)^(3/2) + 15*c^6*sqrt(1/(c^2*x^2) + 1))/c -
15*arccsch(c*x)/x^5) + 1/9*b*((c^4*(1/(c^2*x^2) + 1)^(3/2) - 3*c^4*sqrt(1/(c^2*x^2) + 1))/c - 3*arccsch(c*x)/x
^3)*e - 1/3*a*e/x^3 - 1/5*a*d/x^5

________________________________________________________________________________________

Fricas [A]
time = 0.36, size = 169, normalized size = 1.07 \begin {gather*} -\frac {75 \, a x^{2} \cosh \left (1\right ) + 75 \, a x^{2} \sinh \left (1\right ) + 45 \, a d + 15 \, {\left (5 \, b x^{2} \cosh \left (1\right ) + 5 \, b x^{2} \sinh \left (1\right ) + 3 \, b d\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (24 \, b c^{5} d x^{5} - 12 \, b c^{3} d x^{3} + 9 \, b c d x - 25 \, {\left (2 \, b c^{3} x^{5} - b c x^{3}\right )} \cosh \left (1\right ) - 25 \, {\left (2 \, b c^{3} x^{5} - b c x^{3}\right )} \sinh \left (1\right )\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{225 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)*(a+b*arccsch(c*x))/x^6,x, algorithm="fricas")

[Out]

-1/225*(75*a*x^2*cosh(1) + 75*a*x^2*sinh(1) + 45*a*d + 15*(5*b*x^2*cosh(1) + 5*b*x^2*sinh(1) + 3*b*d)*log((c*x
*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x)) - (24*b*c^5*d*x^5 - 12*b*c^3*d*x^3 + 9*b*c*d*x - 25*(2*b*c^3*x^5 -
b*c*x^3)*cosh(1) - 25*(2*b*c^3*x^5 - b*c*x^3)*sinh(1))*sqrt((c^2*x^2 + 1)/(c^2*x^2)))/x^5

________________________________________________________________________________________

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 ) \left (d + e x^{2}\right )}{x^{6}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)*(a+b*acsch(c*x))/x**6,x)

[Out]

Integral((a + b*acsch(c*x))*(d + e*x**2)/x**6, x)

________________________________________________________________________________________

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((e*x^2+d)*(a+b*arccsch(c*x))/x^6,x, algorithm="giac")

[Out]

integrate((e*x^2 + d)*(b*arccsch(c*x) + a)/x^6, x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (e\,x^2+d\right )\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{x^6} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((d + e*x^2)*(a + b*asinh(1/(c*x))))/x^6,x)

[Out]

int(((d + e*x^2)*(a + b*asinh(1/(c*x))))/x^6, x)

________________________________________________________________________________________