3.1.82 \(\int \frac {(d+e x^2) (a+b \text {csch}^{-1}(c x))}{x^8} \, dx\) [82]

Optimal. Leaf size=205 \[ -\frac {8 b c^5 \left (30 c^2 d-49 e\right ) \sqrt {-1-c^2 x^2}}{3675 \sqrt {-c^2 x^2}}+\frac {b c d \sqrt {-1-c^2 x^2}}{49 x^6 \sqrt {-c^2 x^2}}-\frac {b c \left (30 c^2 d-49 e\right ) \sqrt {-1-c^2 x^2}}{1225 x^4 \sqrt {-c^2 x^2}}+\frac {4 b c^3 \left (30 c^2 d-49 e\right ) \sqrt {-1-c^2 x^2}}{3675 x^2 \sqrt {-c^2 x^2}}-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5} \]

[Out]

-1/7*d*(a+b*arccsch(c*x))/x^7-1/5*e*(a+b*arccsch(c*x))/x^5-8/3675*b*c^5*(30*c^2*d-49*e)*(-c^2*x^2-1)^(1/2)/(-c
^2*x^2)^(1/2)+1/49*b*c*d*(-c^2*x^2-1)^(1/2)/x^6/(-c^2*x^2)^(1/2)-1/1225*b*c*(30*c^2*d-49*e)*(-c^2*x^2-1)^(1/2)
/x^4/(-c^2*x^2)^(1/2)+4/3675*b*c^3*(30*c^2*d-49*e)*(-c^2*x^2-1)^(1/2)/x^2/(-c^2*x^2)^(1/2)

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Rubi [A]
time = 0.08, antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 6, 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 )}{7 x^7}-\frac {e \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}-\frac {b c \sqrt {-c^2 x^2-1} \left (30 c^2 d-49 e\right )}{1225 x^4 \sqrt {-c^2 x^2}}+\frac {b c d \sqrt {-c^2 x^2-1}}{49 x^6 \sqrt {-c^2 x^2}}-\frac {8 b c^5 \sqrt {-c^2 x^2-1} \left (30 c^2 d-49 e\right )}{3675 \sqrt {-c^2 x^2}}+\frac {4 b c^3 \sqrt {-c^2 x^2-1} \left (30 c^2 d-49 e\right )}{3675 x^2 \sqrt {-c^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((d + e*x^2)*(a + b*ArcCsch[c*x]))/x^8,x]

[Out]

(-8*b*c^5*(30*c^2*d - 49*e)*Sqrt[-1 - c^2*x^2])/(3675*Sqrt[-(c^2*x^2)]) + (b*c*d*Sqrt[-1 - c^2*x^2])/(49*x^6*S
qrt[-(c^2*x^2)]) - (b*c*(30*c^2*d - 49*e)*Sqrt[-1 - c^2*x^2])/(1225*x^4*Sqrt[-(c^2*x^2)]) + (4*b*c^3*(30*c^2*d
- 49*e)*Sqrt[-1 - c^2*x^2])/(3675*x^2*Sqrt[-(c^2*x^2)]) - (d*(a + b*ArcCsch[c*x]))/(7*x^7) - (e*(a + b*ArcCsc
h[c*x]))/(5*x^5)

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^8} \, dx &=-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}-\frac {(b c x) \int \frac {-5 d-7 e x^2}{35 x^8 \sqrt {-1-c^2 x^2}} \, dx}{\sqrt {-c^2 x^2}}\\ &=-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}-\frac {(b c x) \int \frac {-5 d-7 e x^2}{x^8 \sqrt {-1-c^2 x^2}} \, dx}{35 \sqrt {-c^2 x^2}}\\ &=\frac {b c d \sqrt {-1-c^2 x^2}}{49 x^6 \sqrt {-c^2 x^2}}-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}-\frac {\left (b c \left (30 c^2 d-49 e\right ) x\right ) \int \frac {1}{x^6 \sqrt {-1-c^2 x^2}} \, dx}{245 \sqrt {-c^2 x^2}}\\ &=\frac {b c d \sqrt {-1-c^2 x^2}}{49 x^6 \sqrt {-c^2 x^2}}-\frac {b c \left (30 c^2 d-49 e\right ) \sqrt {-1-c^2 x^2}}{1225 x^4 \sqrt {-c^2 x^2}}-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}+\frac {\left (4 b c^3 \left (30 c^2 d-49 e\right ) x\right ) \int \frac {1}{x^4 \sqrt {-1-c^2 x^2}} \, dx}{1225 \sqrt {-c^2 x^2}}\\ &=\frac {b c d \sqrt {-1-c^2 x^2}}{49 x^6 \sqrt {-c^2 x^2}}-\frac {b c \left (30 c^2 d-49 e\right ) \sqrt {-1-c^2 x^2}}{1225 x^4 \sqrt {-c^2 x^2}}+\frac {4 b c^3 \left (30 c^2 d-49 e\right ) \sqrt {-1-c^2 x^2}}{3675 x^2 \sqrt {-c^2 x^2}}-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}-\frac {\left (8 b c^5 \left (30 c^2 d-49 e\right ) x\right ) \int \frac {1}{x^2 \sqrt {-1-c^2 x^2}} \, dx}{3675 \sqrt {-c^2 x^2}}\\ &=-\frac {8 b c^5 \left (30 c^2 d-49 e\right ) \sqrt {-1-c^2 x^2}}{3675 \sqrt {-c^2 x^2}}+\frac {b c d \sqrt {-1-c^2 x^2}}{49 x^6 \sqrt {-c^2 x^2}}-\frac {b c \left (30 c^2 d-49 e\right ) \sqrt {-1-c^2 x^2}}{1225 x^4 \sqrt {-c^2 x^2}}+\frac {4 b c^3 \left (30 c^2 d-49 e\right ) \sqrt {-1-c^2 x^2}}{3675 x^2 \sqrt {-c^2 x^2}}-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}\\ \end {align*}

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Mathematica [A]
time = 0.11, size = 109, normalized size = 0.53 \begin {gather*} \frac {-105 a \left (5 d+7 e x^2\right )+b c \sqrt {1+\frac {1}{c^2 x^2}} x \left (49 e x^2 \left (3-4 c^2 x^2+8 c^4 x^4\right )-15 d \left (-5+6 c^2 x^2-8 c^4 x^4+16 c^6 x^6\right )\right )-105 b \left (5 d+7 e x^2\right ) \text {csch}^{-1}(c x)}{3675 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x^2)*(a + b*ArcCsch[c*x]))/x^8,x]

[Out]

(-105*a*(5*d + 7*e*x^2) + b*c*Sqrt[1 + 1/(c^2*x^2)]*x*(49*e*x^2*(3 - 4*c^2*x^2 + 8*c^4*x^4) - 15*d*(-5 + 6*c^2
*x^2 - 8*c^4*x^4 + 16*c^6*x^6)) - 105*b*(5*d + 7*e*x^2)*ArcCsch[c*x])/(3675*x^7)

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Maple [A]
time = 0.25, size = 158, normalized size = 0.77

method	result	size
derivativedivides	\(c^{7} \left (\frac {a \left (-\frac {d}{7 c^{5} x^{7}}-\frac {e}{5 c^{5} x^{5}}\right )}{c^{2}}+\frac {b \left (-\frac {\mathrm {arccsch}\left (c x \right ) d}{7 c^{5} x^{7}}-\frac {\mathrm {arccsch}\left (c x \right ) e}{5 c^{5} x^{5}}-\frac {\left (c^{2} x^{2}+1\right ) \left (240 c^{8} d \,x^{6}-392 c^{6} e \,x^{6}-120 c^{6} d \,x^{4}+196 c^{4} e \,x^{4}+90 c^{4} d \,x^{2}-147 c^{2} e \,x^{2}-75 c^{2} d \right )}{3675 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c^{8} x^{8}}\right )}{c^{2}}\right )\)	\(158\)
default	\(c^{7} \left (\frac {a \left (-\frac {d}{7 c^{5} x^{7}}-\frac {e}{5 c^{5} x^{5}}\right )}{c^{2}}+\frac {b \left (-\frac {\mathrm {arccsch}\left (c x \right ) d}{7 c^{5} x^{7}}-\frac {\mathrm {arccsch}\left (c x \right ) e}{5 c^{5} x^{5}}-\frac {\left (c^{2} x^{2}+1\right ) \left (240 c^{8} d \,x^{6}-392 c^{6} e \,x^{6}-120 c^{6} d \,x^{4}+196 c^{4} e \,x^{4}+90 c^{4} d \,x^{2}-147 c^{2} e \,x^{2}-75 c^{2} d \right )}{3675 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c^{8} x^{8}}\right )}{c^{2}}\right )\)	\(158\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)*(a+b*arccsch(c*x))/x^8,x,method=_RETURNVERBOSE)

[Out]

c^7*(a/c^2*(-1/7*d/c^5/x^7-1/5*e/c^5/x^5)+b/c^2*(-1/7*arccsch(c*x)*d/c^5/x^7-1/5*arccsch(c*x)*e/c^5/x^5-1/3675
*(c^2*x^2+1)*(240*c^8*d*x^6-392*c^6*e*x^6-120*c^6*d*x^4+196*c^4*e*x^4+90*c^4*d*x^2-147*c^2*e*x^2-75*c^2*d)/((c
^2*x^2+1)/c^2/x^2)^(1/2)/c^8/x^8))

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Maxima [A]
time = 0.26, size = 167, normalized size = 0.81 \begin {gather*} \frac {1}{245} \, b d {\left (\frac {5 \, c^{8} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {7}{2}} - 21 \, c^{8} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} + 35 \, c^{8} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 35 \, c^{8} \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c} - \frac {35 \, \operatorname {arcsch}\left (c x\right )}{x^{7}}\right )} + \frac {1}{75} \, b {\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 )} e - \frac {a e}{5 \, x^{5}} - \frac {a d}{7 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)*(a+b*arccsch(c*x))/x^8,x, algorithm="maxima")

[Out]

1/245*b*d*((5*c^8*(1/(c^2*x^2) + 1)^(7/2) - 21*c^8*(1/(c^2*x^2) + 1)^(5/2) + 35*c^8*(1/(c^2*x^2) + 1)^(3/2) -
35*c^8*sqrt(1/(c^2*x^2) + 1))/c - 35*arccsch(c*x)/x^7) + 1/75*b*((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)*e - 1/5*a*e/x^5 - 1/7*a*d/x^7

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Fricas [A]
time = 0.37, size = 196, normalized size = 0.96 \begin {gather*} -\frac {735 \, a x^{2} \cosh \left (1\right ) + 735 \, a x^{2} \sinh \left (1\right ) + 525 \, a d + 105 \, {\left (7 \, b x^{2} \cosh \left (1\right ) + 7 \, b x^{2} \sinh \left (1\right ) + 5 \, b d\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) + {\left (240 \, b c^{7} d x^{7} - 120 \, b c^{5} d x^{5} + 90 \, b c^{3} d x^{3} - 75 \, b c d x - 49 \, {\left (8 \, b c^{5} x^{7} - 4 \, b c^{3} x^{5} + 3 \, b c x^{3}\right )} \cosh \left (1\right ) - 49 \, {\left (8 \, b c^{5} x^{7} - 4 \, b c^{3} x^{5} + 3 \, b c x^{3}\right )} \sinh \left (1\right )\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{3675 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)*(a+b*arccsch(c*x))/x^8,x, algorithm="fricas")

[Out]

-1/3675*(735*a*x^2*cosh(1) + 735*a*x^2*sinh(1) + 525*a*d + 105*(7*b*x^2*cosh(1) + 7*b*x^2*sinh(1) + 5*b*d)*log
((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x)) + (240*b*c^7*d*x^7 - 120*b*c^5*d*x^5 + 90*b*c^3*d*x^3 - 75*b*c
*d*x - 49*(8*b*c^5*x^7 - 4*b*c^3*x^5 + 3*b*c*x^3)*cosh(1) - 49*(8*b*c^5*x^7 - 4*b*c^3*x^5 + 3*b*c*x^3)*sinh(1)
)*sqrt((c^2*x^2 + 1)/(c^2*x^2)))/x^7

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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 ) \left (d + e x^{2}\right )}{x^{8}}\, 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**8,x)

[Out]

Integral((a + b*acsch(c*x))*(d + e*x**2)/x**8, 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((e*x^2+d)*(a+b*arccsch(c*x))/x^8,x, algorithm="giac")

[Out]

integrate((e*x^2 + d)*(b*arccsch(c*x) + a)/x^8, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (e\,x^2+d\right )\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{x^8} \,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^8,x)

[Out]

int(((d + e*x^2)*(a + b*asinh(1/(c*x))))/x^8, x)

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