3.2.4 \(\int \frac {(d+e x^2)^2 (a+b \text {sech}^{-1}(c x))}{x^6} \, dx\) [104]

Optimal. Leaf size=213 \[ \frac {b d^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{25 x^5}+\frac {2 b d \left (6 c^2 d+25 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{225 x^3}+\frac {b \left (24 c^4 d^2+100 c^2 d e+225 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{225 x}-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 x^5}-\frac {2 d e \left (a+b \text {sech}^{-1}(c x)\right )}{3 x^3}-\frac {e^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x} \]

[Out]

-1/5*d^2*(a+b*arcsech(c*x))/x^5-2/3*d*e*(a+b*arcsech(c*x))/x^3-e^2*(a+b*arcsech(c*x))/x+1/25*b*d^2*(1/(c*x+1))
^(1/2)*(c*x+1)^(1/2)*(-c^2*x^2+1)^(1/2)/x^5+2/225*b*d*(6*c^2*d+25*e)*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*(-c^2*x^2
+1)^(1/2)/x^3+1/225*b*(24*c^4*d^2+100*c^2*d*e+225*e^2)*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*(-c^2*x^2+1)^(1/2)/x

________________________________________________________________________________________

Rubi [A]
time = 0.11, antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {276, 6436, 12, 1279, 464, 270} \begin {gather*} -\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 x^5}-\frac {2 d e \left (a+b \text {sech}^{-1}(c x)\right )}{3 x^3}-\frac {e^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x}+\frac {b d^2 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{25 x^5}+\frac {2 b d \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (6 c^2 d+25 e\right )}{225 x^3}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (24 c^4 d^2+100 c^2 d e+225 e^2\right )}{225 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*d^2*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(25*x^5) + (2*b*d*(6*c^2*d + 25*e)*Sqrt[(1 + c*x)
^(-1)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(225*x^3) + (b*(24*c^4*d^2 + 100*c^2*d*e + 225*e^2)*Sqrt[(1 + c*x)^(-1
)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(225*x) - (d^2*(a + b*ArcSech[c*x]))/(5*x^5) - (2*d*e*(a + b*ArcSech[c*x])
)/(3*x^3) - (e^2*(a + b*ArcSech[c*x]))/x

Rule 12

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

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 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

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 1279

Int[((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Wit
h[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, f*x, x], R = PolynomialRemainder[(a + b*x^2 + c*x^4)^p, f*x,
x]}, Simp[R*(f*x)^(m + 1)*((d + e*x^2)^(q + 1)/(d*f*(m + 1))), x] + Dist[1/(d*f^2*(m + 1)), Int[(f*x)^(m + 2)
*(d + e*x^2)^q*ExpandToSum[d*f*(m + 1)*(Qx/x) - e*R*(m + 2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e, f, q},
x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && LtQ[m, -1]

Rule 6436

Int[((a_.) + ArcSech[(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*ArcSech[c*x], u, x] + Dist[b*Sqrt[1 + c*x]*Sqrt[1/(1 + c*x)],
Int[SimplifyIntegrand[u/(x*Sqrt[1 - c*x]*Sqrt[1 + c*x]), 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 )^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x^6} \, dx &=-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 x^5}-\frac {2 d e \left (a+b \text {sech}^{-1}(c x)\right )}{3 x^3}-\frac {e^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x}+\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {-3 d^2-10 d e x^2-15 e^2 x^4}{15 x^6 \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 x^5}-\frac {2 d e \left (a+b \text {sech}^{-1}(c x)\right )}{3 x^3}-\frac {e^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x}+\frac {1}{15} \left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {-3 d^2-10 d e x^2-15 e^2 x^4}{x^6 \sqrt {1-c^2 x^2}} \, dx\\ &=\frac {b d^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{25 x^5}-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 x^5}-\frac {2 d e \left (a+b \text {sech}^{-1}(c x)\right )}{3 x^3}-\frac {e^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x}-\frac {1}{75} \left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {2 d \left (6 c^2 d+25 e\right )+75 e^2 x^2}{x^4 \sqrt {1-c^2 x^2}} \, dx\\ &=\frac {b d^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{25 x^5}+\frac {2 b d \left (6 c^2 d+25 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{225 x^3}-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 x^5}-\frac {2 d e \left (a+b \text {sech}^{-1}(c x)\right )}{3 x^3}-\frac {e^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x}-\frac {1}{225} \left (b \left (24 c^4 d^2+100 c^2 d e+225 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{x^2 \sqrt {1-c^2 x^2}} \, dx\\ &=\frac {b d^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{25 x^5}+\frac {2 b d \left (6 c^2 d+25 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{225 x^3}+\frac {b \left (24 c^4 d^2+100 c^2 d e+225 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{225 x}-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 x^5}-\frac {2 d e \left (a+b \text {sech}^{-1}(c x)\right )}{3 x^3}-\frac {e^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.15, size = 134, normalized size = 0.63 \begin {gather*} \frac {-15 a \left (3 d^2+10 d e x^2+15 e^2 x^4\right )+b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (225 e^2 x^4+50 d e x^2 \left (1+2 c^2 x^2\right )+3 d^2 \left (3+4 c^2 x^2+8 c^4 x^4\right )\right )-15 b \left (3 d^2+10 d e x^2+15 e^2 x^4\right ) \text {sech}^{-1}(c x)}{225 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-15*a*(3*d^2 + 10*d*e*x^2 + 15*e^2*x^4) + b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(225*e^2*x^4 + 50*d*e*x^2*(1
+ 2*c^2*x^2) + 3*d^2*(3 + 4*c^2*x^2 + 8*c^4*x^4)) - 15*b*(3*d^2 + 10*d*e*x^2 + 15*e^2*x^4)*ArcSech[c*x])/(225*
x^5)

________________________________________________________________________________________

Maple [A]
time = 0.25, size = 193, normalized size = 0.91

method	result	size
derivativedivides	\(c^{5} \left (\frac {a \left (-\frac {e^{2}}{c x}-\frac {2 d e}{3 c \,x^{3}}-\frac {d^{2}}{5 c \,x^{5}}\right )}{c^{4}}+\frac {b \left (-\frac {\mathrm {arcsech}\left (c x \right ) e^{2}}{c x}-\frac {2 \,\mathrm {arcsech}\left (c x \right ) d e}{3 c \,x^{3}}-\frac {\mathrm {arcsech}\left (c x \right ) d^{2}}{5 c \,x^{5}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (24 c^{8} d^{2} x^{4}+100 c^{6} d e \,x^{4}+12 c^{6} d^{2} x^{2}+225 c^{4} e^{2} x^{4}+50 c^{4} d e \,x^{2}+9 c^{4} d^{2}\right )}{225 c^{4} x^{4}}\right )}{c^{4}}\right )\)	\(193\)
default	\(c^{5} \left (\frac {a \left (-\frac {e^{2}}{c x}-\frac {2 d e}{3 c \,x^{3}}-\frac {d^{2}}{5 c \,x^{5}}\right )}{c^{4}}+\frac {b \left (-\frac {\mathrm {arcsech}\left (c x \right ) e^{2}}{c x}-\frac {2 \,\mathrm {arcsech}\left (c x \right ) d e}{3 c \,x^{3}}-\frac {\mathrm {arcsech}\left (c x \right ) d^{2}}{5 c \,x^{5}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (24 c^{8} d^{2} x^{4}+100 c^{6} d e \,x^{4}+12 c^{6} d^{2} x^{2}+225 c^{4} e^{2} x^{4}+50 c^{4} d e \,x^{2}+9 c^{4} d^{2}\right )}{225 c^{4} x^{4}}\right )}{c^{4}}\right )\)	\(193\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^5*(a/c^4*(-e^2/c/x-2/3/c*d*e/x^3-1/5/c*d^2/x^5)+b/c^4*(-arcsech(c*x)*e^2/c/x-2/3*arcsech(c*x)/c*d*e/x^3-1/5*
arcsech(c*x)/c*d^2/x^5+1/225*(-(c*x-1)/c/x)^(1/2)/c^4/x^4*((c*x+1)/c/x)^(1/2)*(24*c^8*d^2*x^4+100*c^6*d*e*x^4+
12*c^6*d^2*x^2+225*c^4*e^2*x^4+50*c^4*d*e*x^2+9*c^4*d^2)))

________________________________________________________________________________________

Maxima [A]
time = 0.27, size = 175, normalized size = 0.82 \begin {gather*} \frac {1}{75} \, b d^{2} {\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 {arsech}\left (c x\right )}{x^{5}}\right )} + \frac {2}{9} \, b d {\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 {arsech}\left (c x\right )}{x^{3}}\right )} e + {\left (c \sqrt {\frac {1}{c^{2} x^{2}} - 1} - \frac {\operatorname {arsech}\left (c x\right )}{x}\right )} b e^{2} - \frac {a e^{2}}{x} - \frac {2 \, a d e}{3 \, x^{3}} - \frac {a d^{2}}{5 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 278 vs. \(2 (138) = 276\).
time = 0.45, size = 278, normalized size = 1.31 \begin {gather*} -\frac {225 \, a x^{4} \cosh \left (1\right )^{2} + 225 \, a x^{4} \sinh \left (1\right )^{2} + 150 \, a d x^{2} \cosh \left (1\right ) + 45 \, a d^{2} + 15 \, {\left (15 \, b x^{4} \cosh \left (1\right )^{2} + 15 \, b x^{4} \sinh \left (1\right )^{2} + 10 \, b d x^{2} \cosh \left (1\right ) + 3 \, b d^{2} + 10 \, {\left (3 \, b x^{4} \cosh \left (1\right ) + b d x^{2}\right )} \sinh \left (1\right )\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) + 150 \, {\left (3 \, a x^{4} \cosh \left (1\right ) + a d x^{2}\right )} \sinh \left (1\right ) - {\left (24 \, b c^{5} d^{2} x^{5} + 12 \, b c^{3} d^{2} x^{3} + 225 \, b c x^{5} \cosh \left (1\right )^{2} + 225 \, b c x^{5} \sinh \left (1\right )^{2} + 9 \, b c d^{2} x + 50 \, {\left (2 \, b c^{3} d x^{5} + b c d x^{3}\right )} \cosh \left (1\right ) + 50 \, {\left (2 \, b c^{3} d x^{5} + 9 \, b c x^{5} \cosh \left (1\right ) + b c d 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)^2*(a+b*arcsech(c*x))/x^6,x, algorithm="fricas")

[Out]

-1/225*(225*a*x^4*cosh(1)^2 + 225*a*x^4*sinh(1)^2 + 150*a*d*x^2*cosh(1) + 45*a*d^2 + 15*(15*b*x^4*cosh(1)^2 +
15*b*x^4*sinh(1)^2 + 10*b*d*x^2*cosh(1) + 3*b*d^2 + 10*(3*b*x^4*cosh(1) + b*d*x^2)*sinh(1))*log((c*x*sqrt(-(c^
2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)) + 150*(3*a*x^4*cosh(1) + a*d*x^2)*sinh(1) - (24*b*c^5*d^2*x^5 + 12*b*c^3*d^2
*x^3 + 225*b*c*x^5*cosh(1)^2 + 225*b*c*x^5*sinh(1)^2 + 9*b*c*d^2*x + 50*(2*b*c^3*d*x^5 + b*c*d*x^3)*cosh(1) +
50*(2*b*c^3*d*x^5 + 9*b*c*x^5*cosh(1) + b*c*d*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 {asech}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}}{x^{6}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________