3.14 \(\int \frac{a+b \sec ^{-1}(c x)}{x^7} \, dx\)

Optimal. Leaf size=101 \[ -\frac{a+b \sec ^{-1}(c x)}{6 x^6}+\frac{5 b c^5 \sqrt{1-\frac{1}{c^2 x^2}}}{96 x}+\frac{5 b c^3 \sqrt{1-\frac{1}{c^2 x^2}}}{144 x^3}+\frac{b c \sqrt{1-\frac{1}{c^2 x^2}}}{36 x^5}-\frac{5}{96} b c^6 \csc ^{-1}(c x) \]

[Out]

(b*c*Sqrt[1 - 1/(c^2*x^2)])/(36*x^5) + (5*b*c^3*Sqrt[1 - 1/(c^2*x^2)])/(144*x^3) + (5*b*c^5*Sqrt[1 - 1/(c^2*x^
2)])/(96*x) - (5*b*c^6*ArcCsc[c*x])/96 - (a + b*ArcSec[c*x])/(6*x^6)

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Rubi [A]  time = 0.0588217, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5220, 335, 321, 216} \[ -\frac{a+b \sec ^{-1}(c x)}{6 x^6}+\frac{5 b c^5 \sqrt{1-\frac{1}{c^2 x^2}}}{96 x}+\frac{5 b c^3 \sqrt{1-\frac{1}{c^2 x^2}}}{144 x^3}+\frac{b c \sqrt{1-\frac{1}{c^2 x^2}}}{36 x^5}-\frac{5}{96} b c^6 \csc ^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSec[c*x])/x^7,x]

[Out]

(b*c*Sqrt[1 - 1/(c^2*x^2)])/(36*x^5) + (5*b*c^3*Sqrt[1 - 1/(c^2*x^2)])/(144*x^3) + (5*b*c^5*Sqrt[1 - 1/(c^2*x^
2)])/(96*x) - (5*b*c^6*ArcCsc[c*x])/96 - (a + b*ArcSec[c*x])/(6*x^6)

Rule 5220

Int[((a_.) + ArcSec[(c_.)*(x_)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcSec[c*x]
))/(d*(m + 1)), x] - Dist[(b*d)/(c*(m + 1)), Int[(d*x)^(m - 1)/Sqrt[1 - 1/(c^2*x^2)], x], x] /; FreeQ[{a, b, c
, d, m}, x] && NeQ[m, -1]

Rule 335

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /;
FreeQ[{a, b, p}, x] && ILtQ[n, 0] && IntegerQ[m]

Rule 321

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

Rule 216

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

Rubi steps

\begin{align*} \int \frac{a+b \sec ^{-1}(c x)}{x^7} \, dx &=-\frac{a+b \sec ^{-1}(c x)}{6 x^6}+\frac{b \int \frac{1}{\sqrt{1-\frac{1}{c^2 x^2}} x^8} \, dx}{6 c}\\ &=-\frac{a+b \sec ^{-1}(c x)}{6 x^6}-\frac{b \operatorname{Subst}\left (\int \frac{x^6}{\sqrt{1-\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )}{6 c}\\ &=\frac{b c \sqrt{1-\frac{1}{c^2 x^2}}}{36 x^5}-\frac{a+b \sec ^{-1}(c x)}{6 x^6}-\frac{1}{36} (5 b c) \operatorname{Subst}\left (\int \frac{x^4}{\sqrt{1-\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{b c \sqrt{1-\frac{1}{c^2 x^2}}}{36 x^5}+\frac{5 b c^3 \sqrt{1-\frac{1}{c^2 x^2}}}{144 x^3}-\frac{a+b \sec ^{-1}(c x)}{6 x^6}-\frac{1}{48} \left (5 b c^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{b c \sqrt{1-\frac{1}{c^2 x^2}}}{36 x^5}+\frac{5 b c^3 \sqrt{1-\frac{1}{c^2 x^2}}}{144 x^3}+\frac{5 b c^5 \sqrt{1-\frac{1}{c^2 x^2}}}{96 x}-\frac{a+b \sec ^{-1}(c x)}{6 x^6}-\frac{1}{96} \left (5 b c^5\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{b c \sqrt{1-\frac{1}{c^2 x^2}}}{36 x^5}+\frac{5 b c^3 \sqrt{1-\frac{1}{c^2 x^2}}}{144 x^3}+\frac{5 b c^5 \sqrt{1-\frac{1}{c^2 x^2}}}{96 x}-\frac{5}{96} b c^6 \csc ^{-1}(c x)-\frac{a+b \sec ^{-1}(c x)}{6 x^6}\\ \end{align*}

Mathematica [A]  time = 0.0707693, size = 88, normalized size = 0.87 \[ -\frac{a}{6 x^6}+b \left (\frac{5 c^3}{144 x^3}+\frac{5 c^5}{96 x}+\frac{c}{36 x^5}\right ) \sqrt{\frac{c^2 x^2-1}{c^2 x^2}}-\frac{5}{96} b c^6 \sin ^{-1}\left (\frac{1}{c x}\right )-\frac{b \sec ^{-1}(c x)}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcSec[c*x])/x^7,x]

[Out]

-a/(6*x^6) + b*(c/(36*x^5) + (5*c^3)/(144*x^3) + (5*c^5)/(96*x))*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)] - (b*ArcSec[c*
x])/(6*x^6) - (5*b*c^6*ArcSin[1/(c*x)])/96

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Maple [A]  time = 0.16, size = 174, normalized size = 1.7 \begin{align*} -{\frac{a}{6\,{x}^{6}}}-{\frac{b{\rm arcsec} \left (cx\right )}{6\,{x}^{6}}}-{\frac{5\,{c}^{5}b}{96\,x}\sqrt{{c}^{2}{x}^{2}-1}\arctan \left ({\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}} \right ){\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}+{\frac{5\,{c}^{5}b}{96\,x}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}-{\frac{5\,b{c}^{3}}{288\,{x}^{3}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}-{\frac{cb}{144\,{x}^{5}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}-{\frac{b}{36\,c{x}^{7}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsec(c*x))/x^7,x)

[Out]

-1/6*a/x^6-1/6*b/x^6*arcsec(c*x)-5/96*c^5*b*(c^2*x^2-1)^(1/2)/((c^2*x^2-1)/c^2/x^2)^(1/2)/x*arctan(1/(c^2*x^2-
1)^(1/2))+5/96*c^5*b/((c^2*x^2-1)/c^2/x^2)^(1/2)/x-5/288*c^3*b/((c^2*x^2-1)/c^2/x^2)^(1/2)/x^3-1/144*c*b/((c^2
*x^2-1)/c^2/x^2)^(1/2)/x^5-1/36/c*b/((c^2*x^2-1)/c^2/x^2)^(1/2)/x^7

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Maxima [A]  time = 1.4639, size = 223, normalized size = 2.21 \begin{align*} \frac{1}{288} \, b{\left (\frac{15 \, c^{7} \arctan \left (c x \sqrt{-\frac{1}{c^{2} x^{2}} + 1}\right ) - \frac{15 \, c^{12} x^{5}{\left (-\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{5}{2}} + 40 \, c^{10} x^{3}{\left (-\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{3}{2}} + 33 \, c^{8} x \sqrt{-\frac{1}{c^{2} x^{2}} + 1}}{c^{6} x^{6}{\left (\frac{1}{c^{2} x^{2}} - 1\right )}^{3} - 3 \, c^{4} x^{4}{\left (\frac{1}{c^{2} x^{2}} - 1\right )}^{2} + 3 \, c^{2} x^{2}{\left (\frac{1}{c^{2} x^{2}} - 1\right )} - 1}}{c} - \frac{48 \, \operatorname{arcsec}\left (c x\right )}{x^{6}}\right )} - \frac{a}{6 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsec(c*x))/x^7,x, algorithm="maxima")

[Out]

1/288*b*((15*c^7*arctan(c*x*sqrt(-1/(c^2*x^2) + 1)) - (15*c^12*x^5*(-1/(c^2*x^2) + 1)^(5/2) + 40*c^10*x^3*(-1/
(c^2*x^2) + 1)^(3/2) + 33*c^8*x*sqrt(-1/(c^2*x^2) + 1))/(c^6*x^6*(1/(c^2*x^2) - 1)^3 - 3*c^4*x^4*(1/(c^2*x^2)
- 1)^2 + 3*c^2*x^2*(1/(c^2*x^2) - 1) - 1))/c - 48*arcsec(c*x)/x^6) - 1/6*a/x^6

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Fricas [A]  time = 2.48944, size = 150, normalized size = 1.49 \begin{align*} \frac{3 \,{\left (5 \, b c^{6} x^{6} - 16 \, b\right )} \operatorname{arcsec}\left (c x\right ) +{\left (15 \, b c^{4} x^{4} + 10 \, b c^{2} x^{2} + 8 \, b\right )} \sqrt{c^{2} x^{2} - 1} - 48 \, a}{288 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsec(c*x))/x^7,x, algorithm="fricas")

[Out]

1/288*(3*(5*b*c^6*x^6 - 16*b)*arcsec(c*x) + (15*b*c^4*x^4 + 10*b*c^2*x^2 + 8*b)*sqrt(c^2*x^2 - 1) - 48*a)/x^6

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \operatorname{asec}{\left (c x \right )}}{x^{7}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asec(c*x))/x**7,x)

[Out]

Integral((a + b*asec(c*x))/x**7, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{arcsec}\left (c x\right ) + a}{x^{7}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsec(c*x))/x^7,x, algorithm="giac")

[Out]

integrate((b*arcsec(c*x) + a)/x^7, x)