\(\int \frac {a+b \sec ^{-1}(c x)}{x^7} \, dx\) [14]

Optimal result

Integrand size = 12, antiderivative size = 101 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^7} \, dx=\frac {b c \sqrt {1-\frac {1}{c^2 x^2}}}{36 x^5}+\frac {5 b c^3 \sqrt {1-\frac {1}{c^2 x^2}}}{144 x^3}+\frac {5 b c^5 \sqrt {1-\frac {1}{c^2 x^2}}}{96 x}-\frac {5}{96} b c^6 \csc ^{-1}(c x)-\frac {a+b \sec ^{-1}(c x)}{6 x^6} \] Output:

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.87 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^7} \, dx=-\frac {a}{6 x^6}+b \left (\frac {c}{36 x^5}+\frac {5 c^3}{144 x^3}+\frac {5 c^5}{96 x}\right ) \sqrt {\frac {-1+c^2 x^2}{c^2 x^2}}-\frac {b \sec ^{-1}(c x)}{6 x^6}-\frac {5}{96} b c^6 \arcsin \left (\frac {1}{c x}\right ) \] Input:

Integrate[(a + b*ArcSec[c*x])/x^7,x]
 

Output:

-1/6*a/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
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.27, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5743, 858, 262, 262, 262, 223}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 5743

\(\displaystyle \frac {b \int \frac {1}{\sqrt {1-\frac {1}{c^2 x^2}} x^8}dx}{6 c}-\frac {a+b \sec ^{-1}(c x)}{6 x^6}\)

\(\Big \downarrow \) 858

\(\displaystyle -\frac {b \int \frac {1}{\sqrt {1-\frac {1}{c^2 x^2}} x^6}d\frac {1}{x}}{6 c}-\frac {a+b \sec ^{-1}(c x)}{6 x^6}\)

\(\Big \downarrow \) 262

\(\displaystyle -\frac {b \left (\frac {5}{6} c^2 \int \frac {1}{\sqrt {1-\frac {1}{c^2 x^2}} x^4}d\frac {1}{x}-\frac {c^2 \sqrt {1-\frac {1}{c^2 x^2}}}{6 x^5}\right )}{6 c}-\frac {a+b \sec ^{-1}(c x)}{6 x^6}\)

\(\Big \downarrow \) 262

\(\displaystyle -\frac {b \left (\frac {5}{6} c^2 \left (\frac {3}{4} c^2 \int \frac {1}{\sqrt {1-\frac {1}{c^2 x^2}} x^2}d\frac {1}{x}-\frac {c^2 \sqrt {1-\frac {1}{c^2 x^2}}}{4 x^3}\right )-\frac {c^2 \sqrt {1-\frac {1}{c^2 x^2}}}{6 x^5}\right )}{6 c}-\frac {a+b \sec ^{-1}(c x)}{6 x^6}\)

\(\Big \downarrow \) 262

\(\displaystyle -\frac {b \left (\frac {5}{6} c^2 \left (\frac {3}{4} c^2 \left (\frac {1}{2} c^2 \int \frac {1}{\sqrt {1-\frac {1}{c^2 x^2}}}d\frac {1}{x}-\frac {c^2 \sqrt {1-\frac {1}{c^2 x^2}}}{2 x}\right )-\frac {c^2 \sqrt {1-\frac {1}{c^2 x^2}}}{4 x^3}\right )-\frac {c^2 \sqrt {1-\frac {1}{c^2 x^2}}}{6 x^5}\right )}{6 c}-\frac {a+b \sec ^{-1}(c x)}{6 x^6}\)

\(\Big \downarrow \) 223

\(\displaystyle -\frac {a+b \sec ^{-1}(c x)}{6 x^6}-\frac {b \left (\frac {5}{6} c^2 \left (\frac {3}{4} c^2 \left (\frac {1}{2} c^3 \arcsin \left (\frac {1}{c x}\right )-\frac {c^2 \sqrt {1-\frac {1}{c^2 x^2}}}{2 x}\right )-\frac {c^2 \sqrt {1-\frac {1}{c^2 x^2}}}{4 x^3}\right )-\frac {c^2 \sqrt {1-\frac {1}{c^2 x^2}}}{6 x^5}\right )}{6 c}\)

Input:

Int[(a + b*ArcSec[c*x])/x^7,x]
 

Output:

-1/6*(a + b*ArcSec[c*x])/x^6 - (b*(-1/6*(c^2*Sqrt[1 - 1/(c^2*x^2)])/x^5 + 
(5*c^2*(-1/4*(c^2*Sqrt[1 - 1/(c^2*x^2)])/x^3 + (3*c^2*(-1/2*(c^2*Sqrt[1 - 
1/(c^2*x^2)])/x + (c^3*ArcSin[1/(c*x)])/2))/4))/6))/(6*c)
 

Defintions of rubi rules used

rule 223

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]
 

rule 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, x]
 

rule 858

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] && Int 
egerQ[m]
 

rule 5743

Int[((a_.) + ArcSec[(c_.)*(x_)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Sim 
p[(d*x)^(m + 1)*((a + b*ArcSec[c*x])/(d*(m + 1))), x] - Simp[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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(173\) vs. \(2(85)=170\).

Time = 0.18 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.72

method	result	size
parts	\(-\frac {a}{6 x^{6}}-\frac {\operatorname {arcsec}\left (c x \right ) b}{6 x^{6}}-\frac {5 b \,c^{5} \sqrt {c^{2} x^{2}-1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{96 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {5 b \,c^{3} \left (c^{2} x^{2}-1\right )}{96 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{3}}+\frac {5 b c \left (c^{2} x^{2}-1\right )}{144 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{5}}+\frac {b \left (c^{2} x^{2}-1\right )}{36 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{7}}\)	\(174\)
derivativedivides	\(c^{6} \left (-\frac {a}{6 c^{6} x^{6}}-\frac {b \,\operatorname {arcsec}\left (c x \right )}{6 c^{6} x^{6}}-\frac {5 b \sqrt {c^{2} x^{2}-1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{96 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}+\frac {5 b \left (c^{2} x^{2}-1\right )}{96 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x^{3}}+\frac {5 b \left (c^{2} x^{2}-1\right )}{144 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{5} x^{5}}+\frac {b \left (c^{2} x^{2}-1\right )}{36 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{7} x^{7}}\right )\)	\(186\)
default	\(c^{6} \left (-\frac {a}{6 c^{6} x^{6}}-\frac {b \,\operatorname {arcsec}\left (c x \right )}{6 c^{6} x^{6}}-\frac {5 b \sqrt {c^{2} x^{2}-1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{96 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}+\frac {5 b \left (c^{2} x^{2}-1\right )}{96 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x^{3}}+\frac {5 b \left (c^{2} x^{2}-1\right )}{144 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{5} x^{5}}+\frac {b \left (c^{2} x^{2}-1\right )}{36 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{7} x^{7}}\right )\)	\(186\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.61 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^7} \, dx=\frac {3 \, {\left (5 \, b c^{6} x^{6} - 16 \, b\right )} \operatorname {arcsec}\left (c x\right ) + {\left (15 \, b c^{4} x^{4} + 10 \, b c^{2} x^{2} + 8 \, b\right )} \sqrt {c^{2} x^{2} - 1} - 48 \, a}{288 \, x^{6}} \] Input:

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

Output:

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
 

Sympy [A] (verification not implemented)

Time = 9.99 (sec) , antiderivative size = 241, normalized size of antiderivative = 2.39 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^7} \, dx=- \frac {a}{6 x^{6}} - \frac {b \operatorname {asec}{\left (c x \right )}}{6 x^{6}} + \frac {b \left (\begin {cases} \frac {5 i c^{7} \operatorname {acosh}{\left (\frac {1}{c x} \right )}}{16} - \frac {5 i c^{6}}{16 x \sqrt {-1 + \frac {1}{c^{2} x^{2}}}} + \frac {5 i c^{4}}{48 x^{3} \sqrt {-1 + \frac {1}{c^{2} x^{2}}}} + \frac {i c^{2}}{24 x^{5} \sqrt {-1 + \frac {1}{c^{2} x^{2}}}} + \frac {i}{6 x^{7} \sqrt {-1 + \frac {1}{c^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{c^{2} x^{2}}\right |} > 1 \\- \frac {5 c^{7} \operatorname {asin}{\left (\frac {1}{c x} \right )}}{16} + \frac {5 c^{6}}{16 x \sqrt {1 - \frac {1}{c^{2} x^{2}}}} - \frac {5 c^{4}}{48 x^{3} \sqrt {1 - \frac {1}{c^{2} x^{2}}}} - \frac {c^{2}}{24 x^{5} \sqrt {1 - \frac {1}{c^{2} x^{2}}}} - \frac {1}{6 x^{7} \sqrt {1 - \frac {1}{c^{2} x^{2}}}} & \text {otherwise} \end {cases}\right )}{6 c} \] Input:

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

Output:

-a/(6*x**6) - b*asec(c*x)/(6*x**6) + b*Piecewise((5*I*c**7*acosh(1/(c*x))/ 
16 - 5*I*c**6/(16*x*sqrt(-1 + 1/(c**2*x**2))) + 5*I*c**4/(48*x**3*sqrt(-1 
+ 1/(c**2*x**2))) + I*c**2/(24*x**5*sqrt(-1 + 1/(c**2*x**2))) + I/(6*x**7* 
sqrt(-1 + 1/(c**2*x**2))), 1/Abs(c**2*x**2) > 1), (-5*c**7*asin(1/(c*x))/1 
6 + 5*c**6/(16*x*sqrt(1 - 1/(c**2*x**2))) - 5*c**4/(48*x**3*sqrt(1 - 1/(c* 
*2*x**2))) - c**2/(24*x**5*sqrt(1 - 1/(c**2*x**2))) - 1/(6*x**7*sqrt(1 - 1 
/(c**2*x**2))), True))/(6*c)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.63 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^7} \, dx=\frac {1}{288} \, b {\left (\frac {15 \, c^{7} \arctan \left (c x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}\right ) - \frac {15 \, c^{12} x^{5} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} + 40 \, c^{10} x^{3} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 33 \, c^{8} x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{6} x^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{3} - 3 \, c^{4} x^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} + 3 \, c^{2} x^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} - 1}}{c} - \frac {48 \, \operatorname {arcsec}\left (c x\right )}{x^{6}}\right )} - \frac {a}{6 \, x^{6}} \] Input:

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

Output:

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
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.03 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^7} \, dx=\frac {1}{288} \, {\left (15 \, b c^{5} \arccos \left (\frac {1}{c x}\right ) + \frac {15 \, b c^{4} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{x} + \frac {10 \, b c^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{x^{3}} + \frac {8 \, b \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{x^{5}} - \frac {48 \, b \arccos \left (\frac {1}{c x}\right )}{c x^{6}} - \frac {48 \, a}{c x^{6}}\right )} c \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \sec ^{-1}(c x)}{x^7} \, dx=\int \frac {a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )}{x^7} \,d x \] Input:

int((a + b*acos(1/(c*x)))/x^7,x)
 

Output:

int((a + b*acos(1/(c*x)))/x^7, x)
 

Reduce [F]

\[ \int \frac {a+b \sec ^{-1}(c x)}{x^7} \, dx=\frac {6 \left (\int \frac {\mathit {asec} \left (c x \right )}{x^{7}}d x \right ) b \,x^{6}-a}{6 x^{6}} \] Input:

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

Output:

(6*int(asec(c*x)/x**7,x)*b*x**6 - a)/(6*x**6)