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

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

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 101, normalized size of antiderivative = 1.00, 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 = {5329, 342, 327, 222} \begin {gather*} -\frac {a+b \csc ^{-1}(c x)}{6 x^6}+\frac {5}{96} b c^6 \csc ^{-1}(c x)-\frac {b c \sqrt {1-\frac {1}{c^2 x^2}}}{36 x^5}-\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} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCsc[c*x])/x^7,x]

[Out]

-1/36*(b*c*Sqrt[1 - 1/(c^2*x^2)])/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*ArcCsc[c*x])/(6*x^6)

Rule 222

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 327

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 342

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 5329

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcCsc[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]

Rubi steps

\begin {align*} \int \frac {a+b \csc ^{-1}(c x)}{x^7} \, dx &=-\frac {a+b \csc ^{-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 \csc ^{-1}(c x)}{6 x^6}+\frac {b \text {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 \csc ^{-1}(c x)}{6 x^6}+\frac {1}{36} (5 b c) \text {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 \csc ^{-1}(c x)}{6 x^6}+\frac {1}{48} \left (5 b c^3\right ) \text {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 \csc ^{-1}(c x)}{6 x^6}+\frac {1}{96} \left (5 b c^5\right ) \text {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 \csc ^{-1}(c x)}{6 x^6}\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 88, normalized size = 0.87 \begin {gather*} -\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 \csc ^{-1}(c x)}{6 x^6}+\frac {5}{96} b c^6 \text {ArcSin}\left (\frac {1}{c x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcCsc[c*x])/x^7,x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(185\) vs. \(2(88)=176\).
time = 0.14, size = 186, normalized size = 1.84

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.50, size = 165, normalized size = 1.63 \begin {gather*} -\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 {arccsc}\left (c x\right )}{x^{6}}\right )} - \frac {a}{6 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsc(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*arccsc(c*x)/x^6) - 1/6*a/x^6

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Fricas [A]
time = 0.34, size = 63, normalized size = 0.62 \begin {gather*} \frac {3 \, {\left (5 \, b c^{6} x^{6} - 16 \, b\right )} \operatorname {arccsc}\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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/288*(3*(5*b*c^6*x^6 - 16*b)*arccsc(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 [A]
time = 10.91, size = 243, normalized size = 2.41 \begin {gather*} - \frac {a}{6 x^{6}} - \frac {b \operatorname {acsc}{\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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a/(6*x**6) - b*acsc(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))/16 + 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)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 174 vs. \(2 (85) = 170\).
time = 0.43, size = 174, normalized size = 1.72 \begin {gather*} -\frac {1}{288} \, {\left (48 \, b c^{5} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{3} \arcsin \left (\frac {1}{c x}\right ) + 144 \, b c^{5} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} \arcsin \left (\frac {1}{c x}\right ) + 144 \, b c^{5} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right ) + 33 \, b c^{5} \arcsin \left (\frac {1}{c x}\right ) + \frac {8 \, b c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{x} - \frac {26 \, b c^{4} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}}}{x} + \frac {33 \, b c^{4} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{x} + \frac {48 \, a}{c x^{6}}\right )} c \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/288*(48*b*c^5*(1/(c^2*x^2) - 1)^3*arcsin(1/(c*x)) + 144*b*c^5*(1/(c^2*x^2) - 1)^2*arcsin(1/(c*x)) + 144*b*c
^5*(1/(c^2*x^2) - 1)*arcsin(1/(c*x)) + 33*b*c^5*arcsin(1/(c*x)) + 8*b*c^4*(1/(c^2*x^2) - 1)^2*sqrt(-1/(c^2*x^2
) + 1)/x - 26*b*c^4*(-1/(c^2*x^2) + 1)^(3/2)/x + 33*b*c^4*sqrt(-1/(c^2*x^2) + 1)/x + 48*a/(c*x^6))*c

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )}{x^7} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asin(1/(c*x)))/x^7,x)

[Out]

int((a + b*asin(1/(c*x)))/x^7, x)

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