3.1 \(\int x^6 (a+b \csc ^{-1}(c x)) \, dx\)

Optimal. Leaf size=114 \[ \frac {1}{7} x^7 \left (a+b \csc ^{-1}(c x)\right )+\frac {b x^6 \sqrt {1-\frac {1}{c^2 x^2}}}{42 c}+\frac {5 b \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{112 c^7}+\frac {5 b x^2 \sqrt {1-\frac {1}{c^2 x^2}}}{112 c^5}+\frac {5 b x^4 \sqrt {1-\frac {1}{c^2 x^2}}}{168 c^3} \]

[Out]

1/7*x^7*(a+b*arccsc(c*x))+5/112*b*arctanh((1-1/c^2/x^2)^(1/2))/c^7+5/112*b*x^2*(1-1/c^2/x^2)^(1/2)/c^5+5/168*b
*x^4*(1-1/c^2/x^2)^(1/2)/c^3+1/42*b*x^6*(1-1/c^2/x^2)^(1/2)/c

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Rubi [A]  time = 0.06, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5221, 266, 51, 63, 208} \[ \frac {1}{7} x^7 \left (a+b \csc ^{-1}(c x)\right )+\frac {b x^6 \sqrt {1-\frac {1}{c^2 x^2}}}{42 c}+\frac {5 b x^4 \sqrt {1-\frac {1}{c^2 x^2}}}{168 c^3}+\frac {5 b x^2 \sqrt {1-\frac {1}{c^2 x^2}}}{112 c^5}+\frac {5 b \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{112 c^7} \]

Antiderivative was successfully verified.

[In]

Int[x^6*(a + b*ArcCsc[c*x]),x]

[Out]

(5*b*Sqrt[1 - 1/(c^2*x^2)]*x^2)/(112*c^5) + (5*b*Sqrt[1 - 1/(c^2*x^2)]*x^4)/(168*c^3) + (b*Sqrt[1 - 1/(c^2*x^2
)]*x^6)/(42*c) + (x^7*(a + b*ArcCsc[c*x]))/7 + (5*b*ArcTanh[Sqrt[1 - 1/(c^2*x^2)]])/(112*c^7)

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rule 266

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

Rule 5221

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 x^6 \left (a+b \csc ^{-1}(c x)\right ) \, dx &=\frac {1}{7} x^7 \left (a+b \csc ^{-1}(c x)\right )+\frac {b \int \frac {x^5}{\sqrt {1-\frac {1}{c^2 x^2}}} \, dx}{7 c}\\ &=\frac {1}{7} x^7 \left (a+b \csc ^{-1}(c x)\right )-\frac {b \operatorname {Subst}\left (\int \frac {1}{x^4 \sqrt {1-\frac {x}{c^2}}} \, dx,x,\frac {1}{x^2}\right )}{14 c}\\ &=\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^6}{42 c}+\frac {1}{7} x^7 \left (a+b \csc ^{-1}(c x)\right )-\frac {(5 b) \operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt {1-\frac {x}{c^2}}} \, dx,x,\frac {1}{x^2}\right )}{84 c^3}\\ &=\frac {5 b \sqrt {1-\frac {1}{c^2 x^2}} x^4}{168 c^3}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^6}{42 c}+\frac {1}{7} x^7 \left (a+b \csc ^{-1}(c x)\right )-\frac {(5 b) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1-\frac {x}{c^2}}} \, dx,x,\frac {1}{x^2}\right )}{112 c^5}\\ &=\frac {5 b \sqrt {1-\frac {1}{c^2 x^2}} x^2}{112 c^5}+\frac {5 b \sqrt {1-\frac {1}{c^2 x^2}} x^4}{168 c^3}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^6}{42 c}+\frac {1}{7} x^7 \left (a+b \csc ^{-1}(c x)\right )-\frac {(5 b) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{c^2}}} \, dx,x,\frac {1}{x^2}\right )}{224 c^7}\\ &=\frac {5 b \sqrt {1-\frac {1}{c^2 x^2}} x^2}{112 c^5}+\frac {5 b \sqrt {1-\frac {1}{c^2 x^2}} x^4}{168 c^3}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^6}{42 c}+\frac {1}{7} x^7 \left (a+b \csc ^{-1}(c x)\right )+\frac {(5 b) \operatorname {Subst}\left (\int \frac {1}{c^2-c^2 x^2} \, dx,x,\sqrt {1-\frac {1}{c^2 x^2}}\right )}{112 c^5}\\ &=\frac {5 b \sqrt {1-\frac {1}{c^2 x^2}} x^2}{112 c^5}+\frac {5 b \sqrt {1-\frac {1}{c^2 x^2}} x^4}{168 c^3}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^6}{42 c}+\frac {1}{7} x^7 \left (a+b \csc ^{-1}(c x)\right )+\frac {5 b \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{112 c^7}\\ \end {align*}

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Mathematica [A]  time = 0.14, size = 107, normalized size = 0.94 \[ \frac {a x^7}{7}+\frac {5 b \log \left (x \left (\sqrt {\frac {c^2 x^2-1}{c^2 x^2}}+1\right )\right )}{112 c^7}+b \sqrt {\frac {c^2 x^2-1}{c^2 x^2}} \left (\frac {5 x^2}{112 c^5}+\frac {5 x^4}{168 c^3}+\frac {x^6}{42 c}\right )+\frac {1}{7} b x^7 \csc ^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

Integrate[x^6*(a + b*ArcCsc[c*x]),x]

[Out]

(a*x^7)/7 + b*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)]*((5*x^2)/(112*c^5) + (5*x^4)/(168*c^3) + x^6/(42*c)) + (b*x^7*Arc
Csc[c*x])/7 + (5*b*Log[x*(1 + Sqrt[(-1 + c^2*x^2)/(c^2*x^2)])])/(112*c^7)

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fricas [A]  time = 1.04, size = 115, normalized size = 1.01 \[ \frac {48 \, a c^{7} x^{7} - 96 \, b c^{7} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 48 \, {\left (b c^{7} x^{7} - b c^{7}\right )} \operatorname {arccsc}\left (c x\right ) - 15 \, b \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (8 \, b c^{5} x^{5} + 10 \, b c^{3} x^{3} + 15 \, b c x\right )} \sqrt {c^{2} x^{2} - 1}}{336 \, c^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(a+b*arccsc(c*x)),x, algorithm="fricas")

[Out]

1/336*(48*a*c^7*x^7 - 96*b*c^7*arctan(-c*x + sqrt(c^2*x^2 - 1)) + 48*(b*c^7*x^7 - b*c^7)*arccsc(c*x) - 15*b*lo
g(-c*x + sqrt(c^2*x^2 - 1)) + (8*b*c^5*x^5 + 10*b*c^3*x^3 + 15*b*c*x)*sqrt(c^2*x^2 - 1))/c^7

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giac [B]  time = 1.01, size = 646, normalized size = 5.67 \[ \frac {1}{2688} \, {\left (\frac {3 \, b x^{7} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{7} \arcsin \left (\frac {1}{c x}\right )}{c} + \frac {3 \, a x^{7} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{7}}{c} + \frac {b x^{6} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{6}}{c^{2}} + \frac {21 \, b x^{5} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{5} \arcsin \left (\frac {1}{c x}\right )}{c^{3}} + \frac {21 \, a x^{5} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{5}}{c^{3}} + \frac {9 \, b x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4}}{c^{4}} + \frac {63 \, b x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3} \arcsin \left (\frac {1}{c x}\right )}{c^{5}} + \frac {63 \, a x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}}{c^{5}} + \frac {45 \, b x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}}{c^{6}} + \frac {105 \, b x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )} \arcsin \left (\frac {1}{c x}\right )}{c^{7}} + \frac {105 \, a x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}}{c^{7}} + \frac {120 \, b \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{8}} - \frac {120 \, b \log \left (\frac {1}{{\left | c \right |} {\left | x \right |}}\right )}{c^{8}} + \frac {105 \, b \arcsin \left (\frac {1}{c x}\right )}{c^{9} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} + \frac {105 \, a}{c^{9} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} - \frac {45 \, b}{c^{10} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} + \frac {63 \, b \arcsin \left (\frac {1}{c x}\right )}{c^{11} x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}} + \frac {63 \, a}{c^{11} x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}} - \frac {9 \, b}{c^{12} x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4}} + \frac {21 \, b \arcsin \left (\frac {1}{c x}\right )}{c^{13} x^{5} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{5}} + \frac {21 \, a}{c^{13} x^{5} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{5}} - \frac {b}{c^{14} x^{6} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{6}} + \frac {3 \, b \arcsin \left (\frac {1}{c x}\right )}{c^{15} x^{7} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{7}} + \frac {3 \, a}{c^{15} x^{7} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{7}}\right )} c \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(a+b*arccsc(c*x)),x, algorithm="giac")

[Out]

1/2688*(3*b*x^7*(sqrt(-1/(c^2*x^2) + 1) + 1)^7*arcsin(1/(c*x))/c + 3*a*x^7*(sqrt(-1/(c^2*x^2) + 1) + 1)^7/c +
b*x^6*(sqrt(-1/(c^2*x^2) + 1) + 1)^6/c^2 + 21*b*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5*arcsin(1/(c*x))/c^3 + 21*a*
x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5/c^3 + 9*b*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4/c^4 + 63*b*x^3*(sqrt(-1/(c^2*x
^2) + 1) + 1)^3*arcsin(1/(c*x))/c^5 + 63*a*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3/c^5 + 45*b*x^2*(sqrt(-1/(c^2*x^2
) + 1) + 1)^2/c^6 + 105*b*x*(sqrt(-1/(c^2*x^2) + 1) + 1)*arcsin(1/(c*x))/c^7 + 105*a*x*(sqrt(-1/(c^2*x^2) + 1)
+ 1)/c^7 + 120*b*log(sqrt(-1/(c^2*x^2) + 1) + 1)/c^8 - 120*b*log(1/(abs(c)*abs(x)))/c^8 + 105*b*arcsin(1/(c*x
))/(c^9*x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 105*a/(c^9*x*(sqrt(-1/(c^2*x^2) + 1) + 1)) - 45*b/(c^10*x^2*(sqrt(-1
/(c^2*x^2) + 1) + 1)^2) + 63*b*arcsin(1/(c*x))/(c^11*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 63*a/(c^11*x^3*(sqr
t(-1/(c^2*x^2) + 1) + 1)^3) - 9*b/(c^12*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4) + 21*b*arcsin(1/(c*x))/(c^13*x^5*(
sqrt(-1/(c^2*x^2) + 1) + 1)^5) + 21*a/(c^13*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5) - b/(c^14*x^6*(sqrt(-1/(c^2*x^
2) + 1) + 1)^6) + 3*b*arcsin(1/(c*x))/(c^15*x^7*(sqrt(-1/(c^2*x^2) + 1) + 1)^7) + 3*a/(c^15*x^7*(sqrt(-1/(c^2*
x^2) + 1) + 1)^7))*c

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maple [A]  time = 0.05, size = 177, normalized size = 1.55 \[ \frac {x^{7} a}{7}+\frac {b \,x^{7} \mathrm {arccsc}\left (c x \right )}{7}+\frac {b \,x^{6}}{42 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \,x^{4}}{168 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {5 b \,x^{2}}{336 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {5 b}{112 c^{7} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {5 b \sqrt {c^{2} x^{2}-1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{112 c^{8} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(a+b*arccsc(c*x)),x)

[Out]

1/7*x^7*a+1/7*b*x^7*arccsc(c*x)+1/42/c*b/((c^2*x^2-1)/c^2/x^2)^(1/2)*x^6+1/168/c^3*b/((c^2*x^2-1)/c^2/x^2)^(1/
2)*x^4+5/336/c^5*b/((c^2*x^2-1)/c^2/x^2)^(1/2)*x^2-5/112/c^7*b/((c^2*x^2-1)/c^2/x^2)^(1/2)+5/112/c^8*b*(c^2*x^
2-1)^(1/2)/((c^2*x^2-1)/c^2/x^2)^(1/2)/x*ln(c*x+(c^2*x^2-1)^(1/2))

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maxima [A]  time = 0.34, size = 161, normalized size = 1.41 \[ \frac {1}{7} \, a x^{7} + \frac {1}{672} \, {\left (96 \, x^{7} \operatorname {arccsc}\left (c x\right ) + \frac {\frac {2 \, {\left (15 \, {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} - 40 \, {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 33 \, \sqrt {-\frac {1}{c^{2} x^{2}} + 1}\right )}}{c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{3} + 3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} + 3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{6}} + \frac {15 \, \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{6}} - \frac {15 \, \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{6}}}{c}\right )} b \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(a+b*arccsc(c*x)),x, algorithm="maxima")

[Out]

1/7*a*x^7 + 1/672*(96*x^7*arccsc(c*x) + (2*(15*(-1/(c^2*x^2) + 1)^(5/2) - 40*(-1/(c^2*x^2) + 1)^(3/2) + 33*sqr
t(-1/(c^2*x^2) + 1))/(c^6*(1/(c^2*x^2) - 1)^3 + 3*c^6*(1/(c^2*x^2) - 1)^2 + 3*c^6*(1/(c^2*x^2) - 1) + c^6) + 1
5*log(sqrt(-1/(c^2*x^2) + 1) + 1)/c^6 - 15*log(sqrt(-1/(c^2*x^2) + 1) - 1)/c^6)/c)*b

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^6\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(a + b*asin(1/(c*x))),x)

[Out]

int(x^6*(a + b*asin(1/(c*x))), x)

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sympy [A]  time = 8.92, size = 221, normalized size = 1.94 \[ \frac {a x^{7}}{7} + \frac {b x^{7} \operatorname {acsc}{\left (c x \right )}}{7} + \frac {b \left (\begin {cases} \frac {c x^{7}}{6 \sqrt {c^{2} x^{2} - 1}} + \frac {x^{5}}{24 c \sqrt {c^{2} x^{2} - 1}} + \frac {5 x^{3}}{48 c^{3} \sqrt {c^{2} x^{2} - 1}} - \frac {5 x}{16 c^{5} \sqrt {c^{2} x^{2} - 1}} + \frac {5 \operatorname {acosh}{\left (c x \right )}}{16 c^{6}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- \frac {i c x^{7}}{6 \sqrt {- c^{2} x^{2} + 1}} - \frac {i x^{5}}{24 c \sqrt {- c^{2} x^{2} + 1}} - \frac {5 i x^{3}}{48 c^{3} \sqrt {- c^{2} x^{2} + 1}} + \frac {5 i x}{16 c^{5} \sqrt {- c^{2} x^{2} + 1}} - \frac {5 i \operatorname {asin}{\left (c x \right )}}{16 c^{6}} & \text {otherwise} \end {cases}\right )}{7 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6*(a+b*acsc(c*x)),x)

[Out]

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

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