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

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

[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)

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Rubi [A]  time = 0.0614145, antiderivative size = 114, normalized size of antiderivative = 1., 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 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]

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 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]

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*}

Mathematica [A]  time = 0.142633, size = 107, normalized size = 0.94 \[ \frac{a x^7}{7}+b \sqrt{\frac{c^2 x^2-1}{c^2 x^2}} \left (\frac{5 x^4}{168 c^3}+\frac{5 x^2}{112 c^5}+\frac{x^6}{42 c}\right )+\frac{5 b \log \left (x \left (\sqrt{\frac{c^2 x^2-1}{c^2 x^2}}+1\right )\right )}{112 c^7}+\frac{1}{7} b x^7 \csc ^{-1}(c x) \]

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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Maple [A]  time = 0.178, size = 177, normalized size = 1.6 \begin{align*}{\frac{{x}^{7}a}{7}}+{\frac{b{x}^{7}{\rm arccsc} \left (cx\right )}{7}}+{\frac{b{x}^{6}}{42\,c}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}+{\frac{b{x}^{4}}{168\,{c}^{3}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}+{\frac{5\,b{x}^{2}}{336\,{c}^{5}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}-{\frac{5\,b}{112\,{c}^{7}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}+{\frac{5\,b}{112\,{c}^{8}x}\sqrt{{c}^{2}{x}^{2}-1}\ln \left ( cx+\sqrt{{c}^{2}{x}^{2}-1} \right ){\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(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 = 0.952431, size = 217, normalized size = 1.9 \begin{align*} \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 \end{align*}

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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Fricas [A]  time = 2.70702, size = 275, normalized size = 2.41 \begin{align*} \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}} \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{6} \left (a + b \operatorname{acsc}{\left (c x \right )}\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**6*(a + b*acsc(c*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{arccsc}\left (c x\right ) + a\right )} x^{6}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arccsc(c*x) + a)*x^6, x)