3.82 \(\int \frac{(d+e x^2) (a+b \csc ^{-1}(c x))}{x^8} \, dx\)

Optimal. Leaf size=197 \[ -\frac{d \left (a+b \csc ^{-1}(c x)\right )}{7 x^7}-\frac{e \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}-\frac{8 b c^5 \sqrt{c^2 x^2-1} \left (30 c^2 d+49 e\right )}{3675 \sqrt{c^2 x^2}}-\frac{4 b c^3 \sqrt{c^2 x^2-1} \left (30 c^2 d+49 e\right )}{3675 x^2 \sqrt{c^2 x^2}}-\frac{b c \sqrt{c^2 x^2-1} \left (30 c^2 d+49 e\right )}{1225 x^4 \sqrt{c^2 x^2}}-\frac{b c d \sqrt{c^2 x^2-1}}{49 x^6 \sqrt{c^2 x^2}} \]

[Out]

(-8*b*c^5*(30*c^2*d + 49*e)*Sqrt[-1 + c^2*x^2])/(3675*Sqrt[c^2*x^2]) - (b*c*d*Sqrt[-1 + c^2*x^2])/(49*x^6*Sqrt
[c^2*x^2]) - (b*c*(30*c^2*d + 49*e)*Sqrt[-1 + c^2*x^2])/(1225*x^4*Sqrt[c^2*x^2]) - (4*b*c^3*(30*c^2*d + 49*e)*
Sqrt[-1 + c^2*x^2])/(3675*x^2*Sqrt[c^2*x^2]) - (d*(a + b*ArcCsc[c*x]))/(7*x^7) - (e*(a + b*ArcCsc[c*x]))/(5*x^
5)

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Rubi [A]  time = 0.118038, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {14, 5239, 12, 453, 271, 264} \[ -\frac{d \left (a+b \csc ^{-1}(c x)\right )}{7 x^7}-\frac{e \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}-\frac{8 b c^5 \sqrt{c^2 x^2-1} \left (30 c^2 d+49 e\right )}{3675 \sqrt{c^2 x^2}}-\frac{4 b c^3 \sqrt{c^2 x^2-1} \left (30 c^2 d+49 e\right )}{3675 x^2 \sqrt{c^2 x^2}}-\frac{b c \sqrt{c^2 x^2-1} \left (30 c^2 d+49 e\right )}{1225 x^4 \sqrt{c^2 x^2}}-\frac{b c d \sqrt{c^2 x^2-1}}{49 x^6 \sqrt{c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x^2)*(a + b*ArcCsc[c*x]))/x^8,x]

[Out]

(-8*b*c^5*(30*c^2*d + 49*e)*Sqrt[-1 + c^2*x^2])/(3675*Sqrt[c^2*x^2]) - (b*c*d*Sqrt[-1 + c^2*x^2])/(49*x^6*Sqrt
[c^2*x^2]) - (b*c*(30*c^2*d + 49*e)*Sqrt[-1 + c^2*x^2])/(1225*x^4*Sqrt[c^2*x^2]) - (4*b*c^3*(30*c^2*d + 49*e)*
Sqrt[-1 + c^2*x^2])/(3675*x^2*Sqrt[c^2*x^2]) - (d*(a + b*ArcCsc[c*x]))/(7*x^7) - (e*(a + b*ArcCsc[c*x]))/(5*x^
5)

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 5239

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCsc[c*x], u, x] + Dist[(b*c*x)/Sqrt[c^2*x^2], Int[SimplifyI
ntegrand[u/(x*Sqrt[c^2*x^2 - 1]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && ((IGtQ[p, 0] &&  !(ILtQ
[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 0])) || (IGtQ[(m + 1)/2, 0] &&  !(ILtQ[p, 0] && GtQ[m + 2*p + 3, 0])) || (I
LtQ[(m + 2*p + 1)/2, 0] &&  !ILtQ[(m - 1)/2, 0]))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 453

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

Rule 271

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

Rule 264

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

Rubi steps

\begin{align*} \int \frac{\left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right )}{x^8} \, dx &=-\frac{d \left (a+b \csc ^{-1}(c x)\right )}{7 x^7}-\frac{e \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}+\frac{(b c x) \int \frac{-5 d-7 e x^2}{35 x^8 \sqrt{-1+c^2 x^2}} \, dx}{\sqrt{c^2 x^2}}\\ &=-\frac{d \left (a+b \csc ^{-1}(c x)\right )}{7 x^7}-\frac{e \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}+\frac{(b c x) \int \frac{-5 d-7 e x^2}{x^8 \sqrt{-1+c^2 x^2}} \, dx}{35 \sqrt{c^2 x^2}}\\ &=-\frac{b c d \sqrt{-1+c^2 x^2}}{49 x^6 \sqrt{c^2 x^2}}-\frac{d \left (a+b \csc ^{-1}(c x)\right )}{7 x^7}-\frac{e \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}+\frac{\left (b c \left (-30 c^2 d-49 e\right ) x\right ) \int \frac{1}{x^6 \sqrt{-1+c^2 x^2}} \, dx}{245 \sqrt{c^2 x^2}}\\ &=-\frac{b c d \sqrt{-1+c^2 x^2}}{49 x^6 \sqrt{c^2 x^2}}-\frac{b c \left (30 c^2 d+49 e\right ) \sqrt{-1+c^2 x^2}}{1225 x^4 \sqrt{c^2 x^2}}-\frac{d \left (a+b \csc ^{-1}(c x)\right )}{7 x^7}-\frac{e \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}+\frac{\left (4 b c^3 \left (-30 c^2 d-49 e\right ) x\right ) \int \frac{1}{x^4 \sqrt{-1+c^2 x^2}} \, dx}{1225 \sqrt{c^2 x^2}}\\ &=-\frac{b c d \sqrt{-1+c^2 x^2}}{49 x^6 \sqrt{c^2 x^2}}-\frac{b c \left (30 c^2 d+49 e\right ) \sqrt{-1+c^2 x^2}}{1225 x^4 \sqrt{c^2 x^2}}-\frac{4 b c^3 \left (30 c^2 d+49 e\right ) \sqrt{-1+c^2 x^2}}{3675 x^2 \sqrt{c^2 x^2}}-\frac{d \left (a+b \csc ^{-1}(c x)\right )}{7 x^7}-\frac{e \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}+\frac{\left (8 b c^5 \left (-30 c^2 d-49 e\right ) x\right ) \int \frac{1}{x^2 \sqrt{-1+c^2 x^2}} \, dx}{3675 \sqrt{c^2 x^2}}\\ &=-\frac{8 b c^5 \left (30 c^2 d+49 e\right ) \sqrt{-1+c^2 x^2}}{3675 \sqrt{c^2 x^2}}-\frac{b c d \sqrt{-1+c^2 x^2}}{49 x^6 \sqrt{c^2 x^2}}-\frac{b c \left (30 c^2 d+49 e\right ) \sqrt{-1+c^2 x^2}}{1225 x^4 \sqrt{c^2 x^2}}-\frac{4 b c^3 \left (30 c^2 d+49 e\right ) \sqrt{-1+c^2 x^2}}{3675 x^2 \sqrt{c^2 x^2}}-\frac{d \left (a+b \csc ^{-1}(c x)\right )}{7 x^7}-\frac{e \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}\\ \end{align*}

Mathematica [A]  time = 0.165466, size = 110, normalized size = 0.56 \[ -\frac{105 a \left (5 d+7 e x^2\right )+b c x \sqrt{1-\frac{1}{c^2 x^2}} \left (15 d \left (16 c^6 x^6+8 c^4 x^4+6 c^2 x^2+5\right )+49 e x^2 \left (8 c^4 x^4+4 c^2 x^2+3\right )\right )+105 b \csc ^{-1}(c x) \left (5 d+7 e x^2\right )}{3675 x^7} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x^2)*(a + b*ArcCsc[c*x]))/x^8,x]

[Out]

-(105*a*(5*d + 7*e*x^2) + b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(49*e*x^2*(3 + 4*c^2*x^2 + 8*c^4*x^4) + 15*d*(5 + 6*c^2*
x^2 + 8*c^4*x^4 + 16*c^6*x^6)) + 105*b*(5*d + 7*e*x^2)*ArcCsc[c*x])/(3675*x^7)

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Maple [A]  time = 0.19, size = 158, normalized size = 0.8 \begin{align*}{c}^{7} \left ({\frac{a}{{c}^{2}} \left ( -{\frac{d}{7\,{c}^{5}{x}^{7}}}-{\frac{e}{5\,{c}^{5}{x}^{5}}} \right ) }+{\frac{b}{{c}^{2}} \left ( -{\frac{{\rm arccsc} \left (cx\right )d}{7\,{c}^{5}{x}^{7}}}-{\frac{{\rm arccsc} \left (cx\right )e}{5\,{c}^{5}{x}^{5}}}-{\frac{ \left ({c}^{2}{x}^{2}-1 \right ) \left ( 240\,{c}^{8}d{x}^{6}+392\,{c}^{6}e{x}^{6}+120\,{c}^{6}d{x}^{4}+196\,{c}^{4}e{x}^{4}+90\,{c}^{4}d{x}^{2}+147\,{c}^{2}e{x}^{2}+75\,{c}^{2}d \right ) }{3675\,{c}^{8}{x}^{8}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}} \right ) } \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)*(a+b*arccsc(c*x))/x^8,x)

[Out]

c^7*(a/c^2*(-1/7/c^5*d/x^7-1/5*e/c^5/x^5)+b/c^2*(-1/7*arccsc(c*x)/c^5*d/x^7-1/5*arccsc(c*x)*e/c^5/x^5-1/3675*(
c^2*x^2-1)*(240*c^8*d*x^6+392*c^6*e*x^6+120*c^6*d*x^4+196*c^4*e*x^4+90*c^4*d*x^2+147*c^2*e*x^2+75*c^2*d)/((c^2
*x^2-1)/c^2/x^2)^(1/2)/c^8/x^8))

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Maxima [A]  time = 0.979352, size = 232, normalized size = 1.18 \begin{align*} \frac{1}{245} \, b d{\left (\frac{5 \, c^{8}{\left (-\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{7}{2}} - 21 \, c^{8}{\left (-\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{5}{2}} + 35 \, c^{8}{\left (-\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{3}{2}} - 35 \, c^{8} \sqrt{-\frac{1}{c^{2} x^{2}} + 1}}{c} - \frac{35 \, \operatorname{arccsc}\left (c x\right )}{x^{7}}\right )} - \frac{1}{75} \, b e{\left (\frac{3 \, c^{6}{\left (-\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{5}{2}} - 10 \, c^{6}{\left (-\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{3}{2}} + 15 \, c^{6} \sqrt{-\frac{1}{c^{2} x^{2}} + 1}}{c} + \frac{15 \, \operatorname{arccsc}\left (c x\right )}{x^{5}}\right )} - \frac{a e}{5 \, x^{5}} - \frac{a d}{7 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)*(a+b*arccsc(c*x))/x^8,x, algorithm="maxima")

[Out]

1/245*b*d*((5*c^8*(-1/(c^2*x^2) + 1)^(7/2) - 21*c^8*(-1/(c^2*x^2) + 1)^(5/2) + 35*c^8*(-1/(c^2*x^2) + 1)^(3/2)
- 35*c^8*sqrt(-1/(c^2*x^2) + 1))/c - 35*arccsc(c*x)/x^7) - 1/75*b*e*((3*c^6*(-1/(c^2*x^2) + 1)^(5/2) - 10*c^6
*(-1/(c^2*x^2) + 1)^(3/2) + 15*c^6*sqrt(-1/(c^2*x^2) + 1))/c + 15*arccsc(c*x)/x^5) - 1/5*a*e/x^5 - 1/7*a*d/x^7

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Fricas [A]  time = 2.52314, size = 273, normalized size = 1.39 \begin{align*} -\frac{735 \, a e x^{2} + 525 \, a d + 105 \,{\left (7 \, b e x^{2} + 5 \, b d\right )} \operatorname{arccsc}\left (c x\right ) +{\left (8 \,{\left (30 \, b c^{6} d + 49 \, b c^{4} e\right )} x^{6} + 4 \,{\left (30 \, b c^{4} d + 49 \, b c^{2} e\right )} x^{4} + 3 \,{\left (30 \, b c^{2} d + 49 \, b e\right )} x^{2} + 75 \, b d\right )} \sqrt{c^{2} x^{2} - 1}}{3675 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)*(a+b*arccsc(c*x))/x^8,x, algorithm="fricas")

[Out]

-1/3675*(735*a*e*x^2 + 525*a*d + 105*(7*b*e*x^2 + 5*b*d)*arccsc(c*x) + (8*(30*b*c^6*d + 49*b*c^4*e)*x^6 + 4*(3
0*b*c^4*d + 49*b*c^2*e)*x^4 + 3*(30*b*c^2*d + 49*b*e)*x^2 + 75*b*d)*sqrt(c^2*x^2 - 1))/x^7

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \operatorname{acsc}{\left (c x \right )}\right ) \left (d + e x^{2}\right )}{x^{8}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)*(a+b*acsc(c*x))/x**8,x)

[Out]

Integral((a + b*acsc(c*x))*(d + e*x**2)/x**8, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x^{2} + d\right )}{\left (b \operatorname{arccsc}\left (c x\right ) + a\right )}}{x^{8}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)*(a+b*arccsc(c*x))/x^8,x, algorithm="giac")

[Out]

integrate((e*x^2 + d)*(b*arccsc(c*x) + a)/x^8, x)