3.26 \(\int \frac{\csc ^{-1}(a+b x)}{x^5} \, dx\)

Optimal. Leaf size=239 \[ \frac{\left (3-8 a^2\right ) b^2 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{24 a^2 \left (1-a^2\right )^2 x^2}-\frac{\left (26 a^4-17 a^2+6\right ) b^3 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{24 a^3 \left (1-a^2\right )^3 x}+\frac{b^4 \csc ^{-1}(a+b x)}{4 a^4}+\frac{\left (-8 a^6+8 a^4-7 a^2+2\right ) b^4 \tan ^{-1}\left (\frac{a-\tan \left (\frac{1}{2} \csc ^{-1}(a+b x)\right )}{\sqrt{1-a^2}}\right )}{4 a^4 \left (1-a^2\right )^{7/2}}-\frac{b (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{12 a \left (1-a^2\right ) x^3}-\frac{\csc ^{-1}(a+b x)}{4 x^4} \]

[Out]

-(b*(a + b*x)*Sqrt[1 - (a + b*x)^(-2)])/(12*a*(1 - a^2)*x^3) + ((3 - 8*a^2)*b^2*(a + b*x)*Sqrt[1 - (a + b*x)^(
-2)])/(24*a^2*(1 - a^2)^2*x^2) - ((6 - 17*a^2 + 26*a^4)*b^3*(a + b*x)*Sqrt[1 - (a + b*x)^(-2)])/(24*a^3*(1 - a
^2)^3*x) + (b^4*ArcCsc[a + b*x])/(4*a^4) - ArcCsc[a + b*x]/(4*x^4) + ((2 - 7*a^2 + 8*a^4 - 8*a^6)*b^4*ArcTan[(
a - Tan[ArcCsc[a + b*x]/2])/Sqrt[1 - a^2]])/(4*a^4*(1 - a^2)^(7/2))

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Rubi [A]  time = 0.456579, antiderivative size = 239, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.9, Rules used = {5259, 4427, 3785, 4060, 3919, 3831, 2660, 618, 204} \[ \frac{\left (3-8 a^2\right ) b^2 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{24 a^2 \left (1-a^2\right )^2 x^2}-\frac{\left (26 a^4-17 a^2+6\right ) b^3 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{24 a^3 \left (1-a^2\right )^3 x}+\frac{b^4 \csc ^{-1}(a+b x)}{4 a^4}+\frac{\left (-8 a^6+8 a^4-7 a^2+2\right ) b^4 \tan ^{-1}\left (\frac{a-\tan \left (\frac{1}{2} \csc ^{-1}(a+b x)\right )}{\sqrt{1-a^2}}\right )}{4 a^4 \left (1-a^2\right )^{7/2}}-\frac{b (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{12 a \left (1-a^2\right ) x^3}-\frac{\csc ^{-1}(a+b x)}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Int[ArcCsc[a + b*x]/x^5,x]

[Out]

-(b*(a + b*x)*Sqrt[1 - (a + b*x)^(-2)])/(12*a*(1 - a^2)*x^3) + ((3 - 8*a^2)*b^2*(a + b*x)*Sqrt[1 - (a + b*x)^(
-2)])/(24*a^2*(1 - a^2)^2*x^2) - ((6 - 17*a^2 + 26*a^4)*b^3*(a + b*x)*Sqrt[1 - (a + b*x)^(-2)])/(24*a^3*(1 - a
^2)^3*x) + (b^4*ArcCsc[a + b*x])/(4*a^4) - ArcCsc[a + b*x]/(4*x^4) + ((2 - 7*a^2 + 8*a^4 - 8*a^6)*b^4*ArcTan[(
a - Tan[ArcCsc[a + b*x]/2])/Sqrt[1 - a^2]])/(4*a^4*(1 - a^2)^(7/2))

Rule 5259

Int[((a_.) + ArcCsc[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> -Dist[(d^(m + 1))
^(-1), Subst[Int[(a + b*x)^p*Csc[x]*Cot[x]*(d*e - c*f + f*Csc[x])^m, x], x, ArcCsc[c + d*x]], x] /; FreeQ[{a,
b, c, d, e, f}, x] && IGtQ[p, 0] && IntegerQ[m]

Rule 4427

Int[Cot[(c_.) + (d_.)*(x_)]*Csc[(c_.) + (d_.)*(x_)]*(Csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_.)*((e_.) + (f_.
)*(x_))^(m_.), x_Symbol] :> -Simp[((e + f*x)^m*(a + b*Csc[c + d*x])^(n + 1))/(b*d*(n + 1)), x] + Dist[(f*m)/(b
*d*(n + 1)), Int[(e + f*x)^(m - 1)*(a + b*Csc[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&
IGtQ[m, 0] && NeQ[n, -1]

Rule 3785

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

Rule 4060

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_.) +
(a_))^(m_), x_Symbol] :> Simp[((A*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1))/(a*f*(m + 1
)*(a^2 - b^2)), x] + Dist[1/(a*(m + 1)*(a^2 - b^2)), Int[(a + b*Csc[e + f*x])^(m + 1)*Simp[A*(a^2 - b^2)*(m +
1) - a*(A*b - a*B + b*C)*(m + 1)*Csc[e + f*x] + (A*b^2 - a*b*B + a^2*C)*(m + 2)*Csc[e + f*x]^2, x], x], x] /;
FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]

Rule 3919

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[(c*x)/a,
x] - Dist[(b*c - a*d)/a, Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[
b*c - a*d, 0]

Rule 3831

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[1/b, Int[1/(1 + (a*Sin[e
+ f*x])/b), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 2660

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[(2*e)/d, Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[a^2 - b^2, 0]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\csc ^{-1}(a+b x)}{x^5} \, dx &=-\left (b^4 \operatorname{Subst}\left (\int \frac{x \cot (x) \csc (x)}{(-a+\csc (x))^5} \, dx,x,\csc ^{-1}(a+b x)\right )\right )\\ &=-\frac{\csc ^{-1}(a+b x)}{4 x^4}+\frac{1}{4} b^4 \operatorname{Subst}\left (\int \frac{1}{(-a+\csc (x))^4} \, dx,x,\csc ^{-1}(a+b x)\right )\\ &=-\frac{b (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{12 a \left (1-a^2\right ) x^3}-\frac{\csc ^{-1}(a+b x)}{4 x^4}-\frac{b^4 \operatorname{Subst}\left (\int \frac{3 \left (1-a^2\right )-3 a \csc (x)-2 \csc ^2(x)}{(-a+\csc (x))^3} \, dx,x,\csc ^{-1}(a+b x)\right )}{12 a \left (1-a^2\right )}\\ &=-\frac{b (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{12 a \left (1-a^2\right ) x^3}+\frac{\left (3-8 a^2\right ) b^2 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{24 a^2 \left (1-a^2\right )^2 x^2}-\frac{\csc ^{-1}(a+b x)}{4 x^4}+\frac{b^4 \operatorname{Subst}\left (\int \frac{6 \left (1-a^2\right )^2-2 a \left (1-6 a^2\right ) \csc (x)-\left (3-8 a^2\right ) \csc ^2(x)}{(-a+\csc (x))^2} \, dx,x,\csc ^{-1}(a+b x)\right )}{24 a^2 \left (1-a^2\right )^2}\\ &=-\frac{b (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{12 a \left (1-a^2\right ) x^3}+\frac{\left (3-8 a^2\right ) b^2 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{24 a^2 \left (1-a^2\right )^2 x^2}-\frac{\left (6-17 a^2+26 a^4\right ) b^3 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{24 a^3 \left (1-a^2\right )^3 x}-\frac{\csc ^{-1}(a+b x)}{4 x^4}-\frac{b^4 \operatorname{Subst}\left (\int \frac{6 \left (1-a^2\right )^3-3 a \left (1-2 a^2+6 a^4\right ) \csc (x)}{-a+\csc (x)} \, dx,x,\csc ^{-1}(a+b x)\right )}{24 a^3 \left (1-a^2\right )^3}\\ &=-\frac{b (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{12 a \left (1-a^2\right ) x^3}+\frac{\left (3-8 a^2\right ) b^2 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{24 a^2 \left (1-a^2\right )^2 x^2}-\frac{\left (6-17 a^2+26 a^4\right ) b^3 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{24 a^3 \left (1-a^2\right )^3 x}+\frac{b^4 \csc ^{-1}(a+b x)}{4 a^4}-\frac{\csc ^{-1}(a+b x)}{4 x^4}-\frac{\left (\left (2-7 a^2+8 a^4-8 a^6\right ) b^4\right ) \operatorname{Subst}\left (\int \frac{\csc (x)}{-a+\csc (x)} \, dx,x,\csc ^{-1}(a+b x)\right )}{8 a^4 \left (1-a^2\right )^3}\\ &=-\frac{b (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{12 a \left (1-a^2\right ) x^3}+\frac{\left (3-8 a^2\right ) b^2 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{24 a^2 \left (1-a^2\right )^2 x^2}-\frac{\left (6-17 a^2+26 a^4\right ) b^3 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{24 a^3 \left (1-a^2\right )^3 x}+\frac{b^4 \csc ^{-1}(a+b x)}{4 a^4}-\frac{\csc ^{-1}(a+b x)}{4 x^4}-\frac{\left (\left (2-7 a^2+8 a^4-8 a^6\right ) b^4\right ) \operatorname{Subst}\left (\int \frac{1}{1-a \sin (x)} \, dx,x,\csc ^{-1}(a+b x)\right )}{8 a^4 \left (1-a^2\right )^3}\\ &=-\frac{b (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{12 a \left (1-a^2\right ) x^3}+\frac{\left (3-8 a^2\right ) b^2 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{24 a^2 \left (1-a^2\right )^2 x^2}-\frac{\left (6-17 a^2+26 a^4\right ) b^3 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{24 a^3 \left (1-a^2\right )^3 x}+\frac{b^4 \csc ^{-1}(a+b x)}{4 a^4}-\frac{\csc ^{-1}(a+b x)}{4 x^4}-\frac{\left (\left (2-7 a^2+8 a^4-8 a^6\right ) b^4\right ) \operatorname{Subst}\left (\int \frac{1}{1-2 a x+x^2} \, dx,x,\tan \left (\frac{1}{2} \csc ^{-1}(a+b x)\right )\right )}{4 a^4 \left (1-a^2\right )^3}\\ &=-\frac{b (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{12 a \left (1-a^2\right ) x^3}+\frac{\left (3-8 a^2\right ) b^2 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{24 a^2 \left (1-a^2\right )^2 x^2}-\frac{\left (6-17 a^2+26 a^4\right ) b^3 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{24 a^3 \left (1-a^2\right )^3 x}+\frac{b^4 \csc ^{-1}(a+b x)}{4 a^4}-\frac{\csc ^{-1}(a+b x)}{4 x^4}+\frac{\left (\left (2-7 a^2+8 a^4-8 a^6\right ) b^4\right ) \operatorname{Subst}\left (\int \frac{1}{-4 \left (1-a^2\right )-x^2} \, dx,x,-2 a+2 \tan \left (\frac{1}{2} \csc ^{-1}(a+b x)\right )\right )}{2 a^4 \left (1-a^2\right )^3}\\ &=-\frac{b (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{12 a \left (1-a^2\right ) x^3}+\frac{\left (3-8 a^2\right ) b^2 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{24 a^2 \left (1-a^2\right )^2 x^2}-\frac{\left (6-17 a^2+26 a^4\right ) b^3 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{24 a^3 \left (1-a^2\right )^3 x}+\frac{b^4 \csc ^{-1}(a+b x)}{4 a^4}-\frac{\csc ^{-1}(a+b x)}{4 x^4}+\frac{\left (2-7 a^2+8 a^4-8 a^6\right ) b^4 \tan ^{-1}\left (\frac{a-\tan \left (\frac{1}{2} \csc ^{-1}(a+b x)\right )}{\sqrt{1-a^2}}\right )}{4 a^4 \left (1-a^2\right )^{7/2}}\\ \end{align*}

Mathematica [C]  time = 0.516068, size = 307, normalized size = 1.28 \[ \frac{1}{8} \left (\frac{b \sqrt{\frac{a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}} \left (2 a^5 \left (9 b^2 x^2-2\right )+a^4 b x \left (26 b^2 x^2+7\right )+a^3 \left (2-6 b^2 x^2\right )-a^2 \left (17 b^3 x^3+b x\right )-6 a^6 b x+2 a^7+3 a b^2 x^2+6 b^3 x^3\right )}{3 a^3 \left (a^2-1\right )^3 x^3}+\frac{i \left (8 a^6-8 a^4+7 a^2-2\right ) b^4 \log \left (\frac{16 a^4 \left (a^2-1\right )^3 \left (\sqrt{\frac{a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}} (a+b x)+\frac{i \left (a^2+a b x-1\right )}{\sqrt{1-a^2}}\right )}{\left (8 a^6-8 a^4+7 a^2-2\right ) b^4 x}\right )}{a^4 \left (1-a^2\right )^{7/2}}+\frac{2 b^4 \sin ^{-1}\left (\frac{1}{a+b x}\right )}{a^4}-\frac{2 \csc ^{-1}(a+b x)}{x^4}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCsc[a + b*x]/x^5,x]

[Out]

((b*Sqrt[(-1 + a^2 + 2*a*b*x + b^2*x^2)/(a + b*x)^2]*(2*a^7 - 6*a^6*b*x + 3*a*b^2*x^2 + 6*b^3*x^3 + a^3*(2 - 6
*b^2*x^2) + 2*a^5*(-2 + 9*b^2*x^2) + a^4*b*x*(7 + 26*b^2*x^2) - a^2*(b*x + 17*b^3*x^3)))/(3*a^3*(-1 + a^2)^3*x
^3) - (2*ArcCsc[a + b*x])/x^4 + (2*b^4*ArcSin[(a + b*x)^(-1)])/a^4 + (I*(-2 + 7*a^2 - 8*a^4 + 8*a^6)*b^4*Log[(
16*a^4*(-1 + a^2)^3*((I*(-1 + a^2 + a*b*x))/Sqrt[1 - a^2] + (a + b*x)*Sqrt[(-1 + a^2 + 2*a*b*x + b^2*x^2)/(a +
b*x)^2]))/((-2 + 7*a^2 - 8*a^4 + 8*a^6)*b^4*x)])/(a^4*(1 - a^2)^(7/2)))/8

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Maple [B]  time = 0.24, size = 1172, normalized size = 4.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccsc(b*x+a)/x^5,x)

[Out]

-1/4*arccsc(b*x+a)/x^4+1/4*b^4*(-1+(b*x+a)^2)^(1/2)/((-1+(b*x+a)^2)/(b*x+a)^2)^(1/2)/(b*x+a)*a^2/(a^2-1)^3*arc
tan(1/(-1+(b*x+a)^2)^(1/2))-b^4*(-1+(b*x+a)^2)^(1/2)/((-1+(b*x+a)^2)/(b*x+a)^2)^(1/2)/(b*x+a)*a^4/(a^2-1)^(9/2
)*ln(2*((a^2-1)^(1/2)*(-1+(b*x+a)^2)^(1/2)+a*(b*x+a)-1)/b/x)-3/4*b^4*(-1+(b*x+a)^2)^(1/2)/((-1+(b*x+a)^2)/(b*x
+a)^2)^(1/2)/(b*x+a)/(a^2-1)^3*arctan(1/(-1+(b*x+a)^2)^(1/2))+13/12*b^3*(-1+(b*x+a)^2)/((-1+(b*x+a)^2)/(b*x+a)
^2)^(1/2)/(b*x+a)*a/(a^2-1)^3/x+2*b^4*(-1+(b*x+a)^2)^(1/2)/((-1+(b*x+a)^2)/(b*x+a)^2)^(1/2)/(b*x+a)*a^2/(a^2-1
)^(9/2)*ln(2*((a^2-1)^(1/2)*(-1+(b*x+a)^2)^(1/2)+a*(b*x+a)-1)/b/x)-1/3*b^2*(-1+(b*x+a)^2)/((-1+(b*x+a)^2)/(b*x
+a)^2)^(1/2)/(b*x+a)*a^2/(a^2-1)^3/x^2+3/4*b^4*(-1+(b*x+a)^2)^(1/2)/((-1+(b*x+a)^2)/(b*x+a)^2)^(1/2)/(b*x+a)/a
^2/(a^2-1)^3*arctan(1/(-1+(b*x+a)^2)^(1/2))-17/24*b^3*(-1+(b*x+a)^2)/((-1+(b*x+a)^2)/(b*x+a)^2)^(1/2)/(b*x+a)/
a/(a^2-1)^3/x+1/12*b*(-1+(b*x+a)^2)/((-1+(b*x+a)^2)/(b*x+a)^2)^(1/2)/(b*x+a)*a^3/(a^2-1)^3/x^3-15/8*b^4*(-1+(b
*x+a)^2)^(1/2)/((-1+(b*x+a)^2)/(b*x+a)^2)^(1/2)/(b*x+a)/(a^2-1)^(9/2)*ln(2*((a^2-1)^(1/2)*(-1+(b*x+a)^2)^(1/2)
+a*(b*x+a)-1)/b/x)+11/24*b^2*(-1+(b*x+a)^2)/((-1+(b*x+a)^2)/(b*x+a)^2)^(1/2)/(b*x+a)/(a^2-1)^3/x^2-1/4*b^4*(-1
+(b*x+a)^2)^(1/2)/((-1+(b*x+a)^2)/(b*x+a)^2)^(1/2)/(b*x+a)/a^4/(a^2-1)^3*arctan(1/(-1+(b*x+a)^2)^(1/2))+1/4*b^
3*(-1+(b*x+a)^2)/((-1+(b*x+a)^2)/(b*x+a)^2)^(1/2)/(b*x+a)/a^3/(a^2-1)^3/x-1/6*b*(-1+(b*x+a)^2)/((-1+(b*x+a)^2)
/(b*x+a)^2)^(1/2)/(b*x+a)*a/(a^2-1)^3/x^3+9/8*b^4*(-1+(b*x+a)^2)^(1/2)/((-1+(b*x+a)^2)/(b*x+a)^2)^(1/2)/(b*x+a
)/a^2/(a^2-1)^(9/2)*ln(2*((a^2-1)^(1/2)*(-1+(b*x+a)^2)^(1/2)+a*(b*x+a)-1)/b/x)-1/8*b^2*(-1+(b*x+a)^2)/((-1+(b*
x+a)^2)/(b*x+a)^2)^(1/2)/(b*x+a)/a^2/(a^2-1)^3/x^2+1/12*b*(-1+(b*x+a)^2)/((-1+(b*x+a)^2)/(b*x+a)^2)^(1/2)/(b*x
+a)/a/(a^2-1)^3/x^3-1/4*b^4*(-1+(b*x+a)^2)^(1/2)/((-1+(b*x+a)^2)/(b*x+a)^2)^(1/2)/(b*x+a)/a^4/(a^2-1)^(9/2)*ln
(2*((a^2-1)^(1/2)*(-1+(b*x+a)^2)^(1/2)+a*(b*x+a)-1)/b/x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{x^{4} \int \frac{{\left (b^{2} x + a b\right )} e^{\left (\frac{1}{2} \, \log \left (b x + a + 1\right ) + \frac{1}{2} \, \log \left (b x + a - 1\right )\right )}}{b^{2} x^{6} + 2 \, a b x^{5} +{\left (a^{2} - 1\right )} x^{4} +{\left (b^{2} x^{6} + 2 \, a b x^{5} +{\left (a^{2} - 1\right )} x^{4}\right )}{\left (b x + a + 1\right )}{\left (b x + a - 1\right )}}\,{d x} + \arctan \left (1, \sqrt{b x + a + 1} \sqrt{b x + a - 1}\right )}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccsc(b*x+a)/x^5,x, algorithm="maxima")

[Out]

-1/4*(4*x^4*integrate(1/4*(b^2*x + a*b)*e^(1/2*log(b*x + a + 1) + 1/2*log(b*x + a - 1))/(b^2*x^6 + 2*a*b*x^5 +
(a^2 - 1)*x^4 + (b^2*x^6 + 2*a*b*x^5 + (a^2 - 1)*x^4)*e^(log(b*x + a + 1) + log(b*x + a - 1))), x) + arctan2(
1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)))/x^4

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Fricas [A]  time = 4.24089, size = 1551, normalized size = 6.49 \begin{align*} \left [\frac{3 \,{\left (8 \, a^{6} - 8 \, a^{4} + 7 \, a^{2} - 2\right )} \sqrt{a^{2} - 1} b^{4} x^{4} \log \left (\frac{a^{2} b x + a^{3} + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{\left (a^{2} - \sqrt{a^{2} - 1} a - 1\right )} -{\left (a b x + a^{2} - 1\right )} \sqrt{a^{2} - 1} - a}{x}\right ) - 12 \,{\left (a^{8} - 4 \, a^{6} + 6 \, a^{4} - 4 \, a^{2} + 1\right )} b^{4} x^{4} \arctan \left (-b x - a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) +{\left (26 \, a^{7} - 43 \, a^{5} + 23 \, a^{3} - 6 \, a\right )} b^{4} x^{4} - 6 \,{\left (a^{12} - 4 \, a^{10} + 6 \, a^{8} - 4 \, a^{6} + a^{4}\right )} \operatorname{arccsc}\left (b x + a\right ) +{\left ({\left (26 \, a^{7} - 43 \, a^{5} + 23 \, a^{3} - 6 \, a\right )} b^{3} x^{3} -{\left (8 \, a^{8} - 19 \, a^{6} + 14 \, a^{4} - 3 \, a^{2}\right )} b^{2} x^{2} + 2 \,{\left (a^{9} - 3 \, a^{7} + 3 \, a^{5} - a^{3}\right )} b x\right )} \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}}{24 \,{\left (a^{12} - 4 \, a^{10} + 6 \, a^{8} - 4 \, a^{6} + a^{4}\right )} x^{4}}, \frac{6 \,{\left (8 \, a^{6} - 8 \, a^{4} + 7 \, a^{2} - 2\right )} \sqrt{-a^{2} + 1} b^{4} x^{4} \arctan \left (-\frac{\sqrt{-a^{2} + 1} b x - \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} - 1} \sqrt{-a^{2} + 1}}{a^{2} - 1}\right ) - 12 \,{\left (a^{8} - 4 \, a^{6} + 6 \, a^{4} - 4 \, a^{2} + 1\right )} b^{4} x^{4} \arctan \left (-b x - a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) +{\left (26 \, a^{7} - 43 \, a^{5} + 23 \, a^{3} - 6 \, a\right )} b^{4} x^{4} - 6 \,{\left (a^{12} - 4 \, a^{10} + 6 \, a^{8} - 4 \, a^{6} + a^{4}\right )} \operatorname{arccsc}\left (b x + a\right ) +{\left ({\left (26 \, a^{7} - 43 \, a^{5} + 23 \, a^{3} - 6 \, a\right )} b^{3} x^{3} -{\left (8 \, a^{8} - 19 \, a^{6} + 14 \, a^{4} - 3 \, a^{2}\right )} b^{2} x^{2} + 2 \,{\left (a^{9} - 3 \, a^{7} + 3 \, a^{5} - a^{3}\right )} b x\right )} \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}}{24 \,{\left (a^{12} - 4 \, a^{10} + 6 \, a^{8} - 4 \, a^{6} + a^{4}\right )} x^{4}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccsc(b*x+a)/x^5,x, algorithm="fricas")

[Out]

[1/24*(3*(8*a^6 - 8*a^4 + 7*a^2 - 2)*sqrt(a^2 - 1)*b^4*x^4*log((a^2*b*x + a^3 + sqrt(b^2*x^2 + 2*a*b*x + a^2 -
1)*(a^2 - sqrt(a^2 - 1)*a - 1) - (a*b*x + a^2 - 1)*sqrt(a^2 - 1) - a)/x) - 12*(a^8 - 4*a^6 + 6*a^4 - 4*a^2 +
1)*b^4*x^4*arctan(-b*x - a + sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)) + (26*a^7 - 43*a^5 + 23*a^3 - 6*a)*b^4*x^4 - 6
*(a^12 - 4*a^10 + 6*a^8 - 4*a^6 + a^4)*arccsc(b*x + a) + ((26*a^7 - 43*a^5 + 23*a^3 - 6*a)*b^3*x^3 - (8*a^8 -
19*a^6 + 14*a^4 - 3*a^2)*b^2*x^2 + 2*(a^9 - 3*a^7 + 3*a^5 - a^3)*b*x)*sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1))/((a^1
2 - 4*a^10 + 6*a^8 - 4*a^6 + a^4)*x^4), 1/24*(6*(8*a^6 - 8*a^4 + 7*a^2 - 2)*sqrt(-a^2 + 1)*b^4*x^4*arctan(-(sq
rt(-a^2 + 1)*b*x - sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)*sqrt(-a^2 + 1))/(a^2 - 1)) - 12*(a^8 - 4*a^6 + 6*a^4 - 4*
a^2 + 1)*b^4*x^4*arctan(-b*x - a + sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)) + (26*a^7 - 43*a^5 + 23*a^3 - 6*a)*b^4*x
^4 - 6*(a^12 - 4*a^10 + 6*a^8 - 4*a^6 + a^4)*arccsc(b*x + a) + ((26*a^7 - 43*a^5 + 23*a^3 - 6*a)*b^3*x^3 - (8*
a^8 - 19*a^6 + 14*a^4 - 3*a^2)*b^2*x^2 + 2*(a^9 - 3*a^7 + 3*a^5 - a^3)*b*x)*sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1))
/((a^12 - 4*a^10 + 6*a^8 - 4*a^6 + a^4)*x^4)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acsc}{\left (a + b x \right )}}{x^{5}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acsc(b*x+a)/x**5,x)

[Out]

Integral(acsc(a + b*x)/x**5, x)

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Giac [B]  time = 2.02049, size = 1508, normalized size = 6.31 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccsc(b*x+a)/x^5,x, algorithm="giac")

[Out]

1/12*b*(3*(8*a^6*b^11 - 8*a^4*b^11 + 7*a^2*b^11 - 2*b^11)*arctan(-(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1
))/sqrt(-a^2 + 1))/((a^10*b^8*sgn(b*x + a) - 3*a^8*b^8*sgn(b*x + a) + 3*a^6*b^8*sgn(b*x + a) - a^4*b^8*sgn(b*x
+ a))*sqrt(-a^2 + 1)) - 6*b^3*arctan(-((x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1))*b + a*abs(b))/b)/(a^4*s
gn(b*x + a)) - (18*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1))^5*a^5*b^11 - 52*(x*abs(b) - sqrt(b^2*x^2 + 2
*a*b*x + a^2 - 1))^3*a^7*b^11 + 66*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1))*a^9*b^11 + 6*(x*abs(b) - sqr
t(b^2*x^2 + 2*a*b*x + a^2 - 1))^4*a^6*b^10*abs(b) - 12*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1))^2*a^8*b^
10*abs(b) + 38*a^10*b^10*abs(b) - 6*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1))^5*a^3*b^11 + 76*(x*abs(b) -
sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1))^3*a^5*b^11 - 174*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1))*a^7*b^11 -
18*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1))^4*a^4*b^10*abs(b) + 48*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x +
a^2 - 1))^2*a^6*b^10*abs(b) - 134*a^8*b^10*abs(b) + 3*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1))^5*a*b^11
- 36*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1))^3*a^3*b^11 + 159*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2
- 1))*a^5*b^11 + 18*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1))^4*a^2*b^10*abs(b) - 72*(x*abs(b) - sqrt(b^
2*x^2 + 2*a*b*x + a^2 - 1))^2*a^4*b^10*abs(b) + 180*a^6*b^10*abs(b) + 12*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x +
a^2 - 1))^3*a*b^11 - 60*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1))*a^3*b^11 - 6*(x*abs(b) - sqrt(b^2*x^2 +
2*a*b*x + a^2 - 1))^4*b^10*abs(b) + 48*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1))^2*a^2*b^10*abs(b) - 116
*a^4*b^10*abs(b) + 9*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1))*a*b^11 - 12*(x*abs(b) - sqrt(b^2*x^2 + 2*a
*b*x + a^2 - 1))^2*b^10*abs(b) + 38*a^2*b^10*abs(b) - 6*b^10*abs(b))/((a^9*b^8*sgn(b*x + a) - 3*a^7*b^8*sgn(b*
x + a) + 3*a^5*b^8*sgn(b*x + a) - a^3*b^8*sgn(b*x + a))*((x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1))^2 - a^
2 + 1)^3)) - 1/4*arcsin(1/(b*x + a))/x^4