3.1.26 \(\int \frac {\csc ^{-1}(a+b x)}{x^5} \, dx\) [26]

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

[Out]

1/4*b^4*arccsc(b*x+a)/a^4-1/4*arccsc(b*x+a)/x^4+1/4*(-8*a^6+8*a^4-7*a^2+2)*b^4*arctan((a-tan(1/2*arccsc(b*x+a)
))/(-a^2+1)^(1/2))/a^4/(-a^2+1)^(7/2)-1/12*b*(b*x+a)*(1-1/(b*x+a)^2)^(1/2)/a/(-a^2+1)/x^3+1/24*(-8*a^2+3)*b^2*
(b*x+a)*(1-1/(b*x+a)^2)^(1/2)/a^2/(-a^2+1)^2/x^2-1/24*(26*a^4-17*a^2+6)*b^3*(b*x+a)*(1-1/(b*x+a)^2)^(1/2)/a^3/
(-a^2+1)^3/x

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Rubi [A]
time = 0.32, antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {5367, 4512, 3870, 4145, 4004, 3916, 2739, 632, 210} \begin {gather*} \frac {b^4 \csc ^{-1}(a+b x)}{4 a^4}+\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 {b (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{12 a \left (1-a^2\right ) x^3}+\frac {\left (-8 a^6+8 a^4-7 a^2+2\right ) b^4 \text {ArcTan}\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 {\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 {\csc ^{-1}(a+b x)}{4 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcCsc[a + b*x]/x^5,x]

[Out]

-1/12*(b*(a + b*x)*Sqrt[1 - (a + b*x)^(-2)])/(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 210

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

Rule 632

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 2739

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 3870

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 3916

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

Rule 4004

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 4145

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 4512

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 5367

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]

Rubi steps

\begin {align*} \int \frac {\csc ^{-1}(a+b x)}{x^5} \, dx &=-\left (b^4 \text {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 \text {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 \text {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 \text {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 \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.35, size = 307, normalized size = 1.28 \begin {gather*} \frac {1}{8} \left (\frac {b \sqrt {\frac {-1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}} \left (2 a^7-6 a^6 b x+3 a b^2 x^2+6 b^3 x^3+a^3 \left (2-6 b^2 x^2\right )+2 a^5 \left (-2+9 b^2 x^2\right )+a^4 b x \left (7+26 b^2 x^2\right )-a^2 \left (b x+17 b^3 x^3\right )\right )}{3 a^3 \left (-1+a^2\right )^3 x^3}-\frac {2 \csc ^{-1}(a+b x)}{x^4}+\frac {2 b^4 \text {ArcSin}\left (\frac {1}{a+b x}\right )}{a^4}+\frac {i \left (-2+7 a^2-8 a^4+8 a^6\right ) b^4 \log \left (\frac {16 a^4 \left (-1+a^2\right )^3 \left (\frac {i \left (-1+a^2+a b x\right )}{\sqrt {1-a^2}}+(a+b x) \sqrt {\frac {-1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}}\right )}{\left (-2+7 a^2-8 a^4+8 a^6\right ) b^4 x}\right )}{a^4 \left (1-a^2\right )^{7/2}}\right ) \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(1701\) vs. \(2(215)=430\).
time = 0.67, size = 1702, normalized size = 7.12

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1702\)
default	\(\text {Expression too large to display}\)	\(1702\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^4*(-1/4/b^4/x^4*arccsc(b*x+a)-1/24*((b*x+a)^2-1)^(1/2)*(18*arctan(1/((b*x+a)^2-1)^(1/2))*(a^2-1)^(3/2)*a^7*(
b*x+a)^2+27*ln(2*((a^2-1)^(1/2)*((b*x+a)^2-1)^(1/2)+a*(b*x+a)-1)/b/x)*a^5-6*arctan(1/((b*x+a)^2-1)^(1/2))*(a^2
-1)^(3/2)*a^6*(b*x+a)^3+54*arctan(1/((b*x+a)^2-1)^(1/2))*(a^2-1)^(3/2)*a^6*(b*x+a)-54*arctan(1/((b*x+a)^2-1)^(
1/2))*(a^2-1)^(3/2)*a^5*(b*x+a)^2+18*arctan(1/((b*x+a)^2-1)^(1/2))*(a^2-1)^(3/2)*a^4*(b*x+a)^3+60*(a^2-1)^(3/2
)*((b*x+a)^2-1)^(1/2)*a^6*(b*x+a)-26*(a^2-1)^(3/2)*((b*x+a)^2-1)^(1/2)*a^5*(b*x+a)^2-54*arctan(1/((b*x+a)^2-1)
^(1/2))*(a^2-1)^(3/2)*a^4*(b*x+a)+54*arctan(1/((b*x+a)^2-1)^(1/2))*(a^2-1)^(3/2)*a^3*(b*x+a)^2-18*arctan(1/((b
*x+a)^2-1)^(1/2))*(a^2-1)^(3/2)*a^2*(b*x+a)^3-45*(a^2-1)^(3/2)*((b*x+a)^2-1)^(1/2)*a^4*(b*x+a)+17*(a^2-1)^(3/2
)*((b*x+a)^2-1)^(1/2)*a^3*(b*x+a)^2-18*arctan(1/((b*x+a)^2-1)^(1/2))*(a^2-1)^(3/2)*a*(b*x+a)^2+15*(a^2-1)^(3/2
)*((b*x+a)^2-1)^(1/2)*a^2*(b*x+a)-6*(a^2-1)^(3/2)*((b*x+a)^2-1)^(1/2)*a*(b*x+a)^2-18*arctan(1/((b*x+a)^2-1)^(1
/2))*(a^2-1)^(3/2)*a^8*(b*x+a)-6*ln(2*((a^2-1)^(1/2)*((b*x+a)^2-1)^(1/2)+a*(b*x+a)-1)/b/x)*a^3+6*arctan(1/((b*
x+a)^2-1)^(1/2))*(a^2-1)^(3/2)*(b*x+a)^3-11*(a^2-1)^(3/2)*((b*x+a)^2-1)^(1/2)*a^3+81*ln(2*((a^2-1)^(1/2)*((b*x
+a)^2-1)^(1/2)+a*(b*x+a)-1)/b/x)*a^3*(b*x+a)^2-27*ln(2*((a^2-1)^(1/2)*((b*x+a)^2-1)^(1/2)+a*(b*x+a)-1)/b/x)*a^
2*(b*x+a)^3+72*ln(2*((a^2-1)^(1/2)*((b*x+a)^2-1)^(1/2)+a*(b*x+a)-1)/b/x)*a^10*(b*x+a)-72*ln(2*((a^2-1)^(1/2)*(
(b*x+a)^2-1)^(1/2)+a*(b*x+a)-1)/b/x)*a^9*(b*x+a)^2+24*ln(2*((a^2-1)^(1/2)*((b*x+a)^2-1)^(1/2)+a*(b*x+a)-1)/b/x
)*a^8*(b*x+a)^3-18*arctan(1/((b*x+a)^2-1)^(1/2))*(a^2-1)^(3/2)*a^7-36*(a^2-1)^(3/2)*((b*x+a)^2-1)^(1/2)*a^7-14
4*ln(2*((a^2-1)^(1/2)*((b*x+a)^2-1)^(1/2)+a*(b*x+a)-1)/b/x)*a^8*(b*x+a)+144*ln(2*((a^2-1)^(1/2)*((b*x+a)^2-1)^
(1/2)+a*(b*x+a)-1)/b/x)*a^7*(b*x+a)^2-48*ln(2*((a^2-1)^(1/2)*((b*x+a)^2-1)^(1/2)+a*(b*x+a)-1)/b/x)*a^6*(b*x+a)
^3+18*arctan(1/((b*x+a)^2-1)^(1/2))*(a^2-1)^(3/2)*a^5+32*(a^2-1)^(3/2)*((b*x+a)^2-1)^(1/2)*a^5+135*ln(2*((a^2-
1)^(1/2)*((b*x+a)^2-1)^(1/2)+a*(b*x+a)-1)/b/x)*a^6*(b*x+a)-135*ln(2*((a^2-1)^(1/2)*((b*x+a)^2-1)^(1/2)+a*(b*x+
a)-1)/b/x)*a^5*(b*x+a)^2+6*arctan(1/((b*x+a)^2-1)^(1/2))*(a^2-1)^(3/2)*a^9-18*ln(2*((a^2-1)^(1/2)*((b*x+a)^2-1
)^(1/2)+a*(b*x+a)-1)/b/x)*a*(b*x+a)^2+45*ln(2*((a^2-1)^(1/2)*((b*x+a)^2-1)^(1/2)+a*(b*x+a)-1)/b/x)*a^4*(b*x+a)
^3-6*arctan(1/((b*x+a)^2-1)^(1/2))*(a^2-1)^(3/2)*a^3-24*ln(2*((a^2-1)^(1/2)*((b*x+a)^2-1)^(1/2)+a*(b*x+a)-1)/b
/x)*a^11+48*ln(2*((a^2-1)^(1/2)*((b*x+a)^2-1)^(1/2)+a*(b*x+a)-1)/b/x)*a^9-45*ln(2*((a^2-1)^(1/2)*((b*x+a)^2-1)
^(1/2)+a*(b*x+a)-1)/b/x)*a^7+6*ln(2*((a^2-1)^(1/2)*((b*x+a)^2-1)^(1/2)+a*(b*x+a)-1)/b/x)*(b*x+a)^3+18*ln(2*((a
^2-1)^(1/2)*((b*x+a)^2-1)^(1/2)+a*(b*x+a)-1)/b/x)*a^2*(b*x+a)-81*ln(2*((a^2-1)^(1/2)*((b*x+a)^2-1)^(1/2)+a*(b*
x+a)-1)/b/x)*a^4*(b*x+a)+18*arctan(1/((b*x+a)^2-1)^(1/2))*(a^2-1)^(3/2)*a^2*(b*x+a))/(((b*x+a)^2-1)/(b*x+a)^2)
^(1/2)/(b*x+a)/a^4/(a^2-1)^(9/2)/b^3/x^3)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.44, size = 673, normalized size = 2.82 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {acsc}{\left (a + b x \right )}}{x^{5}}\, dx \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 841 vs. \(2 (209) = 418\).
time = 0.49, size = 841, normalized size = 3.52 \begin {gather*} \frac {1}{12} \, b {\left (\frac {3 \, {\left (8 \, a^{6} b^{3} - 8 \, a^{4} b^{3} + 7 \, a^{2} b^{3} - 2 \, b^{3}\right )} \arctan \left (\frac {{\left (b x + a\right )} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} + a}{\sqrt {-a^{2} + 1}}\right )}{{\left (a^{10} - 3 \, a^{8} + 3 \, a^{6} - a^{4}\right )} \sqrt {-a^{2} + 1}} + \frac {18 \, {\left (b x + a\right )}^{5} a^{5} b^{3} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{5} + 84 \, {\left (b x + a\right )}^{4} a^{6} b^{3} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{4} + 104 \, {\left (b x + a\right )}^{3} a^{7} b^{3} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{3} - 6 \, {\left (b x + a\right )}^{5} a^{3} b^{3} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{5} - 12 \, {\left (b x + a\right )}^{4} a^{4} b^{3} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{4} + 88 \, {\left (b x + a\right )}^{3} a^{5} b^{3} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{3} + 3 \, {\left (b x + a\right )}^{5} a b^{3} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{5} + 228 \, {\left (b x + a\right )}^{2} a^{6} b^{3} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{2} - 3 \, {\left (b x + a\right )}^{4} a^{2} b^{3} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{4} - 78 \, {\left (b x + a\right )}^{3} a^{3} b^{3} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{3} - 114 \, {\left (b x + a\right )}^{2} a^{4} b^{3} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{2} + 6 \, {\left (b x + a\right )}^{4} b^{3} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{4} + 138 \, {\left (b x + a\right )} a^{5} b^{3} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} + 36 \, {\left (b x + a\right )}^{3} a b^{3} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{3} + 24 \, {\left (b x + a\right )}^{2} a^{2} b^{3} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{2} - 96 \, {\left (b x + a\right )} a^{3} b^{3} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} + 26 \, a^{4} b^{3} + 12 \, {\left (b x + a\right )}^{2} b^{3} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{2} + 33 \, {\left (b x + a\right )} a b^{3} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} - 17 \, a^{2} b^{3} + 6 \, b^{3}}{{\left (a^{9} - 3 \, a^{7} + 3 \, a^{5} - a^{3}\right )} {\left ({\left (b x + a\right )}^{2} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{2} + 2 \, {\left (b x + a\right )} a {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} + 1\right )}^{3}} - \frac {3 \, {\left (\frac {4 \, a b^{3}}{b x + a} - \frac {6 \, a^{2} b^{3}}{{\left (b x + a\right )}^{2}} + \frac {4 \, a^{3} b^{3}}{{\left (b x + a\right )}^{3}} - b^{3}\right )} \arcsin \left (-\frac {1}{{\left (b x + a\right )} {\left (\frac {a}{b x + a} - 1\right )} - a}\right )}{a^{4} {\left (\frac {a}{b x + a} - 1\right )}^{4}}\right )} \end {gather*}

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^3 - 8*a^4*b^3 + 7*a^2*b^3 - 2*b^3)*arctan(((b*x + a)*(sqrt(-1/(b*x + a)^2 + 1) - 1) + a)/sq
rt(-a^2 + 1))/((a^10 - 3*a^8 + 3*a^6 - a^4)*sqrt(-a^2 + 1)) + (18*(b*x + a)^5*a^5*b^3*(sqrt(-1/(b*x + a)^2 + 1
) - 1)^5 + 84*(b*x + a)^4*a^6*b^3*(sqrt(-1/(b*x + a)^2 + 1) - 1)^4 + 104*(b*x + a)^3*a^7*b^3*(sqrt(-1/(b*x + a
)^2 + 1) - 1)^3 - 6*(b*x + a)^5*a^3*b^3*(sqrt(-1/(b*x + a)^2 + 1) - 1)^5 - 12*(b*x + a)^4*a^4*b^3*(sqrt(-1/(b*
x + a)^2 + 1) - 1)^4 + 88*(b*x + a)^3*a^5*b^3*(sqrt(-1/(b*x + a)^2 + 1) - 1)^3 + 3*(b*x + a)^5*a*b^3*(sqrt(-1/
(b*x + a)^2 + 1) - 1)^5 + 228*(b*x + a)^2*a^6*b^3*(sqrt(-1/(b*x + a)^2 + 1) - 1)^2 - 3*(b*x + a)^4*a^2*b^3*(sq
rt(-1/(b*x + a)^2 + 1) - 1)^4 - 78*(b*x + a)^3*a^3*b^3*(sqrt(-1/(b*x + a)^2 + 1) - 1)^3 - 114*(b*x + a)^2*a^4*
b^3*(sqrt(-1/(b*x + a)^2 + 1) - 1)^2 + 6*(b*x + a)^4*b^3*(sqrt(-1/(b*x + a)^2 + 1) - 1)^4 + 138*(b*x + a)*a^5*
b^3*(sqrt(-1/(b*x + a)^2 + 1) - 1) + 36*(b*x + a)^3*a*b^3*(sqrt(-1/(b*x + a)^2 + 1) - 1)^3 + 24*(b*x + a)^2*a^
2*b^3*(sqrt(-1/(b*x + a)^2 + 1) - 1)^2 - 96*(b*x + a)*a^3*b^3*(sqrt(-1/(b*x + a)^2 + 1) - 1) + 26*a^4*b^3 + 12
*(b*x + a)^2*b^3*(sqrt(-1/(b*x + a)^2 + 1) - 1)^2 + 33*(b*x + a)*a*b^3*(sqrt(-1/(b*x + a)^2 + 1) - 1) - 17*a^2
*b^3 + 6*b^3)/((a^9 - 3*a^7 + 3*a^5 - a^3)*((b*x + a)^2*(sqrt(-1/(b*x + a)^2 + 1) - 1)^2 + 2*(b*x + a)*a*(sqrt
(-1/(b*x + a)^2 + 1) - 1) + 1)^3) - 3*(4*a*b^3/(b*x + a) - 6*a^2*b^3/(b*x + a)^2 + 4*a^3*b^3/(b*x + a)^3 - b^3
)*arcsin(-1/((b*x + a)*(a/(b*x + a) - 1) - a))/(a^4*(a/(b*x + a) - 1)^4))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\mathrm {asin}\left (\frac {1}{a+b\,x}\right )}{x^5} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asin(1/(a + b*x))/x^5,x)

[Out]

int(asin(1/(a + b*x))/x^5, x)

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