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

Optimal result

Integrand size = 10, antiderivative size = 239 \[ \int \frac {\csc ^{-1}(a+b x)}{x^5} \, dx=-\frac {b (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{12 a \left (1-a^2\right ) x^3}+\frac {\left (3-8 a^2\right ) b^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{24 a^2 \left (1-a^2\right )^2 x^2}-\frac {\left (6-17 a^2+26 a^4\right ) b^3 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{24 a^3 \left (1-a^2\right )^3 x}+\frac {b^4 \csc ^{-1}(a+b x)}{4 a^4}-\frac {\csc ^{-1}(a+b x)}{4 x^4}+\frac {\left (2-7 a^2+8 a^4-8 a^6\right ) b^4 \arctan \left (\frac {a-\tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )}{\sqrt {1-a^2}}\right )}{4 a^4 \left (1-a^2\right )^{7/2}} \] Output:

-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+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*arc 
csc(b*x+a)))/(-a^2+1)^(1/2))/a^4/(-a^2+1)^(7/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.33 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.28 \[ \int \frac {\csc ^{-1}(a+b x)}{x^5} \, dx=\frac {1}{8} \left (\frac {b \sqrt {\frac {-1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}} \left (2 a^7-6 a^6 b x+3 a b^2 x^2+6 b^3 x^3+a^3 \left (2-6 b^2 x^2\right )+2 a^5 \left (-2+9 b^2 x^2\right )+a^4 b x \left (7+26 b^2 x^2\right )-a^2 \left (b x+17 b^3 x^3\right )\right )}{3 a^3 \left (-1+a^2\right )^3 x^3}-\frac {2 \csc ^{-1}(a+b x)}{x^4}+\frac {2 b^4 \arcsin \left (\frac {1}{a+b x}\right )}{a^4}+\frac {i \left (-2+7 a^2-8 a^4+8 a^6\right ) b^4 \log \left (\frac {16 a^4 \left (-1+a^2\right )^3 \left (\frac {i \left (-1+a^2+a b x\right )}{\sqrt {1-a^2}}+(a+b x) \sqrt {\frac {-1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}}\right )}{\left (-2+7 a^2-8 a^4+8 a^6\right ) b^4 x}\right )}{a^4 \left (1-a^2\right )^{7/2}}\right ) \] Input:

Integrate[ArcCsc[a + b*x]/x^5,x]
 

Output:

((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
 

Rubi [A] (verified)

Time = 1.42 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.28, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.800, Rules used = {5782, 25, 4927, 3042, 4272, 3042, 4548, 3042, 4548, 27, 3042, 4407, 3042, 4318, 3042, 3139, 1083, 217}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 5782

\(\displaystyle -b^4 \int \frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^5 x^5}d\csc ^{-1}(a+b x)\)

\(\Big \downarrow \) 25

\(\displaystyle b^4 \int -\frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^5 x^5}d\csc ^{-1}(a+b x)\)

\(\Big \downarrow \) 4927

\(\displaystyle -b^4 \left (\frac {\csc ^{-1}(a+b x)}{4 b^4 x^4}-\frac {1}{4} \int \frac {1}{b^4 x^4}d\csc ^{-1}(a+b x)\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -b^4 \left (\frac {\csc ^{-1}(a+b x)}{4 b^4 x^4}-\frac {1}{4} \int \frac {1}{\left (a-\csc \left (\csc ^{-1}(a+b x)\right )\right )^4}d\csc ^{-1}(a+b x)\right )\)

\(\Big \downarrow \) 4272

\(\displaystyle -b^4 \left (\frac {1}{4} \left (\frac {(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{3 a \left (1-a^2\right ) b^3 x^3}-\frac {\int -\frac {-2 (a+b x)^2-3 a (a+b x)+3 \left (1-a^2\right )}{b^3 x^3}d\csc ^{-1}(a+b x)}{3 a \left (1-a^2\right )}\right )+\frac {\csc ^{-1}(a+b x)}{4 b^4 x^4}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -b^4 \left (\frac {1}{4} \left (\frac {(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{3 a \left (1-a^2\right ) b^3 x^3}-\frac {\int \frac {-2 \csc \left (\csc ^{-1}(a+b x)\right )^2-3 a \csc \left (\csc ^{-1}(a+b x)\right )+3 \left (1-a^2\right )}{\left (a-\csc \left (\csc ^{-1}(a+b x)\right )\right )^3}d\csc ^{-1}(a+b x)}{3 a \left (1-a^2\right )}\right )+\frac {\csc ^{-1}(a+b x)}{4 b^4 x^4}\right )\)

\(\Big \downarrow \) 4548

\(\displaystyle -b^4 \left (\frac {1}{4} \left (\frac {(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{3 a \left (1-a^2\right ) b^3 x^3}-\frac {\frac {\int \frac {6 \left (1-a^2\right )^2-\left (3-8 a^2\right ) (a+b x)^2-2 a \left (1-6 a^2\right ) (a+b x)}{b^2 x^2}d\csc ^{-1}(a+b x)}{2 a \left (1-a^2\right )}+\frac {\left (3-8 a^2\right ) \sqrt {1-\frac {1}{(a+b x)^2}} (a+b x)}{2 a \left (1-a^2\right ) b^2 x^2}}{3 a \left (1-a^2\right )}\right )+\frac {\csc ^{-1}(a+b x)}{4 b^4 x^4}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -b^4 \left (\frac {1}{4} \left (\frac {(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{3 a \left (1-a^2\right ) b^3 x^3}-\frac {\frac {\int \frac {6 \left (1-a^2\right )^2-\left (3-8 a^2\right ) \csc \left (\csc ^{-1}(a+b x)\right )^2-2 a \left (1-6 a^2\right ) \csc \left (\csc ^{-1}(a+b x)\right )}{\left (a-\csc \left (\csc ^{-1}(a+b x)\right )\right )^2}d\csc ^{-1}(a+b x)}{2 a \left (1-a^2\right )}+\frac {\left (3-8 a^2\right ) \sqrt {1-\frac {1}{(a+b x)^2}} (a+b x)}{2 a \left (1-a^2\right ) b^2 x^2}}{3 a \left (1-a^2\right )}\right )+\frac {\csc ^{-1}(a+b x)}{4 b^4 x^4}\right )\)

\(\Big \downarrow \) 4548

\(\displaystyle -b^4 \left (\frac {1}{4} \left (\frac {(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{3 a \left (1-a^2\right ) b^3 x^3}-\frac {\frac {\frac {\int -\frac {3 \left (2 \left (1-a^2\right )^3-a \left (6 a^4-2 a^2+1\right ) (a+b x)\right )}{b x}d\csc ^{-1}(a+b x)}{a \left (1-a^2\right )}-\frac {\left (26 a^4-17 a^2+6\right ) (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{a \left (1-a^2\right ) b x}}{2 a \left (1-a^2\right )}+\frac {\left (3-8 a^2\right ) \sqrt {1-\frac {1}{(a+b x)^2}} (a+b x)}{2 a \left (1-a^2\right ) b^2 x^2}}{3 a \left (1-a^2\right )}\right )+\frac {\csc ^{-1}(a+b x)}{4 b^4 x^4}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle -b^4 \left (\frac {1}{4} \left (\frac {(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{3 a \left (1-a^2\right ) b^3 x^3}-\frac {\frac {\frac {3 \int -\frac {2 \left (1-a^2\right )^3-a \left (6 a^4-2 a^2+1\right ) (a+b x)}{b x}d\csc ^{-1}(a+b x)}{a \left (1-a^2\right )}-\frac {\left (26 a^4-17 a^2+6\right ) (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{a \left (1-a^2\right ) b x}}{2 a \left (1-a^2\right )}+\frac {\left (3-8 a^2\right ) \sqrt {1-\frac {1}{(a+b x)^2}} (a+b x)}{2 a \left (1-a^2\right ) b^2 x^2}}{3 a \left (1-a^2\right )}\right )+\frac {\csc ^{-1}(a+b x)}{4 b^4 x^4}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -b^4 \left (\frac {1}{4} \left (\frac {(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{3 a \left (1-a^2\right ) b^3 x^3}-\frac {\frac {\frac {3 \int \frac {2 \left (1-a^2\right )^3-a \left (6 a^4-2 a^2+1\right ) \csc \left (\csc ^{-1}(a+b x)\right )}{a-\csc \left (\csc ^{-1}(a+b x)\right )}d\csc ^{-1}(a+b x)}{a \left (1-a^2\right )}-\frac {\left (26 a^4-17 a^2+6\right ) (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{a \left (1-a^2\right ) b x}}{2 a \left (1-a^2\right )}+\frac {\left (3-8 a^2\right ) \sqrt {1-\frac {1}{(a+b x)^2}} (a+b x)}{2 a \left (1-a^2\right ) b^2 x^2}}{3 a \left (1-a^2\right )}\right )+\frac {\csc ^{-1}(a+b x)}{4 b^4 x^4}\right )\)

\(\Big \downarrow \) 4407

\(\displaystyle -b^4 \left (\frac {1}{4} \left (\frac {(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{3 a \left (1-a^2\right ) b^3 x^3}-\frac {\frac {\frac {3 \left (\frac {\left (-8 a^6+8 a^4-7 a^2+2\right ) \int -\frac {a+b x}{b x}d\csc ^{-1}(a+b x)}{a}+\frac {2 \left (1-a^2\right )^3 \csc ^{-1}(a+b x)}{a}\right )}{a \left (1-a^2\right )}-\frac {\left (26 a^4-17 a^2+6\right ) (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{a \left (1-a^2\right ) b x}}{2 a \left (1-a^2\right )}+\frac {\left (3-8 a^2\right ) \sqrt {1-\frac {1}{(a+b x)^2}} (a+b x)}{2 a \left (1-a^2\right ) b^2 x^2}}{3 a \left (1-a^2\right )}\right )+\frac {\csc ^{-1}(a+b x)}{4 b^4 x^4}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -b^4 \left (\frac {1}{4} \left (\frac {(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{3 a \left (1-a^2\right ) b^3 x^3}-\frac {\frac {\frac {3 \left (\frac {\left (-8 a^6+8 a^4-7 a^2+2\right ) \int \frac {\csc \left (\csc ^{-1}(a+b x)\right )}{a-\csc \left (\csc ^{-1}(a+b x)\right )}d\csc ^{-1}(a+b x)}{a}+\frac {2 \left (1-a^2\right )^3 \csc ^{-1}(a+b x)}{a}\right )}{a \left (1-a^2\right )}-\frac {\left (26 a^4-17 a^2+6\right ) (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{a \left (1-a^2\right ) b x}}{2 a \left (1-a^2\right )}+\frac {\left (3-8 a^2\right ) \sqrt {1-\frac {1}{(a+b x)^2}} (a+b x)}{2 a \left (1-a^2\right ) b^2 x^2}}{3 a \left (1-a^2\right )}\right )+\frac {\csc ^{-1}(a+b x)}{4 b^4 x^4}\right )\)

\(\Big \downarrow \) 4318

\(\displaystyle -b^4 \left (\frac {1}{4} \left (\frac {(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{3 a \left (1-a^2\right ) b^3 x^3}-\frac {\frac {\frac {3 \left (\frac {2 \left (1-a^2\right )^3 \csc ^{-1}(a+b x)}{a}-\frac {\left (-8 a^6+8 a^4-7 a^2+2\right ) \int \frac {1}{1-\frac {a}{a+b x}}d\csc ^{-1}(a+b x)}{a}\right )}{a \left (1-a^2\right )}-\frac {\left (26 a^4-17 a^2+6\right ) (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{a \left (1-a^2\right ) b x}}{2 a \left (1-a^2\right )}+\frac {\left (3-8 a^2\right ) \sqrt {1-\frac {1}{(a+b x)^2}} (a+b x)}{2 a \left (1-a^2\right ) b^2 x^2}}{3 a \left (1-a^2\right )}\right )+\frac {\csc ^{-1}(a+b x)}{4 b^4 x^4}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -b^4 \left (\frac {1}{4} \left (\frac {(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{3 a \left (1-a^2\right ) b^3 x^3}-\frac {\frac {\frac {3 \left (\frac {2 \left (1-a^2\right )^3 \csc ^{-1}(a+b x)}{a}-\frac {\left (-8 a^6+8 a^4-7 a^2+2\right ) \int \frac {1}{1-a \sin \left (\csc ^{-1}(a+b x)\right )}d\csc ^{-1}(a+b x)}{a}\right )}{a \left (1-a^2\right )}-\frac {\left (26 a^4-17 a^2+6\right ) (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{a \left (1-a^2\right ) b x}}{2 a \left (1-a^2\right )}+\frac {\left (3-8 a^2\right ) \sqrt {1-\frac {1}{(a+b x)^2}} (a+b x)}{2 a \left (1-a^2\right ) b^2 x^2}}{3 a \left (1-a^2\right )}\right )+\frac {\csc ^{-1}(a+b x)}{4 b^4 x^4}\right )\)

\(\Big \downarrow \) 3139

\(\displaystyle -b^4 \left (\frac {1}{4} \left (\frac {(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{3 a \left (1-a^2\right ) b^3 x^3}-\frac {\frac {\frac {3 \left (\frac {2 \left (1-a^2\right )^3 \csc ^{-1}(a+b x)}{a}-\frac {2 \left (-8 a^6+8 a^4-7 a^2+2\right ) \int \frac {1}{\tan ^2\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-2 a \tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )+1}d\tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )}{a}\right )}{a \left (1-a^2\right )}-\frac {\left (26 a^4-17 a^2+6\right ) (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{a \left (1-a^2\right ) b x}}{2 a \left (1-a^2\right )}+\frac {\left (3-8 a^2\right ) \sqrt {1-\frac {1}{(a+b x)^2}} (a+b x)}{2 a \left (1-a^2\right ) b^2 x^2}}{3 a \left (1-a^2\right )}\right )+\frac {\csc ^{-1}(a+b x)}{4 b^4 x^4}\right )\)

\(\Big \downarrow \) 1083

\(\displaystyle -b^4 \left (\frac {1}{4} \left (\frac {(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{3 a \left (1-a^2\right ) b^3 x^3}-\frac {\frac {\frac {3 \left (\frac {4 \left (-8 a^6+8 a^4-7 a^2+2\right ) \int \frac {1}{-\left (2 \tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-2 a\right )^2-4 \left (1-a^2\right )}d\left (2 \tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-2 a\right )}{a}+\frac {2 \left (1-a^2\right )^3 \csc ^{-1}(a+b x)}{a}\right )}{a \left (1-a^2\right )}-\frac {\left (26 a^4-17 a^2+6\right ) (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{a \left (1-a^2\right ) b x}}{2 a \left (1-a^2\right )}+\frac {\left (3-8 a^2\right ) \sqrt {1-\frac {1}{(a+b x)^2}} (a+b x)}{2 a \left (1-a^2\right ) b^2 x^2}}{3 a \left (1-a^2\right )}\right )+\frac {\csc ^{-1}(a+b x)}{4 b^4 x^4}\right )\)

\(\Big \downarrow \) 217

\(\displaystyle -b^4 \left (\frac {1}{4} \left (\frac {(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{3 a \left (1-a^2\right ) b^3 x^3}-\frac {\frac {\left (3-8 a^2\right ) \sqrt {1-\frac {1}{(a+b x)^2}} (a+b x)}{2 a \left (1-a^2\right ) b^2 x^2}+\frac {\frac {3 \left (\frac {2 \left (1-a^2\right )^3 \csc ^{-1}(a+b x)}{a}-\frac {2 \left (-8 a^6+8 a^4-7 a^2+2\right ) \arctan \left (\frac {2 \tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-2 a}{2 \sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}\right )}{a \left (1-a^2\right )}-\frac {\left (26 a^4-17 a^2+6\right ) (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{a \left (1-a^2\right ) b x}}{2 a \left (1-a^2\right )}}{3 a \left (1-a^2\right )}\right )+\frac {\csc ^{-1}(a+b x)}{4 b^4 x^4}\right )\)

Input:

Int[ArcCsc[a + b*x]/x^5,x]
 

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

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 1083

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

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3139

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre 
eFactors[Tan[(c + d*x)/2], x]}, Simp[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 4272

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] + Sim 
p[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] && Integ 
erQ[2*n]
 

rule 4318

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

rule 4407

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + 
 (a_)), x_Symbol] :> Simp[c*(x/a), x] - Simp[(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 4548

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] + Simp[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 4927

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] + Simp[f*(m/(b*d*( 
n + 1)))   Int[(e + f*x)^(m - 1)*(a + b*Csc[c + d*x])^(n + 1), x], x] /; Fr 
eeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
 

rule 5782

Int[((a_.) + ArcCsc[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m 
_.), x_Symbol] :> Simp[-(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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(794\) vs. \(2(215)=430\).

Time = 0.49 (sec) , antiderivative size = 795, normalized size of antiderivative = 3.33

method	result	size
parts	\(-\frac {\operatorname {arccsc}\left (b x +a \right )}{4 x^{4}}+\frac {b \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \left (6 \left (a^{2}-1\right )^{\frac {3}{2}} \arctan \left (\frac {1}{\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}\right ) a^{6} b^{3} x^{3}-24 \ln \left (\frac {2 a^{2}-2+2 a b x +2 \sqrt {a^{2}-1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}{x}\right ) a^{8} b^{3} x^{3}-18 \left (a^{2}-1\right )^{\frac {3}{2}} \arctan \left (\frac {1}{\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}\right ) a^{4} b^{3} x^{3}+26 \left (a^{2}-1\right )^{\frac {3}{2}} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, a^{5} b^{2} x^{2}+48 \ln \left (\frac {2 a^{2}-2+2 a b x +2 \sqrt {a^{2}-1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}{x}\right ) a^{6} b^{3} x^{3}+18 \left (a^{2}-1\right )^{\frac {3}{2}} \arctan \left (\frac {1}{\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}\right ) a^{2} b^{3} x^{3}-8 \left (a^{2}-1\right )^{\frac {3}{2}} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, a^{6} b x +2 \left (a^{2}-1\right )^{\frac {3}{2}} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, a^{7}-17 \left (a^{2}-1\right )^{\frac {3}{2}} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, a^{3} b^{2} x^{2}-45 \ln \left (\frac {2 a^{2}-2+2 a b x +2 \sqrt {a^{2}-1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}{x}\right ) a^{4} b^{3} x^{3}-6 b^{3} \arctan \left (\frac {1}{\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}\right ) x^{3} \left (a^{2}-1\right )^{\frac {3}{2}}+11 \left (a^{2}-1\right )^{\frac {3}{2}} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, a^{4} b x -4 \left (a^{2}-1\right )^{\frac {3}{2}} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, a^{5}+6 \left (a^{2}-1\right )^{\frac {3}{2}} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, a \,b^{2} x^{2}+27 \ln \left (\frac {2 a^{2}-2+2 a b x +2 \sqrt {a^{2}-1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}{x}\right ) a^{2} b^{3} x^{3}-3 \left (a^{2}-1\right )^{\frac {3}{2}} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, a^{2} b x +2 \left (a^{2}-1\right )^{\frac {3}{2}} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, a^{3}-6 b^{3} \ln \left (\frac {2 a^{2}-2+2 a b x +2 \sqrt {a^{2}-1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}{x}\right ) x^{3}\right )}{24 \sqrt {\frac {b^{2} x^{2}+2 a b x +a^{2}-1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) a^{4} \left (a^{2}-1\right )^{\frac {9}{2}} x^{3}}\)	\(795\)
derivativedivides	\(\text {Expression too large to display}\)	\(1702\)
default	\(\text {Expression too large to display}\)	\(1702\)

Input:

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

Output:

-1/4*arccsc(b*x+a)/x^4+1/24*b*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)*(6*(a^2-1)^(3/ 
2)*arctan(1/(b^2*x^2+2*a*b*x+a^2-1)^(1/2))*a^6*b^3*x^3-24*ln(2*(a*b*x+(a^2 
-1)^(1/2)*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)+a^2-1)/x)*a^8*b^3*x^3-18*(a^2-1)^( 
3/2)*arctan(1/(b^2*x^2+2*a*b*x+a^2-1)^(1/2))*a^4*b^3*x^3+26*(a^2-1)^(3/2)* 
(b^2*x^2+2*a*b*x+a^2-1)^(1/2)*a^5*b^2*x^2+48*ln(2*(a*b*x+(a^2-1)^(1/2)*(b^ 
2*x^2+2*a*b*x+a^2-1)^(1/2)+a^2-1)/x)*a^6*b^3*x^3+18*(a^2-1)^(3/2)*arctan(1 
/(b^2*x^2+2*a*b*x+a^2-1)^(1/2))*a^2*b^3*x^3-8*(a^2-1)^(3/2)*(b^2*x^2+2*a*b 
*x+a^2-1)^(1/2)*a^6*b*x+2*(a^2-1)^(3/2)*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)*a^7- 
17*(a^2-1)^(3/2)*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)*a^3*b^2*x^2-45*ln(2*(a*b*x+ 
(a^2-1)^(1/2)*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)+a^2-1)/x)*a^4*b^3*x^3-6*b^3*ar 
ctan(1/(b^2*x^2+2*a*b*x+a^2-1)^(1/2))*x^3*(a^2-1)^(3/2)+11*(a^2-1)^(3/2)*( 
b^2*x^2+2*a*b*x+a^2-1)^(1/2)*a^4*b*x-4*(a^2-1)^(3/2)*(b^2*x^2+2*a*b*x+a^2- 
1)^(1/2)*a^5+6*(a^2-1)^(3/2)*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)*a*b^2*x^2+27*ln 
(2*(a*b*x+(a^2-1)^(1/2)*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)+a^2-1)/x)*a^2*b^3*x^ 
3-3*(a^2-1)^(3/2)*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)*a^2*b*x+2*(a^2-1)^(3/2)*(b 
^2*x^2+2*a*b*x+a^2-1)^(1/2)*a^3-6*b^3*ln(2*(a*b*x+(a^2-1)^(1/2)*(b^2*x^2+2 
*a*b*x+a^2-1)^(1/2)+a^2-1)/x)*x^3)/((b^2*x^2+2*a*b*x+a^2-1)/(b*x+a)^2)^(1/ 
2)/(b*x+a)/a^4/(a^2-1)^(9/2)/x^3
 

Fricas [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 673, normalized size of antiderivative = 2.82 \[ \int \frac {\csc ^{-1}(a+b x)}{x^5} \, dx=\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 ] \] Input:

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

Output:

[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)*a 
rccsc(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), 1/ 
24*(6*(8*a^6 - 8*a^4 + 7*a^2 - 2)*sqrt(-a^2 + 1)*b^4*x^4*arctan(-(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)) 
- 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)]
 

Sympy [F]

\[ \int \frac {\csc ^{-1}(a+b x)}{x^5} \, dx=\int \frac {\operatorname {acsc}{\left (a + b x \right )}}{x^{5}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {\csc ^{-1}(a+b x)}{x^5} \, dx=\int { \frac {\operatorname {arccsc}\left (b x + a\right )}{x^{5}} \,d x } \] Input:

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

Output:

-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
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 841 vs. \(2 (209) = 418\).

Time = 0.19 (sec) , antiderivative size = 841, normalized size of antiderivative = 3.52 \[ \int \frac {\csc ^{-1}(a+b x)}{x^5} \, dx=\text {Too large to display} \] Input:

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

Output:

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)*(s 
qrt(-1/(b*x + a)^2 + 1) - 1) + a)/sqrt(-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*(sqrt(-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*(sqr 
t(-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))/...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\csc ^{-1}(a+b x)}{x^5} \, dx=\int \frac {\mathrm {asin}\left (\frac {1}{a+b\,x}\right )}{x^5} \,d x \] Input:

int(asin(1/(a + b*x))/x^5,x)
 

Output:

int(asin(1/(a + b*x))/x^5, x)
 

Reduce [F]

\[ \int \frac {\csc ^{-1}(a+b x)}{x^5} \, dx=\int \frac {\mathit {acsc} \left (b x +a \right )}{x^{5}}d x \] Input:

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

Output:

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