Integrand size = 10, antiderivative size = 239 \[ \int \frac {\csc ^{-1}(a+b x)}{x^5} \, dx=-\frac {b (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{12 a \left (1-a^2\right ) x^3}+\frac {\left (3-8 a^2\right ) b^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{24 a^2 \left (1-a^2\right )^2 x^2}-\frac {\left (6-17 a^2+26 a^4\right ) b^3 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{24 a^3 \left (1-a^2\right )^3 x}+\frac {b^4 \csc ^{-1}(a+b x)}{4 a^4}-\frac {\csc ^{-1}(a+b x)}{4 x^4}+\frac {\left (2-7 a^2+8 a^4-8 a^6\right ) b^4 \arctan \left (\frac {a-\tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )}{\sqrt {1-a^2}}\right )}{4 a^4 \left (1-a^2\right )^{7/2}} \]
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Time = 0.33 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {5367, 4512, 3870, 4145, 4004, 3916, 2739, 632, 210} \[ \int \frac {\csc ^{-1}(a+b x)}{x^5} \, dx=\frac {b^4 \csc ^{-1}(a+b x)}{4 a^4}+\frac {\left (3-8 a^2\right ) b^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{24 a^2 \left (1-a^2\right )^2 x^2}-\frac {b (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{12 a \left (1-a^2\right ) x^3}+\frac {\left (-8 a^6+8 a^4-7 a^2+2\right ) b^4 \arctan \left (\frac {a-\tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )}{\sqrt {1-a^2}}\right )}{4 a^4 \left (1-a^2\right )^{7/2}}-\frac {\left (26 a^4-17 a^2+6\right ) b^3 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{24 a^3 \left (1-a^2\right )^3 x}-\frac {\csc ^{-1}(a+b x)}{4 x^4} \]
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Rule 210
Rule 632
Rule 2739
Rule 3870
Rule 3916
Rule 4004
Rule 4145
Rule 4512
Rule 5367
Rubi steps \begin{align*} \text {integral}& = -\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 \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}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.36 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.28 \[ \int \frac {\csc ^{-1}(a+b x)}{x^5} \, dx=\frac {1}{8} \left (\frac {b \sqrt {\frac {-1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}} \left (2 a^7-6 a^6 b x+3 a b^2 x^2+6 b^3 x^3+a^3 \left (2-6 b^2 x^2\right )+2 a^5 \left (-2+9 b^2 x^2\right )+a^4 b x \left (7+26 b^2 x^2\right )-a^2 \left (b x+17 b^3 x^3\right )\right )}{3 a^3 \left (-1+a^2\right )^3 x^3}-\frac {2 \csc ^{-1}(a+b x)}{x^4}+\frac {2 b^4 \arcsin \left (\frac {1}{a+b x}\right )}{a^4}+\frac {i \left (-2+7 a^2-8 a^4+8 a^6\right ) b^4 \log \left (\frac {16 a^4 \left (-1+a^2\right )^3 \left (\frac {i \left (-1+a^2+a b x\right )}{\sqrt {1-a^2}}+(a+b x) \sqrt {\frac {-1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}}\right )}{\left (-2+7 a^2-8 a^4+8 a^6\right ) b^4 x}\right )}{a^4 \left (1-a^2\right )^{7/2}}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(794\) vs. \(2(215)=430\).
Time = 0.72 (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 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \left (a^{2}-1\right )^{\frac {3}{2}} 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}-8 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \left (a^{2}-1\right )^{\frac {3}{2}} a^{6} b x +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}+2 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \left (a^{2}-1\right )^{\frac {3}{2}} a^{7}-17 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \left (a^{2}-1\right )^{\frac {3}{2}} 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}+11 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \left (a^{2}-1\right )^{\frac {3}{2}} a^{4} b x -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}}-4 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \left (a^{2}-1\right )^{\frac {3}{2}} a^{5}+6 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \left (a^{2}-1\right )^{\frac {3}{2}} 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 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \left (a^{2}-1\right )^{\frac {3}{2}} a^{2} b x +2 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \left (a^{2}-1\right )^{\frac {3}{2}} 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 | \(\text {Expression too large to display}\) | \(1702\) |
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Time = 0.37 (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 ] \]
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\[ \int \frac {\csc ^{-1}(a+b x)}{x^5} \, dx=\int \frac {\operatorname {acsc}{\left (a + b x \right )}}{x^{5}}\, dx \]
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\[ \int \frac {\csc ^{-1}(a+b x)}{x^5} \, dx=\int { \frac {\operatorname {arccsc}\left (b x + a\right )}{x^{5}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 841 vs. \(2 (209) = 418\).
Time = 0.37 (sec) , antiderivative size = 841, normalized size of antiderivative = 3.52 \[ \int \frac {\csc ^{-1}(a+b x)}{x^5} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\csc ^{-1}(a+b x)}{x^5} \, dx=\int \frac {\mathrm {asin}\left (\frac {1}{a+b\,x}\right )}{x^5} \,d x \]
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