Optimal. Leaf size=239 \[ \frac {b^4 \csc ^{-1}(a+b x)}{4 a^4}+\frac {\left (3-8 a^2\right ) b^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{24 a^2 \left (1-a^2\right )^2 x^2}-\frac {b (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{12 a \left (1-a^2\right ) x^3}+\frac {\left (-8 a^6+8 a^4-7 a^2+2\right ) b^4 \tan ^{-1}\left (\frac {a-\tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )}{\sqrt {1-a^2}}\right )}{4 a^4 \left (1-a^2\right )^{7/2}}-\frac {\left (26 a^4-17 a^2+6\right ) b^3 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{24 a^3 \left (1-a^2\right )^3 x}-\frac {\csc ^{-1}(a+b x)}{4 x^4} \]
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Rubi [A] time = 0.46, 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 = {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.
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Rule 204
Rule 618
Rule 2660
Rule 3785
Rule 3831
Rule 3919
Rule 4060
Rule 4427
Rule 5259
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*}
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Mathematica [C] time = 0.54, size = 307, normalized size = 1.28 \[ \frac {1}{8} \left (\frac {2 b^4 \sin ^{-1}\left (\frac {1}{a+b x}\right )}{a^4}+\frac {i \left (8 a^6-8 a^4+7 a^2-2\right ) b^4 \log \left (\frac {16 a^4 \left (a^2-1\right )^3 \left (\sqrt {\frac {a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}} (a+b x)+\frac {i \left (a^2+a b x-1\right )}{\sqrt {1-a^2}}\right )}{\left (8 a^6-8 a^4+7 a^2-2\right ) b^4 x}\right )}{a^4 \left (1-a^2\right )^{7/2}}+\frac {b \sqrt {\frac {a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}} \left (2 a^7-6 a^6 b x+2 a^5 \left (9 b^2 x^2-2\right )+a^4 b x \left (26 b^2 x^2+7\right )+a^3 \left (2-6 b^2 x^2\right )-a^2 \left (17 b^3 x^3+b x\right )+3 a b^2 x^2+6 b^3 x^3\right )}{3 a^3 \left (a^2-1\right )^3 x^3}-\frac {2 \csc ^{-1}(a+b x)}{x^4}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 673, normalized size = 2.82 \[ \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 ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 841, normalized size = 3.52 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 1172, normalized size = 4.90 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {x^{4} \int \frac {{\left (b^{2} x + a b\right )} e^{\left (\frac {1}{2} \, \log \left (b x + a + 1\right ) + \frac {1}{2} \, \log \left (b x + a - 1\right )\right )}}{b^{2} x^{6} + 2 \, a b x^{5} + {\left (a^{2} - 1\right )} x^{4} + {\left (b^{2} x^{6} + 2 \, a b x^{5} + {\left (a^{2} - 1\right )} x^{4}\right )} {\left (b x + a + 1\right )} {\left (b x + a - 1\right )}}\,{d x} + \arctan \left (1, \sqrt {b x + a + 1} \sqrt {b x + a - 1}\right )}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\mathrm {asin}\left (\frac {1}{a+b\,x}\right )}{x^5} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {acsc}{\left (a + b x \right )}}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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