Integrand size = 14, antiderivative size = 208 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^5} \, dx=\frac {3 b^3 c \sqrt {1-\frac {1}{c^2 x^2}}}{128 x^3}+\frac {45 b^3 c^3 \sqrt {1-\frac {1}{c^2 x^2}}}{256 x}-\frac {45}{256} b^3 c^4 \csc ^{-1}(c x)+\frac {3 b^2 \left (a+b \csc ^{-1}(c x)\right )}{32 x^4}+\frac {9 b^2 c^2 \left (a+b \csc ^{-1}(c x)\right )}{32 x^2}-\frac {3 b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{16 x^3}-\frac {9 b c^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{32 x}+\frac {3}{32} c^4 \left (a+b \csc ^{-1}(c x)\right )^3-\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{4 x^4} \]
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Time = 0.13 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5331, 4489, 3392, 32, 2715, 8} \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^5} \, dx=\frac {9 b^2 c^2 \left (a+b \csc ^{-1}(c x)\right )}{32 x^2}+\frac {3 b^2 \left (a+b \csc ^{-1}(c x)\right )}{32 x^4}+\frac {3}{32} c^4 \left (a+b \csc ^{-1}(c x)\right )^3-\frac {3 b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{16 x^3}-\frac {9 b c^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{32 x}-\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{4 x^4}-\frac {45}{256} b^3 c^4 \csc ^{-1}(c x)+\frac {3 b^3 c \sqrt {1-\frac {1}{c^2 x^2}}}{128 x^3}+\frac {45 b^3 c^3 \sqrt {1-\frac {1}{c^2 x^2}}}{256 x} \]
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Rule 8
Rule 32
Rule 2715
Rule 3392
Rule 4489
Rule 5331
Rubi steps \begin{align*} \text {integral}& = -\left (c^4 \text {Subst}\left (\int (a+b x)^3 \cos (x) \sin ^3(x) \, dx,x,\csc ^{-1}(c x)\right )\right ) \\ & = -\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{4 x^4}+\frac {1}{4} \left (3 b c^4\right ) \text {Subst}\left (\int (a+b x)^2 \sin ^4(x) \, dx,x,\csc ^{-1}(c x)\right ) \\ & = \frac {3 b^2 \left (a+b \csc ^{-1}(c x)\right )}{32 x^4}-\frac {3 b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{16 x^3}-\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{4 x^4}+\frac {1}{16} \left (9 b c^4\right ) \text {Subst}\left (\int (a+b x)^2 \sin ^2(x) \, dx,x,\csc ^{-1}(c x)\right )-\frac {1}{32} \left (3 b^3 c^4\right ) \text {Subst}\left (\int \sin ^4(x) \, dx,x,\csc ^{-1}(c x)\right ) \\ & = \frac {3 b^3 c \sqrt {1-\frac {1}{c^2 x^2}}}{128 x^3}+\frac {3 b^2 \left (a+b \csc ^{-1}(c x)\right )}{32 x^4}+\frac {9 b^2 c^2 \left (a+b \csc ^{-1}(c x)\right )}{32 x^2}-\frac {3 b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{16 x^3}-\frac {9 b c^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{32 x}-\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{4 x^4}+\frac {1}{32} \left (9 b c^4\right ) \text {Subst}\left (\int (a+b x)^2 \, dx,x,\csc ^{-1}(c x)\right )-\frac {1}{128} \left (9 b^3 c^4\right ) \text {Subst}\left (\int \sin ^2(x) \, dx,x,\csc ^{-1}(c x)\right )-\frac {1}{32} \left (9 b^3 c^4\right ) \text {Subst}\left (\int \sin ^2(x) \, dx,x,\csc ^{-1}(c x)\right ) \\ & = \frac {3 b^3 c \sqrt {1-\frac {1}{c^2 x^2}}}{128 x^3}+\frac {45 b^3 c^3 \sqrt {1-\frac {1}{c^2 x^2}}}{256 x}+\frac {3 b^2 \left (a+b \csc ^{-1}(c x)\right )}{32 x^4}+\frac {9 b^2 c^2 \left (a+b \csc ^{-1}(c x)\right )}{32 x^2}-\frac {3 b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{16 x^3}-\frac {9 b c^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{32 x}+\frac {3}{32} c^4 \left (a+b \csc ^{-1}(c x)\right )^3-\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{4 x^4}-\frac {1}{256} \left (9 b^3 c^4\right ) \text {Subst}\left (\int 1 \, dx,x,\csc ^{-1}(c x)\right )-\frac {1}{64} \left (9 b^3 c^4\right ) \text {Subst}\left (\int 1 \, dx,x,\csc ^{-1}(c x)\right ) \\ & = \frac {3 b^3 c \sqrt {1-\frac {1}{c^2 x^2}}}{128 x^3}+\frac {45 b^3 c^3 \sqrt {1-\frac {1}{c^2 x^2}}}{256 x}-\frac {45}{256} b^3 c^4 \csc ^{-1}(c x)+\frac {3 b^2 \left (a+b \csc ^{-1}(c x)\right )}{32 x^4}+\frac {9 b^2 c^2 \left (a+b \csc ^{-1}(c x)\right )}{32 x^2}-\frac {3 b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{16 x^3}-\frac {9 b c^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{32 x}+\frac {3}{32} c^4 \left (a+b \csc ^{-1}(c x)\right )^3-\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{4 x^4} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.36 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^5} \, dx=\frac {-64 a^3+24 a b^2-48 a^2 b c \sqrt {1-\frac {1}{c^2 x^2}} x+6 b^3 c \sqrt {1-\frac {1}{c^2 x^2}} x+72 a b^2 c^2 x^2-72 a^2 b c^3 \sqrt {1-\frac {1}{c^2 x^2}} x^3+45 b^3 c^3 \sqrt {1-\frac {1}{c^2 x^2}} x^3+24 b \left (-8 a^2+b^2 \left (1+3 c^2 x^2\right )-2 a b c \sqrt {1-\frac {1}{c^2 x^2}} x \left (2+3 c^2 x^2\right )\right ) \csc ^{-1}(c x)-24 b^2 \left (b c \sqrt {1-\frac {1}{c^2 x^2}} x \left (2+3 c^2 x^2\right )+a \left (8-3 c^4 x^4\right )\right ) \csc ^{-1}(c x)^2+8 b^3 \left (-8+3 c^4 x^4\right ) \csc ^{-1}(c x)^3+9 b \left (8 a^2-5 b^2\right ) c^4 x^4 \arcsin \left (\frac {1}{c x}\right )}{256 x^4} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(478\) vs. \(2(182)=364\).
Time = 1.66 (sec) , antiderivative size = 479, normalized size of antiderivative = 2.30
method | result | size |
parts | \(-\frac {a^{3}}{4 x^{4}}+b^{3} c^{4} \left (-\frac {\operatorname {arccsc}\left (c x \right )^{3}}{4 c^{4} x^{4}}+\frac {3 \operatorname {arccsc}\left (c x \right )^{2} \left (3 c^{3} x^{3} \operatorname {arccsc}\left (c x \right )-3 c^{2} x^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}-2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )}{32 c^{3} x^{3}}+\frac {3 \,\operatorname {arccsc}\left (c x \right )}{32 c^{4} x^{4}}+\frac {3 \left (3 c^{2} x^{2}+2\right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{256 c^{3} x^{3}}+\frac {27 \,\operatorname {arccsc}\left (c x \right )}{256}-\frac {9 \left (c^{2} x^{2}-1\right ) \operatorname {arccsc}\left (c x \right )}{32 c^{2} x^{2}}+\frac {9 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{64 c x}-\frac {3 \operatorname {arccsc}\left (c x \right )^{3}}{16}\right )+3 a \,b^{2} c^{4} \left (-\frac {\operatorname {arccsc}\left (c x \right )^{2}}{4 c^{4} x^{4}}+\frac {\operatorname {arccsc}\left (c x \right ) \left (3 c^{3} x^{3} \operatorname {arccsc}\left (c x \right )-3 c^{2} x^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}-2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )}{16 c^{3} x^{3}}-\frac {3 \operatorname {arccsc}\left (c x \right )^{2}}{32}+\frac {\left (3 c^{2} x^{2}+2\right )^{2}}{128 c^{4} x^{4}}\right )-\frac {3 a^{2} b \,\operatorname {arccsc}\left (c x \right )}{4 x^{4}}+\frac {9 a^{2} b \,c^{3} \sqrt {c^{2} x^{2}-1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{32 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}-\frac {9 a^{2} b c \left (c^{2} x^{2}-1\right )}{32 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{3}}-\frac {3 a^{2} b \left (c^{2} x^{2}-1\right )}{16 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{5}}\) | \(479\) |
derivativedivides | \(c^{4} \left (-\frac {a^{3}}{4 c^{4} x^{4}}+b^{3} \left (-\frac {\operatorname {arccsc}\left (c x \right )^{3}}{4 c^{4} x^{4}}+\frac {3 \operatorname {arccsc}\left (c x \right )^{2} \left (3 c^{3} x^{3} \operatorname {arccsc}\left (c x \right )-3 c^{2} x^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}-2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )}{32 c^{3} x^{3}}+\frac {3 \,\operatorname {arccsc}\left (c x \right )}{32 c^{4} x^{4}}+\frac {3 \left (3 c^{2} x^{2}+2\right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{256 c^{3} x^{3}}+\frac {27 \,\operatorname {arccsc}\left (c x \right )}{256}-\frac {9 \left (c^{2} x^{2}-1\right ) \operatorname {arccsc}\left (c x \right )}{32 c^{2} x^{2}}+\frac {9 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{64 c x}-\frac {3 \operatorname {arccsc}\left (c x \right )^{3}}{16}\right )+3 a \,b^{2} \left (-\frac {\operatorname {arccsc}\left (c x \right )^{2}}{4 c^{4} x^{4}}+\frac {\operatorname {arccsc}\left (c x \right ) \left (3 c^{3} x^{3} \operatorname {arccsc}\left (c x \right )-3 c^{2} x^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}-2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )}{16 c^{3} x^{3}}-\frac {3 \operatorname {arccsc}\left (c x \right )^{2}}{32}+\frac {\left (3 c^{2} x^{2}+2\right )^{2}}{128 c^{4} x^{4}}\right )-\frac {3 a^{2} b \,\operatorname {arccsc}\left (c x \right )}{4 c^{4} x^{4}}+\frac {9 a^{2} b \sqrt {c^{2} x^{2}-1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{32 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}-\frac {9 a^{2} b \left (c^{2} x^{2}-1\right )}{32 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x^{3}}-\frac {3 a^{2} b \left (c^{2} x^{2}-1\right )}{16 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{5} x^{5}}\right )\) | \(485\) |
default | \(c^{4} \left (-\frac {a^{3}}{4 c^{4} x^{4}}+b^{3} \left (-\frac {\operatorname {arccsc}\left (c x \right )^{3}}{4 c^{4} x^{4}}+\frac {3 \operatorname {arccsc}\left (c x \right )^{2} \left (3 c^{3} x^{3} \operatorname {arccsc}\left (c x \right )-3 c^{2} x^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}-2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )}{32 c^{3} x^{3}}+\frac {3 \,\operatorname {arccsc}\left (c x \right )}{32 c^{4} x^{4}}+\frac {3 \left (3 c^{2} x^{2}+2\right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{256 c^{3} x^{3}}+\frac {27 \,\operatorname {arccsc}\left (c x \right )}{256}-\frac {9 \left (c^{2} x^{2}-1\right ) \operatorname {arccsc}\left (c x \right )}{32 c^{2} x^{2}}+\frac {9 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{64 c x}-\frac {3 \operatorname {arccsc}\left (c x \right )^{3}}{16}\right )+3 a \,b^{2} \left (-\frac {\operatorname {arccsc}\left (c x \right )^{2}}{4 c^{4} x^{4}}+\frac {\operatorname {arccsc}\left (c x \right ) \left (3 c^{3} x^{3} \operatorname {arccsc}\left (c x \right )-3 c^{2} x^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}-2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )}{16 c^{3} x^{3}}-\frac {3 \operatorname {arccsc}\left (c x \right )^{2}}{32}+\frac {\left (3 c^{2} x^{2}+2\right )^{2}}{128 c^{4} x^{4}}\right )-\frac {3 a^{2} b \,\operatorname {arccsc}\left (c x \right )}{4 c^{4} x^{4}}+\frac {9 a^{2} b \sqrt {c^{2} x^{2}-1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{32 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}-\frac {9 a^{2} b \left (c^{2} x^{2}-1\right )}{32 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x^{3}}-\frac {3 a^{2} b \left (c^{2} x^{2}-1\right )}{16 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{5} x^{5}}\right )\) | \(485\) |
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Time = 0.27 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^5} \, dx=\frac {72 \, a b^{2} c^{2} x^{2} + 8 \, {\left (3 \, b^{3} c^{4} x^{4} - 8 \, b^{3}\right )} \operatorname {arccsc}\left (c x\right )^{3} - 64 \, a^{3} + 24 \, a b^{2} + 24 \, {\left (3 \, a b^{2} c^{4} x^{4} - 8 \, a b^{2}\right )} \operatorname {arccsc}\left (c x\right )^{2} + 3 \, {\left (3 \, {\left (8 \, a^{2} b - 5 \, b^{3}\right )} c^{4} x^{4} + 24 \, b^{3} c^{2} x^{2} - 64 \, a^{2} b + 8 \, b^{3}\right )} \operatorname {arccsc}\left (c x\right ) - 3 \, {\left (3 \, {\left (8 \, a^{2} b - 5 \, b^{3}\right )} c^{2} x^{2} + 16 \, a^{2} b - 2 \, b^{3} + 8 \, {\left (3 \, b^{3} c^{2} x^{2} + 2 \, b^{3}\right )} \operatorname {arccsc}\left (c x\right )^{2} + 16 \, {\left (3 \, a b^{2} c^{2} x^{2} + 2 \, a b^{2}\right )} \operatorname {arccsc}\left (c x\right )\right )} \sqrt {c^{2} x^{2} - 1}}{256 \, x^{4}} \]
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\[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^5} \, dx=\int \frac {\left (a + b \operatorname {acsc}{\left (c x \right )}\right )^{3}}{x^{5}}\, dx \]
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\[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^5} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}^{3}}{x^{5}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 576 vs. \(2 (182) = 364\).
Time = 0.32 (sec) , antiderivative size = 576, normalized size of antiderivative = 2.77 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^5} \, dx=-\frac {1}{256} \, {\left (64 \, b^{3} c^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} \arcsin \left (\frac {1}{c x}\right )^{3} + 192 \, a b^{2} c^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} \arcsin \left (\frac {1}{c x}\right )^{2} + 128 \, b^{3} c^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right )^{3} + 192 \, a^{2} b c^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} \arcsin \left (\frac {1}{c x}\right ) - 24 \, b^{3} c^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} \arcsin \left (\frac {1}{c x}\right ) + 384 \, a b^{2} c^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right )^{2} + 40 \, b^{3} c^{3} \arcsin \left (\frac {1}{c x}\right )^{3} - 24 \, a b^{2} c^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} + 384 \, a^{2} b c^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right ) - 120 \, b^{3} c^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right ) + 120 \, a b^{2} c^{3} \arcsin \left (\frac {1}{c x}\right )^{2} - \frac {48 \, b^{3} c^{2} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} \arcsin \left (\frac {1}{c x}\right )^{2}}{x} - 120 \, a b^{2} c^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + 120 \, a^{2} b c^{3} \arcsin \left (\frac {1}{c x}\right ) - 51 \, b^{3} c^{3} \arcsin \left (\frac {1}{c x}\right ) - \frac {96 \, a b^{2} c^{2} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} \arcsin \left (\frac {1}{c x}\right )}{x} + \frac {120 \, b^{3} c^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \arcsin \left (\frac {1}{c x}\right )^{2}}{x} - 51 \, a b^{2} c^{3} - \frac {48 \, a^{2} b c^{2} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}}}{x} + \frac {6 \, b^{3} c^{2} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}}}{x} + \frac {240 \, a b^{2} c^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \arcsin \left (\frac {1}{c x}\right )}{x} + \frac {120 \, a^{2} b c^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{x} - \frac {51 \, b^{3} c^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{x} + \frac {64 \, a^{3}}{c x^{4}}\right )} c \]
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Timed out. \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^5} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}^3}{x^5} \,d x \]
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