Optimal. Leaf size=134 \[ \frac {3}{16} a b c^4 \csc ^{-1}(c x)-\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )}{8 x^3}-\frac {3 b c^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )}{16 x}-\frac {\left (a+b \csc ^{-1}(c x)\right )^2}{4 x^4}+\frac {3}{32} b^2 c^4 \csc ^{-1}(c x)^2+\frac {3 b^2 c^2}{32 x^2}+\frac {b^2}{32 x^4} \]
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Rubi [A] time = 0.11, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {5223, 4404, 3310} \[ -\frac {3 b c^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )}{16 x}-\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )}{8 x^3}+\frac {3}{16} a b c^4 \csc ^{-1}(c x)-\frac {\left (a+b \csc ^{-1}(c x)\right )^2}{4 x^4}+\frac {3 b^2 c^2}{32 x^2}+\frac {3}{32} b^2 c^4 \csc ^{-1}(c x)^2+\frac {b^2}{32 x^4} \]
Antiderivative was successfully verified.
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Rule 3310
Rule 4404
Rule 5223
Rubi steps
\begin {align*} \int \frac {\left (a+b \csc ^{-1}(c x)\right )^2}{x^5} \, dx &=-\left (c^4 \operatorname {Subst}\left (\int (a+b x)^2 \cos (x) \sin ^3(x) \, dx,x,\csc ^{-1}(c x)\right )\right )\\ &=-\frac {\left (a+b \csc ^{-1}(c x)\right )^2}{4 x^4}+\frac {1}{2} \left (b c^4\right ) \operatorname {Subst}\left (\int (a+b x) \sin ^4(x) \, dx,x,\csc ^{-1}(c x)\right )\\ &=\frac {b^2}{32 x^4}-\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )}{8 x^3}-\frac {\left (a+b \csc ^{-1}(c x)\right )^2}{4 x^4}+\frac {1}{8} \left (3 b c^4\right ) \operatorname {Subst}\left (\int (a+b x) \sin ^2(x) \, dx,x,\csc ^{-1}(c x)\right )\\ &=\frac {b^2}{32 x^4}+\frac {3 b^2 c^2}{32 x^2}-\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )}{8 x^3}-\frac {3 b c^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )}{16 x}-\frac {\left (a+b \csc ^{-1}(c x)\right )^2}{4 x^4}+\frac {1}{16} \left (3 b c^4\right ) \operatorname {Subst}\left (\int (a+b x) \, dx,x,\csc ^{-1}(c x)\right )\\ &=\frac {b^2}{32 x^4}+\frac {3 b^2 c^2}{32 x^2}+\frac {3}{16} a b c^4 \csc ^{-1}(c x)+\frac {3}{32} b^2 c^4 \csc ^{-1}(c x)^2-\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )}{8 x^3}-\frac {3 b c^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )}{16 x}-\frac {\left (a+b \csc ^{-1}(c x)\right )^2}{4 x^4}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 148, normalized size = 1.10 \[ \frac {-8 a^2+6 a b c^4 x^4 \sin ^{-1}\left (\frac {1}{c x}\right )-4 a b c x \sqrt {1-\frac {1}{c^2 x^2}}-2 b \csc ^{-1}(c x) \left (8 a+b c x \sqrt {1-\frac {1}{c^2 x^2}} \left (3 c^2 x^2+2\right )\right )-6 a b c^3 x^3 \sqrt {1-\frac {1}{c^2 x^2}}+b^2 \left (3 c^4 x^4-8\right ) \csc ^{-1}(c x)^2+3 b^2 c^2 x^2+b^2}{32 x^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 120, normalized size = 0.90 \[ \frac {3 \, b^{2} c^{2} x^{2} + {\left (3 \, b^{2} c^{4} x^{4} - 8 \, b^{2}\right )} \operatorname {arccsc}\left (c x\right )^{2} - 8 \, a^{2} + b^{2} + 2 \, {\left (3 \, a b c^{4} x^{4} - 8 \, a b\right )} \operatorname {arccsc}\left (c x\right ) - 2 \, {\left (3 \, a b c^{2} x^{2} + 2 \, a b + {\left (3 \, b^{2} c^{2} x^{2} + 2 \, b^{2}\right )} \operatorname {arccsc}\left (c x\right )\right )} \sqrt {c^{2} x^{2} - 1}}{32 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 304, normalized size = 2.27 \[ -\frac {1}{256} \, {\left (64 \, b^{2} c^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} \arcsin \left (\frac {1}{c x}\right )^{2} + 128 \, a b c^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} \arcsin \left (\frac {1}{c x}\right ) + 128 \, b^{2} c^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right )^{2} - 8 \, b^{2} c^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} + 256 \, a b c^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right ) + 40 \, b^{2} c^{3} \arcsin \left (\frac {1}{c x}\right )^{2} - 40 \, b^{2} c^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + 80 \, a b c^{3} \arcsin \left (\frac {1}{c x}\right ) - \frac {32 \, b^{2} c^{2} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} \arcsin \left (\frac {1}{c x}\right )}{x} - 17 \, b^{2} c^{3} - \frac {32 \, a b c^{2} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}}}{x} + \frac {80 \, b^{2} c^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \arcsin \left (\frac {1}{c x}\right )}{x} + \frac {80 \, a b c^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{x} + \frac {64 \, a^{2}}{c x^{4}}\right )} c \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.46, size = 265, normalized size = 1.98 \[ -\frac {a^{2}}{4 x^{4}}-\frac {b^{2} \mathrm {arccsc}\left (c x \right )^{2}}{4 x^{4}}+\frac {3 b^{2} c^{4} \mathrm {arccsc}\left (c x \right )^{2}}{32}-\frac {3 c^{3} b^{2} \mathrm {arccsc}\left (c x \right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{16 x}-\frac {c \,b^{2} \mathrm {arccsc}\left (c x \right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{8 x^{3}}+\frac {b^{2}}{32 x^{4}}+\frac {3 b^{2} c^{2}}{32 x^{2}}-\frac {a b \,\mathrm {arccsc}\left (c x \right )}{2 x^{4}}+\frac {3 c^{3} a b \sqrt {c^{2} x^{2}-1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{16 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}-\frac {3 c^{3} a b}{16 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {c a b}{16 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{3}}+\frac {a b}{8 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{5}} \]
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 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}^2}{x^5} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {acsc}{\left (c x \right )}\right )^{2}}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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