Optimal. Leaf size=208 \[ \frac{9 b^2 c^2 \left (a+b \csc ^{-1}(c x)\right )}{32 x^2}+\frac{3 b^2 \left (a+b \csc ^{-1}(c x)\right )}{32 x^4}-\frac{9 b c^3 \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{32 x}-\frac{3 b c \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{16 x^3}+\frac{3}{32} c^4 \left (a+b \csc ^{-1}(c x)\right )^3-\frac{\left (a+b \csc ^{-1}(c x)\right )^3}{4 x^4}+\frac{45 b^3 c^3 \sqrt{1-\frac{1}{c^2 x^2}}}{256 x}+\frac{3 b^3 c \sqrt{1-\frac{1}{c^2 x^2}}}{128 x^3}-\frac{45}{256} b^3 c^4 \csc ^{-1}(c x) \]
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Rubi [A] time = 0.175948, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {5223, 4404, 3311, 32, 2635, 8} \[ \frac{9 b^2 c^2 \left (a+b \csc ^{-1}(c x)\right )}{32 x^2}+\frac{3 b^2 \left (a+b \csc ^{-1}(c x)\right )}{32 x^4}-\frac{9 b c^3 \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{32 x}-\frac{3 b c \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{16 x^3}+\frac{3}{32} c^4 \left (a+b \csc ^{-1}(c x)\right )^3-\frac{\left (a+b \csc ^{-1}(c x)\right )^3}{4 x^4}+\frac{45 b^3 c^3 \sqrt{1-\frac{1}{c^2 x^2}}}{256 x}+\frac{3 b^3 c \sqrt{1-\frac{1}{c^2 x^2}}}{128 x^3}-\frac{45}{256} b^3 c^4 \csc ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 5223
Rule 4404
Rule 3311
Rule 32
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \frac{\left (a+b \csc ^{-1}(c x)\right )^3}{x^5} \, dx &=-\left (c^4 \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{Subst}\left (\int (a+b x)^2 \, dx,x,\csc ^{-1}(c x)\right )-\frac{1}{128} \left (9 b^3 c^4\right ) \operatorname{Subst}\left (\int \sin ^2(x) \, dx,x,\csc ^{-1}(c x)\right )-\frac{1}{32} \left (9 b^3 c^4\right ) \operatorname{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 ) \operatorname{Subst}\left (\int 1 \, dx,x,\csc ^{-1}(c x)\right )-\frac{1}{64} \left (9 b^3 c^4\right ) \operatorname{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*}
Mathematica [A] time = 0.339767, size = 283, normalized size = 1.36 \[ \frac{9 b c^4 x^4 \left (8 a^2-5 b^2\right ) \sin ^{-1}\left (\frac{1}{c x}\right )+24 b \csc ^{-1}(c x) \left (-8 a^2-2 a b c x \sqrt{1-\frac{1}{c^2 x^2}} \left (3 c^2 x^2+2\right )+b^2 \left (3 c^2 x^2+1\right )\right )-72 a^2 b c^3 x^3 \sqrt{1-\frac{1}{c^2 x^2}}-48 a^2 b c x \sqrt{1-\frac{1}{c^2 x^2}}-64 a^3+72 a b^2 c^2 x^2-24 b^2 \csc ^{-1}(c x)^2 \left (a \left (8-3 c^4 x^4\right )+b c x \sqrt{1-\frac{1}{c^2 x^2}} \left (3 c^2 x^2+2\right )\right )+24 a b^2+45 b^3 c^3 x^3 \sqrt{1-\frac{1}{c^2 x^2}}+6 b^3 c x \sqrt{1-\frac{1}{c^2 x^2}}+8 b^3 \left (3 c^4 x^4-8\right ) \csc ^{-1}(c x)^3}{256 x^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.268, size = 472, normalized size = 2.3 \begin{align*} -{\frac{{a}^{3}}{4\,{x}^{4}}}-{\frac{{b}^{3} \left ({\rm arccsc} \left (cx\right ) \right ) ^{3}}{4\,{x}^{4}}}+{\frac{3\,{c}^{4}{b}^{3} \left ({\rm arccsc} \left (cx\right ) \right ) ^{3}}{32}}-{\frac{9\,{c}^{3}{b}^{3} \left ({\rm arccsc} \left (cx\right ) \right ) ^{2}}{32\,x}\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}-{\frac{3\,c{b}^{3} \left ({\rm arccsc} \left (cx\right ) \right ) ^{2}}{16\,{x}^{3}}\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}+{\frac{3\,{b}^{3}{\rm arccsc} \left (cx\right )}{32\,{x}^{4}}}+{\frac{45\,{c}^{3}{b}^{3}}{256\,x}\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}+{\frac{3\,c{b}^{3}}{128\,{x}^{3}}\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}-{\frac{45\,{b}^{3}{c}^{4}{\rm arccsc} \left (cx\right )}{256}}+{\frac{9\,{b}^{3}{c}^{2}{\rm arccsc} \left (cx\right )}{32\,{x}^{2}}}-{\frac{3\,a{b}^{2} \left ({\rm arccsc} \left (cx\right ) \right ) ^{2}}{4\,{x}^{4}}}+{\frac{9\,{c}^{4}a{b}^{2} \left ({\rm arccsc} \left (cx\right ) \right ) ^{2}}{32}}-{\frac{9\,a{c}^{3}{b}^{2}{\rm arccsc} \left (cx\right )}{16\,x}\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}-{\frac{3\,a{b}^{2}c{\rm arccsc} \left (cx\right )}{8\,{x}^{3}}\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}+{\frac{3\,a{b}^{2}}{32\,{x}^{4}}}+{\frac{9\,{c}^{2}a{b}^{2}}{32\,{x}^{2}}}-{\frac{3\,{a}^{2}b{\rm arccsc} \left (cx\right )}{4\,{x}^{4}}}+{\frac{9\,{c}^{3}{a}^{2}b}{32\,x}\sqrt{{c}^{2}{x}^{2}-1}\arctan \left ({\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}} \right ){\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}-{\frac{9\,{c}^{3}{a}^{2}b}{32\,x}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}+{\frac{3\,{a}^{2}cb}{32\,{x}^{3}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}+{\frac{3\,{a}^{2}b}{16\,c{x}^{5}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.38485, size = 512, normalized size = 2.46 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \operatorname{acsc}{\left (c x \right )}\right )^{3}}{x^{5}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arccsc}\left (c x\right ) + a\right )}^{3}}{x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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