Optimal. Leaf size=101 \[ -\frac{a+b \csc ^{-1}(c x)}{6 x^6}-\frac{5 b c^5 \sqrt{1-\frac{1}{c^2 x^2}}}{96 x}-\frac{5 b c^3 \sqrt{1-\frac{1}{c^2 x^2}}}{144 x^3}-\frac{b c \sqrt{1-\frac{1}{c^2 x^2}}}{36 x^5}+\frac{5}{96} b c^6 \csc ^{-1}(c x) \]
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Rubi [A] time = 0.0624712, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5221, 335, 321, 216} \[ -\frac{a+b \csc ^{-1}(c x)}{6 x^6}-\frac{5 b c^5 \sqrt{1-\frac{1}{c^2 x^2}}}{96 x}-\frac{5 b c^3 \sqrt{1-\frac{1}{c^2 x^2}}}{144 x^3}-\frac{b c \sqrt{1-\frac{1}{c^2 x^2}}}{36 x^5}+\frac{5}{96} b c^6 \csc ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 5221
Rule 335
Rule 321
Rule 216
Rubi steps
\begin{align*} \int \frac{a+b \csc ^{-1}(c x)}{x^7} \, dx &=-\frac{a+b \csc ^{-1}(c x)}{6 x^6}-\frac{b \int \frac{1}{\sqrt{1-\frac{1}{c^2 x^2}} x^8} \, dx}{6 c}\\ &=-\frac{a+b \csc ^{-1}(c x)}{6 x^6}+\frac{b \operatorname{Subst}\left (\int \frac{x^6}{\sqrt{1-\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )}{6 c}\\ &=-\frac{b c \sqrt{1-\frac{1}{c^2 x^2}}}{36 x^5}-\frac{a+b \csc ^{-1}(c x)}{6 x^6}+\frac{1}{36} (5 b c) \operatorname{Subst}\left (\int \frac{x^4}{\sqrt{1-\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{b c \sqrt{1-\frac{1}{c^2 x^2}}}{36 x^5}-\frac{5 b c^3 \sqrt{1-\frac{1}{c^2 x^2}}}{144 x^3}-\frac{a+b \csc ^{-1}(c x)}{6 x^6}+\frac{1}{48} \left (5 b c^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{b c \sqrt{1-\frac{1}{c^2 x^2}}}{36 x^5}-\frac{5 b c^3 \sqrt{1-\frac{1}{c^2 x^2}}}{144 x^3}-\frac{5 b c^5 \sqrt{1-\frac{1}{c^2 x^2}}}{96 x}-\frac{a+b \csc ^{-1}(c x)}{6 x^6}+\frac{1}{96} \left (5 b c^5\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{b c \sqrt{1-\frac{1}{c^2 x^2}}}{36 x^5}-\frac{5 b c^3 \sqrt{1-\frac{1}{c^2 x^2}}}{144 x^3}-\frac{5 b c^5 \sqrt{1-\frac{1}{c^2 x^2}}}{96 x}+\frac{5}{96} b c^6 \csc ^{-1}(c x)-\frac{a+b \csc ^{-1}(c x)}{6 x^6}\\ \end{align*}
Mathematica [A] time = 0.10211, size = 88, normalized size = 0.87 \[ -\frac{a}{6 x^6}+b \left (-\frac{5 c^3}{144 x^3}-\frac{5 c^5}{96 x}-\frac{c}{36 x^5}\right ) \sqrt{\frac{c^2 x^2-1}{c^2 x^2}}+\frac{5}{96} b c^6 \sin ^{-1}\left (\frac{1}{c x}\right )-\frac{b \csc ^{-1}(c x)}{6 x^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.177, size = 174, normalized size = 1.7 \begin{align*} -{\frac{a}{6\,{x}^{6}}}-{\frac{b{\rm arccsc} \left (cx\right )}{6\,{x}^{6}}}+{\frac{5\,{c}^{5}b}{96\,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{5\,{c}^{5}b}{96\,x}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}+{\frac{5\,{c}^{3}b}{288\,{x}^{3}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}+{\frac{cb}{144\,{x}^{5}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}+{\frac{b}{36\,c{x}^{7}}{\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 [A] time = 1.47346, size = 223, normalized size = 2.21 \begin{align*} -\frac{1}{288} \, b{\left (\frac{15 \, c^{7} \arctan \left (c x \sqrt{-\frac{1}{c^{2} x^{2}} + 1}\right ) - \frac{15 \, c^{12} x^{5}{\left (-\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{5}{2}} + 40 \, c^{10} x^{3}{\left (-\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{3}{2}} + 33 \, c^{8} x \sqrt{-\frac{1}{c^{2} x^{2}} + 1}}{c^{6} x^{6}{\left (\frac{1}{c^{2} x^{2}} - 1\right )}^{3} - 3 \, c^{4} x^{4}{\left (\frac{1}{c^{2} x^{2}} - 1\right )}^{2} + 3 \, c^{2} x^{2}{\left (\frac{1}{c^{2} x^{2}} - 1\right )} - 1}}{c} + \frac{48 \, \operatorname{arccsc}\left (c x\right )}{x^{6}}\right )} - \frac{a}{6 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.24553, size = 150, normalized size = 1.49 \begin{align*} \frac{3 \,{\left (5 \, b c^{6} x^{6} - 16 \, b\right )} \operatorname{arccsc}\left (c x\right ) -{\left (15 \, b c^{4} x^{4} + 10 \, b c^{2} x^{2} + 8 \, b\right )} \sqrt{c^{2} x^{2} - 1} - 48 \, a}{288 \, x^{6}} \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{a + b \operatorname{acsc}{\left (c x \right )}}{x^{7}}\, 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{b \operatorname{arccsc}\left (c x\right ) + a}{x^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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