Optimal. Leaf size=69 \[ \frac {1}{2} c^6 x^3 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} \csc ^{-1}\left (c^2 x^2\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))-\frac {1}{2} x \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x)) \]
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Rubi [A] time = 0.06, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5552, 5550, 335, 275, 288, 216} \[ \frac {1}{2} c^6 x^3 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} \csc ^{-1}\left (c^2 x^2\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))-\frac {1}{2} x \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x)) \]
Antiderivative was successfully verified.
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Rule 216
Rule 275
Rule 288
Rule 335
Rule 5550
Rule 5552
Rubi steps
\begin {align*} \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x^4} \, dx &=c^3 \operatorname {Subst}\left (\int \frac {\text {csch}^{\frac {3}{2}}(2 \log (x))}{x^4} \, dx,x,c x\right )\\ &=\left (c^6 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {1}{x^4}\right )^{3/2} x^7} \, dx,x,c x\right )\\ &=-\left (\left (c^6 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))\right ) \operatorname {Subst}\left (\int \frac {x^5}{\left (1-x^4\right )^{3/2}} \, dx,x,\frac {1}{c x}\right )\right )\\ &=-\left (\frac {1}{2} \left (c^6 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (1-x^2\right )^{3/2}} \, dx,x,\frac {1}{c^2 x^2}\right )\right )\\ &=-\frac {1}{2} \left (c^4-\frac {1}{x^4}\right ) x \text {csch}^{\frac {3}{2}}(2 \log (c x))+\frac {1}{2} \left (c^6 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,\frac {1}{c^2 x^2}\right )\\ &=-\frac {1}{2} \left (c^4-\frac {1}{x^4}\right ) x \text {csch}^{\frac {3}{2}}(2 \log (c x))+\frac {1}{2} c^6 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \csc ^{-1}\left (c^2 x^2\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))\\ \end {align*}
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Mathematica [C] time = 0.11, size = 53, normalized size = 0.77 \[ -\frac {\sqrt {2} c^2 \sqrt {\frac {c^2 x^2}{c^4 x^4-1}} \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};1-c^4 x^4\right )}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 78, normalized size = 1.13 \[ -\frac {\sqrt {2} c^{3} x \arctan \left (\frac {{\left (c^{4} x^{4} - 1\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} - 1}}}{c x}\right ) + \sqrt {2} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} - 1}} c^{2}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.15, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {csch}\left (2 \ln \left (c x \right )\right )^{\frac {3}{2}}}{x^{4}}\, dx \]
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 \[ \int \frac {\operatorname {csch}\left (2 \, \log \left (c x\right )\right )^{\frac {3}{2}}}{x^{4}}\,{d x} \]
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 \[ \int \frac {{\left (\frac {1}{\mathrm {sinh}\left (2\,\ln \left (c\,x\right )\right )}\right )}^{3/2}}{x^4} \,d x \]
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 \[ \int \frac {\operatorname {csch}^{\frac {3}{2}}{\left (2 \log {\left (c x \right )} \right )}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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