Optimal. Leaf size=66 \[ \frac {1}{2} x \left (c^4+\frac {1}{x^4}\right ) \text {sech}^{\frac {3}{2}}(2 \log (c x))-\frac {1}{2} c^6 x^3 \left (\frac {1}{c^4 x^4}+1\right )^{3/2} \text {csch}^{-1}\left (c^2 x^2\right ) \text {sech}^{\frac {3}{2}}(2 \log (c x)) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 66, 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 = {5551, 5549, 335, 275, 288, 215} \[ \frac {1}{2} x \left (c^4+\frac {1}{x^4}\right ) \text {sech}^{\frac {3}{2}}(2 \log (c x))-\frac {1}{2} c^6 x^3 \left (\frac {1}{c^4 x^4}+1\right )^{3/2} \text {csch}^{-1}\left (c^2 x^2\right ) \text {sech}^{\frac {3}{2}}(2 \log (c x)) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 215
Rule 275
Rule 288
Rule 335
Rule 5549
Rule 5551
Rubi steps
\begin {align*} \int \frac {\text {sech}^{\frac {3}{2}}(2 \log (c x))}{x^4} \, dx &=c^3 \operatorname {Subst}\left (\int \frac {\text {sech}^{\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 {sech}^{\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 {sech}^{\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 {sech}^{\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 {sech}^{\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 {sech}^{\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 {sech}^{\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 \text {csch}^{-1}\left (c^2 x^2\right ) \text {sech}^{\frac {3}{2}}(2 \log (c x))\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.11, size = 51, normalized size = 0.77 \[ \frac {\sqrt {2} c^2 \sqrt {\frac {c^2 x^2}{c^4 x^4+1}} \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};c^4 x^4+1\right )}{x} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.44, size = 93, normalized size = 1.41 \[ \frac {\sqrt {2} c^{3} x \log \left (\frac {c^{5} x^{5} + 2 \, c x - 2 \, {\left (c^{4} x^{4} + 1\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} + 1}}}{c x^{5}}\right ) + 2 \, \sqrt {2} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} + 1}} c^{2}}{2 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.18, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {sech}\left (2 \ln \left (c x \right )\right )^{\frac {3}{2}}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {sech}\left (2 \, \log \left (c x\right )\right )^{\frac {3}{2}}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (\frac {1}{\mathrm {cosh}\left (2\,\ln \left (c\,x\right )\right )}\right )}^{3/2}}{x^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {sech}^{\frac {3}{2}}{\left (2 \log {\left (c x \right )} \right )}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________