Optimal. Leaf size=94 \[ \frac {b \sqrt {1-c x}}{4 x^2 \sqrt {\frac {1}{1+c x}}}-\frac {a+b \text {sech}^{-1}(c x)}{2 x^2}+\frac {1}{4} b c^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \tanh ^{-1}\left (\sqrt {1-c x} \sqrt {1+c x}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6418, 105, 12,
94, 214} \begin {gather*} -\frac {a+b \text {sech}^{-1}(c x)}{2 x^2}+\frac {1}{4} b c^2 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \tanh ^{-1}\left (\sqrt {1-c x} \sqrt {c x+1}\right )+\frac {b \sqrt {1-c x}}{4 x^2 \sqrt {\frac {1}{c x+1}}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 94
Rule 105
Rule 214
Rule 6418
Rubi steps
\begin {align*} \int \frac {a+b \text {sech}^{-1}(c x)}{x^3} \, dx &=-\frac {a+b \text {sech}^{-1}(c x)}{2 x^2}-\frac {1}{2} \left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{x^3 \sqrt {1-c x} \sqrt {1+c x}} \, dx\\ &=\frac {b \sqrt {1-c x}}{4 x^2 \sqrt {\frac {1}{1+c x}}}-\frac {a+b \text {sech}^{-1}(c x)}{2 x^2}-\frac {1}{4} \left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {c^2}{x \sqrt {1-c x} \sqrt {1+c x}} \, dx\\ &=\frac {b \sqrt {1-c x}}{4 x^2 \sqrt {\frac {1}{1+c x}}}-\frac {a+b \text {sech}^{-1}(c x)}{2 x^2}-\frac {1}{4} \left (b c^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{x \sqrt {1-c x} \sqrt {1+c x}} \, dx\\ &=\frac {b \sqrt {1-c x}}{4 x^2 \sqrt {\frac {1}{1+c x}}}-\frac {a+b \text {sech}^{-1}(c x)}{2 x^2}+\frac {1}{4} \left (b c^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {1}{c-c x^2} \, dx,x,\sqrt {1-c x} \sqrt {1+c x}\right )\\ &=\frac {b \sqrt {1-c x}}{4 x^2 \sqrt {\frac {1}{1+c x}}}-\frac {a+b \text {sech}^{-1}(c x)}{2 x^2}+\frac {1}{4} b c^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \tanh ^{-1}\left (\sqrt {1-c x} \sqrt {1+c x}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.05, size = 117, normalized size = 1.24 \begin {gather*} -\frac {a}{2 x^2}+b \left (\frac {1}{4 x^2}+\frac {c}{4 x}\right ) \sqrt {\frac {1-c x}{1+c x}}-\frac {b \text {sech}^{-1}(c x)}{2 x^2}-\frac {1}{4} b c^2 \log (x)+\frac {1}{4} b c^2 \log \left (1+\sqrt {\frac {1-c x}{1+c x}}+c x \sqrt {\frac {1-c x}{1+c x}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.17, size = 112, normalized size = 1.19
method | result | size |
derivativedivides | \(c^{2} \left (-\frac {a}{2 c^{2} x^{2}}+b \left (-\frac {\mathrm {arcsech}\left (c x \right )}{2 c^{2} x^{2}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (\arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}\right )}{4 c x \sqrt {-c^{2} x^{2}+1}}\right )\right )\) | \(112\) |
default | \(c^{2} \left (-\frac {a}{2 c^{2} x^{2}}+b \left (-\frac {\mathrm {arcsech}\left (c x \right )}{2 c^{2} x^{2}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (\arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}\right )}{4 c x \sqrt {-c^{2} x^{2}+1}}\right )\right )\) | \(112\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.26, size = 105, normalized size = 1.12 \begin {gather*} -\frac {1}{8} \, b {\left (\frac {\frac {2 \, c^{4} x \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{2} x^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} - 1} - c^{3} \log \left (c x \sqrt {\frac {1}{c^{2} x^{2}} - 1} + 1\right ) + c^{3} \log \left (c x \sqrt {\frac {1}{c^{2} x^{2}} - 1} - 1\right )}{c} + \frac {4 \, \operatorname {arsech}\left (c x\right )}{x^{2}}\right )} - \frac {a}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.37, size = 77, normalized size = 0.82 \begin {gather*} \frac {b c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + {\left (b c^{2} x^{2} - 2 \, b\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - 2 \, a}{4 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b \operatorname {asech}{\left (c x \right )}}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.46, size = 61, normalized size = 0.65 \begin {gather*} \frac {b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\,\left (\frac {c^2\,x}{4}-\frac {1}{2\,x}\right )}{x}-\frac {a}{2\,x^2}+\frac {b\,c\,\sqrt {\frac {1}{c\,x}-1}\,\sqrt {\frac {1}{c\,x}+1}}{4\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________