Optimal. Leaf size=49 \[ 2 b c \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{x}-\frac {2 b^2}{x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {6286, 3296, 2637} \[ 2 b c \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{x}-\frac {2 b^2}{x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2637
Rule 3296
Rule 6286
Rubi steps
\begin {align*} \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{x^2} \, dx &=-\left (c \operatorname {Subst}\left (\int (a+b x)^2 \cosh (x) \, dx,x,\text {csch}^{-1}(c x)\right )\right )\\ &=-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{x}+(2 b c) \operatorname {Subst}\left (\int (a+b x) \sinh (x) \, dx,x,\text {csch}^{-1}(c x)\right )\\ &=2 b c \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{x}-\left (2 b^2 c\right ) \operatorname {Subst}\left (\int \cosh (x) \, dx,x,\text {csch}^{-1}(c x)\right )\\ &=-\frac {2 b^2}{x}+2 b c \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{x}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.17, size = 70, normalized size = 1.43 \[ -\frac {a^2-2 a b c x \sqrt {\frac {1}{c^2 x^2}+1}+2 b \text {csch}^{-1}(c x) \left (a-b c x \sqrt {\frac {1}{c^2 x^2}+1}\right )+b^2 \text {csch}^{-1}(c x)^2+2 b^2}{x} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.07, size = 139, normalized size = 2.84 \[ \frac {2 \, a b c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - b^{2} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} - a^{2} - 2 \, b^{2} + 2 \, {\left (b^{2} c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - a b\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}^{2}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \,\mathrm {arccsch}\left (c x \right )\right )^{2}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.33, size = 78, normalized size = 1.59 \[ 2 \, {\left (c \sqrt {\frac {1}{c^{2} x^{2}} + 1} - \frac {\operatorname {arcsch}\left (c x\right )}{x}\right )} a b + 2 \, {\left (c \sqrt {\frac {1}{c^{2} x^{2}} + 1} \operatorname {arcsch}\left (c x\right ) - \frac {1}{x}\right )} b^{2} - \frac {b^{2} \operatorname {arcsch}\left (c x\right )^{2}}{x} - \frac {a^{2}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}^2}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {acsch}{\left (c x \right )}\right )^{2}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________