Optimal. Leaf size=85 \[ \frac {(c+d x) \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{d}+\frac {4 b \tanh ^{-1}\left (e^{\text {csch}^{-1}(c+d x)}\right ) \left (a+b \text {csch}^{-1}(c+d x)\right )}{d}+\frac {2 b^2 \text {Li}_2\left (-e^{\text {csch}^{-1}(c+d x)}\right )}{d}-\frac {2 b^2 \text {Li}_2\left (e^{\text {csch}^{-1}(c+d x)}\right )}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6316, 6280, 5452, 4182, 2279, 2391} \[ \frac {2 b^2 \text {PolyLog}\left (2,-e^{\text {csch}^{-1}(c+d x)}\right )}{d}-\frac {2 b^2 \text {PolyLog}\left (2,e^{\text {csch}^{-1}(c+d x)}\right )}{d}+\frac {(c+d x) \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{d}+\frac {4 b \tanh ^{-1}\left (e^{\text {csch}^{-1}(c+d x)}\right ) \left (a+b \text {csch}^{-1}(c+d x)\right )}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2279
Rule 2391
Rule 4182
Rule 5452
Rule 6280
Rule 6316
Rubi steps
\begin {align*} \int \left (a+b \text {csch}^{-1}(c+d x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \left (a+b \text {csch}^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int (a+b x)^2 \coth (x) \text {csch}(x) \, dx,x,\text {csch}^{-1}(c+d x)\right )}{d}\\ &=\frac {(c+d x) \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{d}-\frac {(2 b) \operatorname {Subst}\left (\int (a+b x) \text {csch}(x) \, dx,x,\text {csch}^{-1}(c+d x)\right )}{d}\\ &=\frac {(c+d x) \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{d}+\frac {4 b \left (a+b \text {csch}^{-1}(c+d x)\right ) \tanh ^{-1}\left (e^{\text {csch}^{-1}(c+d x)}\right )}{d}+\frac {\left (2 b^2\right ) \operatorname {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {csch}^{-1}(c+d x)\right )}{d}-\frac {\left (2 b^2\right ) \operatorname {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {csch}^{-1}(c+d x)\right )}{d}\\ &=\frac {(c+d x) \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{d}+\frac {4 b \left (a+b \text {csch}^{-1}(c+d x)\right ) \tanh ^{-1}\left (e^{\text {csch}^{-1}(c+d x)}\right )}{d}+\frac {\left (2 b^2\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {csch}^{-1}(c+d x)}\right )}{d}-\frac {\left (2 b^2\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {csch}^{-1}(c+d x)}\right )}{d}\\ &=\frac {(c+d x) \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{d}+\frac {4 b \left (a+b \text {csch}^{-1}(c+d x)\right ) \tanh ^{-1}\left (e^{\text {csch}^{-1}(c+d x)}\right )}{d}+\frac {2 b^2 \text {Li}_2\left (-e^{\text {csch}^{-1}(c+d x)}\right )}{d}-\frac {2 b^2 \text {Li}_2\left (e^{\text {csch}^{-1}(c+d x)}\right )}{d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.26, size = 160, normalized size = 1.88 \[ \frac {a^2 c+a^2 d x+2 a b (c+d x) \text {csch}^{-1}(c+d x)-2 a b \log \left (\tanh \left (\frac {1}{2} \text {csch}^{-1}(c+d x)\right )\right )-2 b^2 \text {Li}_2\left (-e^{-\text {csch}^{-1}(c+d x)}\right )+2 b^2 \text {Li}_2\left (e^{-\text {csch}^{-1}(c+d x)}\right )+b^2 c \text {csch}^{-1}(c+d x)^2+b^2 d x \text {csch}^{-1}(c+d x)^2-2 b^2 \text {csch}^{-1}(c+d x) \log \left (1-e^{-\text {csch}^{-1}(c+d x)}\right )+2 b^2 \text {csch}^{-1}(c+d x) \log \left (e^{-\text {csch}^{-1}(c+d x)}+1\right )}{d} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{2} \operatorname {arcsch}\left (d x + c\right )^{2} + 2 \, a b \operatorname {arcsch}\left (d x + c\right ) + a^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \operatorname {arcsch}\left (d x + c\right ) + a\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \left (a +b \,\mathrm {arccsch}\left (d x +c \right )\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ {\left (x \log \left (\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} + 1\right )^{2} - \int -\frac {{\left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}^{\frac {3}{2}} \log \left (d x + c\right )^{2} + {\left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )} \log \left (d x + c\right )^{2} - 2 \, {\left ({\left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )} \log \left (d x + c\right ) + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} {\left (d^{2} x^{2} + c d x + {\left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )} \log \left (d x + c\right )\right )}\right )} \log \left (\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} + 1\right )}{d^{2} x^{2} + 2 \, c d x + c^{2} + {\left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}^{\frac {3}{2}} + 1}\,{d x}\right )} b^{2} + a^{2} x + \frac {{\left (2 \, {\left (d x + c\right )} \operatorname {arcsch}\left (d x + c\right ) + \log \left (\sqrt {\frac {1}{{\left (d x + c\right )}^{2}} + 1} + 1\right ) - \log \left (\sqrt {\frac {1}{{\left (d x + c\right )}^{2}} + 1} - 1\right )\right )} a b}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a+b\,\mathrm {asinh}\left (\frac {1}{c+d\,x}\right )\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \operatorname {acsch}{\left (c + d x \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________