Optimal. Leaf size=56 \[ -\frac {2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt {\pi }}-\frac {b \text {PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )}{\sqrt {\pi }}+\frac {b \text {PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )}{\sqrt {\pi }} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.08, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {5816, 4267,
2317, 2438} \begin {gather*} -\frac {2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {\pi }}-\frac {b \text {Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{\sqrt {\pi }}+\frac {b \text {Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt {\pi }} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2317
Rule 2438
Rule 4267
Rule 5816
Rubi steps
\begin {align*} \int \frac {a+b \sinh ^{-1}(c x)}{x \sqrt {\pi +c^2 \pi x^2}} \, dx &=\frac {\text {Subst}\left (\int (a+b x) \text {csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt {\pi }}\\ &=-\frac {2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt {\pi }}-\frac {b \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt {\pi }}+\frac {b \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt {\pi }}\\ &=-\frac {2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt {\pi }}-\frac {b \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{\sqrt {\pi }}+\frac {b \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{\sqrt {\pi }}\\ &=-\frac {2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt {\pi }}-\frac {b \text {Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{\sqrt {\pi }}+\frac {b \text {Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt {\pi }}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.10, size = 96, normalized size = 1.71 \begin {gather*} \frac {b \sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-b \sinh ^{-1}(c x) \log \left (1+e^{-\sinh ^{-1}(c x)}\right )+a \log (x)-a \log \left (\pi \left (1+\sqrt {1+c^2 x^2}\right )\right )+b \text {PolyLog}\left (2,-e^{-\sinh ^{-1}(c x)}\right )-b \text {PolyLog}\left (2,e^{-\sinh ^{-1}(c x)}\right )}{\sqrt {\pi }} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 1.10, size = 72, normalized size = 1.29
method | result | size |
default | \(-\frac {a \arctanh \left (\frac {\sqrt {\pi }}{\sqrt {\pi \,c^{2} x^{2}+\pi }}\right )}{\sqrt {\pi }}+\frac {b \left (4 \dilog \left (\frac {1}{c x +\sqrt {c^{2} x^{2}+1}}\right )-\dilog \left (\frac {1}{\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}}\right )\right )}{2 \sqrt {\pi }}\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [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]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {a}{x \sqrt {c^{2} x^{2} + 1}}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{x \sqrt {c^{2} x^{2} + 1}}\, dx}{\sqrt {\pi }} \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 [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{x\,\sqrt {\Pi \,c^2\,x^2+\Pi }} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________