Optimal. Leaf size=98 \[ \frac {x \cosh \left (\frac {a}{2 b}\right ) \text {Chi}\left (\frac {a+b \cosh ^{-1}\left (1+d x^2\right )}{2 b}\right )}{\sqrt {2} b \sqrt {d x^2}}-\frac {x \sinh \left (\frac {a}{2 b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}\left (1+d x^2\right )}{2 b}\right )}{\sqrt {2} b \sqrt {d x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {6002}
\begin {gather*} \frac {x \cosh \left (\frac {a}{2 b}\right ) \text {Chi}\left (\frac {a+b \cosh ^{-1}\left (d x^2+1\right )}{2 b}\right )}{\sqrt {2} b \sqrt {d x^2}}-\frac {x \sinh \left (\frac {a}{2 b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}\left (d x^2+1\right )}{2 b}\right )}{\sqrt {2} b \sqrt {d x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 6002
Rubi steps
\begin {align*} \int \frac {1}{a+b \cosh ^{-1}\left (1+d x^2\right )} \, dx &=\frac {x \cosh \left (\frac {a}{2 b}\right ) \text {Chi}\left (\frac {a+b \cosh ^{-1}\left (1+d x^2\right )}{2 b}\right )}{\sqrt {2} b \sqrt {d x^2}}-\frac {x \sinh \left (\frac {a}{2 b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}\left (1+d x^2\right )}{2 b}\right )}{\sqrt {2} b \sqrt {d x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.10, size = 118, normalized size = 1.20 \begin {gather*} \frac {x \sinh \left (\frac {1}{2} \cosh ^{-1}\left (1+d x^2\right )\right ) \left (\cosh \left (\frac {a}{2 b}\right ) \text {Chi}\left (\frac {a+b \cosh ^{-1}\left (1+d x^2\right )}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}\left (1+d x^2\right )}{2 b}\right )\right )}{b \sqrt {d x^2} \sqrt {\frac {d x^2}{2+d x^2}} \sqrt {2+d x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {1}{a +b \,\mathrm {arccosh}\left (d \,x^{2}+1\right )}\, dx\]
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*} \int \frac {1}{a + b \operatorname {acosh}{\left (d x^{2} + 1 \right )}}\, 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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{a+b\,\mathrm {acosh}\left (d\,x^2+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________