Optimal. Leaf size=55 \[ \frac {\left (a+b x^n\right ) \cosh ^{-1}\left (a+b x^n\right )}{b n}-\frac {\sqrt {a+b x^n-1} \sqrt {a+b x^n+1}}{b n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6715, 5864, 5654, 74} \[ \frac {\left (a+b x^n\right ) \cosh ^{-1}\left (a+b x^n\right )}{b n}-\frac {\sqrt {a+b x^n-1} \sqrt {a+b x^n+1}}{b n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 74
Rule 5654
Rule 5864
Rule 6715
Rubi steps
\begin {align*} \int x^{-1+n} \cosh ^{-1}\left (a+b x^n\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \cosh ^{-1}(a+b x) \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \cosh ^{-1}(x) \, dx,x,a+b x^n\right )}{b n}\\ &=\frac {\left (a+b x^n\right ) \cosh ^{-1}\left (a+b x^n\right )}{b n}-\frac {\operatorname {Subst}\left (\int \frac {x}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,a+b x^n\right )}{b n}\\ &=-\frac {\sqrt {-1+a+b x^n} \sqrt {1+a+b x^n}}{b n}+\frac {\left (a+b x^n\right ) \cosh ^{-1}\left (a+b x^n\right )}{b n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 50, normalized size = 0.91 \[ \frac {\left (a+b x^n\right ) \cosh ^{-1}\left (a+b x^n\right )-\sqrt {a+b x^n-1} \sqrt {a+b x^n+1}}{b n} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.71, size = 152, normalized size = 2.76 \[ \frac {{\left (b \cosh \left (n \log \relax (x)\right ) + b \sinh \left (n \log \relax (x)\right ) + a\right )} \log \left (b \cosh \left (n \log \relax (x)\right ) + b \sinh \left (n \log \relax (x)\right ) + a + \sqrt {\frac {2 \, a b + {\left (a^{2} + b^{2} - 1\right )} \cosh \left (n \log \relax (x)\right ) - {\left (a^{2} - b^{2} - 1\right )} \sinh \left (n \log \relax (x)\right )}{\cosh \left (n \log \relax (x)\right ) - \sinh \left (n \log \relax (x)\right )}}\right ) - \sqrt {\frac {2 \, a b + {\left (a^{2} + b^{2} - 1\right )} \cosh \left (n \log \relax (x)\right ) - {\left (a^{2} - b^{2} - 1\right )} \sinh \left (n \log \relax (x)\right )}{\cosh \left (n \log \relax (x)\right ) - \sinh \left (n \log \relax (x)\right )}}}{b n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.73, size = 124, normalized size = 2.25 \[ -\frac {b {\left (\frac {a \log \left ({\left | -a b - {\left (x^{n} {\left | b \right |} - \sqrt {b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2} - 1}\right )} {\left | b \right |} \right |}\right )}{b {\left | b \right |}} + \frac {\sqrt {b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2} - 1}}{b^{2}}\right )} - x^{n} \log \left (b x^{n} + a + \sqrt {b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2} - 1}\right )}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int x^{-1+n} \mathrm {arccosh}\left (a +b \,x^{n}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.30, size = 39, normalized size = 0.71 \[ \frac {{\left (b x^{n} + a\right )} \operatorname {arcosh}\left (b x^{n} + a\right ) - \sqrt {{\left (b x^{n} + a\right )}^{2} - 1}}{b n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.07, size = 303, normalized size = 5.51 \[ \frac {x^n\,\mathrm {acosh}\left (a+b\,x^n\right )}{n}-\frac {\frac {4\,a\,{\left (\sqrt {a-1}-\sqrt {a+b\,x^n-1}\right )}^3}{b\,{\left (\sqrt {a+1}-\sqrt {a+b\,x^n+1}\right )}^3}+\frac {4\,a\,\left (\sqrt {a-1}-\sqrt {a+b\,x^n-1}\right )}{b\,\left (\sqrt {a+1}-\sqrt {a+b\,x^n+1}\right )}-\frac {8\,{\left (\sqrt {a-1}-\sqrt {a+b\,x^n-1}\right )}^2\,\sqrt {a-1}\,\sqrt {a+1}}{b\,{\left (\sqrt {a+1}-\sqrt {a+b\,x^n+1}\right )}^2}}{n\,\left (\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x^n-1}\right )}^4}{{\left (\sqrt {a+1}-\sqrt {a+b\,x^n+1}\right )}^4}-\frac {2\,{\left (\sqrt {a-1}-\sqrt {a+b\,x^n-1}\right )}^2}{{\left (\sqrt {a+1}-\sqrt {a+b\,x^n+1}\right )}^2}+1\right )}+\frac {4\,a\,\mathrm {atanh}\left (\frac {\sqrt {a-1}-\sqrt {a+b\,x^n-1}}{\sqrt {a+1}-\sqrt {a+b\,x^n+1}}\right )}{b\,n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 58.15, size = 76, normalized size = 1.38 \[ \begin {cases} \log {\relax (x )} \operatorname {acosh}{\relax (a )} & \text {for}\: b = 0 \wedge n = 0 \\\frac {x^{n} \operatorname {acosh}{\relax (a )}}{n} & \text {for}\: b = 0 \\\log {\relax (x )} \operatorname {acosh}{\left (a + b \right )} & \text {for}\: n = 0 \\\frac {a \operatorname {acosh}{\left (a + b x^{n} \right )}}{b n} + \frac {x^{n} \operatorname {acosh}{\left (a + b x^{n} \right )}}{n} - \frac {\sqrt {a^{2} + 2 a b x^{n} + b^{2} x^{2 n} - 1}}{b n} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________