Integrand size = 14, antiderivative size = 55 \[ \int x^{-1+n} \text {arccosh}\left (a+b x^n\right ) \, dx=-\frac {\sqrt {-1+a+b x^n} \sqrt {1+a+b x^n}}{b n}+\frac {\left (a+b x^n\right ) \text {arccosh}\left (a+b x^n\right )}{b n} \] Output:
-(-1+a+b*x^n)^(1/2)*(1+a+b*x^n)^(1/2)/b/n+(a+b*x^n)*arccosh(a+b*x^n)/b/n
Time = 0.03 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.91 \[ \int x^{-1+n} \text {arccosh}\left (a+b x^n\right ) \, dx=\frac {-\sqrt {-1+a+b x^n} \sqrt {1+a+b x^n}+\left (a+b x^n\right ) \text {arccosh}\left (a+b x^n\right )}{b n} \] Input:
Integrate[x^(-1 + n)*ArcCosh[a + b*x^n],x]
Output:
(-(Sqrt[-1 + a + b*x^n]*Sqrt[1 + a + b*x^n]) + (a + b*x^n)*ArcCosh[a + b*x ^n])/(b*n)
Time = 0.31 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.91, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {7266, 6410, 6294, 83}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x^{n-1} \text {arccosh}\left (a+b x^n\right ) \, dx\) |
\(\Big \downarrow \) 7266 |
\(\displaystyle \frac {\int \text {arccosh}\left (b x^n+a\right )dx^n}{n}\) |
\(\Big \downarrow \) 6410 |
\(\displaystyle \frac {\int \text {arccosh}\left (b x^n+a\right )d\left (b x^n+a\right )}{b n}\) |
\(\Big \downarrow \) 6294 |
\(\displaystyle \frac {\left (a+b x^n\right ) \text {arccosh}\left (a+b x^n\right )-\int \frac {b x^n+a}{\sqrt {b x^n+a-1} \sqrt {b x^n+a+1}}d\left (b x^n+a\right )}{b n}\) |
\(\Big \downarrow \) 83 |
\(\displaystyle \frac {\left (a+b x^n\right ) \text {arccosh}\left (a+b x^n\right )-\sqrt {a+b x^n-1} \sqrt {a+b x^n+1}}{b n}\) |
Input:
Int[x^(-1 + n)*ArcCosh[a + b*x^n],x]
Output:
(-(Sqrt[-1 + a + b*x^n]*Sqrt[1 + a + b*x^n]) + (a + b*x^n)*ArcCosh[a + b*x ^n])/(b*n)
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] && EqQ[a*d*f *(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*A rcCosh[c*x])^n, x] - Simp[b*c*n Int[x*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt [1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[1/d Subst[Int[(a + b*ArcCosh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d , n}, x]
Int[(u_)*(x_)^(m_.), x_Symbol] :> Simp[1/(m + 1) Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /; FreeQ[m, x] && NeQ[m, -1] && Function OfQ[x^(m + 1), u, x]
\[\int x^{-1+n} \operatorname {arccosh}\left (a +b \,x^{n}\right )d x\]
Input:
int(x^(-1+n)*arccosh(a+b*x^n),x)
Output:
int(x^(-1+n)*arccosh(a+b*x^n),x)
Leaf count of result is larger than twice the leaf count of optimal. 152 vs. \(2 (51) = 102\).
Time = 0.15 (sec) , antiderivative size = 152, normalized size of antiderivative = 2.76 \[ \int x^{-1+n} \text {arccosh}\left (a+b x^n\right ) \, dx=\frac {{\left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a\right )} \log \left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a + \sqrt {\frac {2 \, a b + {\left (a^{2} + b^{2} - 1\right )} \cosh \left (n \log \left (x\right )\right ) - {\left (a^{2} - b^{2} - 1\right )} \sinh \left (n \log \left (x\right )\right )}{\cosh \left (n \log \left (x\right )\right ) - \sinh \left (n \log \left (x\right )\right )}}\right ) - \sqrt {\frac {2 \, a b + {\left (a^{2} + b^{2} - 1\right )} \cosh \left (n \log \left (x\right )\right ) - {\left (a^{2} - b^{2} - 1\right )} \sinh \left (n \log \left (x\right )\right )}{\cosh \left (n \log \left (x\right )\right ) - \sinh \left (n \log \left (x\right )\right )}}}{b n} \] Input:
integrate(x^(-1+n)*arccosh(a+b*x^n),x, algorithm="fricas")
Output:
((b*cosh(n*log(x)) + b*sinh(n*log(x)) + a)*log(b*cosh(n*log(x)) + b*sinh(n *log(x)) + a + sqrt((2*a*b + (a^2 + b^2 - 1)*cosh(n*log(x)) - (a^2 - b^2 - 1)*sinh(n*log(x)))/(cosh(n*log(x)) - sinh(n*log(x))))) - sqrt((2*a*b + (a ^2 + b^2 - 1)*cosh(n*log(x)) - (a^2 - b^2 - 1)*sinh(n*log(x)))/(cosh(n*log (x)) - sinh(n*log(x)))))/(b*n)
\[ \int x^{-1+n} \text {arccosh}\left (a+b x^n\right ) \, dx=\int x^{n - 1} \operatorname {acosh}{\left (a + b x^{n} \right )}\, dx \] Input:
integrate(x**(-1+n)*acosh(a+b*x**n),x)
Output:
Integral(x**(n - 1)*acosh(a + b*x**n), x)
Time = 0.03 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.71 \[ \int x^{-1+n} \text {arccosh}\left (a+b x^n\right ) \, dx=\frac {{\left (b x^{n} + a\right )} \operatorname {arcosh}\left (b x^{n} + a\right ) - \sqrt {{\left (b x^{n} + a\right )}^{2} - 1}}{b n} \] Input:
integrate(x^(-1+n)*arccosh(a+b*x^n),x, algorithm="maxima")
Output:
((b*x^n + a)*arccosh(b*x^n + a) - sqrt((b*x^n + a)^2 - 1))/(b*n)
Leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (51) = 102\).
Time = 0.15 (sec) , antiderivative size = 124, normalized size of antiderivative = 2.25 \[ \int x^{-1+n} \text {arccosh}\left (a+b x^n\right ) \, dx=-\frac {b {\left (\frac {a \log \left ({\left | -a b - {\left (x^{n} {\left | b \right |} - \sqrt {b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2} - 1}\right )} {\left | b \right |} \right |}\right )}{b {\left | b \right |}} + \frac {\sqrt {b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2} - 1}}{b^{2}}\right )} - x^{n} \log \left (b x^{n} + a + \sqrt {b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2} - 1}\right )}{n} \] Input:
integrate(x^(-1+n)*arccosh(a+b*x^n),x, algorithm="giac")
Output:
-(b*(a*log(abs(-a*b - (x^n*abs(b) - sqrt(b^2*x^(2*n) + 2*a*b*x^n + a^2 - 1 ))*abs(b)))/(b*abs(b)) + sqrt(b^2*x^(2*n) + 2*a*b*x^n + a^2 - 1)/b^2) - x^ n*log(b*x^n + a + sqrt(b^2*x^(2*n) + 2*a*b*x^n + a^2 - 1)))/n
Time = 3.29 (sec) , antiderivative size = 303, normalized size of antiderivative = 5.51 \[ \int x^{-1+n} \text {arccosh}\left (a+b x^n\right ) \, dx=\frac {x^n\,\mathrm {acosh}\left (a+b\,x^n\right )}{n}-\frac {\frac {4\,a\,{\left (\sqrt {a-1}-\sqrt {a+b\,x^n-1}\right )}^3}{b\,{\left (\sqrt {a+1}-\sqrt {a+b\,x^n+1}\right )}^3}+\frac {4\,a\,\left (\sqrt {a-1}-\sqrt {a+b\,x^n-1}\right )}{b\,\left (\sqrt {a+1}-\sqrt {a+b\,x^n+1}\right )}-\frac {8\,{\left (\sqrt {a-1}-\sqrt {a+b\,x^n-1}\right )}^2\,\sqrt {a-1}\,\sqrt {a+1}}{b\,{\left (\sqrt {a+1}-\sqrt {a+b\,x^n+1}\right )}^2}}{n\,\left (\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x^n-1}\right )}^4}{{\left (\sqrt {a+1}-\sqrt {a+b\,x^n+1}\right )}^4}-\frac {2\,{\left (\sqrt {a-1}-\sqrt {a+b\,x^n-1}\right )}^2}{{\left (\sqrt {a+1}-\sqrt {a+b\,x^n+1}\right )}^2}+1\right )}+\frac {4\,a\,\mathrm {atanh}\left (\frac {\sqrt {a-1}-\sqrt {a+b\,x^n-1}}{\sqrt {a+1}-\sqrt {a+b\,x^n+1}}\right )}{b\,n} \] Input:
int(x^(n - 1)*acosh(a + b*x^n),x)
Output:
(x^n*acosh(a + b*x^n))/n - ((4*a*((a - 1)^(1/2) - (a + b*x^n - 1)^(1/2))^3 )/(b*((a + 1)^(1/2) - (a + b*x^n + 1)^(1/2))^3) + (4*a*((a - 1)^(1/2) - (a + b*x^n - 1)^(1/2)))/(b*((a + 1)^(1/2) - (a + b*x^n + 1)^(1/2))) - (8*((a - 1)^(1/2) - (a + b*x^n - 1)^(1/2))^2*(a - 1)^(1/2)*(a + 1)^(1/2))/(b*((a + 1)^(1/2) - (a + b*x^n + 1)^(1/2))^2))/(n*(((a - 1)^(1/2) - (a + b*x^n - 1)^(1/2))^4/((a + 1)^(1/2) - (a + b*x^n + 1)^(1/2))^4 - (2*((a - 1)^(1/2) - (a + b*x^n - 1)^(1/2))^2)/((a + 1)^(1/2) - (a + b*x^n + 1)^(1/2))^2 + 1 )) + (4*a*atanh(((a - 1)^(1/2) - (a + b*x^n - 1)^(1/2))/((a + 1)^(1/2) - ( a + b*x^n + 1)^(1/2))))/(b*n)
\[ \int x^{-1+n} \text {arccosh}\left (a+b x^n\right ) \, dx=\int \frac {x^{n} \mathit {acosh} \left (x^{n} b +a \right )}{x}d x \] Input:
int(x^(-1+n)*acosh(a+b*x^n),x)
Output:
int((x**n*acosh(x**n*b + a))/x,x)