Integrand size = 18, antiderivative size = 71 \[ \int x^{-1+\frac {n}{2}} \sinh \left (a+b x^n\right ) \, dx=-\frac {e^{-a} \sqrt {\pi } \text {erf}\left (\sqrt {b} x^{n/2}\right )}{2 \sqrt {b} n}+\frac {e^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x^{n/2}\right )}{2 \sqrt {b} n} \] Output:
-1/2*Pi^(1/2)*erf(b^(1/2)*x^(1/2*n))/b^(1/2)/exp(a)/n+1/2*exp(a)*Pi^(1/2)* erfi(b^(1/2)*x^(1/2*n))/b^(1/2)/n
Time = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.83 \[ \int x^{-1+\frac {n}{2}} \sinh \left (a+b x^n\right ) \, dx=\frac {e^{-a} \sqrt {\pi } \left (-\text {erf}\left (\sqrt {b} x^{n/2}\right )+e^{2 a} \text {erfi}\left (\sqrt {b} x^{n/2}\right )\right )}{2 \sqrt {b} n} \] Input:
Integrate[x^(-1 + n/2)*Sinh[a + b*x^n],x]
Output:
(Sqrt[Pi]*(-Erf[Sqrt[b]*x^(n/2)] + E^(2*a)*Erfi[Sqrt[b]*x^(n/2)]))/(2*Sqrt [b]*E^a*n)
Time = 0.30 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5879, 5821, 2633, 2634}
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.
\(\displaystyle \int x^{\frac {n}{2}-1} \sinh \left (a+b x^n\right ) \, dx\) |
\(\Big \downarrow \) 5879 |
\(\displaystyle \frac {2 \int \sinh \left (b x^n+a\right )dx^{n/2}}{n}\) |
\(\Big \downarrow \) 5821 |
\(\displaystyle \frac {2 \left (\frac {1}{2} \int e^{b x^n+a}dx^{n/2}-\frac {1}{2} \int e^{-b x^n-a}dx^{n/2}\right )}{n}\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle \frac {2 \left (\frac {\sqrt {\pi } e^a \text {erfi}\left (\sqrt {b} x^{n/2}\right )}{4 \sqrt {b}}-\frac {1}{2} \int e^{-b x^n-a}dx^{n/2}\right )}{n}\) |
\(\Big \downarrow \) 2634 |
\(\displaystyle \frac {2 \left (\frac {\sqrt {\pi } e^a \text {erfi}\left (\sqrt {b} x^{n/2}\right )}{4 \sqrt {b}}-\frac {\sqrt {\pi } e^{-a} \text {erf}\left (\sqrt {b} x^{n/2}\right )}{4 \sqrt {b}}\right )}{n}\) |
Input:
Int[x^(-1 + n/2)*Sinh[a + b*x^n],x]
Output:
(2*(-1/4*(Sqrt[Pi]*Erf[Sqrt[b]*x^(n/2)])/(Sqrt[b]*E^a) + (E^a*Sqrt[Pi]*Erf i[Sqrt[b]*x^(n/2)])/(4*Sqrt[b])))/n
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ F, a, b, c, d}, x] && PosQ[b]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F], 2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; Fr eeQ[{F, a, b, c, d}, x] && NegQ[b]
Int[Sinh[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> Simp[1/2 Int[E^(c + d*x^n ), x], x] - Simp[1/2 Int[E^(-c - d*x^n), x], x] /; FreeQ[{c, d}, x] && IG tQ[n, 1]
Int[(x_)^(m_.)*((a_.) + (b_.)*Sinh[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbo l] :> Simp[1/(m + 1) Subst[Int[(a + b*Sinh[c + d*x^Simplify[n/(m + 1)]])^ p, x], x, x^(m + 1)], x] /; FreeQ[{a, b, c, d, m, n}, x] && IntegerQ[p] && NeQ[m, -1] && IGtQ[Simplify[n/(m + 1)], 0] && !IntegerQ[n]
Time = 0.19 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.76
method | result | size |
risch | \(-\frac {{\mathrm e}^{-a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, x^{\frac {n}{2}}\right )}{2 n \sqrt {b}}+\frac {{\mathrm e}^{a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-b}\, x^{\frac {n}{2}}\right )}{2 n \sqrt {-b}}\) | \(54\) |
meijerg | \(\frac {\sqrt {2}\, \sqrt {\pi }\, \left (\frac {\sqrt {i b}\, \sqrt {2}\, \operatorname {erf}\left (\sqrt {b}\, x^{\frac {n}{2}}\right )}{2 \sqrt {b}}+\frac {\sqrt {i b}\, \sqrt {2}\, \operatorname {erfi}\left (\sqrt {b}\, x^{\frac {n}{2}}\right )}{2 \sqrt {b}}\right ) \sinh \left (a \right )}{2 \sqrt {i b}\, n}-\frac {i \sqrt {2}\, \sqrt {\pi }\, \left (-\frac {\sqrt {2}\, \left (i b \right )^{\frac {3}{2}} \operatorname {erf}\left (\sqrt {b}\, x^{\frac {n}{2}}\right )}{2 b^{\frac {3}{2}}}+\frac {\sqrt {2}\, \left (i b \right )^{\frac {3}{2}} \operatorname {erfi}\left (\sqrt {b}\, x^{\frac {n}{2}}\right )}{2 b^{\frac {3}{2}}}\right ) \cosh \left (a \right )}{2 \sqrt {i b}\, n}\) | \(139\) |
Input:
int(x^(-1+1/2*n)*sinh(a+b*x^n),x,method=_RETURNVERBOSE)
Output:
-1/2/n*exp(-a)*Pi^(1/2)/b^(1/2)*erf(b^(1/2)*x^(1/2*n))+1/2/n*exp(a)*Pi^(1/ 2)/(-b)^(1/2)*erf((-b)^(1/2)*x^(1/2*n))
Time = 0.12 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.37 \[ \int x^{-1+\frac {n}{2}} \sinh \left (a+b x^n\right ) \, dx=-\frac {\sqrt {\pi } \sqrt {-b} {\left (\cosh \left (a\right ) + \sinh \left (a\right )\right )} \operatorname {erf}\left (\sqrt {-b} x \cosh \left (\frac {1}{2} \, {\left (n - 2\right )} \log \left (x\right )\right ) + \sqrt {-b} x \sinh \left (\frac {1}{2} \, {\left (n - 2\right )} \log \left (x\right )\right )\right ) + \sqrt {\pi } \sqrt {b} {\left (\cosh \left (a\right ) - \sinh \left (a\right )\right )} \operatorname {erf}\left (\sqrt {b} x \cosh \left (\frac {1}{2} \, {\left (n - 2\right )} \log \left (x\right )\right ) + \sqrt {b} x \sinh \left (\frac {1}{2} \, {\left (n - 2\right )} \log \left (x\right )\right )\right )}{2 \, b n} \] Input:
integrate(x^(-1+1/2*n)*sinh(a+b*x^n),x, algorithm="fricas")
Output:
-1/2*(sqrt(pi)*sqrt(-b)*(cosh(a) + sinh(a))*erf(sqrt(-b)*x*cosh(1/2*(n - 2 )*log(x)) + sqrt(-b)*x*sinh(1/2*(n - 2)*log(x))) + sqrt(pi)*sqrt(b)*(cosh( a) - sinh(a))*erf(sqrt(b)*x*cosh(1/2*(n - 2)*log(x)) + sqrt(b)*x*sinh(1/2* (n - 2)*log(x))))/(b*n)
\[ \int x^{-1+\frac {n}{2}} \sinh \left (a+b x^n\right ) \, dx=\int x^{\frac {n}{2} - 1} \sinh {\left (a + b x^{n} \right )}\, dx \] Input:
integrate(x**(-1+1/2*n)*sinh(a+b*x**n),x)
Output:
Integral(x**(n/2 - 1)*sinh(a + b*x**n), x)
Time = 0.12 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.97 \[ \int x^{-1+\frac {n}{2}} \sinh \left (a+b x^n\right ) \, dx=-\frac {\sqrt {\pi } x^{\frac {1}{2} \, n} {\left (\operatorname {erf}\left (\sqrt {b x^{n}}\right ) - 1\right )} e^{\left (-a\right )}}{2 \, \sqrt {b x^{n}} n} + \frac {\sqrt {\pi } x^{\frac {1}{2} \, n} {\left (\operatorname {erf}\left (\sqrt {-b x^{n}}\right ) - 1\right )} e^{a}}{2 \, \sqrt {-b x^{n}} n} \] Input:
integrate(x^(-1+1/2*n)*sinh(a+b*x^n),x, algorithm="maxima")
Output:
-1/2*sqrt(pi)*x^(1/2*n)*(erf(sqrt(b*x^n)) - 1)*e^(-a)/(sqrt(b*x^n)*n) + 1/ 2*sqrt(pi)*x^(1/2*n)*(erf(sqrt(-b*x^n)) - 1)*e^a/(sqrt(-b*x^n)*n)
Time = 0.13 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.75 \[ \int x^{-1+\frac {n}{2}} \sinh \left (a+b x^n\right ) \, dx=\frac {\frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {b} \sqrt {x^{n}}\right ) e^{\left (-a\right )}}{\sqrt {b}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-b} \sqrt {x^{n}}\right ) e^{a}}{\sqrt {-b}}}{2 \, n} \] Input:
integrate(x^(-1+1/2*n)*sinh(a+b*x^n),x, algorithm="giac")
Output:
1/2*(sqrt(pi)*erf(-sqrt(b)*sqrt(x^n))*e^(-a)/sqrt(b) - sqrt(pi)*erf(-sqrt( -b)*sqrt(x^n))*e^a/sqrt(-b))/n
Timed out. \[ \int x^{-1+\frac {n}{2}} \sinh \left (a+b x^n\right ) \, dx=\int x^{\frac {n}{2}-1}\,\mathrm {sinh}\left (a+b\,x^n\right ) \,d x \] Input:
int(x^(n/2 - 1)*sinh(a + b*x^n),x)
Output:
int(x^(n/2 - 1)*sinh(a + b*x^n), x)
\[ \int x^{-1+\frac {n}{2}} \sinh \left (a+b x^n\right ) \, dx=\int \frac {x^{\frac {n}{2}} \sinh \left (x^{n} b +a \right )}{x}d x \] Input:
int(x^(-1+1/2*n)*sinh(a+b*x^n),x)
Output:
int((x**(n/2)*sinh(x**n*b + a))/x,x)