3.53 \(\int x^{-1+\frac {n}{2}} \cosh (a+b x^n) \, dx\)

Optimal. Leaf size=71 \[ \frac {\sqrt {\pi } e^{-a} \text {erf}\left (\sqrt {b} x^{n/2}\right )}{2 \sqrt {b} n}+\frac {\sqrt {\pi } e^a \text {erfi}\left (\sqrt {b} x^{n/2}\right )}{2 \sqrt {b} n} \]

[Out]

1/2*erf(x^(1/2*n)*b^(1/2))*Pi^(1/2)/exp(a)/n/b^(1/2)+1/2*exp(a)*erfi(x^(1/2*n)*b^(1/2))*Pi^(1/2)/n/b^(1/2)

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Rubi [A]  time = 0.04, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5357, 5299, 2204, 2205} \[ \frac {\sqrt {\pi } e^{-a} \text {Erf}\left (\sqrt {b} x^{n/2}\right )}{2 \sqrt {b} n}+\frac {\sqrt {\pi } e^a \text {Erfi}\left (\sqrt {b} x^{n/2}\right )}{2 \sqrt {b} n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 + n/2)*Cosh[a + b*x^n],x]

[Out]

(Sqrt[Pi]*Erf[Sqrt[b]*x^(n/2)])/(2*Sqrt[b]*E^a*n) + (E^a*Sqrt[Pi]*Erfi[Sqrt[b]*x^(n/2)])/(2*Sqrt[b]*n)

Rule 2204

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]

Rule 2205

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] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 5299

Int[Cosh[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> Dist[1/2, Int[E^(c + d*x^n), x], x] + Dist[1/2, Int[E^(-c - d*
x^n), x], x] /; FreeQ[{c, d}, x] && IGtQ[n, 1]

Rule 5357

Int[((a_.) + Cosh[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[(a
+ b*Cosh[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]

Rubi steps

\begin {align*} \int x^{-1+\frac {n}{2}} \cosh \left (a+b x^n\right ) \, dx &=\frac {2 \operatorname {Subst}\left (\int \cosh \left (a+b x^2\right ) \, dx,x,x^{n/2}\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int e^{-a-b x^2} \, dx,x,x^{n/2}\right )}{n}+\frac {\operatorname {Subst}\left (\int e^{a+b x^2} \, dx,x,x^{n/2}\right )}{n}\\ &=\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}\\ \end {align*}

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Mathematica [A]  time = 0.78, size = 60, normalized size = 0.85 \[ \frac {\sqrt {\pi } \left ((\cosh (a)-\sinh (a)) \text {erf}\left (\sqrt {b} x^{n/2}\right )+(\sinh (a)+\cosh (a)) \text {erfi}\left (\sqrt {b} x^{n/2}\right )\right )}{2 \sqrt {b} n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 + n/2)*Cosh[a + b*x^n],x]

[Out]

(Sqrt[Pi]*(Erf[Sqrt[b]*x^(n/2)]*(Cosh[a] - Sinh[a]) + Erfi[Sqrt[b]*x^(n/2)]*(Cosh[a] + Sinh[a])))/(2*Sqrt[b]*n
)

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fricas [A]  time = 0.48, size = 98, normalized size = 1.38 \[ -\frac {\sqrt {\pi } \sqrt {-b} {\left (\cosh \relax (a) + \sinh \relax (a)\right )} \operatorname {erf}\left (\sqrt {-b} x \cosh \left (\frac {1}{2} \, {\left (n - 2\right )} \log \relax (x)\right ) + \sqrt {-b} x \sinh \left (\frac {1}{2} \, {\left (n - 2\right )} \log \relax (x)\right )\right ) - \sqrt {\pi } \sqrt {b} {\left (\cosh \relax (a) - \sinh \relax (a)\right )} \operatorname {erf}\left (\sqrt {b} x \cosh \left (\frac {1}{2} \, {\left (n - 2\right )} \log \relax (x)\right ) + \sqrt {b} x \sinh \left (\frac {1}{2} \, {\left (n - 2\right )} \log \relax (x)\right )\right )}{2 \, b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+1/2*n)*cosh(a+b*x^n),x, algorithm="fricas")

[Out]

-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)

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giac [A]  time = 0.20, size = 52, normalized size = 0.73 \[ -\frac {\frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {b} \sqrt {x^{n}}\right ) e^{\left (-a\right )}}{\sqrt {b}} + \frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-b} \sqrt {x^{n}}\right ) e^{a}}{\sqrt {-b}}}{2 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+1/2*n)*cosh(a+b*x^n),x, algorithm="giac")

[Out]

-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

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maple [A]  time = 0.11, size = 54, normalized size = 0.76 \[ \frac {{\mathrm e}^{-a} \sqrt {\pi }\, \erf \left (x^{\frac {n}{2}} \sqrt {b}\right )}{2 n \sqrt {b}}+\frac {{\mathrm e}^{a} \sqrt {\pi }\, \erf \left (\sqrt {-b}\, x^{\frac {n}{2}}\right )}{2 n \sqrt {-b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1+1/2*n)*cosh(a+b*x^n),x)

[Out]

1/2/n*exp(-a)*Pi^(1/2)/b^(1/2)*erf(x^(1/2*n)*b^(1/2))+1/2/n*exp(a)*Pi^(1/2)/(-b)^(1/2)*erf((-b)^(1/2)*x^(1/2*n
))

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maxima [A]  time = 0.40, size = 69, normalized size = 0.97 \[ \frac {\sqrt {\pi } x^{\frac {1}{2} \, n} {\left (\operatorname {erf}\left (\sqrt {b x^{n}}\right ) - 1\right )} e^{\left (-a\right )}}{2 \, \sqrt {b x^{n}} n} + \frac {\sqrt {\pi } x^{\frac {1}{2} \, n} {\left (\operatorname {erf}\left (\sqrt {-b x^{n}}\right ) - 1\right )} e^{a}}{2 \, \sqrt {-b x^{n}} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+1/2*n)*cosh(a+b*x^n),x, algorithm="maxima")

[Out]

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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^{\frac {n}{2}-1}\,\mathrm {cosh}\left (a+b\,x^n\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(n/2 - 1)*cosh(a + b*x^n),x)

[Out]

int(x^(n/2 - 1)*cosh(a + b*x^n), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{\frac {n}{2} - 1} \cosh {\left (a + b x^{n} \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-1+1/2*n)*cosh(a+b*x**n),x)

[Out]

Integral(x**(n/2 - 1)*cosh(a + b*x**n), x)

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