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

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

[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)

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Rubi [A]  time = 0.0447823, antiderivative size = 71, normalized size of antiderivative = 1., 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 = {5356, 5298, 2204, 2205} \[ \frac{\sqrt{\pi } e^a \text{Erfi}\left (\sqrt{b} x^{n/2}\right )}{2 \sqrt{b} n}-\frac{\sqrt{\pi } e^{-a} \text{Erf}\left (\sqrt{b} x^{n/2}\right )}{2 \sqrt{b} n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 + n/2)*Sinh[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 5356

Int[(x_)^(m_.)*((a_.) + (b_.)*Sinh[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[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]

Rule 5298

Int[Sinh[(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 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]

Rubi steps

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

Mathematica [A]  time = 1.50246, size = 60, normalized size = 0.85 \[ \frac{\sqrt{\pi } \left ((\sinh (a)-\cosh (a)) \text{Erf}\left (\sqrt{b} x^{n/2}\right )+(\sinh (a)+\cosh (a)) \text{Erfi}\left (\sqrt{b} x^{n/2}\right )\right )}{2 \sqrt{b} n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 + n/2)*Sinh[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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Maple [A]  time = 0.057, size = 54, normalized size = 0.8 \begin{align*} -{\frac{{{\rm e}^{-a}}\sqrt{\pi }}{2\,n}{\it Erf} \left ({x}^{{\frac{n}{2}}}\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}+{\frac{{{\rm e}^{a}}\sqrt{\pi }}{2\,n}{\it Erf} \left ( \sqrt{-b}{x}^{{\frac{n}{2}}} \right ){\frac{1}{\sqrt{-b}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1+1/2*n)*sinh(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 = 1.3147, size = 93, normalized size = 1.31 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+1/2*n)*sinh(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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Fricas [A]  time = 1.95006, size = 333, normalized size = 4.69 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+1/2*n)*sinh(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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{\frac{n}{2} - 1} \sinh{\left (a + b x^{n} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{\frac{1}{2} \, n - 1} \sinh \left (b x^{n} + a\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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