3.43 \(\int (e x)^{-1+n} (b \cosh (c+d x^n))^p \, dx\)

Optimal. Leaf size=95 \[ -\frac{x^{-n} (e x)^n \sinh \left (c+d x^n\right ) \left (b \cosh \left (c+d x^n\right )\right )^{p+1} \, _2F_1\left (\frac{1}{2},\frac{p+1}{2};\frac{p+3}{2};\cosh ^2\left (d x^n+c\right )\right )}{b d e n (p+1) \sqrt{-\sinh ^2\left (c+d x^n\right )}} \]

[Out]

-(((e*x)^n*(b*Cosh[c + d*x^n])^(1 + p)*Hypergeometric2F1[1/2, (1 + p)/2, (3 + p)/2, Cosh[c + d*x^n]^2]*Sinh[c
+ d*x^n])/(b*d*e*n*(1 + p)*x^n*Sqrt[-Sinh[c + d*x^n]^2]))

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Rubi [A]  time = 0.10428, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {5323, 5321, 2643} \[ -\frac{x^{-n} (e x)^n \sinh \left (c+d x^n\right ) \left (b \cosh \left (c+d x^n\right )\right )^{p+1} \, _2F_1\left (\frac{1}{2},\frac{p+1}{2};\frac{p+3}{2};\cosh ^2\left (d x^n+c\right )\right )}{b d e n (p+1) \sqrt{-\sinh ^2\left (c+d x^n\right )}} \]

Antiderivative was successfully verified.

[In]

Int[(e*x)^(-1 + n)*(b*Cosh[c + d*x^n])^p,x]

[Out]

-(((e*x)^n*(b*Cosh[c + d*x^n])^(1 + p)*Hypergeometric2F1[1/2, (1 + p)/2, (3 + p)/2, Cosh[c + d*x^n]^2]*Sinh[c
+ d*x^n])/(b*d*e*n*(1 + p)*x^n*Sqrt[-Sinh[c + d*x^n]^2]))

Rule 5323

Int[((a_.) + Cosh[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*((e_)*(x_))^(m_), x_Symbol] :> Dist[(e^IntPart[m]*(e*x
)^FracPart[m])/x^FracPart[m], Int[x^m*(a + b*Cosh[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x]
&& IntegerQ[Simplify[(m + 1)/n]]

Rule 5321

Int[((a_.) + Cosh[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simpli
fy[(m + 1)/n] - 1)*(a + b*Cosh[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Sim
plify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0]))

Rule 2643

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1)*Hypergeomet
ric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2])/(b*d*(n + 1)*Sqrt[Cos[c + d*x]^2]), x] /; FreeQ[{b, c, d, n
}, x] &&  !IntegerQ[2*n]

Rubi steps

\begin{align*} \int (e x)^{-1+n} \left (b \cosh \left (c+d x^n\right )\right )^p \, dx &=\frac{\left (x^{-n} (e x)^n\right ) \int x^{-1+n} \left (b \cosh \left (c+d x^n\right )\right )^p \, dx}{e}\\ &=\frac{\left (x^{-n} (e x)^n\right ) \operatorname{Subst}\left (\int (b \cosh (c+d x))^p \, dx,x,x^n\right )}{e n}\\ &=-\frac{x^{-n} (e x)^n \left (b \cosh \left (c+d x^n\right )\right )^{1+p} \, _2F_1\left (\frac{1}{2},\frac{1+p}{2};\frac{3+p}{2};\cosh ^2\left (c+d x^n\right )\right ) \sinh \left (c+d x^n\right )}{b d e n (1+p) \sqrt{-\sinh ^2\left (c+d x^n\right )}}\\ \end{align*}

Mathematica [A]  time = 0.140312, size = 94, normalized size = 0.99 \[ -\frac{x^{-n} (e x)^n \sinh \left (2 \left (c+d x^n\right )\right ) \left (b \cosh \left (c+d x^n\right )\right )^p \, _2F_1\left (\frac{1}{2},\frac{p+1}{2};\frac{p+3}{2};\cosh ^2\left (d x^n+c\right )\right )}{2 d e n (p+1) \sqrt{-\sinh ^2\left (c+d x^n\right )}} \]

Antiderivative was successfully verified.

[In]

Integrate[(e*x)^(-1 + n)*(b*Cosh[c + d*x^n])^p,x]

[Out]

-((e*x)^n*(b*Cosh[c + d*x^n])^p*Hypergeometric2F1[1/2, (1 + p)/2, (3 + p)/2, Cosh[c + d*x^n]^2]*Sinh[2*(c + d*
x^n)])/(2*d*e*n*(1 + p)*x^n*Sqrt[-Sinh[c + d*x^n]^2])

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Maple [F]  time = 0.802, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{-1+n} \left ( b\cosh \left ( c+d{x}^{n} \right ) \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^(-1+n)*(b*cosh(c+d*x^n))^p,x)

[Out]

int((e*x)^(-1+n)*(b*cosh(c+d*x^n))^p,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{n - 1} \left (b \cosh \left (d x^{n} + c\right )\right )^{p}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^(-1+n)*(b*cosh(c+d*x^n))^p,x, algorithm="maxima")

[Out]

integrate((e*x)^(n - 1)*(b*cosh(d*x^n + c))^p, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (e x\right )^{n - 1} \left (b \cosh \left (d x^{n} + c\right )\right )^{p}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^(-1+n)*(b*cosh(c+d*x^n))^p,x, algorithm="fricas")

[Out]

integral((e*x)^(n - 1)*(b*cosh(d*x^n + c))^p, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**(-1+n)*(b*cosh(c+d*x**n))**p,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{n - 1} \left (b \cosh \left (d x^{n} + c\right )\right )^{p}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^(-1+n)*(b*cosh(c+d*x^n))^p,x, algorithm="giac")

[Out]

integrate((e*x)^(n - 1)*(b*cosh(d*x^n + c))^p, x)