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)*hypergeom([1/2, 1/2+1/2*p],[3/2+1/2*p],cosh(c+d*x^n)^2)*sinh(c+d*x^n)/b/d/e/n
/(1+p)/(x^n)/(-sinh(c+d*x^n)^2)^(1/2)

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Rubi [A]  time = 0.10, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, 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 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]

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 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]]

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*}

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Mathematica [A]  time = 0.15, 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]

-1/2*((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)])/(d*e*n*(1 + p)*x^n*Sqrt[-Sinh[c + d*x^n]^2])

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fricas [F]  time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (e x\right )^{n - 1} \left (b \cosh \left (d x^{n} + c\right )\right )^{p}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e x\right )^{n - 1} \left (b \cosh \left (d x^{n} + c\right )\right )^{p}\,{d x} \]

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)

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maple [F]  time = 1.18, size = 0, normalized size = 0.00 \[ \int \left (e x \right )^{-1+n} \left (b \cosh \left (c +d \,x^{n}\right )\right )^{p}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e x\right )^{n - 1} \left (b \cosh \left (d x^{n} + c\right )\right )^{p}\,{d x} \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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]

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

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