3.1.23 \(\int (b \cosh (c+d x))^n \, dx\) [23]

Optimal. Leaf size=71 \[ -\frac {(b \cosh (c+d x))^{1+n} \, _2F_1\left (\frac {1}{2},\frac {1+n}{2};\frac {3+n}{2};\cosh ^2(c+d x)\right ) \sinh (c+d x)}{b d (1+n) \sqrt {-\sinh ^2(c+d x)}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2722} \begin {gather*} -\frac {\sinh (c+d x) (b \cosh (c+d x))^{n+1} \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\cosh ^2(c+d x)\right )}{b d (n+1) \sqrt {-\sinh ^2(c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*Cosh[c + d*x])^n,x]

[Out]

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

Rule 2722

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*((b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1
)*Sqrt[Cos[c + d*x]^2]))*Hypergeometric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2], x] /; FreeQ[{b, c, d, n
}, x] &&  !IntegerQ[2*n]

Rubi steps

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

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Mathematica [A]
time = 0.05, size = 65, normalized size = 0.92 \begin {gather*} \frac {(b \cosh (c+d x))^n \coth (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1+n}{2};\frac {3+n}{2};\cosh ^2(c+d x)\right ) \sqrt {-\sinh ^2(c+d x)}}{d (1+n)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*Cosh[c + d*x])^n,x]

[Out]

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

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Maple [F]
time = 0.71, size = 0, normalized size = 0.00 \[\int \left (b \cosh \left (d x +c \right )\right )^{n}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*cosh(d*x+c))^n,x)

[Out]

int((b*cosh(d*x+c))^n,x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*cosh(d*x + c))^n, x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*cosh(d*x + c))^n, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (b \cosh {\left (c + d x \right )}\right )^{n}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*cosh(d*x+c))**n,x)

[Out]

Integral((b*cosh(c + d*x))**n, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*cosh(d*x + c))^n, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (b\,\mathrm {cosh}\left (c+d\,x\right )\right )}^n \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*cosh(c + d*x))^n,x)

[Out]

int((b*cosh(c + d*x))^n, x)

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