3.23 \(\int (b \cosh (c+d x))^n \, dx\)

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

[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 = 0.02, 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 = {2643} \[ -\frac {\sinh (c+d x) (b \cosh (c+d x))^{n+1} \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\cosh ^2(c+d x)\right )}{b d (n+1) \sqrt {-\sinh ^2(c+d x)}} \]

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 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 (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 = 0.07, size = 65, normalized size = 0.92 \[ \frac {\sqrt {-\sinh ^2(c+d x)} \coth (c+d x) (b \cosh (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\cosh ^2(c+d x)\right )}{d (n+1)} \]

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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fricas [F]  time = 1.00, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (b \cosh \left (d x + c\right )\right )^{n}, x\right ) \]

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

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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maple [F]  time = 0.27, 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 \[ \int \left (b \cosh \left (d x + c\right )\right )^{n}\,{d x} \]

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

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

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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