3.1.39 \(\int (-i \sinh (c+d x))^n \, dx\) [39]

Optimal. Leaf size=72 \[ \frac {i \cosh (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1+n}{2};\frac {3+n}{2};-\sinh ^2(c+d x)\right ) (-i \sinh (c+d x))^{1+n}}{d (1+n) \sqrt {\cosh ^2(c+d x)}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2722} \begin {gather*} \frac {i \cosh (c+d x) (-i \sinh (c+d x))^{n+1} \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};-\sinh ^2(c+d x)\right )}{d (n+1) \sqrt {\cosh ^2(c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-I)*Sinh[c + d*x])^n,x]

[Out]

(I*Cosh[c + d*x]*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, -Sinh[c + d*x]^2]*((-I)*Sinh[c + d*x])^(1 + n))/
(d*(1 + n)*Sqrt[Cosh[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 (-i \sinh (c+d x))^n \, dx &=\frac {i \cosh (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1+n}{2};\frac {3+n}{2};-\sinh ^2(c+d x)\right ) (-i \sinh (c+d x))^{1+n}}{d (1+n) \sqrt {\cosh ^2(c+d x)}}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 67, normalized size = 0.93 \begin {gather*} \frac {\sqrt {\cosh ^2(c+d x)} \, _2F_1\left (\frac {1}{2},\frac {1+n}{2};\frac {3+n}{2};-\sinh ^2(c+d x)\right ) (-i \sinh (c+d x))^n \tanh (c+d x)}{d (1+n)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-I)*Sinh[c + d*x])^n,x]

[Out]

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

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Maple [F]
time = 0.27, size = 0, normalized size = 0.00 \[\int \left (-i \sinh \left (d x +c \right )\right )^{n}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-I*sinh(d*x+c))^n,x)

[Out]

int((-I*sinh(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((-I*sinh(d*x+c))^n,x, algorithm="maxima")

[Out]

integrate((-I*sinh(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((-I*sinh(d*x+c))^n,x, algorithm="fricas")

[Out]

integral((1/2*(-I*e^(2*d*x + 2*c) + I)*e^(-d*x - c))^n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-I*sinh(d*x+c))**n,x)

[Out]

Integral((-I*sinh(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((-I*sinh(d*x+c))^n,x, algorithm="giac")

[Out]

integrate((-I*sinh(d*x + c))^n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-sinh(c + d*x)*1i)^n,x)

[Out]

int((-sinh(c + d*x)*1i)^n, x)

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