3.10.75 \(\int e^{n \sinh (c (a+b x))} \cosh (a c+b c x) \, dx\) [975]

Optimal. Leaf size=23 \[ \frac {e^{n \sinh (a c+b c x)}}{b c n} \]

[Out]

exp(n*sinh(b*c*x+a*c))/b/c/n

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Rubi [A]
time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {4421, 2225} \begin {gather*} \frac {e^{n \sinh (a c+b c x)}}{b c n} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(n*Sinh[c*(a + b*x)])*Cosh[a*c + b*c*x],x]

[Out]

E^(n*Sinh[a*c + b*c*x])/(b*c*n)

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 4421

Int[Cosh[(c_.)*((a_.) + (b_.)*(x_))]*(u_), x_Symbol] :> With[{d = FreeFactors[Sinh[c*(a + b*x)], x]}, Dist[d/(
b*c), Subst[Int[SubstFor[1, Sinh[c*(a + b*x)]/d, u, x], x], x, Sinh[c*(a + b*x)]/d], x] /; FunctionOfQ[Sinh[c*
(a + b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x]

Rubi steps

\begin {align*} \int e^{n \sinh (c (a+b x))} \cosh (a c+b c x) \, dx &=\frac {\text {Subst}\left (\int e^{n x} \, dx,x,\sinh (a c+b c x)\right )}{b c}\\ &=\frac {e^{n \sinh (a c+b c x)}}{b c n}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 22, normalized size = 0.96 \begin {gather*} \frac {e^{n \sinh (c (a+b x))}}{b c n} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(n*Sinh[c*(a + b*x)])*Cosh[a*c + b*c*x],x]

[Out]

E^(n*Sinh[c*(a + b*x)])/(b*c*n)

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Maple [A]
time = 11.76, size = 39, normalized size = 1.70

method	result	size
risch	\(\frac {{\mathrm e}^{-\frac {n \left (-{\mathrm e}^{c \left (b x +a \right )}+{\mathrm e}^{-c \left (b x +a \right )}\right )}{2}}}{n b c}\)	\(35\)
default	\(\frac {\frac {\sinh \left (n \sinh \left (c \left (b x +a \right )\right )\right )}{n}+\frac {\cosh \left (n \sinh \left (c \left (b x +a \right )\right )\right )}{n}}{b c}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*sinh(c*(b*x+a)))*cosh(b*c*x+a*c),x,method=_RETURNVERBOSE)

[Out]

1/b/c*(1/n*sinh(n*sinh(c*(b*x+a)))+cosh(n*sinh(c*(b*x+a)))/n)

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Maxima [A]
time = 0.28, size = 22, normalized size = 0.96 \begin {gather*} \frac {e^{\left (n \sinh \left (b c x + a c\right )\right )}}{b c n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*sinh(c*(b*x+a)))*cosh(b*c*x+a*c),x, algorithm="maxima")

[Out]

e^(n*sinh(b*c*x + a*c))/(b*c*n)

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Fricas [A]
time = 0.38, size = 35, normalized size = 1.52 \begin {gather*} \frac {\cosh \left (n \sinh \left (b c x + a c\right )\right ) + \sinh \left (n \sinh \left (b c x + a c\right )\right )}{b c n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*sinh(c*(b*x+a)))*cosh(b*c*x+a*c),x, algorithm="fricas")

[Out]

(cosh(n*sinh(b*c*x + a*c)) + sinh(n*sinh(b*c*x + a*c)))/(b*c*n)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (17) = 34\).
time = 0.57, size = 48, normalized size = 2.09 \begin {gather*} \begin {cases} x & \text {for}\: b = 0 \wedge c = 0 \wedge n = 0 \\\frac {\sinh {\left (a c + b c x \right )}}{b c} & \text {for}\: n = 0 \\x & \text {for}\: c = 0 \\x e^{n \sinh {\left (a c \right )}} \cosh {\left (a c \right )} & \text {for}\: b = 0 \\\frac {e^{n \sinh {\left (a c + b c x \right )}}}{b c n} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*sinh(c*(b*x+a)))*cosh(b*c*x+a*c),x)

[Out]

Piecewise((x, Eq(b, 0) & Eq(c, 0) & Eq(n, 0)), (sinh(a*c + b*c*x)/(b*c), Eq(n, 0)), (x, Eq(c, 0)), (x*exp(n*si
nh(a*c))*cosh(a*c), Eq(b, 0)), (exp(n*sinh(a*c + b*c*x))/(b*c*n), True))

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Giac [A]
time = 0.41, size = 38, normalized size = 1.65 \begin {gather*} \frac {e^{\left (\frac {1}{2} \, n e^{\left (b c x + a c\right )} - \frac {1}{2} \, n e^{\left (-b c x - a c\right )}\right )}}{b c n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*sinh(c*(b*x+a)))*cosh(b*c*x+a*c),x, algorithm="giac")

[Out]

e^(1/2*n*e^(b*c*x + a*c) - 1/2*n*e^(-b*c*x - a*c))/(b*c*n)

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Mupad [B]
time = 1.69, size = 38, normalized size = 1.65 \begin {gather*} \frac {{\mathrm {e}}^{\frac {n\,{\mathrm {e}}^{b\,c\,x}\,{\mathrm {e}}^{a\,c}}{2}}\,{\mathrm {e}}^{-\frac {n\,{\mathrm {e}}^{-b\,c\,x}\,{\mathrm {e}}^{-a\,c}}{2}}}{b\,c\,n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*sinh(c*(a + b*x)))*cosh(a*c + b*c*x),x)

[Out]

(exp((n*exp(b*c*x)*exp(a*c))/2)*exp(-(n*exp(-b*c*x)*exp(-a*c))/2))/(b*c*n)

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