3.3.54 \(\int \cosh (a+b x) \sinh (a+b x) \, dx\) [254]

Optimal. Leaf size=15 \[ \frac {\sinh ^2(a+b x)}{2 b} \]

[Out]

1/2*sinh(b*x+a)^2/b

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Rubi [A]
time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2644, 30} \begin {gather*} \frac {\sinh ^2(a+b x)}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*x]*Sinh[a + b*x],x]

[Out]

Sinh[a + b*x]^2/(2*b)

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2644

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(a*f), Subst[Int[
x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, m, n])

Rubi steps

\begin {align*} \int \cosh (a+b x) \sinh (a+b x) \, dx &=-\frac {\text {Subst}(\int x \, dx,x,i \sinh (a+b x))}{b}\\ &=\frac {\sinh ^2(a+b x)}{2 b}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(37\) vs. \(2(15)=30\).
time = 0.01, size = 37, normalized size = 2.47 \begin {gather*} \frac {1}{2} \left (\frac {\cosh (2 a) \cosh (2 b x)}{2 b}+\frac {\sinh (2 a) \sinh (2 b x)}{2 b}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cosh[a + b*x]*Sinh[a + b*x],x]

[Out]

((Cosh[2*a]*Cosh[2*b*x])/(2*b) + (Sinh[2*a]*Sinh[2*b*x])/(2*b))/2

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Maple [A]
time = 0.36, size = 14, normalized size = 0.93

method	result	size
derivativedivides	\(\frac {\cosh ^{2}\left (b x +a \right )}{2 b}\)	\(14\)
default	\(\frac {\cosh ^{2}\left (b x +a \right )}{2 b}\)	\(14\)
risch	\(\frac {{\mathrm e}^{2 b x +2 a}}{8 b}+\frac {{\mathrm e}^{-2 b x -2 a}}{8 b}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*cosh(b*x+a)^2/b

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Maxima [A]
time = 0.28, size = 13, normalized size = 0.87 \begin {gather*} \frac {\cosh \left (b x + a\right )^{2}}{2 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*cosh(b*x + a)^2/b

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Fricas [A]
time = 0.35, size = 22, normalized size = 1.47 \begin {gather*} \frac {\cosh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{2}}{4 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(cosh(b*x + a)^2 + sinh(b*x + a)^2)/b

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Sympy [A]
time = 0.06, size = 19, normalized size = 1.27 \begin {gather*} \begin {cases} \frac {\cosh ^{2}{\left (a + b x \right )}}{2 b} & \text {for}\: b \neq 0 \\x \sinh {\left (a \right )} \cosh {\left (a \right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)*sinh(b*x+a),x)

[Out]

Piecewise((cosh(a + b*x)**2/(2*b), Ne(b, 0)), (x*sinh(a)*cosh(a), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 29 vs. \(2 (13) = 26\).
time = 0.39, size = 29, normalized size = 1.93 \begin {gather*} \frac {e^{\left (2 \, b x + 2 \, a\right )}}{8 \, b} + \frac {e^{\left (-2 \, b x - 2 \, a\right )}}{8 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*e^(2*b*x + 2*a)/b + 1/8*e^(-2*b*x - 2*a)/b

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Mupad [B]
time = 1.43, size = 13, normalized size = 0.87 \begin {gather*} \frac {{\mathrm {cosh}\left (a+b\,x\right )}^2}{2\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a + b*x)*sinh(a + b*x),x)

[Out]

cosh(a + b*x)^2/(2*b)

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