### 3.254 $$\int \cosh (a+b x) \sinh (a+b x) \, dx$$

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

[Out]

Sinh[a + b*x]^2/(2*b)

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Rubi [A]  time = 0.0137316, antiderivative size = 15, normalized size of antiderivative = 1., 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 = {2564, 30} $\frac{\sinh ^2(a+b x)}{2 b}$

Antiderivative was successfully veriﬁed.

[In]

Int[Cosh[a + b*x]*Sinh[a + b*x],x]

[Out]

Sinh[a + b*x]^2/(2*b)

Rule 2564

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])

Rule 30

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

Rubi steps

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

Mathematica [B]  time = 0.0096956, size = 37, normalized size = 2.47 $\frac{1}{2} \left (\frac{\sinh (2 a) \sinh (2 b x)}{2 b}+\frac{\cosh (2 a) \cosh (2 b x)}{2 b}\right )$

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., size = 14, normalized size = 0.9 \begin{align*}{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{2\,b}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.0907, size = 18, normalized size = 1.2 \begin{align*} \frac{\cosh \left (b x + a\right )^{2}}{2 \, b} \end{align*}

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 = 1.99963, size = 58, normalized size = 3.87 \begin{align*} \frac{\cosh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{2}}{4 \, b} \end{align*}

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 = 0.22203, size = 19, normalized size = 1.27 \begin{align*} \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{align*}

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 [A]  time = 1.12654, size = 32, normalized size = 2.13 \begin{align*} \frac{e^{\left (2 \, b x + 2 \, a\right )} + e^{\left (-2 \, b x - 2 \, a\right )}}{8 \, b} \end{align*}

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) + e^(-2*b*x - 2*a))/b