### 3.9 $$\int \cosh (a+b x) \sinh ^n(a+b x) \, dx$$

Optimal. Leaf size=19 $\frac{\sinh ^{n+1}(a+b x)}{b (n+1)}$

[Out]

Sinh[a + b*x]^(1 + n)/(b*(1 + n))

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Rubi [A]  time = 0.0225391, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.133, Rules used = {2564, 30} $\frac{\sinh ^{n+1}(a+b x)}{b (n+1)}$

Antiderivative was successfully veriﬁed.

[In]

Int[Cosh[a + b*x]*Sinh[a + b*x]^n,x]

[Out]

Sinh[a + b*x]^(1 + n)/(b*(1 + n))

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 ^n(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int x^n \, dx,x,\sinh (a+b x)\right )}{b}\\ &=\frac{\sinh ^{1+n}(a+b x)}{b (1+n)}\\ \end{align*}

Mathematica [A]  time = 0.0096959, size = 19, normalized size = 1. $\frac{\sinh ^{n+1}(a+b x)}{b (n+1)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

Sinh[a + b*x]^(1 + n)/(b*(1 + n))

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Maple [A]  time = 0.014, size = 20, normalized size = 1.1 \begin{align*}{\frac{ \left ( \sinh \left ( bx+a \right ) \right ) ^{n+1}}{b \left ( n+1 \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.20277, size = 193, normalized size = 10.16 \begin{align*} \frac{\cosh \left (n \log \left (\sinh \left (b x + a\right )\right )\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right ) \sinh \left (n \log \left (\sinh \left (b x + a\right )\right )\right )}{{\left (b n + b\right )} \cosh \left (b x + a\right )^{2} -{\left (b n + b\right )} \sinh \left (b x + a\right )^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(cosh(n*log(sinh(b*x + a)))*sinh(b*x + a) + sinh(b*x + a)*sinh(n*log(sinh(b*x + a))))/((b*n + b)*cosh(b*x + a)
^2 - (b*n + b)*sinh(b*x + a)^2)

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Sympy [A]  time = 1.66918, size = 49, normalized size = 2.58 \begin{align*} \begin{cases} \frac{x \cosh{\left (a \right )}}{\sinh{\left (a \right )}} & \text{for}\: b = 0 \wedge n = -1 \\x \sinh ^{n}{\left (a \right )} \cosh{\left (a \right )} & \text{for}\: b = 0 \\\frac{\log{\left (\sinh{\left (a + b x \right )} \right )}}{b} & \text{for}\: n = -1 \\\frac{\sinh{\left (a + b x \right )} \sinh ^{n}{\left (a + b x \right )}}{b n + b} & \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)**n,x)

[Out]

Piecewise((x*cosh(a)/sinh(a), Eq(b, 0) & Eq(n, -1)), (x*sinh(a)**n*cosh(a), Eq(b, 0)), (log(sinh(a + b*x))/b,
Eq(n, -1)), (sinh(a + b*x)*sinh(a + b*x)**n/(b*n + b), True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sinh \left (b x + a\right )^{n} \cosh \left (b x + a\right )\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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