### 3.12 $$\int \cosh ^m(a+b x) \sinh (a+b x) \, dx$$

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

[Out]

Cosh[a + b*x]^(1 + m)/(b*(1 + m))

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Rubi [A]  time = 0.0267734, 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 = {2565, 30} $\frac{\cosh ^{m+1}(a+b x)}{b (m+1)}$

Antiderivative was successfully veriﬁed.

[In]

Int[Cosh[a + b*x]^m*Sinh[a + b*x],x]

[Out]

Cosh[a + b*x]^(1 + m)/(b*(1 + m))

Rule 2565

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(a*f)^(-1), Subst[
Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2]
&&  !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[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 ^m(a+b x) \sinh (a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int x^m \, dx,x,\cosh (a+b x)\right )}{b}\\ &=\frac{\cosh ^{1+m}(a+b x)}{b (1+m)}\\ \end{align*}

Mathematica [A]  time = 0.0087856, size = 19, normalized size = 1. $\frac{\cosh ^{m+1}(a+b x)}{b (m+1)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

Cosh[a + b*x]^(1 + m)/(b*(1 + m))

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Maple [A]  time = 0.012, size = 20, normalized size = 1.1 \begin{align*}{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{1+m}}{b \left ( 1+m \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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)^m*sinh(b*x+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.1134, size = 193, normalized size = 10.16 \begin{align*} \frac{\cosh \left (b x + a\right ) \cosh \left (m \log \left (\cosh \left (b x + a\right )\right )\right ) + \cosh \left (b x + a\right ) \sinh \left (m \log \left (\cosh \left (b x + a\right )\right )\right )}{{\left (b m + b\right )} \cosh \left (b x + a\right )^{2} -{\left (b m + 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)^m*sinh(b*x+a),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.78465, size = 49, normalized size = 2.58 \begin{align*} \begin{cases} \frac{x \sinh{\left (a \right )}}{\cosh{\left (a \right )}} & \text{for}\: b = 0 \wedge m = -1 \\x \sinh{\left (a \right )} \cosh ^{m}{\left (a \right )} & \text{for}\: b = 0 \\\frac{\log{\left (\cosh{\left (a + b x \right )} \right )}}{b} & \text{for}\: m = -1 \\\frac{\cosh{\left (a + b x \right )} \cosh ^{m}{\left (a + b x \right )}}{b m + b} & \text{otherwise} \end{cases} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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