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

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Rubi [A]  time = 0.02, antiderivative size = 19, normalized size of antiderivative = 1.00, 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 30

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

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])

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*}

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Mathematica [A]  time = 0.01, size = 19, normalized size = 1.00 \[ \frac {\sinh ^{n+1}(a+b x)}{b (n+1)} \]

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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fricas [B]  time = 0.50, size = 68, normalized size = 3.58 \[ \frac {\cosh \left (n \log \left (\sinh \left (b x + a\right )\right )\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right ) \sinh \left (n \log \left (\sinh \left (b x + a\right )\right )\right )}{{\left (b n + b\right )} \cosh \left (b x + a\right )^{2} - {\left (b n + b\right )} \sinh \left (b x + a\right )^{2}} \]

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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giac [B]  time = 0.29, size = 96, normalized size = 5.05 \[ \frac {e^{\left (3 \, b x + n \log \left (\frac {1}{2} \, {\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-b x - a\right )}\right ) + 3 \, a\right )} - e^{\left (b x + n \log \left (\frac {1}{2} \, {\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-b x - a\right )}\right ) + a\right )}}{2 \, {\left (b n e^{\left (2 \, b x + 2 \, a\right )} + b e^{\left (2 \, b x + 2 \, a\right )}\right )}} \]

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]

1/2*(e^(3*b*x + n*log(1/2*(e^(2*b*x + 2*a) - 1)*e^(-b*x - a)) + 3*a) - e^(b*x + n*log(1/2*(e^(2*b*x + 2*a) - 1
)*e^(-b*x - a)) + a))/(b*n*e^(2*b*x + 2*a) + b*e^(2*b*x + 2*a))

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maple [A]  time = 0.12, size = 20, normalized size = 1.05 \[ \frac {\sinh ^{n +1}\left (b x +a \right )}{b \left (n +1\right )} \]

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 [A]  time = 0.38, size = 19, normalized size = 1.00 \[ \frac {\sinh \left (b x + a\right )^{n + 1}}{b {\left (n + 1\right )}} \]

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]

sinh(b*x + a)^(n + 1)/(b*(n + 1))

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mupad [B]  time = 1.55, size = 19, normalized size = 1.00 \[ \frac {{\mathrm {sinh}\left (a+b\,x\right )}^{n+1}}{b\,\left (n+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a + b*x)*sinh(a + b*x)^n,x)

[Out]

sinh(a + b*x)^(n + 1)/(b*(n + 1))

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sympy [A]  time = 1.25, size = 49, normalized size = 2.58 \[ \begin {cases} \frac {x \cosh {\relax (a )}}{\sinh {\relax (a )}} & \text {for}\: b = 0 \wedge n = -1 \\x \sinh ^{n}{\relax (a )} \cosh {\relax (a )} & \text {for}\: b = 0 \\\frac {\log {\left (\sinh {\left (a + b x \right )} \right )}}{b} & \text {for}\: n = -1 \\\frac {\sinh {\left (a + b x \right )} \sinh ^{n}{\left (a + b x \right )}}{b n + b} & \text {otherwise} \end {cases} \]

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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