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

________________________________________________________________________________________

Rubi [A]  time = 0.03, 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 = {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 30

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

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])

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 = 0.01, size = 19, normalized size = 1.00 \[ \frac {\cosh ^{m+1}(a+b x)}{b (m+1)} \]

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))

________________________________________________________________________________________

fricas [B]  time = 0.61, size = 68, normalized size = 3.58 \[ \frac {\cosh \left (b x + a\right ) \cosh \left (m \log \left (\cosh \left (b x + a\right )\right )\right ) + \cosh \left (b x + a\right ) \sinh \left (m \log \left (\cosh \left (b x + a\right )\right )\right )}{{\left (b m + b\right )} \cosh \left (b x + a\right )^{2} - {\left (b m + b\right )} \sinh \left (b x + a\right )^{2}} \]

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)

________________________________________________________________________________________

giac [B]  time = 0.21, size = 94, normalized size = 4.95 \[ \frac {e^{\left (3 \, b x + m \log \left (\frac {1}{2} \, {\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )} e^{\left (-b x - a\right )}\right ) + 3 \, a\right )} + e^{\left (b x + m \log \left (\frac {1}{2} \, {\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )} e^{\left (-b x - a\right )}\right ) + a\right )}}{2 \, {\left (b m e^{\left (2 \, b x + 2 \, a\right )} + b e^{\left (2 \, b x + 2 \, a\right )}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.07, size = 20, normalized size = 1.05 \[ \frac {\cosh ^{1+m}\left (b x +a \right )}{b \left (1+m \right )} \]

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)

________________________________________________________________________________________

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

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]

cosh(b*x + a)^(m + 1)/(b*(m + 1))

________________________________________________________________________________________

mupad [B]  time = 1.58, size = 19, normalized size = 1.00 \[ \frac {{\mathrm {cosh}\left (a+b\,x\right )}^{m+1}}{b\,\left (m+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a + b*x)^m*sinh(a + b*x),x)

[Out]

cosh(a + b*x)^(m + 1)/(b*(m + 1))

________________________________________________________________________________________

sympy [A]  time = 1.26, size = 49, normalized size = 2.58 \[ \begin {cases} \frac {x \sinh {\relax (a )}}{\cosh {\relax (a )}} & \text {for}\: b = 0 \wedge m = -1 \\x \sinh {\relax (a )} \cosh ^{m}{\relax (a )} & \text {for}\: b = 0 \\\frac {\log {\left (\cosh {\left (a + b x \right )} \right )}}{b} & \text {for}\: m = -1 \\\frac {\cosh {\left (a + b x \right )} \cosh ^{m}{\left (a + b x \right )}}{b m + b} & \text {otherwise} \end {cases} \]

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))

________________________________________________________________________________________