3.2.99 \(\int \cosh (m x) \sinh (x) \, dx\) [199]

Optimal. Leaf size=35 \[ \frac {\cosh ((1-m) x)}{2 (1-m)}+\frac {\cosh ((1+m) x)}{2 (1+m)} \]

[Out]

1/2*cosh((1-m)*x)/(1-m)+1/2*cosh((1+m)*x)/(1+m)

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Rubi [A]
time = 0.03, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5737, 2718} \begin {gather*} \frac {\cosh ((1-m) x)}{2 (1-m)}+\frac {\cosh ((m+1) x)}{2 (m+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cosh[m*x]*Sinh[x],x]

[Out]

Cosh[(1 - m)*x]/(2*(1 - m)) + Cosh[(1 + m)*x]/(2*(1 + m))

Rule 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 5737

Int[Cosh[w_]^(q_.)*Sinh[v_]^(p_.), x_Symbol] :> Int[ExpandTrigReduce[Sinh[v]^p*Cosh[w]^q, x], x] /; IGtQ[p, 0]
&& IGtQ[q, 0] && ((PolynomialQ[v, x] && PolynomialQ[w, x]) || (BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/
w], x]))

Rubi steps

\begin {align*} \int \cosh (m x) \sinh (x) \, dx &=\int \left (\frac {1}{2} \sinh ((1-m) x)+\frac {1}{2} \sinh ((1+m) x)\right ) \, dx\\ &=\frac {1}{2} \int \sinh ((1-m) x) \, dx+\frac {1}{2} \int \sinh ((1+m) x) \, dx\\ &=\frac {\cosh ((1-m) x)}{2 (1-m)}+\frac {\cosh ((1+m) x)}{2 (1+m)}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 25, normalized size = 0.71 \begin {gather*} \frac {-\cosh (x) \cosh (m x)+m \sinh (x) \sinh (m x)}{-1+m^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cosh[m*x]*Sinh[x],x]

[Out]

(-(Cosh[x]*Cosh[m*x]) + m*Sinh[x]*Sinh[m*x])/(-1 + m^2)

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Maple [A]
time = 0.37, size = 28, normalized size = 0.80

method	result	size
default	\(-\frac {\cosh \left (\left (-1+m \right ) x \right )}{2 \left (-1+m \right )}+\frac {\cosh \left (\left (1+m \right ) x \right )}{2+2 m}\)	\(28\)
risch	\(\frac {\left (m \,{\mathrm e}^{2 x}-{\mathrm e}^{2 x}-m -1\right ) {\mathrm e}^{\left (-1+m \right ) x}}{4 \left (1+m \right ) \left (-1+m \right )}-\frac {\left (m \,{\mathrm e}^{2 x}+{\mathrm e}^{2 x}-m +1\right ) {\mathrm e}^{-\left (1+m \right ) x}}{4 \left (1+m \right ) \left (-1+m \right )}\)	\(71\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(m*x)*sinh(x),x,method=_RETURNVERBOSE)

[Out]

-1/2/(-1+m)*cosh((-1+m)*x)+1/2*cosh((1+m)*x)/(1+m)

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(m*x)*sinh(x),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(m-2>0)', see `assume?` for mor
e details)Is

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Fricas [A]
time = 0.34, size = 42, normalized size = 1.20 \begin {gather*} \frac {m \sinh \left (m x\right ) \sinh \left (x\right ) - \cosh \left (m x\right ) \cosh \left (x\right )}{{\left (m^{2} - 1\right )} \cosh \left (x\right )^{2} - {\left (m^{2} - 1\right )} \sinh \left (x\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(m*x)*sinh(x),x, algorithm="fricas")

[Out]

(m*sinh(m*x)*sinh(x) - cosh(m*x)*cosh(x))/((m^2 - 1)*cosh(x)^2 - (m^2 - 1)*sinh(x)^2)

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Sympy [A]
time = 0.19, size = 37, normalized size = 1.06 \begin {gather*} \begin {cases} \frac {\cosh ^{2}{\left (x \right )}}{2} & \text {for}\: m = -1 \vee m = 1 \\\frac {m \sinh {\left (x \right )} \sinh {\left (m x \right )}}{m^{2} - 1} - \frac {\cosh {\left (x \right )} \cosh {\left (m x \right )}}{m^{2} - 1} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(m*x)*sinh(x),x)

[Out]

Piecewise((cosh(x)**2/2, Eq(m, -1) | Eq(m, 1)), (m*sinh(x)*sinh(m*x)/(m**2 - 1) - cosh(x)*cosh(m*x)/(m**2 - 1)
, True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (28) = 56\).
time = 0.39, size = 59, normalized size = 1.69 \begin {gather*} \frac {e^{\left (m x + x\right )}}{4 \, {\left (m + 1\right )}} - \frac {e^{\left (m x - x\right )}}{4 \, {\left (m - 1\right )}} - \frac {e^{\left (-m x + x\right )}}{4 \, {\left (m - 1\right )}} + \frac {e^{\left (-m x - x\right )}}{4 \, {\left (m + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(m*x)*sinh(x),x, algorithm="giac")

[Out]

1/4*e^(m*x + x)/(m + 1) - 1/4*e^(m*x - x)/(m - 1) - 1/4*e^(-m*x + x)/(m - 1) + 1/4*e^(-m*x - x)/(m + 1)

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Mupad [B]
time = 1.44, size = 26, normalized size = 0.74 \begin {gather*} -\frac {\mathrm {cosh}\left (m\,x\right )\,\mathrm {cosh}\left (x\right )-m\,\mathrm {sinh}\left (m\,x\right )\,\mathrm {sinh}\left (x\right )}{m^2-1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(m*x)*sinh(x),x)

[Out]

-(cosh(m*x)*cosh(x) - m*sinh(m*x)*sinh(x))/(m^2 - 1)

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