3.2.79 \(\int \cosh (c+d x) \sinh (a+b x) \, dx\) [179]

Optimal. Leaf size=43 \[ \frac {\cosh (a-c+(b-d) x)}{2 (b-d)}+\frac {\cosh (a+c+(b+d) x)}{2 (b+d)} \]

[Out]

1/2*cosh(a-c+(b-d)*x)/(b-d)+1/2*cosh(a+c+(b+d)*x)/(b+d)

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Rubi [A]
time = 0.03, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {5737, 2718} \begin {gather*} \frac {\cosh (a+x (b-d)-c)}{2 (b-d)}+\frac {\cosh (a+x (b+d)+c)}{2 (b+d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cosh[c + d*x]*Sinh[a + b*x],x]

[Out]

Cosh[a - c + (b - d)*x]/(2*(b - d)) + Cosh[a + c + (b + d)*x]/(2*(b + d))

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 (c+d x) \sinh (a+b x) \, dx &=\int \left (\frac {1}{2} \sinh (a-c+(b-d) x)+\frac {1}{2} \sinh (a+c+(b+d) x)\right ) \, dx\\ &=\frac {1}{2} \int \sinh (a-c+(b-d) x) \, dx+\frac {1}{2} \int \sinh (a+c+(b+d) x) \, dx\\ &=\frac {\cosh (a-c+(b-d) x)}{2 (b-d)}+\frac {\cosh (a+c+(b+d) x)}{2 (b+d)}\\ \end {align*}

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Mathematica [A]
time = 0.15, size = 43, normalized size = 1.00 \begin {gather*} \frac {\cosh (a-c+(b-d) x)}{2 (b-d)}+\frac {\cosh (a+c+(b+d) x)}{2 (b+d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cosh[c + d*x]*Sinh[a + b*x],x]

[Out]

Cosh[a - c + (b - d)*x]/(2*(b - d)) + Cosh[a + c + (b + d)*x]/(2*(b + d))

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Maple [A]
time = 0.60, size = 40, normalized size = 0.93

method	result	size
default	\(\frac {\cosh \left (a -c +\left (b -d \right ) x \right )}{2 b -2 d}+\frac {\cosh \left (a +c +\left (b +d \right ) x \right )}{2 b +2 d}\)	\(40\)
risch	\(\frac {\left (b \,{\mathrm e}^{2 b x +2 a}-{\mathrm e}^{2 b x +2 a} d +b +d \right ) {\mathrm e}^{-b x +d x -a +c}}{4 \left (b +d \right ) \left (b -d \right )}+\frac {\left (b \,{\mathrm e}^{2 b x +2 a}+{\mathrm e}^{2 b x +2 a} d +b -d \right ) {\mathrm e}^{-b x -d x -a -c}}{4 \left (b +d \right ) \left (b -d \right )}\)	\(112\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(d*x+c)*sinh(b*x+a),x,method=_RETURNVERBOSE)

[Out]

1/2*cosh(a-c+(b-d)*x)/(b-d)+1/2*cosh(a+c+(b+d)*x)/(b+d)

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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(d*x+c)*sinh(b*x+a),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(-d/b>0)', see `assume?` for mo
re details)I

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Fricas [A]
time = 0.38, size = 71, normalized size = 1.65 \begin {gather*} \frac {b \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) - d \sinh \left (b x + a\right ) \sinh \left (d x + c\right )}{{\left (b^{2} - d^{2}\right )} \cosh \left (b x + a\right )^{2} - {\left (b^{2} - d^{2}\right )} \sinh \left (b x + a\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)*sinh(b*x+a),x, algorithm="fricas")

[Out]

(b*cosh(b*x + a)*cosh(d*x + c) - d*sinh(b*x + a)*sinh(d*x + c))/((b^2 - d^2)*cosh(b*x + a)^2 - (b^2 - d^2)*sin
h(b*x + a)^2)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 153 vs. \(2 (32) = 64\).
time = 0.36, size = 153, normalized size = 3.56 \begin {gather*} \begin {cases} x \sinh {\left (a \right )} \cosh {\left (c \right )} & \text {for}\: b = 0 \wedge d = 0 \\\frac {x \sinh {\left (a - d x \right )} \cosh {\left (c + d x \right )}}{2} + \frac {x \sinh {\left (c + d x \right )} \cosh {\left (a - d x \right )}}{2} - \frac {\cosh {\left (a - d x \right )} \cosh {\left (c + d x \right )}}{2 d} & \text {for}\: b = - d \\\frac {x \sinh {\left (a + d x \right )} \cosh {\left (c + d x \right )}}{2} - \frac {x \sinh {\left (c + d x \right )} \cosh {\left (a + d x \right )}}{2} + \frac {\cosh {\left (a + d x \right )} \cosh {\left (c + d x \right )}}{2 d} & \text {for}\: b = d \\\frac {b \cosh {\left (a + b x \right )} \cosh {\left (c + d x \right )}}{b^{2} - d^{2}} - \frac {d \sinh {\left (a + b x \right )} \sinh {\left (c + d x \right )}}{b^{2} - d^{2}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)*sinh(b*x+a),x)

[Out]

Piecewise((x*sinh(a)*cosh(c), Eq(b, 0) & Eq(d, 0)), (x*sinh(a - d*x)*cosh(c + d*x)/2 + x*sinh(c + d*x)*cosh(a
- d*x)/2 - cosh(a - d*x)*cosh(c + d*x)/(2*d), Eq(b, -d)), (x*sinh(a + d*x)*cosh(c + d*x)/2 - x*sinh(c + d*x)*c
osh(a + d*x)/2 + cosh(a + d*x)*cosh(c + d*x)/(2*d), Eq(b, d)), (b*cosh(a + b*x)*cosh(c + d*x)/(b**2 - d**2) -
d*sinh(a + b*x)*sinh(c + d*x)/(b**2 - d**2), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (39) = 78\).
time = 0.40, size = 85, normalized size = 1.98 \begin {gather*} \frac {e^{\left (b x + d x + a + c\right )}}{4 \, {\left (b + d\right )}} + \frac {e^{\left (b x - d x + a - c\right )}}{4 \, {\left (b - d\right )}} + \frac {e^{\left (-b x + d x - a + c\right )}}{4 \, {\left (b - d\right )}} + \frac {e^{\left (-b x - d x - a - c\right )}}{4 \, {\left (b + d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)*sinh(b*x+a),x, algorithm="giac")

[Out]

1/4*e^(b*x + d*x + a + c)/(b + d) + 1/4*e^(b*x - d*x + a - c)/(b - d) + 1/4*e^(-b*x + d*x - a + c)/(b - d) + 1
/4*e^(-b*x - d*x - a - c)/(b + d)

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Mupad [B]
time = 0.14, size = 42, normalized size = 0.98 \begin {gather*} \frac {b\,\mathrm {cosh}\left (a+b\,x\right )\,\mathrm {cosh}\left (c+d\,x\right )-d\,\mathrm {sinh}\left (a+b\,x\right )\,\mathrm {sinh}\left (c+d\,x\right )}{b^2-d^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(c + d*x)*sinh(a + b*x),x)

[Out]

(b*cosh(a + b*x)*cosh(c + d*x) - d*sinh(a + b*x)*sinh(c + d*x))/(b^2 - d^2)

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