3.180 \(\int \cosh ^2(c+d x) \sinh (a+b x) \, dx\)

Optimal. Leaf size=62 \[ \frac{\cosh (a+x (b-2 d)-2 c)}{4 (b-2 d)}+\frac{\cosh (a+x (b+2 d)+2 c)}{4 (b+2 d)}+\frac{\cosh (a+b x)}{2 b} \]

[Out]

Cosh[a + b*x]/(2*b) + Cosh[a - 2*c + (b - 2*d)*x]/(4*(b - 2*d)) + Cosh[a + 2*c + (b + 2*d)*x]/(4*(b + 2*d))

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Rubi [A]  time = 0.0570709, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {5618, 2638} \[ \frac{\cosh (a+x (b-2 d)-2 c)}{4 (b-2 d)}+\frac{\cosh (a+x (b+2 d)+2 c)}{4 (b+2 d)}+\frac{\cosh (a+b x)}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[c + d*x]^2*Sinh[a + b*x],x]

[Out]

Cosh[a + b*x]/(2*b) + Cosh[a - 2*c + (b - 2*d)*x]/(4*(b - 2*d)) + Cosh[a + 2*c + (b + 2*d)*x]/(4*(b + 2*d))

Rule 5618

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]))

Rule 2638

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

Rubi steps

\begin{align*} \int \cosh ^2(c+d x) \sinh (a+b x) \, dx &=\int \left (\frac{1}{2} \sinh (a+b x)+\frac{1}{4} \sinh (a-2 c+(b-2 d) x)+\frac{1}{4} \sinh (a+2 c+(b+2 d) x)\right ) \, dx\\ &=\frac{1}{4} \int \sinh (a-2 c+(b-2 d) x) \, dx+\frac{1}{4} \int \sinh (a+2 c+(b+2 d) x) \, dx+\frac{1}{2} \int \sinh (a+b x) \, dx\\ &=\frac{\cosh (a+b x)}{2 b}+\frac{\cosh (a-2 c+(b-2 d) x)}{4 (b-2 d)}+\frac{\cosh (a+2 c+(b+2 d) x)}{4 (b+2 d)}\\ \end{align*}

Mathematica [A]  time = 0.677511, size = 69, normalized size = 1.11 \[ \frac{1}{4} \left (\frac{\cosh (a+b x-2 c-2 d x)}{b-2 d}+\frac{\cosh (a+b x+2 c+2 d x)}{b+2 d}+\frac{2 \sinh (a) \sinh (b x)}{b}+\frac{2 \cosh (a) \cosh (b x)}{b}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[c + d*x]^2*Sinh[a + b*x],x]

[Out]

((2*Cosh[a]*Cosh[b*x])/b + Cosh[a - 2*c + b*x - 2*d*x]/(b - 2*d) + Cosh[a + 2*c + b*x + 2*d*x]/(b + 2*d) + (2*
Sinh[a]*Sinh[b*x])/b)/4

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Maple [A]  time = 0.007, size = 57, normalized size = 0.9 \begin{align*}{\frac{\cosh \left ( bx+a \right ) }{2\,b}}+{\frac{\cosh \left ( a-2\,c+ \left ( b-2\,d \right ) x \right ) }{4\,b-8\,d}}+{\frac{\cosh \left ( a+2\,c+ \left ( b+2\,d \right ) x \right ) }{4\,b+8\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(d*x+c)^2*sinh(b*x+a),x)

[Out]

1/2*cosh(b*x+a)/b+1/4*cosh(a-2*c+(b-2*d)*x)/(b-2*d)+1/4*cosh(a+2*c+(b+2*d)*x)/(b+2*d)

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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(d*x+c)^2*sinh(b*x+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.85953, size = 304, normalized size = 4.9 \begin{align*} \frac{b^{2} \cosh \left (b x + a\right ) \cosh \left (d x + c\right )^{2} - 4 \, b d \cosh \left (d x + c\right ) \sinh \left (b x + a\right ) \sinh \left (d x + c\right ) + b^{2} \cosh \left (b x + a\right ) \sinh \left (d x + c\right )^{2} +{\left (b^{2} - 4 \, d^{2}\right )} \cosh \left (b x + a\right )}{2 \,{\left ({\left (b^{3} - 4 \, b d^{2}\right )} \cosh \left (b x + a\right )^{2} -{\left (b^{3} - 4 \, b d^{2}\right )} \sinh \left (b x + a\right )^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^2*sinh(b*x+a),x, algorithm="fricas")

[Out]

1/2*(b^2*cosh(b*x + a)*cosh(d*x + c)^2 - 4*b*d*cosh(d*x + c)*sinh(b*x + a)*sinh(d*x + c) + b^2*cosh(b*x + a)*s
inh(d*x + c)^2 + (b^2 - 4*d^2)*cosh(b*x + a))/((b^3 - 4*b*d^2)*cosh(b*x + a)^2 - (b^3 - 4*b*d^2)*sinh(b*x + a)
^2)

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Sympy [A]  time = 8.54137, size = 403, normalized size = 6.5 \begin{align*} \begin{cases} x \sinh{\left (a \right )} \cosh ^{2}{\left (c \right )} & \text{for}\: b = 0 \wedge d = 0 \\\left (- \frac{x \sinh ^{2}{\left (c + d x \right )}}{2} + \frac{x \cosh ^{2}{\left (c + d x \right )}}{2} + \frac{\sinh{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{2 d}\right ) \sinh{\left (a \right )} & \text{for}\: b = 0 \\\frac{x \sinh{\left (a - 2 d x \right )} \sinh ^{2}{\left (c + d x \right )}}{4} + \frac{x \sinh{\left (a - 2 d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} + \frac{x \sinh{\left (c + d x \right )} \cosh{\left (a - 2 d x \right )} \cosh{\left (c + d x \right )}}{2} + \frac{\sinh ^{2}{\left (c + d x \right )} \cosh{\left (a - 2 d x \right )}}{8 d} - \frac{3 \cosh{\left (a - 2 d x \right )} \cosh ^{2}{\left (c + d x \right )}}{8 d} & \text{for}\: b = - 2 d \\\frac{x \sinh{\left (a + 2 d x \right )} \sinh ^{2}{\left (c + d x \right )}}{4} + \frac{x \sinh{\left (a + 2 d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} - \frac{x \sinh{\left (c + d x \right )} \cosh{\left (a + 2 d x \right )} \cosh{\left (c + d x \right )}}{2} + \frac{3 \sinh{\left (a + 2 d x \right )} \sinh{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{4 d} - \frac{\sinh ^{2}{\left (c + d x \right )} \cosh{\left (a + 2 d x \right )}}{2 d} & \text{for}\: b = 2 d \\\frac{b^{2} \cosh{\left (a + b x \right )} \cosh ^{2}{\left (c + d x \right )}}{b^{3} - 4 b d^{2}} - \frac{2 b d \sinh{\left (a + b x \right )} \sinh{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{b^{3} - 4 b d^{2}} + \frac{2 d^{2} \sinh ^{2}{\left (c + d x \right )} \cosh{\left (a + b x \right )}}{b^{3} - 4 b d^{2}} - \frac{2 d^{2} \cosh{\left (a + b x \right )} \cosh ^{2}{\left (c + d x \right )}}{b^{3} - 4 b d^{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)**2*sinh(b*x+a),x)

[Out]

Piecewise((x*sinh(a)*cosh(c)**2, Eq(b, 0) & Eq(d, 0)), ((-x*sinh(c + d*x)**2/2 + x*cosh(c + d*x)**2/2 + sinh(c
+ d*x)*cosh(c + d*x)/(2*d))*sinh(a), Eq(b, 0)), (x*sinh(a - 2*d*x)*sinh(c + d*x)**2/4 + x*sinh(a - 2*d*x)*cos
h(c + d*x)**2/4 + x*sinh(c + d*x)*cosh(a - 2*d*x)*cosh(c + d*x)/2 + sinh(c + d*x)**2*cosh(a - 2*d*x)/(8*d) - 3
*cosh(a - 2*d*x)*cosh(c + d*x)**2/(8*d), Eq(b, -2*d)), (x*sinh(a + 2*d*x)*sinh(c + d*x)**2/4 + x*sinh(a + 2*d*
x)*cosh(c + d*x)**2/4 - x*sinh(c + d*x)*cosh(a + 2*d*x)*cosh(c + d*x)/2 + 3*sinh(a + 2*d*x)*sinh(c + d*x)*cosh
(c + d*x)/(4*d) - sinh(c + d*x)**2*cosh(a + 2*d*x)/(2*d), Eq(b, 2*d)), (b**2*cosh(a + b*x)*cosh(c + d*x)**2/(b
**3 - 4*b*d**2) - 2*b*d*sinh(a + b*x)*sinh(c + d*x)*cosh(c + d*x)/(b**3 - 4*b*d**2) + 2*d**2*sinh(c + d*x)**2*
cosh(a + b*x)/(b**3 - 4*b*d**2) - 2*d**2*cosh(a + b*x)*cosh(c + d*x)**2/(b**3 - 4*b*d**2), True))

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Giac [B]  time = 1.23527, size = 162, normalized size = 2.61 \begin{align*} \frac{e^{\left (b x + 2 \, d x + a + 2 \, c\right )}}{8 \,{\left (b + 2 \, d\right )}} + \frac{e^{\left (b x - 2 \, d x + a - 2 \, c\right )}}{8 \,{\left (b - 2 \, d\right )}} + \frac{e^{\left (b x + a\right )}}{4 \, b} + \frac{e^{\left (-b x + 2 \, d x - a + 2 \, c\right )}}{8 \,{\left (b - 2 \, d\right )}} + \frac{e^{\left (-b x - 2 \, d x - a - 2 \, c\right )}}{8 \,{\left (b + 2 \, d\right )}} + \frac{e^{\left (-b x - a\right )}}{4 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^2*sinh(b*x+a),x, algorithm="giac")

[Out]

1/8*e^(b*x + 2*d*x + a + 2*c)/(b + 2*d) + 1/8*e^(b*x - 2*d*x + a - 2*c)/(b - 2*d) + 1/4*e^(b*x + a)/b + 1/8*e^
(-b*x + 2*d*x - a + 2*c)/(b - 2*d) + 1/8*e^(-b*x - 2*d*x - a - 2*c)/(b + 2*d) + 1/4*e^(-b*x - a)/b