3.954 \(\int e^{c+d x} \cosh ^2(a+b x) \sinh (a+b x) \, dx\)

Optimal. Leaf size=127 \[ -\frac{d e^{c+d x} \sinh (a+b x)}{4 \left (b^2-d^2\right )}-\frac{d e^{c+d x} \sinh (3 a+3 b x)}{4 \left (9 b^2-d^2\right )}+\frac{b e^{c+d x} \cosh (a+b x)}{4 \left (b^2-d^2\right )}+\frac{3 b e^{c+d x} \cosh (3 a+3 b x)}{4 \left (9 b^2-d^2\right )} \]

[Out]

(b*E^(c + d*x)*Cosh[a + b*x])/(4*(b^2 - d^2)) + (3*b*E^(c + d*x)*Cosh[3*a + 3*b*x])/(4*(9*b^2 - d^2)) - (d*E^(
c + d*x)*Sinh[a + b*x])/(4*(b^2 - d^2)) - (d*E^(c + d*x)*Sinh[3*a + 3*b*x])/(4*(9*b^2 - d^2))

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Rubi [A]  time = 0.0901656, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {5509, 5474} \[ -\frac{d e^{c+d x} \sinh (a+b x)}{4 \left (b^2-d^2\right )}-\frac{d e^{c+d x} \sinh (3 a+3 b x)}{4 \left (9 b^2-d^2\right )}+\frac{b e^{c+d x} \cosh (a+b x)}{4 \left (b^2-d^2\right )}+\frac{3 b e^{c+d x} \cosh (3 a+3 b x)}{4 \left (9 b^2-d^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[E^(c + d*x)*Cosh[a + b*x]^2*Sinh[a + b*x],x]

[Out]

(b*E^(c + d*x)*Cosh[a + b*x])/(4*(b^2 - d^2)) + (3*b*E^(c + d*x)*Cosh[3*a + 3*b*x])/(4*(9*b^2 - d^2)) - (d*E^(
c + d*x)*Sinh[a + b*x])/(4*(b^2 - d^2)) - (d*E^(c + d*x)*Sinh[3*a + 3*b*x])/(4*(9*b^2 - d^2))

Rule 5509

Int[Cosh[(f_.) + (g_.)*(x_)]^(n_.)*(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sinh[(d_.) + (e_.)*(x_)]^(m_.), x_Symbol]
:> Int[ExpandTrigReduce[F^(c*(a + b*x)), Sinh[d + e*x]^m*Cosh[f + g*x]^n, x], x] /; FreeQ[{F, a, b, c, d, e,
f, g}, x] && IGtQ[m, 0] && IGtQ[n, 0]

Rule 5474

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sinh[(d_.) + (e_.)*(x_)], x_Symbol] :> -Simp[(b*c*Log[F]*F^(c*(a + b*x))
*Sinh[d + e*x])/(e^2 - b^2*c^2*Log[F]^2), x] + Simp[(e*F^(c*(a + b*x))*Cosh[d + e*x])/(e^2 - b^2*c^2*Log[F]^2)
, x] /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2 - b^2*c^2*Log[F]^2, 0]

Rubi steps

\begin{align*} \int e^{c+d x} \cosh ^2(a+b x) \sinh (a+b x) \, dx &=\int \left (\frac{1}{4} e^{c+d x} \sinh (a+b x)+\frac{1}{4} e^{c+d x} \sinh (3 a+3 b x)\right ) \, dx\\ &=\frac{1}{4} \int e^{c+d x} \sinh (a+b x) \, dx+\frac{1}{4} \int e^{c+d x} \sinh (3 a+3 b x) \, dx\\ &=\frac{b e^{c+d x} \cosh (a+b x)}{4 \left (b^2-d^2\right )}+\frac{3 b e^{c+d x} \cosh (3 a+3 b x)}{4 \left (9 b^2-d^2\right )}-\frac{d e^{c+d x} \sinh (a+b x)}{4 \left (b^2-d^2\right )}-\frac{d e^{c+d x} \sinh (3 a+3 b x)}{4 \left (9 b^2-d^2\right )}\\ \end{align*}

Mathematica [A]  time = 0.524209, size = 80, normalized size = 0.63 \[ \frac{1}{4} e^{c+d x} \left (\frac{3 b \cosh (3 (a+b x))-d \sinh (3 (a+b x))}{9 b^2-d^2}+\frac{b \cosh (a+b x)-d \sinh (a+b x)}{(b-d) (b+d)}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(E^(c + d*x)*((b*Cosh[a + b*x] - d*Sinh[a + b*x])/((b - d)*(b + d)) + (3*b*Cosh[3*(a + b*x)] - d*Sinh[3*(a + b
*x)])/(9*b^2 - d^2)))/4

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Maple [A]  time = 0.007, size = 178, normalized size = 1.4 \begin{align*} -{\frac{\sinh \left ( a-c+ \left ( b-d \right ) x \right ) }{8\,b-8\,d}}+{\frac{\sinh \left ( a+c+ \left ( b+d \right ) x \right ) }{8\,b+8\,d}}-{\frac{\sinh \left ( 3\,a-c+ \left ( 3\,b-d \right ) x \right ) }{24\,b-8\,d}}+{\frac{\sinh \left ( 3\,a+c+ \left ( 3\,b+d \right ) x \right ) }{24\,b+8\,d}}+{\frac{\cosh \left ( a-c+ \left ( b-d \right ) x \right ) }{8\,b-8\,d}}+{\frac{\cosh \left ( a+c+ \left ( b+d \right ) x \right ) }{8\,b+8\,d}}+{\frac{\cosh \left ( 3\,a-c+ \left ( 3\,b-d \right ) x \right ) }{24\,b-8\,d}}+{\frac{\cosh \left ( 3\,a+c+ \left ( 3\,b+d \right ) x \right ) }{24\,b+8\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*sinh(a-c+(b-d)*x)/(b-d)+1/8*sinh(a+c+(b+d)*x)/(b+d)-1/8*sinh(3*a-c+(3*b-d)*x)/(3*b-d)+1/8*sinh(3*a+c+(3*b
+d)*x)/(3*b+d)+1/8*cosh(a-c+(b-d)*x)/(b-d)+1/8*cosh(a+c+(b+d)*x)/(b+d)+1/8*cosh(3*a-c+(3*b-d)*x)/(3*b-d)+1/8*c
osh(3*a+c+(3*b+d)*x)/(3*b+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(exp(d*x+c)*cosh(b*x+a)^2*sinh(b*x+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.13763, size = 886, normalized size = 6.98 \begin{align*} \frac{9 \,{\left (b^{3} - b d^{2}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{2} -{\left (b^{2} d - d^{3}\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{3} -{\left (9 \, b^{2} d - d^{3} + 3 \,{\left (b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right ) +{\left (3 \,{\left (b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{3} +{\left (9 \, b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )\right )} \cosh \left (d x + c\right ) +{\left (3 \,{\left (b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{3} + 9 \,{\left (b^{3} - b d^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} -{\left (b^{2} d - d^{3}\right )} \sinh \left (b x + a\right )^{3} +{\left (9 \, b^{3} - b d^{2}\right )} \cosh \left (b x + a\right ) -{\left (9 \, b^{2} d - d^{3} + 3 \,{\left (b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )\right )} \sinh \left (d x + c\right )}{4 \,{\left ({\left (9 \, b^{4} - 10 \, b^{2} d^{2} + d^{4}\right )} \cosh \left (b x + a\right )^{4} - 2 \,{\left (9 \, b^{4} - 10 \, b^{2} d^{2} + d^{4}\right )} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} +{\left (9 \, b^{4} - 10 \, b^{2} d^{2} + d^{4}\right )} \sinh \left (b x + a\right )^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(9*(b^3 - b*d^2)*cosh(b*x + a)*cosh(d*x + c)*sinh(b*x + a)^2 - (b^2*d - d^3)*cosh(d*x + c)*sinh(b*x + a)^3
- (9*b^2*d - d^3 + 3*(b^2*d - d^3)*cosh(b*x + a)^2)*cosh(d*x + c)*sinh(b*x + a) + (3*(b^3 - b*d^2)*cosh(b*x +
a)^3 + (9*b^3 - b*d^2)*cosh(b*x + a))*cosh(d*x + c) + (3*(b^3 - b*d^2)*cosh(b*x + a)^3 + 9*(b^3 - b*d^2)*cosh
(b*x + a)*sinh(b*x + a)^2 - (b^2*d - d^3)*sinh(b*x + a)^3 + (9*b^3 - b*d^2)*cosh(b*x + a) - (9*b^2*d - d^3 + 3
*(b^2*d - d^3)*cosh(b*x + a)^2)*sinh(b*x + a))*sinh(d*x + c))/((9*b^4 - 10*b^2*d^2 + d^4)*cosh(b*x + a)^4 - 2*
(9*b^4 - 10*b^2*d^2 + d^4)*cosh(b*x + a)^2*sinh(b*x + a)^2 + (9*b^4 - 10*b^2*d^2 + d^4)*sinh(b*x + a)^4)

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Sympy [A]  time = 86.4629, size = 994, normalized size = 7.83 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x*exp(c)*sinh(a)*cosh(a)**2, Eq(b, 0) & Eq(d, 0)), (-x*exp(c)*exp(d*x)*sinh(a - d*x)**3/8 - x*exp(c
)*exp(d*x)*sinh(a - d*x)**2*cosh(a - d*x)/8 + x*exp(c)*exp(d*x)*sinh(a - d*x)*cosh(a - d*x)**2/8 + x*exp(c)*ex
p(d*x)*cosh(a - d*x)**3/8 - exp(c)*exp(d*x)*sinh(a - d*x)**2*cosh(a - d*x)/(8*d) - exp(c)*exp(d*x)*sinh(a - d*
x)*cosh(a - d*x)**2/(8*d) - exp(c)*exp(d*x)*cosh(a - d*x)**3/(4*d), Eq(b, -d)), (x*exp(c)*exp(d*x)*sinh(a - d*
x/3)**3/8 + 3*x*exp(c)*exp(d*x)*sinh(a - d*x/3)**2*cosh(a - d*x/3)/8 + 3*x*exp(c)*exp(d*x)*sinh(a - d*x/3)*cos
h(a - d*x/3)**2/8 + x*exp(c)*exp(d*x)*cosh(a - d*x/3)**3/8 - 3*exp(c)*exp(d*x)*sinh(a - d*x/3)**3/(8*d) - 3*ex
p(c)*exp(d*x)*sinh(a - d*x/3)**2*cosh(a - d*x/3)/(4*d) - exp(c)*exp(d*x)*cosh(a - d*x/3)**3/(8*d), Eq(b, -d/3)
), (x*exp(c)*exp(d*x)*sinh(a + d*x/3)**3/8 - 3*x*exp(c)*exp(d*x)*sinh(a + d*x/3)**2*cosh(a + d*x/3)/8 + 3*x*ex
p(c)*exp(d*x)*sinh(a + d*x/3)*cosh(a + d*x/3)**2/8 - x*exp(c)*exp(d*x)*cosh(a + d*x/3)**3/8 - 3*exp(c)*exp(d*x
)*sinh(a + d*x/3)**2*cosh(a + d*x/3)/(8*d) + 9*exp(c)*exp(d*x)*sinh(a + d*x/3)*cosh(a + d*x/3)**2/(8*d) - exp(
c)*exp(d*x)*cosh(a + d*x/3)**3/(4*d), Eq(b, d/3)), (-x*exp(c)*exp(d*x)*sinh(a + d*x)**3/8 + x*exp(c)*exp(d*x)*
sinh(a + d*x)**2*cosh(a + d*x)/8 + x*exp(c)*exp(d*x)*sinh(a + d*x)*cosh(a + d*x)**2/8 - x*exp(c)*exp(d*x)*cosh
(a + d*x)**3/8 + exp(c)*exp(d*x)*sinh(a + d*x)**2*cosh(a + d*x)/(8*d) - exp(c)*exp(d*x)*sinh(a + d*x)*cosh(a +
d*x)**2/(8*d) + exp(c)*exp(d*x)*cosh(a + d*x)**3/(4*d), Eq(b, d)), (3*b**3*exp(c)*exp(d*x)*cosh(a + b*x)**3/(
9*b**4 - 10*b**2*d**2 + d**4) + 2*b**2*d*exp(c)*exp(d*x)*sinh(a + b*x)**3/(9*b**4 - 10*b**2*d**2 + d**4) - 3*b
**2*d*exp(c)*exp(d*x)*sinh(a + b*x)*cosh(a + b*x)**2/(9*b**4 - 10*b**2*d**2 + d**4) - 2*b*d**2*exp(c)*exp(d*x)
*sinh(a + b*x)**2*cosh(a + b*x)/(9*b**4 - 10*b**2*d**2 + d**4) - b*d**2*exp(c)*exp(d*x)*cosh(a + b*x)**3/(9*b*
*4 - 10*b**2*d**2 + d**4) + d**3*exp(c)*exp(d*x)*sinh(a + b*x)*cosh(a + b*x)**2/(9*b**4 - 10*b**2*d**2 + d**4)
, True))

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Giac [A]  time = 1.17696, size = 116, normalized size = 0.91 \begin{align*} \frac{e^{\left (3 \, b x + d x + 3 \, a + c\right )}}{8 \,{\left (3 \, b + d\right )}} + \frac{e^{\left (b x + d x + a + c\right )}}{8 \,{\left (b + d\right )}} + \frac{e^{\left (-b x + d x - a + c\right )}}{8 \,{\left (b - d\right )}} + \frac{e^{\left (-3 \, b x + d x - 3 \, a + c\right )}}{8 \,{\left (3 \, b - d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*e^(3*b*x + d*x + 3*a + c)/(3*b + d) + 1/8*e^(b*x + d*x + a + c)/(b + d) + 1/8*e^(-b*x + d*x - a + c)/(b -
d) + 1/8*e^(-3*b*x + d*x - 3*a + c)/(3*b - d)