### 3.953 $$\int e^{c+d x} \cosh ^2(a+b x) \sinh ^2(a+b x) \, dx$$

Optimal. Leaf size=83 $\frac{b e^{c+d x} \sinh (4 a+4 b x)}{2 \left (16 b^2-d^2\right )}-\frac{d e^{c+d x} \cosh (4 a+4 b x)}{8 \left (16 b^2-d^2\right )}-\frac{e^{c+d x}}{8 d}$

[Out]

-E^(c + d*x)/(8*d) - (d*E^(c + d*x)*Cosh[4*a + 4*b*x])/(8*(16*b^2 - d^2)) + (b*E^(c + d*x)*Sinh[4*a + 4*b*x])/
(2*(16*b^2 - d^2))

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Rubi [A]  time = 0.0792998, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.125, Rules used = {5509, 2194, 5475} $\frac{b e^{c+d x} \sinh (4 a+4 b x)}{2 \left (16 b^2-d^2\right )}-\frac{d e^{c+d x} \cosh (4 a+4 b x)}{8 \left (16 b^2-d^2\right )}-\frac{e^{c+d x}}{8 d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-E^(c + d*x)/(8*d) - (d*E^(c + d*x)*Cosh[4*a + 4*b*x])/(8*(16*b^2 - d^2)) + (b*E^(c + d*x)*Sinh[4*a + 4*b*x])/
(2*(16*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 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 5475

Int[Cosh[(d_.) + (e_.)*(x_)]*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbol] :> -Simp[(b*c*Log[F]*F^(c*(a + b*x))
*Cosh[d + e*x])/(e^2 - b^2*c^2*Log[F]^2), x] + Simp[(e*F^(c*(a + b*x))*Sinh[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 ^2(a+b x) \, dx &=\int \left (-\frac{1}{8} e^{c+d x}+\frac{1}{8} e^{c+d x} \cosh (4 a+4 b x)\right ) \, dx\\ &=-\left (\frac{1}{8} \int e^{c+d x} \, dx\right )+\frac{1}{8} \int e^{c+d x} \cosh (4 a+4 b x) \, dx\\ &=-\frac{e^{c+d x}}{8 d}-\frac{d e^{c+d x} \cosh (4 a+4 b x)}{8 \left (16 b^2-d^2\right )}+\frac{b e^{c+d x} \sinh (4 a+4 b x)}{2 \left (16 b^2-d^2\right )}\\ \end{align*}

Mathematica [A]  time = 0.355294, size = 58, normalized size = 0.7 $\frac{e^{c+d x} \left (d^2 \cosh (4 (a+b x))-4 b d \sinh (4 (a+b x))+16 b^2-d^2\right )}{8 \left (d^3-16 b^2 d\right )}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(E^(c + d*x)*(16*b^2 - d^2 + d^2*Cosh[4*(a + b*x)] - 4*b*d*Sinh[4*(a + b*x)]))/(8*(-16*b^2*d + d^3))

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Maple [A]  time = 0.012, size = 124, normalized size = 1.5 \begin{align*} -{\frac{\sinh \left ( dx+c \right ) }{8\,d}}+{\frac{\sinh \left ( \left ( 4\,b-d \right ) x+4\,a-c \right ) }{64\,b-16\,d}}+{\frac{\sinh \left ( \left ( 4\,b+d \right ) x+4\,a+c \right ) }{64\,b+16\,d}}-{\frac{\cosh \left ( dx+c \right ) }{8\,d}}-{\frac{\cosh \left ( \left ( 4\,b-d \right ) x+4\,a-c \right ) }{64\,b-16\,d}}+{\frac{\cosh \left ( \left ( 4\,b+d \right ) x+4\,a+c \right ) }{64\,b+16\,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)^2,x)

[Out]

-1/8*sinh(d*x+c)/d+1/16/(4*b-d)*sinh((4*b-d)*x+4*a-c)+1/16/(4*b+d)*sinh((4*b+d)*x+4*a+c)-1/8*cosh(d*x+c)/d-1/1
6*cosh((4*b-d)*x+4*a-c)/(4*b-d)+1/16*cosh((4*b+d)*x+4*a+c)/(4*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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.98238, size = 770, normalized size = 9.28 \begin{align*} \frac{16 \, b d \cosh \left (b x + a\right )^{3} \cosh \left (d x + c\right ) \sinh \left (b x + a\right ) - 6 \, d^{2} \cosh \left (b x + a\right )^{2} \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{2} + 16 \, b d \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{3} - d^{2} \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{4} -{\left (d^{2} \cosh \left (b x + a\right )^{4} + 16 \, b^{2} - d^{2}\right )} \cosh \left (d x + c\right ) -{\left (d^{2} \cosh \left (b x + a\right )^{4} - 16 \, b d \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right ) + 6 \, d^{2} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} - 16 \, b d \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + d^{2} \sinh \left (b x + a\right )^{4} + 16 \, b^{2} - d^{2}\right )} \sinh \left (d x + c\right )}{8 \,{\left ({\left (16 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )^{4} - 2 \,{\left (16 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} +{\left (16 \, b^{2} d - d^{3}\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)^2,x, algorithm="fricas")

[Out]

1/8*(16*b*d*cosh(b*x + a)^3*cosh(d*x + c)*sinh(b*x + a) - 6*d^2*cosh(b*x + a)^2*cosh(d*x + c)*sinh(b*x + a)^2
+ 16*b*d*cosh(b*x + a)*cosh(d*x + c)*sinh(b*x + a)^3 - d^2*cosh(d*x + c)*sinh(b*x + a)^4 - (d^2*cosh(b*x + a)^
4 + 16*b^2 - d^2)*cosh(d*x + c) - (d^2*cosh(b*x + a)^4 - 16*b*d*cosh(b*x + a)^3*sinh(b*x + a) + 6*d^2*cosh(b*x
+ a)^2*sinh(b*x + a)^2 - 16*b*d*cosh(b*x + a)*sinh(b*x + a)^3 + d^2*sinh(b*x + a)^4 + 16*b^2 - d^2)*sinh(d*x
+ c))/((16*b^2*d - d^3)*cosh(b*x + a)^4 - 2*(16*b^2*d - d^3)*cosh(b*x + a)^2*sinh(b*x + a)^2 + (16*b^2*d - d^3
)*sinh(b*x + a)^4)

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Sympy [A]  time = 150.693, size = 831, normalized size = 10.01 \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)**2,x)

[Out]

Piecewise((x*exp(c)*sinh(a)**2*cosh(a)**2, Eq(b, 0) & Eq(d, 0)), (x*exp(c)*exp(d*x)*sinh(a - d*x/4)**4/16 + x*
exp(c)*exp(d*x)*sinh(a - d*x/4)**3*cosh(a - d*x/4)/4 + 3*x*exp(c)*exp(d*x)*sinh(a - d*x/4)**2*cosh(a - d*x/4)*
*2/8 + x*exp(c)*exp(d*x)*sinh(a - d*x/4)*cosh(a - d*x/4)**3/4 + x*exp(c)*exp(d*x)*cosh(a - d*x/4)**4/16 + exp(
c)*exp(d*x)*sinh(a - d*x/4)**3*cosh(a - d*x/4)/(4*d) + exp(c)*exp(d*x)*sinh(a - d*x/4)**2*cosh(a - d*x/4)**2/d
+ exp(c)*exp(d*x)*sinh(a - d*x/4)*cosh(a - d*x/4)**3/(4*d), Eq(b, -d/4)), (x*exp(c)*exp(d*x)*sinh(a + d*x/4)*
*4/16 - x*exp(c)*exp(d*x)*sinh(a + d*x/4)**3*cosh(a + d*x/4)/4 + 3*x*exp(c)*exp(d*x)*sinh(a + d*x/4)**2*cosh(a
+ d*x/4)**2/8 - x*exp(c)*exp(d*x)*sinh(a + d*x/4)*cosh(a + d*x/4)**3/4 + x*exp(c)*exp(d*x)*cosh(a + d*x/4)**4
/16 - exp(c)*exp(d*x)*sinh(a + d*x/4)**4/(8*d) + exp(c)*exp(d*x)*sinh(a + d*x/4)**3*cosh(a + d*x/4)/(4*d) + ex
p(c)*exp(d*x)*sinh(a + d*x/4)**2*cosh(a + d*x/4)**2/(4*d) + exp(c)*exp(d*x)*sinh(a + d*x/4)*cosh(a + d*x/4)**3
/(4*d) - exp(c)*exp(d*x)*cosh(a + d*x/4)**4/(8*d), Eq(b, d/4)), ((-x*sinh(a + b*x)**4/8 + x*sinh(a + b*x)**2*c
osh(a + b*x)**2/4 - x*cosh(a + b*x)**4/8 + sinh(a + b*x)**3*cosh(a + b*x)/(8*b) + sinh(a + b*x)*cosh(a + b*x)*
*3/(8*b))*exp(c), Eq(d, 0)), (-2*b**2*exp(c)*exp(d*x)*sinh(a + b*x)**4/(16*b**2*d - d**3) + 4*b**2*exp(c)*exp(
d*x)*sinh(a + b*x)**2*cosh(a + b*x)**2/(16*b**2*d - d**3) - 2*b**2*exp(c)*exp(d*x)*cosh(a + b*x)**4/(16*b**2*d
- d**3) + 2*b*d*exp(c)*exp(d*x)*sinh(a + b*x)**3*cosh(a + b*x)/(16*b**2*d - d**3) + 2*b*d*exp(c)*exp(d*x)*sin
h(a + b*x)*cosh(a + b*x)**3/(16*b**2*d - d**3) - d**2*exp(c)*exp(d*x)*sinh(a + b*x)**2*cosh(a + b*x)**2/(16*b*
*2*d - d**3), True))

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Giac [A]  time = 1.18579, size = 78, normalized size = 0.94 \begin{align*} \frac{e^{\left (4 \, b x + d x + 4 \, a + c\right )}}{16 \,{\left (4 \, b + d\right )}} - \frac{e^{\left (-4 \, b x + d x - 4 \, a + c\right )}}{16 \,{\left (4 \, b - d\right )}} - \frac{e^{\left (d x + c\right )}}{8 \, d} \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)^2,x, algorithm="giac")

[Out]

1/16*e^(4*b*x + d*x + 4*a + c)/(4*b + d) - 1/16*e^(-4*b*x + d*x - 4*a + c)/(4*b - d) - 1/8*e^(d*x + c)/d