### 3.962 $$\int e^{c+d x} \cosh ^3(a+b x) \, dx$$

Optimal. Leaf size=144 $\frac{6 b^3 e^{c+d x} \sinh (a+b x)}{-10 b^2 d^2+9 b^4+d^4}-\frac{d e^{c+d x} \cosh ^3(a+b x)}{9 b^2-d^2}-\frac{6 b^2 d e^{c+d x} \cosh (a+b x)}{-10 b^2 d^2+9 b^4+d^4}+\frac{3 b e^{c+d x} \sinh (a+b x) \cosh ^2(a+b x)}{9 b^2-d^2}$

[Out]

(-6*b^2*d*E^(c + d*x)*Cosh[a + b*x])/(9*b^4 - 10*b^2*d^2 + d^4) - (d*E^(c + d*x)*Cosh[a + b*x]^3)/(9*b^2 - d^2
) + (6*b^3*E^(c + d*x)*Sinh[a + b*x])/(9*b^4 - 10*b^2*d^2 + d^4) + (3*b*E^(c + d*x)*Cosh[a + b*x]^2*Sinh[a + b
*x])/(9*b^2 - d^2)

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Rubi [A]  time = 0.0566672, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.125, Rules used = {5477, 5475} $\frac{6 b^3 e^{c+d x} \sinh (a+b x)}{-10 b^2 d^2+9 b^4+d^4}-\frac{d e^{c+d x} \cosh ^3(a+b x)}{9 b^2-d^2}-\frac{6 b^2 d e^{c+d x} \cosh (a+b x)}{-10 b^2 d^2+9 b^4+d^4}+\frac{3 b e^{c+d x} \sinh (a+b x) \cosh ^2(a+b x)}{9 b^2-d^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[E^(c + d*x)*Cosh[a + b*x]^3,x]

[Out]

(-6*b^2*d*E^(c + d*x)*Cosh[a + b*x])/(9*b^4 - 10*b^2*d^2 + d^4) - (d*E^(c + d*x)*Cosh[a + b*x]^3)/(9*b^2 - d^2
) + (6*b^3*E^(c + d*x)*Sinh[a + b*x])/(9*b^4 - 10*b^2*d^2 + d^4) + (3*b*E^(c + d*x)*Cosh[a + b*x]^2*Sinh[a + b
*x])/(9*b^2 - d^2)

Rule 5477

Int[Cosh[(d_.) + (e_.)*(x_)]^(n_)*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbol] :> -Simp[(b*c*Log[F]*F^(c*(a +
b*x))*Cosh[d + e*x]^n)/(e^2*n^2 - b^2*c^2*Log[F]^2), x] + (Dist[(n*(n - 1)*e^2)/(e^2*n^2 - b^2*c^2*Log[F]^2),
Int[F^(c*(a + b*x))*Cosh[d + e*x]^(n - 2), x], x] + Simp[(e*n*F^(c*(a + b*x))*Sinh[d + e*x]*Cosh[d + e*x]^(n -
1))/(e^2*n^2 - b^2*c^2*Log[F]^2), x]) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*n^2 - b^2*c^2*Log[F]^2, 0] &
& GtQ[n, 1]

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 ^3(a+b x) \, dx &=-\frac{d e^{c+d x} \cosh ^3(a+b x)}{9 b^2-d^2}+\frac{3 b e^{c+d x} \cosh ^2(a+b x) \sinh (a+b x)}{9 b^2-d^2}+\frac{\left (6 b^2\right ) \int e^{c+d x} \cosh (a+b x) \, dx}{9 b^2-d^2}\\ &=-\frac{6 b^2 d e^{c+d x} \cosh (a+b x)}{9 b^4-10 b^2 d^2+d^4}-\frac{d e^{c+d x} \cosh ^3(a+b x)}{9 b^2-d^2}+\frac{6 b^3 e^{c+d x} \sinh (a+b x)}{9 b^4-10 b^2 d^2+d^4}+\frac{3 b e^{c+d x} \cosh ^2(a+b x) \sinh (a+b x)}{9 b^2-d^2}\\ \end{align*}

Mathematica [A]  time = 0.444232, size = 106, normalized size = 0.74 $\frac{e^{c+d x} \left (3 d \left (d^2-9 b^2\right ) \cosh (a+b x)+\left (d^3-b^2 d\right ) \cosh (3 (a+b x))+6 b \sinh (a+b x) \left (\left (b^2-d^2\right ) \cosh (2 (a+b x))+5 b^2-d^2\right )\right )}{4 \left (-10 b^2 d^2+9 b^4+d^4\right )}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[E^(c + d*x)*Cosh[a + b*x]^3,x]

[Out]

(E^(c + d*x)*(3*d*(-9*b^2 + d^2)*Cosh[a + b*x] + (-(b^2*d) + d^3)*Cosh[3*(a + b*x)] + 6*b*(5*b^2 - d^2 + (b^2
- d^2)*Cosh[2*(a + b*x)])*Sinh[a + b*x]))/(4*(9*b^4 - 10*b^2*d^2 + d^4))

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Maple [A]  time = 0.006, size = 178, normalized size = 1.2 \begin{align*}{\frac{3\,\sinh \left ( a-c+ \left ( b-d \right ) x \right ) }{8\,b-8\,d}}+{\frac{3\,\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{3\,\cosh \left ( a-c+ \left ( b-d \right ) x \right ) }{8\,b-8\,d}}+{\frac{3\,\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)^3,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.12696, size = 898, normalized size = 6.24 \begin{align*} -\frac{3 \,{\left (b^{2} d - d^{3}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{2} - 3 \,{\left (b^{3} - b d^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{3} - 3 \,{\left (9 \, b^{3} - b d^{2} + 3 \,{\left (b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right ) +{\left ({\left (b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )^{3} + 3 \,{\left (9 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )\right )} \cosh \left (d x + c\right ) +{\left ({\left (b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )^{3} + 3 \,{\left (b^{2} d - d^{3}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - 3 \,{\left (b^{3} - b d^{2}\right )} \sinh \left (b x + a\right )^{3} + 3 \,{\left (9 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right ) - 3 \,{\left (9 \, b^{3} - b d^{2} + 3 \,{\left (b^{3} - b d^{2}\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)^3,x, algorithm="fricas")

[Out]

-1/4*(3*(b^2*d - d^3)*cosh(b*x + a)*cosh(d*x + c)*sinh(b*x + a)^2 - 3*(b^3 - b*d^2)*cosh(d*x + c)*sinh(b*x + a
)^3 - 3*(9*b^3 - b*d^2 + 3*(b^3 - b*d^2)*cosh(b*x + a)^2)*cosh(d*x + c)*sinh(b*x + a) + ((b^2*d - d^3)*cosh(b*
x + a)^3 + 3*(9*b^2*d - d^3)*cosh(b*x + a))*cosh(d*x + c) + ((b^2*d - d^3)*cosh(b*x + a)^3 + 3*(b^2*d - d^3)*c
osh(b*x + a)*sinh(b*x + a)^2 - 3*(b^3 - b*d^2)*sinh(b*x + a)^3 + 3*(9*b^2*d - d^3)*cosh(b*x + a) - 3*(9*b^3 -
b*d^2 + 3*(b^3 - b*d^2)*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 = 82.6233, size = 1114, normalized size = 7.74 \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)**3,x)

[Out]

Piecewise((x*exp(c)*cosh(a)**3, Eq(b, 0) & Eq(d, 0)), (-3*x*exp(c)*exp(d*x)*sinh(a - d*x)**3/8 - 3*x*exp(c)*ex
p(d*x)*sinh(a - d*x)**2*cosh(a - d*x)/8 + 3*x*exp(c)*exp(d*x)*sinh(a - d*x)*cosh(a - d*x)**2/8 + 3*x*exp(c)*ex
p(d*x)*cosh(a - d*x)**3/8 + 2*exp(c)*exp(d*x)*sinh(a - d*x)**3/(3*d) + 7*exp(c)*exp(d*x)*sinh(a - d*x)**2*cosh
(a - d*x)/(24*d) - 25*exp(c)*exp(d*x)*sinh(a - d*x)*cosh(a - d*x)**2/(24*d) - 5*exp(c)*exp(d*x)*cosh(a - d*x)*
*3/(12*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)*cosh(a - d*x/3)**2/8 + x*exp(c)*exp(d*x)*cosh(a - d*x/3)**3/
8 - 11*exp(c)*exp(d*x)*sinh(a - d*x/3)**3/(8*d) - 15*exp(c)*exp(d*x)*sinh(a - d*x/3)**2*cosh(a - d*x/3)/(4*d)
- 3*exp(c)*exp(d*x)*sinh(a - d*x/3)*cosh(a - d*x/3)**2/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
*exp(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 - 2*exp(c)*exp(
d*x)*sinh(a + d*x/3)**3/d + 51*exp(c)*exp(d*x)*sinh(a + d*x/3)**2*cosh(a + d*x/3)/(8*d) - 57*exp(c)*exp(d*x)*s
inh(a + d*x/3)*cosh(a + d*x/3)**2/(8*d) + 13*exp(c)*exp(d*x)*cosh(a + d*x/3)**3/(4*d), Eq(b, d/3)), (3*x*exp(c
)*exp(d*x)*sinh(a + d*x)**3/8 - 3*x*exp(c)*exp(d*x)*sinh(a + d*x)**2*cosh(a + d*x)/8 - 3*x*exp(c)*exp(d*x)*sin
h(a + d*x)*cosh(a + d*x)**2/8 + 3*x*exp(c)*exp(d*x)*cosh(a + d*x)**3/8 - 2*exp(c)*exp(d*x)*sinh(a + d*x)**3/(3
*d) + 7*exp(c)*exp(d*x)*sinh(a + d*x)**2*cosh(a + d*x)/(24*d) + 25*exp(c)*exp(d*x)*sinh(a + d*x)*cosh(a + d*x)
**2/(24*d) - 5*exp(c)*exp(d*x)*cosh(a + d*x)**3/(12*d), Eq(b, d)), (-6*b**3*exp(c)*exp(d*x)*sinh(a + b*x)**3/(
9*b**4 - 10*b**2*d**2 + d**4) + 9*b**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) + 6*b**2*d*exp(c)*exp(d*x)*sinh(a + b*x)**2*cosh(a + b*x)/(9*b**4 - 10*b**2*d**2 + d**4) - 7*b**2*d*exp
(c)*exp(d*x)*cosh(a + b*x)**3/(9*b**4 - 10*b**2*d**2 + d**4) - 3*b*d**2*exp(c)*exp(d*x)*sinh(a + b*x)*cosh(a +
b*x)**2/(9*b**4 - 10*b**2*d**2 + d**4) + d**3*exp(c)*exp(d*x)*cosh(a + b*x)**3/(9*b**4 - 10*b**2*d**2 + d**4)
, True))

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Giac [A]  time = 1.17066, size = 116, normalized size = 0.81 \begin{align*} \frac{e^{\left (3 \, b x + d x + 3 \, a + c\right )}}{8 \,{\left (3 \, b + d\right )}} + \frac{3 \, e^{\left (b x + d x + a + c\right )}}{8 \,{\left (b + d\right )}} - \frac{3 \, 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)^3,x, algorithm="giac")

[Out]

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