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

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

[Out]

-6*b^2*d*exp(d*x+c)*cosh(b*x+a)/(9*b^4-10*b^2*d^2+d^4)-d*exp(d*x+c)*cosh(b*x+a)^3/(9*b^2-d^2)+6*b^3*exp(d*x+c)
*sinh(b*x+a)/(9*b^4-10*b^2*d^2+d^4)+3*b*exp(d*x+c)*cosh(b*x+a)^2*sinh(b*x+a)/(9*b^2-d^2)

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Rubi [A]  time = 0.06, antiderivative size = 144, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.48, size = 106, normalized size = 0.74 \[ \frac {e^{c+d x} \left (\left (d^3-b^2 d\right ) \cosh (3 (a+b x))+3 d \left (d^2-9 b^2\right ) \cosh (a+b x)+6 b \sinh (a+b x) \left (\left (b^2-d^2\right ) \cosh (2 (a+b x))+5 b^2-d^2\right )\right )}{4 \left (9 b^4-10 b^2 d^2+d^4\right )} \]

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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fricas [B]  time = 0.45, size = 381, normalized size = 2.65 \[ -\frac {3 \, {\left (b^{2} d - d^{3}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{2} - 3 \, {\left (b^{3} - b d^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{3} - 3 \, {\left (9 \, b^{3} - b d^{2} + 3 \, {\left (b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right ) + {\left ({\left (b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )^{3} + 3 \, {\left (9 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )\right )} \cosh \left (d x + c\right ) + {\left ({\left (b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )^{3} + 3 \, {\left (b^{2} d - d^{3}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - 3 \, {\left (b^{3} - b d^{2}\right )} \sinh \left (b x + a\right )^{3} + 3 \, {\left (9 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right ) - 3 \, {\left (9 \, b^{3} - b d^{2} + 3 \, {\left (b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )\right )} \sinh \left (d x + c\right )}{4 \, {\left ({\left (9 \, b^{4} - 10 \, b^{2} d^{2} + d^{4}\right )} \cosh \left (b x + a\right )^{4} - 2 \, {\left (9 \, b^{4} - 10 \, b^{2} d^{2} + d^{4}\right )} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} + {\left (9 \, b^{4} - 10 \, b^{2} d^{2} + d^{4}\right )} \sinh \left (b x + a\right )^{4}\right )}} \]

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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giac [A]  time = 0.13, size = 86, normalized size = 0.60 \[ \frac {e^{\left (3 \, b x + d x + 3 \, a + c\right )}}{8 \, {\left (3 \, b + d\right )}} + \frac {3 \, e^{\left (b x + d x + a + c\right )}}{8 \, {\left (b + d\right )}} - \frac {3 \, e^{\left (-b x + d x - a + c\right )}}{8 \, {\left (b - d\right )}} - \frac {e^{\left (-3 \, b x + d x - 3 \, a + c\right )}}{8 \, {\left (3 \, b - d\right )}} \]

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)

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maple [A]  time = 0.40, size = 178, normalized size = 1.24 \[ \frac {3 \sinh \left (a -c +\left (b -d \right ) x \right )}{8 \left (b -d \right )}+\frac {3 \sinh \left (a +c +\left (b +d \right ) x \right )}{8 \left (b +d \right )}+\frac {\sinh \left (3 a -c +\left (3 b -d \right ) x \right )}{24 b -8 d}+\frac {\sinh \left (3 a +c +\left (3 b +d \right ) x \right )}{24 b +8 d}-\frac {3 \cosh \left (a -c +\left (b -d \right ) x \right )}{8 \left (b -d \right )}+\frac {3 \cosh \left (a +c +\left (b +d \right ) x \right )}{8 \left (b +d \right )}-\frac {\cosh \left (3 a -c +\left (3 b -d \right ) x \right )}{8 \left (3 b -d \right )}+\frac {\cosh \left (3 a +c +\left (3 b +d \right ) x \right )}{24 b +8 d} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

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 >> 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)Is -d/b equal to -1?

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mupad [B]  time = 2.20, size = 125, normalized size = 0.87 \[ \frac {{\mathrm {e}}^{c+d\,x}\,\left (9\,b^3\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )-6\,b^3\,{\mathrm {sinh}\left (a+b\,x\right )}^3-7\,b^2\,d\,{\mathrm {cosh}\left (a+b\,x\right )}^3+6\,b^2\,d\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2-3\,b\,d^2\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )+d^3\,{\mathrm {cosh}\left (a+b\,x\right )}^3\right )}{9\,b^4-10\,b^2\,d^2+d^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a + b*x)^3*exp(c + d*x),x)

[Out]

(exp(c + d*x)*(d^3*cosh(a + b*x)^3 - 6*b^3*sinh(a + b*x)^3 - 7*b^2*d*cosh(a + b*x)^3 + 9*b^3*cosh(a + b*x)^2*s
inh(a + b*x) - 3*b*d^2*cosh(a + b*x)^2*sinh(a + b*x) + 6*b^2*d*cosh(a + b*x)*sinh(a + b*x)^2))/(9*b^4 + d^4 -
10*b^2*d^2)

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sympy [A]  time = 42.81, size = 1046, normalized size = 7.26 \[ \text {result too large to display} \]

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 + 5*exp(c)*exp(d*x)*sinh(a - d*x)**3/(8*d) + exp(c)*exp(d*x)*sinh(a - d*x)**2*cosh(a
- d*x)/(4*d) - exp(c)*exp(d*x)*sinh(a - d*x)*cosh(a - d*x)**2/d - 3*exp(c)*exp(d*x)*cosh(a - d*x)**3/(8*d), E
q(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 - exp(c)*e
xp(d*x)*sinh(a - d*x/3)**3/(8*d) + 3*exp(c)*exp(d*x)*sinh(a - d*x/3)*cosh(a - d*x/3)**2/(4*d) + 9*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)*sin
h(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)), (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)*sinh(a + d*x)*cosh(a + d*x)**2/8 + 3*x*exp(c)*exp(d*x)*cosh(a + d*x)**3/8 - 3
*exp(c)*exp(d*x)*sinh(a + d*x)**3/(8*d) + 3*exp(c)*exp(d*x)*sinh(a + d*x)*cosh(a + d*x)**2/(4*d) - exp(c)*exp(
d*x)*cosh(a + d*x)**3/(8*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)*ex
p(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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