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

Optimal. Leaf size=95 \[ -\frac {d e^{c+d x} \cosh ^2(a+b x)}{4 b^2-d^2}+\frac {2 b e^{c+d x} \sinh (a+b x) \cosh (a+b x)}{4 b^2-d^2}+\frac {2 b^2 e^{c+d x}}{d \left (4 b^2-d^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 95, 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, 2194} \[ -\frac {d e^{c+d x} \cosh ^2(a+b x)}{4 b^2-d^2}+\frac {2 b e^{c+d x} \sinh (a+b x) \cosh (a+b x)}{4 b^2-d^2}+\frac {2 b^2 e^{c+d x}}{d \left (4 b^2-d^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[E^(c + d*x)*Cosh[a + b*x]^2,x]

[Out]

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

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

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Mathematica [A]  time = 0.15, size = 55, normalized size = 0.58 \[ \frac {e^{c+d x} \left (d^2 \cosh (2 (a+b x))-2 b d \sinh (2 (a+b x))-4 b^2+d^2\right )}{2 d^3-8 b^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.45, size = 176, normalized size = 1.85 \[ \frac {4 \, b d \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) \sinh \left (b x + a\right ) - d^{2} \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{2} - {\left (d^{2} \cosh \left (b x + a\right )^{2} - 4 \, b^{2} + d^{2}\right )} \cosh \left (d x + c\right ) - {\left (d^{2} \cosh \left (b x + a\right )^{2} - 4 \, b d \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + d^{2} \sinh \left (b x + a\right )^{2} - 4 \, b^{2} + d^{2}\right )} \sinh \left (d x + c\right )}{2 \, {\left ({\left (4 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )^{2} - {\left (4 \, b^{2} d - d^{3}\right )} \sinh \left (b x + a\right )^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.12, size = 58, normalized size = 0.61 \[ \frac {e^{\left (2 \, b x + d x + 2 \, a + c\right )}}{4 \, {\left (2 \, b + d\right )}} - \frac {e^{\left (-2 \, b x + d x - 2 \, a + c\right )}}{4 \, {\left (2 \, b - d\right )}} + \frac {e^{\left (d x + c\right )}}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.26, size = 124, normalized size = 1.31 \[ \frac {\sinh \left (d x +c \right )}{2 d}+\frac {\sinh \left (2 a -c +\left (2 b -d \right ) x \right )}{8 b -4 d}+\frac {\sinh \left (2 a +c +\left (2 b +d \right ) x \right )}{8 b +4 d}+\frac {\cosh \left (d x +c \right )}{2 d}-\frac {\cosh \left (2 a -c +\left (2 b -d \right ) x \right )}{4 \left (2 b -d \right )}+\frac {\cosh \left (2 a +c +\left (2 b +d \right ) x \right )}{8 b +4 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sinh(d*x+c)/d+1/4*sinh(2*a-c+(2*b-d)*x)/(2*b-d)+1/4*sinh(2*a+c+(2*b+d)*x)/(2*b+d)+1/2*cosh(d*x+c)/d-1/4*co
sh(2*a-c+(2*b-d)*x)/(2*b-d)+1/4*cosh(2*a+c+(2*b+d)*x)/(2*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)^2,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(1-d/b>0)', see `assume?` for m
ore details)Is 1-d/b equal to -1?

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mupad [B]  time = 0.27, size = 68, normalized size = 0.72 \[ \frac {2\,b^2\,{\mathrm {e}}^{c+d\,x}-d^2\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,{\mathrm {e}}^{c+d\,x}+2\,b\,d\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {e}}^{c+d\,x}\,\mathrm {sinh}\left (a+b\,x\right )}{4\,b^2\,d-d^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a + b*x)^2*exp(c + d*x),x)

[Out]

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

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sympy [A]  time = 8.57, size = 432, normalized size = 4.55 \[ \begin {cases} x e^{c} \cosh ^{2}{\relax (a )} & \text {for}\: b = 0 \wedge d = 0 \\\frac {x e^{c} e^{d x} \sinh ^{2}{\left (a - \frac {d x}{2} \right )}}{4} + \frac {x e^{c} e^{d x} \sinh {\left (a - \frac {d x}{2} \right )} \cosh {\left (a - \frac {d x}{2} \right )}}{2} + \frac {x e^{c} e^{d x} \cosh ^{2}{\left (a - \frac {d x}{2} \right )}}{4} + \frac {e^{c} e^{d x} \sinh {\left (a - \frac {d x}{2} \right )} \cosh {\left (a - \frac {d x}{2} \right )}}{2 d} + \frac {e^{c} e^{d x} \cosh ^{2}{\left (a - \frac {d x}{2} \right )}}{d} & \text {for}\: b = - \frac {d}{2} \\\frac {x e^{c} e^{d x} \sinh ^{2}{\left (a + \frac {d x}{2} \right )}}{4} - \frac {x e^{c} e^{d x} \sinh {\left (a + \frac {d x}{2} \right )} \cosh {\left (a + \frac {d x}{2} \right )}}{2} + \frac {x e^{c} e^{d x} \cosh ^{2}{\left (a + \frac {d x}{2} \right )}}{4} - \frac {e^{c} e^{d x} \sinh {\left (a + \frac {d x}{2} \right )} \cosh {\left (a + \frac {d x}{2} \right )}}{2 d} + \frac {e^{c} e^{d x} \cosh ^{2}{\left (a + \frac {d x}{2} \right )}}{d} & \text {for}\: b = \frac {d}{2} \\\left (- \frac {x \sinh ^{2}{\left (a + b x \right )}}{2} + \frac {x \cosh ^{2}{\left (a + b x \right )}}{2} + \frac {\sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{2 b}\right ) e^{c} & \text {for}\: d = 0 \\- \frac {2 b^{2} e^{c} e^{d x} \sinh ^{2}{\left (a + b x \right )}}{4 b^{2} d - d^{3}} + \frac {2 b^{2} e^{c} e^{d x} \cosh ^{2}{\left (a + b x \right )}}{4 b^{2} d - d^{3}} + \frac {2 b d e^{c} e^{d x} \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{4 b^{2} d - d^{3}} - \frac {d^{2} e^{c} e^{d x} \cosh ^{2}{\left (a + b x \right )}}{4 b^{2} d - d^{3}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x*exp(c)*cosh(a)**2, Eq(b, 0) & Eq(d, 0)), (x*exp(c)*exp(d*x)*sinh(a - d*x/2)**2/4 + x*exp(c)*exp(d
*x)*sinh(a - d*x/2)*cosh(a - d*x/2)/2 + x*exp(c)*exp(d*x)*cosh(a - d*x/2)**2/4 + exp(c)*exp(d*x)*sinh(a - d*x/
2)*cosh(a - d*x/2)/(2*d) + exp(c)*exp(d*x)*cosh(a - d*x/2)**2/d, Eq(b, -d/2)), (x*exp(c)*exp(d*x)*sinh(a + d*x
/2)**2/4 - x*exp(c)*exp(d*x)*sinh(a + d*x/2)*cosh(a + d*x/2)/2 + x*exp(c)*exp(d*x)*cosh(a + d*x/2)**2/4 - exp(
c)*exp(d*x)*sinh(a + d*x/2)*cosh(a + d*x/2)/(2*d) + exp(c)*exp(d*x)*cosh(a + d*x/2)**2/d, Eq(b, d/2)), ((-x*si
nh(a + b*x)**2/2 + x*cosh(a + b*x)**2/2 + sinh(a + b*x)*cosh(a + b*x)/(2*b))*exp(c), Eq(d, 0)), (-2*b**2*exp(c
)*exp(d*x)*sinh(a + b*x)**2/(4*b**2*d - d**3) + 2*b**2*exp(c)*exp(d*x)*cosh(a + b*x)**2/(4*b**2*d - d**3) + 2*
b*d*exp(c)*exp(d*x)*sinh(a + b*x)*cosh(a + b*x)/(4*b**2*d - d**3) - d**2*exp(c)*exp(d*x)*cosh(a + b*x)**2/(4*b
**2*d - d**3), True))

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