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

Optimal. Leaf size=49 \[ -\frac {e^{-a-b x}}{4 b}+\frac {e^{a+b x}}{2 b}+\frac {e^{3 a+3 b x}}{12 b} \]

[Out]

-1/4*exp(-b*x-a)/b+1/2*exp(b*x+a)/b+1/12*exp(3*b*x+3*a)/b

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Rubi [A]  time = 0.03, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2282, 12, 270} \[ -\frac {e^{-a-b x}}{4 b}+\frac {e^{a+b x}}{2 b}+\frac {e^{3 a+3 b x}}{12 b} \]

Antiderivative was successfully verified.

[In]

Int[E^(a + b*x)*Cosh[a + b*x]^2,x]

[Out]

-E^(-a - b*x)/(4*b) + E^(a + b*x)/(2*b) + E^(3*a + 3*b*x)/(12*b)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rubi steps

\begin {align*} \int e^{a+b x} \cosh ^2(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^2}{4 x^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^2}{x^2} \, dx,x,e^{a+b x}\right )}{4 b}\\ &=\frac {\operatorname {Subst}\left (\int \left (2+\frac {1}{x^2}+x^2\right ) \, dx,x,e^{a+b x}\right )}{4 b}\\ &=-\frac {e^{-a-b x}}{4 b}+\frac {e^{a+b x}}{2 b}+\frac {e^{3 a+3 b x}}{12 b}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 39, normalized size = 0.80 \[ \frac {e^{-a-b x} \left (6 e^{2 (a+b x)}+e^{4 (a+b x)}-3\right )}{12 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(E^(-a - b*x)*(-3 + 6*E^(2*(a + b*x)) + E^(4*(a + b*x))))/(12*b)

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fricas [A]  time = 0.94, size = 54, normalized size = 1.10 \[ -\frac {\cosh \left (b x + a\right )^{2} - 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 3}{6 \, {\left (b \cosh \left (b x + a\right ) - b \sinh \left (b x + a\right )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.14, size = 34, normalized size = 0.69 \[ \frac {e^{\left (3 \, b x + 3 \, a\right )} + 6 \, e^{\left (b x + a\right )} - 3 \, e^{\left (-b x - a\right )}}{12 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(e^(3*b*x + 3*a) + 6*e^(b*x + a) - 3*e^(-b*x - a))/b

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maple [A]  time = 0.21, size = 35, normalized size = 0.71 \[ \frac {\frac {\left (\cosh ^{3}\left (b x +a \right )\right )}{3}+\left (\frac {2}{3}+\frac {\left (\cosh ^{2}\left (b x +a \right )\right )}{3}\right ) \sinh \left (b x +a \right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b*(1/3*cosh(b*x+a)^3+(2/3+1/3*cosh(b*x+a)^2)*sinh(b*x+a))

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maxima [A]  time = 0.31, size = 40, normalized size = 0.82 \[ \frac {e^{\left (3 \, b x + 3 \, a\right )}}{12 \, b} + \frac {e^{\left (b x + a\right )}}{2 \, b} - \frac {e^{\left (-b x - a\right )}}{4 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*e^(3*b*x + 3*a)/b + 1/2*e^(b*x + a)/b - 1/4*e^(-b*x - a)/b

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mupad [B]  time = 0.97, size = 34, normalized size = 0.69 \[ \frac {6\,{\mathrm {e}}^{a+b\,x}-3\,{\mathrm {e}}^{-a-b\,x}+{\mathrm {e}}^{3\,a+3\,b\,x}}{12\,b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a + b*x)^2*exp(a + b*x),x)

[Out]

(6*exp(a + b*x) - 3*exp(- a - b*x) + exp(3*a + 3*b*x))/(12*b)

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sympy [A]  time = 4.34, size = 78, normalized size = 1.59 \[ \begin {cases} - \frac {2 e^{a} e^{b x} \sinh ^{2}{\left (a + b x \right )}}{3 b} + \frac {2 e^{a} e^{b x} \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{3 b} + \frac {e^{a} e^{b x} \cosh ^{2}{\left (a + b x \right )}}{3 b} & \text {for}\: b \neq 0 \\x e^{a} \cosh ^{2}{\relax (a )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-2*exp(a)*exp(b*x)*sinh(a + b*x)**2/(3*b) + 2*exp(a)*exp(b*x)*sinh(a + b*x)*cosh(a + b*x)/(3*b) + e
xp(a)*exp(b*x)*cosh(a + b*x)**2/(3*b), Ne(b, 0)), (x*exp(a)*cosh(a)**2, True))

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