3.10.13 \(\int e^{a+b x} \cosh ^3(a+b x) \sinh ^3(a+b x) \, dx\) [913]

Optimal. Leaf size=69 \[ \frac {e^{-5 a-5 b x}}{320 b}-\frac {3 e^{-a-b x}}{64 b}-\frac {e^{3 a+3 b x}}{64 b}+\frac {e^{7 a+7 b x}}{448 b} \]

[Out]

1/320*exp(-5*b*x-5*a)/b-3/64*exp(-b*x-a)/b-1/64*exp(3*b*x+3*a)/b+1/448*exp(7*b*x+7*a)/b

________________________________________________________________________________________

Rubi [A]
time = 0.04, antiderivative size = 69, normalized size of antiderivative = 1.00, 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 = {2320, 12, 276} \begin {gather*} \frac {e^{-5 a-5 b x}}{320 b}-\frac {3 e^{-a-b x}}{64 b}-\frac {e^{3 a+3 b x}}{64 b}+\frac {e^{7 a+7 b x}}{448 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(a + b*x)*Cosh[a + b*x]^3*Sinh[a + b*x]^3,x]

[Out]

E^(-5*a - 5*b*x)/(320*b) - (3*E^(-a - b*x))/(64*b) - E^(3*a + 3*b*x)/(64*b) + E^(7*a + 7*b*x)/(448*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 276

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 2320

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 ^3(a+b x) \sinh ^3(a+b x) \, dx &=\frac {\text {Subst}\left (\int \frac {\left (-1+x^4\right )^3}{64 x^6} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {\text {Subst}\left (\int \frac {\left (-1+x^4\right )^3}{x^6} \, dx,x,e^{a+b x}\right )}{64 b}\\ &=\frac {\text {Subst}\left (\int \left (-\frac {1}{x^6}+\frac {3}{x^2}-3 x^2+x^6\right ) \, dx,x,e^{a+b x}\right )}{64 b}\\ &=\frac {e^{-5 a-5 b x}}{320 b}-\frac {3 e^{-a-b x}}{64 b}-\frac {e^{3 a+3 b x}}{64 b}+\frac {e^{7 a+7 b x}}{448 b}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.03, size = 51, normalized size = 0.74 \begin {gather*} \frac {e^{-5 (a+b x)} \left (7-105 e^{4 (a+b x)}-35 e^{8 (a+b x)}+5 e^{12 (a+b x)}\right )}{2240 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(7 - 105*E^(4*(a + b*x)) - 35*E^(8*(a + b*x)) + 5*E^(12*(a + b*x)))/(2240*b*E^(5*(a + b*x)))

________________________________________________________________________________________

Maple [A]
time = 1.80, size = 108, normalized size = 1.57

method	result	size
risch	\(\frac {{\mathrm e}^{-5 b x -5 a}}{320 b}-\frac {3 \,{\mathrm e}^{-b x -a}}{64 b}-\frac {{\mathrm e}^{3 b x +3 a}}{64 b}+\frac {{\mathrm e}^{7 b x +7 a}}{448 b}\)	\(58\)
default	\(\frac {3 \sinh \left (b x +a \right )}{64 b}-\frac {\sinh \left (3 b x +3 a \right )}{64 b}-\frac {\sinh \left (5 b x +5 a \right )}{320 b}+\frac {\sinh \left (7 b x +7 a \right )}{448 b}-\frac {3 \cosh \left (b x +a \right )}{64 b}-\frac {\cosh \left (3 b x +3 a \right )}{64 b}+\frac {\cosh \left (5 b x +5 a \right )}{320 b}+\frac {\cosh \left (7 b x +7 a \right )}{448 b}\)	\(108\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(b*x+a)*cosh(b*x+a)^3*sinh(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

3/64*sinh(b*x+a)/b-1/64/b*sinh(3*b*x+3*a)-1/320/b*sinh(5*b*x+5*a)+1/448/b*sinh(7*b*x+7*a)-3/64*cosh(b*x+a)/b-1
/64*cosh(3*b*x+3*a)/b+1/320*cosh(5*b*x+5*a)/b+1/448*cosh(7*b*x+7*a)/b

________________________________________________________________________________________

Maxima [A]
time = 0.26, size = 54, normalized size = 0.78 \begin {gather*} -\frac {{\left (15 \, e^{\left (4 \, b x + 4 \, a\right )} - 1\right )} e^{\left (-5 \, b x - 5 \, a\right )}}{320 \, b} + \frac {e^{\left (7 \, b x + 7 \, a\right )} - 7 \, e^{\left (3 \, b x + 3 \, a\right )}}{448 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(b*x+a)*cosh(b*x+a)^3*sinh(b*x+a)^3,x, algorithm="maxima")

[Out]

-1/320*(15*e^(4*b*x + 4*a) - 1)*e^(-5*b*x - 5*a)/b + 1/448*(e^(7*b*x + 7*a) - 7*e^(3*b*x + 3*a))/b

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 154 vs. \(2 (57) = 114\).
time = 0.33, size = 154, normalized size = 2.23 \begin {gather*} \frac {3 \, \cosh \left (b x + a\right )^{6} - 10 \, \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right )^{3} + 45 \, \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{4} - 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + 3 \, \sinh \left (b x + a\right )^{6} + 5 \, {\left (9 \, \cosh \left (b x + a\right )^{4} - 7\right )} \sinh \left (b x + a\right )^{2} - 35 \, \cosh \left (b x + a\right )^{2} - {\left (3 \, \cosh \left (b x + a\right )^{5} - 35 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{560 \, {\left (b \cosh \left (b x + a\right ) - b \sinh \left (b x + a\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(b*x+a)*cosh(b*x+a)^3*sinh(b*x+a)^3,x, algorithm="fricas")

[Out]

1/560*(3*cosh(b*x + a)^6 - 10*cosh(b*x + a)^3*sinh(b*x + a)^3 + 45*cosh(b*x + a)^2*sinh(b*x + a)^4 - 3*cosh(b*
x + a)*sinh(b*x + a)^5 + 3*sinh(b*x + a)^6 + 5*(9*cosh(b*x + a)^4 - 7)*sinh(b*x + a)^2 - 35*cosh(b*x + a)^2 -
(3*cosh(b*x + a)^5 - 35*cosh(b*x + a))*sinh(b*x + a))/(b*cosh(b*x + a) - b*sinh(b*x + a))

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 202 vs. \(2 (54) = 108\).
time = 18.10, size = 202, normalized size = 2.93 \begin {gather*} \begin {cases} - \frac {2 e^{a} e^{b x} \sinh ^{6}{\left (a + b x \right )}}{35 b} + \frac {2 e^{a} e^{b x} \sinh ^{5}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{35 b} + \frac {e^{a} e^{b x} \sinh ^{4}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{7 b} - \frac {e^{a} e^{b x} \sinh ^{3}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{7 b} + \frac {e^{a} e^{b x} \sinh ^{2}{\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{7 b} + \frac {2 e^{a} e^{b x} \sinh {\left (a + b x \right )} \cosh ^{5}{\left (a + b x \right )}}{35 b} - \frac {2 e^{a} e^{b x} \cosh ^{6}{\left (a + b x \right )}}{35 b} & \text {for}\: b \neq 0 \\x e^{a} \sinh ^{3}{\left (a \right )} \cosh ^{3}{\left (a \right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(b*x+a)*cosh(b*x+a)**3*sinh(b*x+a)**3,x)

[Out]

Piecewise((-2*exp(a)*exp(b*x)*sinh(a + b*x)**6/(35*b) + 2*exp(a)*exp(b*x)*sinh(a + b*x)**5*cosh(a + b*x)/(35*b
) + exp(a)*exp(b*x)*sinh(a + b*x)**4*cosh(a + b*x)**2/(7*b) - exp(a)*exp(b*x)*sinh(a + b*x)**3*cosh(a + b*x)**
3/(7*b) + exp(a)*exp(b*x)*sinh(a + b*x)**2*cosh(a + b*x)**4/(7*b) + 2*exp(a)*exp(b*x)*sinh(a + b*x)*cosh(a + b
*x)**5/(35*b) - 2*exp(a)*exp(b*x)*cosh(a + b*x)**6/(35*b), Ne(b, 0)), (x*exp(a)*sinh(a)**3*cosh(a)**3, True))

________________________________________________________________________________________

Giac [A]
time = 0.41, size = 52, normalized size = 0.75 \begin {gather*} -\frac {7 \, {\left (15 \, e^{\left (4 \, b x + 4 \, a\right )} - 1\right )} e^{\left (-5 \, b x - 5 \, a\right )} - 5 \, e^{\left (7 \, b x + 7 \, a\right )} + 35 \, e^{\left (3 \, b x + 3 \, a\right )}}{2240 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(b*x+a)*cosh(b*x+a)^3*sinh(b*x+a)^3,x, algorithm="giac")

[Out]

-1/2240*(7*(15*e^(4*b*x + 4*a) - 1)*e^(-5*b*x - 5*a) - 5*e^(7*b*x + 7*a) + 35*e^(3*b*x + 3*a))/b

________________________________________________________________________________________

Mupad [B]
time = 2.10, size = 50, normalized size = 0.72 \begin {gather*} -\frac {105\,{\mathrm {e}}^{-a-b\,x}+35\,{\mathrm {e}}^{3\,a+3\,b\,x}-7\,{\mathrm {e}}^{-5\,a-5\,b\,x}-5\,{\mathrm {e}}^{7\,a+7\,b\,x}}{2240\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a + b*x)^3*exp(a + b*x)*sinh(a + b*x)^3,x)

[Out]

-(105*exp(- a - b*x) + 35*exp(3*a + 3*b*x) - 7*exp(- 5*a - 5*b*x) - 5*exp(7*a + 7*b*x))/(2240*b)

________________________________________________________________________________________