3.272 \(\int \cosh ^3(a+b x) \sinh (a+b x) \, dx\)

Optimal. Leaf size=15 \[ \frac{\cosh ^4(a+b x)}{4 b} \]

[Out]

Cosh[a + b*x]^4/(4*b)

________________________________________________________________________________________

Rubi [A]  time = 0.0202069, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2565, 30} \[ \frac{\cosh ^4(a+b x)}{4 b} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*x]^3*Sinh[a + b*x],x]

[Out]

Cosh[a + b*x]^4/(4*b)

Rule 2565

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(a*f)^(-1), Subst[
Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2]
&&  !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0024895, size = 15, normalized size = 1. \[ \frac{\cosh ^4(a+b x)}{4 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Cosh[a + b*x]^4/(4*b)

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 14, normalized size = 0.9 \begin{align*}{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{4}}{4\,b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*cosh(b*x+a)^4/b

________________________________________________________________________________________

Maxima [A]  time = 1.06118, size = 18, normalized size = 1.2 \begin{align*} \frac{\cosh \left (b x + a\right )^{4}}{4 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*cosh(b*x + a)^4/b

________________________________________________________________________________________

Fricas [B]  time = 1.66286, size = 146, normalized size = 9.73 \begin{align*} \frac{\cosh \left (b x + a\right )^{4} + \sinh \left (b x + a\right )^{4} + 2 \,{\left (3 \, \cosh \left (b x + a\right )^{2} + 2\right )} \sinh \left (b x + a\right )^{2} + 4 \, \cosh \left (b x + a\right )^{2}}{32 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*(cosh(b*x + a)^4 + sinh(b*x + a)^4 + 2*(3*cosh(b*x + a)^2 + 2)*sinh(b*x + a)^2 + 4*cosh(b*x + a)^2)/b

________________________________________________________________________________________

Sympy [A]  time = 1.05049, size = 20, normalized size = 1.33 \begin{align*} \begin{cases} \frac{\cosh ^{4}{\left (a + b x \right )}}{4 b} & \text{for}\: b \neq 0 \\x \sinh{\left (a \right )} \cosh ^{3}{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((cosh(a + b*x)**4/(4*b), Ne(b, 0)), (x*sinh(a)*cosh(a)**3, True))

________________________________________________________________________________________

Giac [B]  time = 1.14367, size = 66, normalized size = 4.4 \begin{align*} \frac{{\left (e^{\left (2 \, b x + 2 \, a\right )} + e^{\left (-2 \, b x - 2 \, a\right )}\right )}^{2} + 4 \, e^{\left (2 \, b x + 2 \, a\right )} + 4 \, e^{\left (-2 \, b x - 2 \, a\right )}}{64 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*((e^(2*b*x + 2*a) + e^(-2*b*x - 2*a))^2 + 4*e^(2*b*x + 2*a) + 4*e^(-2*b*x - 2*a))/b