3.2 \(\int \sinh ^2(a+b x) \, dx\)

Optimal. Leaf size=25 \[ \frac{\sinh (a+b x) \cosh (a+b x)}{2 b}-\frac{x}{2} \]

[Out]

-x/2 + (Cosh[a + b*x]*Sinh[a + b*x])/(2*b)

________________________________________________________________________________________

Rubi [A]  time = 0.0091, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2635, 8} \[ \frac{\sinh (a+b x) \cosh (a+b x)}{2 b}-\frac{x}{2} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[a + b*x]^2,x]

[Out]

-x/2 + (Cosh[a + b*x]*Sinh[a + b*x])/(2*b)

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \sinh ^2(a+b x) \, dx &=\frac{\cosh (a+b x) \sinh (a+b x)}{2 b}-\frac{\int 1 \, dx}{2}\\ &=-\frac{x}{2}+\frac{\cosh (a+b x) \sinh (a+b x)}{2 b}\\ \end{align*}

Mathematica [A]  time = 0.024184, size = 23, normalized size = 0.92 \[ \frac{\sinh (2 (a+b x))-2 (a+b x)}{4 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[a + b*x]^2,x]

[Out]

(-2*(a + b*x) + Sinh[2*(a + b*x)])/(4*b)

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 27, normalized size = 1.1 \begin{align*}{\frac{1}{b} \left ({\frac{\cosh \left ( bx+a \right ) \sinh \left ( bx+a \right ) }{2}}-{\frac{bx}{2}}-{\frac{a}{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(b*x+a)^2,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.07968, size = 43, normalized size = 1.72 \begin{align*} -\frac{1}{2} \, x + \frac{e^{\left (2 \, b x + 2 \, a\right )}}{8 \, b} - \frac{e^{\left (-2 \, b x - 2 \, a\right )}}{8 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*x + 1/8*e^(2*b*x + 2*a)/b - 1/8*e^(-2*b*x - 2*a)/b

________________________________________________________________________________________

Fricas [A]  time = 2.05974, size = 59, normalized size = 2.36 \begin{align*} -\frac{b x - \cosh \left (b x + a\right ) \sinh \left (b x + a\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.26181, size = 46, normalized size = 1.84 \begin{align*} \begin{cases} \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} & \text{for}\: b \neq 0 \\x \sinh ^{2}{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(b*x+a)**2,x)

[Out]

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

________________________________________________________________________________________

Giac [B]  time = 1.32558, size = 65, normalized size = 2.6 \begin{align*} -\frac{4 \, b x -{\left (2 \, e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-2 \, b x - 2 \, a\right )} + 4 \, a - e^{\left (2 \, b x + 2 \, a\right )}}{8 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(4*b*x - (2*e^(2*b*x + 2*a) - 1)*e^(-2*b*x - 2*a) + 4*a - e^(2*b*x + 2*a))/b