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

Optimal. Leaf size=46 \[ \frac{\sinh (a+b x) \cosh ^3(a+b x)}{4 b}-\frac{\sinh (a+b x) \cosh (a+b x)}{8 b}-\frac{x}{8} \]

[Out]

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

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Rubi [A]  time = 0.0435783, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {2568, 2635, 8} \[ \frac{\sinh (a+b x) \cosh ^3(a+b x)}{4 b}-\frac{\sinh (a+b x) \cosh (a+b x)}{8 b}-\frac{x}{8} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*x]^2*Sinh[a + b*x]^2,x]

[Out]

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

Rule 2568

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

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 \cosh ^2(a+b x) \sinh ^2(a+b x) \, dx &=\frac{\cosh ^3(a+b x) \sinh (a+b x)}{4 b}-\frac{1}{4} \int \cosh ^2(a+b x) \, dx\\ &=-\frac{\cosh (a+b x) \sinh (a+b x)}{8 b}+\frac{\cosh ^3(a+b x) \sinh (a+b x)}{4 b}-\frac{\int 1 \, dx}{8}\\ &=-\frac{x}{8}-\frac{\cosh (a+b x) \sinh (a+b x)}{8 b}+\frac{\cosh ^3(a+b x) \sinh (a+b x)}{4 b}\\ \end{align*}

Mathematica [A]  time = 0.0295206, size = 23, normalized size = 0.5 \[ \frac{\sinh (4 (a+b x))-4 (a+b x)}{32 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[a + b*x]^2*Sinh[a + b*x]^2,x]

[Out]

(-4*(a + b*x) + Sinh[4*(a + b*x)])/(32*b)

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Maple [A]  time = 0.008, size = 43, normalized size = 0.9 \begin{align*}{\frac{1}{b} \left ({\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{3}\sinh \left ( bx+a \right ) }{4}}-{\frac{\cosh \left ( bx+a \right ) \sinh \left ( bx+a \right ) }{8}}-{\frac{bx}{8}}-{\frac{a}{8}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b*(1/4*cosh(b*x+a)^3*sinh(b*x+a)-1/8*cosh(b*x+a)*sinh(b*x+a)-1/8*b*x-1/8*a)

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Maxima [A]  time = 1.01234, size = 53, normalized size = 1.15 \begin{align*} -\frac{b x + a}{8 \, b} + \frac{e^{\left (4 \, b x + 4 \, a\right )}}{64 \, b} - \frac{e^{\left (-4 \, b x - 4 \, a\right )}}{64 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(b*x + a)/b + 1/64*e^(4*b*x + 4*a)/b - 1/64*e^(-4*b*x - 4*a)/b

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Fricas [A]  time = 2.05852, size = 104, normalized size = 2.26 \begin{align*} \frac{\cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right ) + \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} - b x}{8 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(cosh(b*x + a)^3*sinh(b*x + a) + cosh(b*x + a)*sinh(b*x + a)^3 - b*x)/b

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Sympy [A]  time = 1.118, size = 92, normalized size = 2. \begin{align*} \begin{cases} - \frac{x \sinh ^{4}{\left (a + b x \right )}}{8} + \frac{x \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{4} - \frac{x \cosh ^{4}{\left (a + b x \right )}}{8} + \frac{\sinh ^{3}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{8 b} + \frac{\sinh{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{8 b} & \text{for}\: b \neq 0 \\x \sinh ^{2}{\left (a \right )} \cosh ^{2}{\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)**2*sinh(b*x+a)**2,x)

[Out]

Piecewise((-x*sinh(a + b*x)**4/8 + x*sinh(a + b*x)**2*cosh(a + b*x)**2/4 - x*cosh(a + b*x)**4/8 + sinh(a + b*x
)**3*cosh(a + b*x)/(8*b) + sinh(a + b*x)*cosh(a + b*x)**3/(8*b), Ne(b, 0)), (x*sinh(a)**2*cosh(a)**2, True))

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Giac [A]  time = 1.16746, size = 65, normalized size = 1.41 \begin{align*} -\frac{8 \, b x -{\left (2 \, e^{\left (4 \, b x + 4 \, a\right )} - 1\right )} e^{\left (-4 \, b x - 4 \, a\right )} + 8 \, a - e^{\left (4 \, b x + 4 \, a\right )}}{64 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/64*(8*b*x - (2*e^(4*b*x + 4*a) - 1)*e^(-4*b*x - 4*a) + 8*a - e^(4*b*x + 4*a))/b