3.1.15 \(\int \cosh ^2(a+b x) \sinh ^2(a+b x) \, dx\) [15]

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

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 46, normalized size of antiderivative = 1.00, 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 = {2648, 2715, 8} \begin {gather*} \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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/8*x - (Cosh[a + b*x]*Sinh[a + b*x])/(8*b) + (Cosh[a + b*x]^3*Sinh[a + b*x])/(4*b)

Rule 8

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

Rule 2648

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 2715

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] && Integ
erQ[2*n]

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*}

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Mathematica [A]
time = 0.02, size = 23, normalized size = 0.50 \begin {gather*} \frac {-4 (a+b x)+\sinh (4 (a+b x))}{32 b} \end {gather*}

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 = 0.60, size = 43, normalized size = 0.93

method	result	size
risch	\(-\frac {x}{8}+\frac {{\mathrm e}^{4 b x +4 a}}{64 b}-\frac {{\mathrm e}^{-4 b x -4 a}}{64 b}\)	\(33\)
derivativedivides	\(\frac {\frac {\left (\cosh ^{3}\left (b x +a \right )\right ) \sinh \left (b x +a \right )}{4}-\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{8}-\frac {b x}{8}-\frac {a}{8}}{b}\)	\(43\)
default	\(\frac {\frac {\left (\cosh ^{3}\left (b x +a \right )\right ) \sinh \left (b x +a \right )}{4}-\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{8}-\frac {b x}{8}-\frac {a}{8}}{b}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.26, size = 39, normalized size = 0.85 \begin {gather*} -\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 {gather*}

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 = 0.39, size = 40, normalized size = 0.87 \begin {gather*} \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 {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (37) = 74\).
time = 0.18, size = 92, normalized size = 2.00 \begin {gather*} \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 {gather*}

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 = 0.39, size = 32, normalized size = 0.70 \begin {gather*} -\frac {1}{8} \, x + \frac {e^{\left (4 \, b x + 4 \, a\right )}}{64 \, b} - \frac {e^{\left (-4 \, b x - 4 \, a\right )}}{64 \, b} \end {gather*}

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]

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

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Mupad [B]
time = 0.11, size = 18, normalized size = 0.39 \begin {gather*} \frac {\mathrm {sinh}\left (4\,a+4\,b\,x\right )}{32\,b}-\frac {x}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sinh(4*a + 4*b*x)/(32*b) - x/8

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