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

Optimal. Leaf size=41 \[ -\frac {\cosh (4 a+4 b x)}{128 b^2}+\frac {x \sinh (4 a+4 b x)}{32 b}-\frac {x^2}{16} \]

[Out]

-1/16*x^2-1/128*cosh(4*b*x+4*a)/b^2+1/32*x*sinh(4*b*x+4*a)/b

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Rubi [A]  time = 0.05, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5448, 3296, 2638} \[ -\frac {\cosh (4 a+4 b x)}{128 b^2}+\frac {x \sinh (4 a+4 b x)}{32 b}-\frac {x^2}{16} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-x^2/16 - Cosh[4*a + 4*b*x]/(128*b^2) + (x*Sinh[4*a + 4*b*x])/(32*b)

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 5448

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int x \cosh ^2(a+b x) \sinh ^2(a+b x) \, dx &=\int \left (-\frac {x}{8}+\frac {1}{8} x \cosh (4 a+4 b x)\right ) \, dx\\ &=-\frac {x^2}{16}+\frac {1}{8} \int x \cosh (4 a+4 b x) \, dx\\ &=-\frac {x^2}{16}+\frac {x \sinh (4 a+4 b x)}{32 b}-\frac {\int \sinh (4 a+4 b x) \, dx}{32 b}\\ &=-\frac {x^2}{16}-\frac {\cosh (4 a+4 b x)}{128 b^2}+\frac {x \sinh (4 a+4 b x)}{32 b}\\ \end {align*}

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Mathematica [A]  time = 0.14, size = 41, normalized size = 1.00 \[ -\frac {-8 a^2-4 b x \sinh (4 (a+b x))+\cosh (4 (a+b x))+8 b^2 x^2}{128 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/128*(-8*a^2 + 8*b^2*x^2 + Cosh[4*(a + b*x)] - 4*b*x*Sinh[4*(a + b*x)])/b^2

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fricas [B]  time = 0.72, size = 88, normalized size = 2.15 \[ \frac {16 \, b x \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right ) + 16 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} - 8 \, b^{2} x^{2} - \cosh \left (b x + a\right )^{4} - 6 \, \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} - \sinh \left (b x + a\right )^{4}}{128 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/128*(16*b*x*cosh(b*x + a)^3*sinh(b*x + a) + 16*b*x*cosh(b*x + a)*sinh(b*x + a)^3 - 8*b^2*x^2 - cosh(b*x + a)
^4 - 6*cosh(b*x + a)^2*sinh(b*x + a)^2 - sinh(b*x + a)^4)/b^2

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giac [A]  time = 0.14, size = 46, normalized size = 1.12 \[ -\frac {1}{16} \, x^{2} + \frac {{\left (4 \, b x - 1\right )} e^{\left (4 \, b x + 4 \, a\right )}}{256 \, b^{2}} - \frac {{\left (4 \, b x + 1\right )} e^{\left (-4 \, b x - 4 \, a\right )}}{256 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*x^2 + 1/256*(4*b*x - 1)*e^(4*b*x + 4*a)/b^2 - 1/256*(4*b*x + 1)*e^(-4*b*x - 4*a)/b^2

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maple [B]  time = 0.07, size = 116, normalized size = 2.83 \[ \frac {\frac {\left (b x +a \right ) \sinh \left (b x +a \right ) \left (\cosh ^{3}\left (b x +a \right )\right )}{4}-\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{8}-\frac {\left (b x +a \right )^{2}}{16}-\frac {\left (\cosh ^{4}\left (b x +a \right )\right )}{16}+\frac {\left (\cosh ^{2}\left (b x +a \right )\right )}{16}-a \left (\frac {\left (\cosh ^{3}\left (b x +a \right )\right ) \sinh \left (b x +a \right )}{4}-\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{8}-\frac {b x}{8}-\frac {a}{8}\right )}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b^2*(1/4*(b*x+a)*sinh(b*x+a)*cosh(b*x+a)^3-1/8*(b*x+a)*cosh(b*x+a)*sinh(b*x+a)-1/16*(b*x+a)^2-1/16*cosh(b*x+
a)^4+1/16*cosh(b*x+a)^2-a*(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 = 0.38, size = 51, normalized size = 1.24 \[ -\frac {1}{16} \, x^{2} + \frac {{\left (4 \, b x e^{\left (4 \, a\right )} - e^{\left (4 \, a\right )}\right )} e^{\left (4 \, b x\right )}}{256 \, b^{2}} - \frac {{\left (4 \, b x + 1\right )} e^{\left (-4 \, b x - 4 \, a\right )}}{256 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*x^2 + 1/256*(4*b*x*e^(4*a) - e^(4*a))*e^(4*b*x)/b^2 - 1/256*(4*b*x + 1)*e^(-4*b*x - 4*a)/b^2

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mupad [B]  time = 1.76, size = 36, normalized size = 0.88 \[ -\frac {\frac {\mathrm {cosh}\left (4\,a+4\,b\,x\right )}{128}-\frac {b\,x\,\mathrm {sinh}\left (4\,a+4\,b\,x\right )}{32}}{b^2}-\frac {x^2}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- (cosh(4*a + 4*b*x)/128 - (b*x*sinh(4*a + 4*b*x))/32)/b^2 - x^2/16

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sympy [A]  time = 1.71, size = 131, normalized size = 3.20 \[ \begin {cases} - \frac {x^{2} \sinh ^{4}{\left (a + b x \right )}}{16} + \frac {x^{2} \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{8} - \frac {x^{2} \cosh ^{4}{\left (a + b x \right )}}{16} + \frac {x \sinh ^{3}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{8 b} + \frac {x \sinh {\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{8 b} - \frac {\sinh ^{4}{\left (a + b x \right )}}{32 b^{2}} - \frac {\cosh ^{4}{\left (a + b x \right )}}{32 b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{2} \sinh ^{2}{\relax (a )} \cosh ^{2}{\relax (a )}}{2} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-x**2*sinh(a + b*x)**4/16 + x**2*sinh(a + b*x)**2*cosh(a + b*x)**2/8 - x**2*cosh(a + b*x)**4/16 + x
*sinh(a + b*x)**3*cosh(a + b*x)/(8*b) + x*sinh(a + b*x)*cosh(a + b*x)**3/(8*b) - sinh(a + b*x)**4/(32*b**2) -
cosh(a + b*x)**4/(32*b**2), Ne(b, 0)), (x**2*sinh(a)**2*cosh(a)**2/2, True))

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