3.3.83 \(\int x \cosh (a+b x) \sinh ^2(a+b x) \, dx\) [283]

Optimal. Leaf size=45 \[ \frac {\cosh (a+b x)}{3 b^2}-\frac {\cosh ^3(a+b x)}{9 b^2}+\frac {x \sinh ^3(a+b x)}{3 b} \]

[Out]

1/3*cosh(b*x+a)/b^2-1/9*cosh(b*x+a)^3/b^2+1/3*x*sinh(b*x+a)^3/b

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Rubi [A]
time = 0.03, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5480, 2713} \begin {gather*} -\frac {\cosh ^3(a+b x)}{9 b^2}+\frac {\cosh (a+b x)}{3 b^2}+\frac {x \sinh ^3(a+b x)}{3 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Cosh[a + b*x]/(3*b^2) - Cosh[a + b*x]^3/(9*b^2) + (x*Sinh[a + b*x]^3)/(3*b)

Rule 2713

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rule 5480

Int[Cosh[(a_.) + (b_.)*(x_)^(n_.)]*(x_)^(m_.)*Sinh[(a_.) + (b_.)*(x_)^(n_.)]^(p_.), x_Symbol] :> Simp[x^(m - n
+ 1)*(Sinh[a + b*x^n]^(p + 1)/(b*n*(p + 1))), x] - Dist[(m - n + 1)/(b*n*(p + 1)), Int[x^(m - n)*Sinh[a + b*x
^n]^(p + 1), x], x] /; FreeQ[{a, b, p}, x] && LtQ[0, n, m + 1] && NeQ[p, -1]

Rubi steps

\begin {align*} \int x \cosh (a+b x) \sinh ^2(a+b x) \, dx &=\frac {x \sinh ^3(a+b x)}{3 b}-\frac {\int \sinh ^3(a+b x) \, dx}{3 b}\\ &=\frac {x \sinh ^3(a+b x)}{3 b}+\frac {\text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh (a+b x)\right )}{3 b^2}\\ &=\frac {\cosh (a+b x)}{3 b^2}-\frac {\cosh ^3(a+b x)}{9 b^2}+\frac {x \sinh ^3(a+b x)}{3 b}\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 38, normalized size = 0.84 \begin {gather*} \frac {9 \cosh (a+b x)-\cosh (3 (a+b x))+12 b x \sinh ^3(a+b x)}{36 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(9*Cosh[a + b*x] - Cosh[3*(a + b*x)] + 12*b*x*Sinh[a + b*x]^3)/(36*b^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(83\) vs. \(2(39)=78\).
time = 1.57, size = 84, normalized size = 1.87

method	result	size
risch	\(\frac {\left (3 b x -1\right ) {\mathrm e}^{3 b x +3 a}}{72 b^{2}}-\frac {\left (b x -1\right ) {\mathrm e}^{b x +a}}{8 b^{2}}+\frac {\left (b x +1\right ) {\mathrm e}^{-b x -a}}{8 b^{2}}-\frac {\left (3 b x +1\right ) {\mathrm e}^{-3 b x -3 a}}{72 b^{2}}\)	\(77\)
default	\(-\frac {\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )-a \sinh \left (b x +a \right )}{4 b^{2}}+\frac {\left (3 b x +3 a \right ) \sinh \left (3 b x +3 a \right )-\cosh \left (3 b x +3 a \right )-3 a \sinh \left (3 b x +3 a \right )}{36 b^{2}}\)	\(84\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4/b^2*((b*x+a)*sinh(b*x+a)-cosh(b*x+a)-a*sinh(b*x+a))+1/36/b^2*((3*b*x+3*a)*sinh(3*b*x+3*a)-cosh(3*b*x+3*a)
-3*a*sinh(3*b*x+3*a))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (39) = 78\).
time = 0.28, size = 84, normalized size = 1.87 \begin {gather*} \frac {{\left (3 \, b x e^{\left (3 \, a\right )} - e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )}}{72 \, b^{2}} - \frac {{\left (b x e^{a} - e^{a}\right )} e^{\left (b x\right )}}{8 \, b^{2}} + \frac {{\left (b x + 1\right )} e^{\left (-b x - a\right )}}{8 \, b^{2}} - \frac {{\left (3 \, b x + 1\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{72 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/72*(3*b*x*e^(3*a) - e^(3*a))*e^(3*b*x)/b^2 - 1/8*(b*x*e^a - e^a)*e^(b*x)/b^2 + 1/8*(b*x + 1)*e^(-b*x - a)/b^
2 - 1/72*(3*b*x + 1)*e^(-3*b*x - 3*a)/b^2

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Fricas [A]
time = 0.37, size = 76, normalized size = 1.69 \begin {gather*} \frac {3 \, b x \sinh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )^{3} - 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + 9 \, {\left (b x \cosh \left (b x + a\right )^{2} - b x\right )} \sinh \left (b x + a\right ) + 9 \, \cosh \left (b x + a\right )}{36 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/36*(3*b*x*sinh(b*x + a)^3 - cosh(b*x + a)^3 - 3*cosh(b*x + a)*sinh(b*x + a)^2 + 9*(b*x*cosh(b*x + a)^2 - b*x
)*sinh(b*x + a) + 9*cosh(b*x + a))/b^2

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Sympy [A]
time = 0.18, size = 61, normalized size = 1.36 \begin {gather*} \begin {cases} \frac {x \sinh ^{3}{\left (a + b x \right )}}{3 b} - \frac {\sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{3 b^{2}} + \frac {2 \cosh ^{3}{\left (a + b x \right )}}{9 b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{2} \sinh ^{2}{\left (a \right )} \cosh {\left (a \right )}}{2} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x*sinh(a + b*x)**3/(3*b) - sinh(a + b*x)**2*cosh(a + b*x)/(3*b**2) + 2*cosh(a + b*x)**3/(9*b**2), N
e(b, 0)), (x**2*sinh(a)**2*cosh(a)/2, True))

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Giac [A]
time = 0.38, size = 76, normalized size = 1.69 \begin {gather*} \frac {{\left (3 \, b x - 1\right )} e^{\left (3 \, b x + 3 \, a\right )}}{72 \, b^{2}} - \frac {{\left (b x - 1\right )} e^{\left (b x + a\right )}}{8 \, b^{2}} + \frac {{\left (b x + 1\right )} e^{\left (-b x - a\right )}}{8 \, b^{2}} - \frac {{\left (3 \, b x + 1\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{72 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/72*(3*b*x - 1)*e^(3*b*x + 3*a)/b^2 - 1/8*(b*x - 1)*e^(b*x + a)/b^2 + 1/8*(b*x + 1)*e^(-b*x - a)/b^2 - 1/72*(
3*b*x + 1)*e^(-3*b*x - 3*a)/b^2

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Mupad [B]
time = 1.68, size = 44, normalized size = 0.98 \begin {gather*} \frac {2\,{\mathrm {cosh}\left (a+b\,x\right )}^3-3\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2+3\,b\,x\,{\mathrm {sinh}\left (a+b\,x\right )}^3}{9\,b^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*cosh(a + b*x)^3 - 3*cosh(a + b*x)*sinh(a + b*x)^2 + 3*b*x*sinh(a + b*x)^3)/(9*b^2)

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