3.3.60 \(\int x^3 \cosh ^2(a+b x) \sinh (a+b x) \, dx\) [260]

Optimal. Leaf size=117 \[ \frac {4 x \cosh (a+b x)}{3 b^3}+\frac {2 x \cosh ^3(a+b x)}{9 b^3}+\frac {x^3 \cosh ^3(a+b x)}{3 b}-\frac {14 \sinh (a+b x)}{9 b^4}-\frac {2 x^2 \sinh (a+b x)}{3 b^2}-\frac {x^2 \cosh ^2(a+b x) \sinh (a+b x)}{3 b^2}-\frac {2 \sinh ^3(a+b x)}{27 b^4} \]

[Out]

4/3*x*cosh(b*x+a)/b^3+2/9*x*cosh(b*x+a)^3/b^3+1/3*x^3*cosh(b*x+a)^3/b-14/9*sinh(b*x+a)/b^4-2/3*x^2*sinh(b*x+a)
/b^2-1/3*x^2*cosh(b*x+a)^2*sinh(b*x+a)/b^2-2/27*sinh(b*x+a)^3/b^4

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Rubi [A]
time = 0.09, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {5481, 3392, 3377, 2717, 2713} \begin {gather*} -\frac {2 \sinh ^3(a+b x)}{27 b^4}-\frac {14 \sinh (a+b x)}{9 b^4}+\frac {2 x \cosh ^3(a+b x)}{9 b^3}+\frac {4 x \cosh (a+b x)}{3 b^3}-\frac {2 x^2 \sinh (a+b x)}{3 b^2}-\frac {x^2 \sinh (a+b x) \cosh ^2(a+b x)}{3 b^2}+\frac {x^3 \cosh ^3(a+b x)}{3 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*Cosh[a + b*x]^2*Sinh[a + b*x],x]

[Out]

(4*x*Cosh[a + b*x])/(3*b^3) + (2*x*Cosh[a + b*x]^3)/(9*b^3) + (x^3*Cosh[a + b*x]^3)/(3*b) - (14*Sinh[a + b*x])
/(9*b^4) - (2*x^2*Sinh[a + b*x])/(3*b^2) - (x^2*Cosh[a + b*x]^2*Sinh[a + b*x])/(3*b^2) - (2*Sinh[a + b*x]^3)/(
27*b^4)

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 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3377

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 3392

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*m*(c + d*x)^(m - 1)*((
b*Sin[e + f*x])^n/(f^2*n^2)), x] + (Dist[b^2*((n - 1)/n), Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - D
ist[d^2*m*((m - 1)/(f^2*n^2)), Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x] - Simp[b*(c + d*x)^m*Cos[e + f
*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]

Rule 5481

Int[Cosh[(a_.) + (b_.)*(x_)^(n_.)]^(p_.)*(x_)^(m_.)*Sinh[(a_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Simp[x^(m - n
+ 1)*(Cosh[a + b*x^n]^(p + 1)/(b*n*(p + 1))), x] - Dist[(m - n + 1)/(b*n*(p + 1)), Int[x^(m - n)*Cosh[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^3 \cosh ^2(a+b x) \sinh (a+b x) \, dx &=\frac {x^3 \cosh ^3(a+b x)}{3 b}-\frac {\int x^2 \cosh ^3(a+b x) \, dx}{b}\\ &=\frac {2 x \cosh ^3(a+b x)}{9 b^3}+\frac {x^3 \cosh ^3(a+b x)}{3 b}-\frac {x^2 \cosh ^2(a+b x) \sinh (a+b x)}{3 b^2}-\frac {2 \int \cosh ^3(a+b x) \, dx}{9 b^3}-\frac {2 \int x^2 \cosh (a+b x) \, dx}{3 b}\\ &=\frac {2 x \cosh ^3(a+b x)}{9 b^3}+\frac {x^3 \cosh ^3(a+b x)}{3 b}-\frac {2 x^2 \sinh (a+b x)}{3 b^2}-\frac {x^2 \cosh ^2(a+b x) \sinh (a+b x)}{3 b^2}-\frac {(2 i) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-i \sinh (a+b x)\right )}{9 b^4}+\frac {4 \int x \sinh (a+b x) \, dx}{3 b^2}\\ &=\frac {4 x \cosh (a+b x)}{3 b^3}+\frac {2 x \cosh ^3(a+b x)}{9 b^3}+\frac {x^3 \cosh ^3(a+b x)}{3 b}-\frac {2 \sinh (a+b x)}{9 b^4}-\frac {2 x^2 \sinh (a+b x)}{3 b^2}-\frac {x^2 \cosh ^2(a+b x) \sinh (a+b x)}{3 b^2}-\frac {2 \sinh ^3(a+b x)}{27 b^4}-\frac {4 \int \cosh (a+b x) \, dx}{3 b^3}\\ &=\frac {4 x \cosh (a+b x)}{3 b^3}+\frac {2 x \cosh ^3(a+b x)}{9 b^3}+\frac {x^3 \cosh ^3(a+b x)}{3 b}-\frac {14 \sinh (a+b x)}{9 b^4}-\frac {2 x^2 \sinh (a+b x)}{3 b^2}-\frac {x^2 \cosh ^2(a+b x) \sinh (a+b x)}{3 b^2}-\frac {2 \sinh ^3(a+b x)}{27 b^4}\\ \end {align*}

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Mathematica [A]
time = 0.27, size = 86, normalized size = 0.74 \begin {gather*} \frac {27 b x \left (6+b^2 x^2\right ) \cosh (a+b x)+\left (6 b x+9 b^3 x^3\right ) \cosh (3 (a+b x))-2 \left (82+45 b^2 x^2+\left (2+9 b^2 x^2\right ) \cosh (2 (a+b x))\right ) \sinh (a+b x)}{108 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*Cosh[a + b*x]^2*Sinh[a + b*x],x]

[Out]

(27*b*x*(6 + b^2*x^2)*Cosh[a + b*x] + (6*b*x + 9*b^3*x^3)*Cosh[3*(a + b*x)] - 2*(82 + 45*b^2*x^2 + (2 + 9*b^2*
x^2)*Cosh[2*(a + b*x)])*Sinh[a + b*x])/(108*b^4)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(313\) vs. \(2(103)=206\).
time = 0.97, size = 314, normalized size = 2.68

method	result	size
risch	\(\frac {\left (9 b^{3} x^{3}-9 b^{2} x^{2}+6 b x -2\right ) {\mathrm e}^{3 b x +3 a}}{216 b^{4}}+\frac {\left (b^{3} x^{3}-3 b^{2} x^{2}+6 b x -6\right ) {\mathrm e}^{b x +a}}{8 b^{4}}+\frac {\left (b^{3} x^{3}+3 b^{2} x^{2}+6 b x +6\right ) {\mathrm e}^{-b x -a}}{8 b^{4}}+\frac {\left (9 b^{3} x^{3}+9 b^{2} x^{2}+6 b x +2\right ) {\mathrm e}^{-3 b x -3 a}}{216 b^{4}}\)	\(141\)
default	\(\frac {\left (b x +a \right )^{3} \cosh \left (b x +a \right )-3 \left (b x +a \right )^{2} \sinh \left (b x +a \right )+6 \left (b x +a \right ) \cosh \left (b x +a \right )-6 \sinh \left (b x +a \right )-3 a \left (\left (b x +a \right )^{2} \cosh \left (b x +a \right )-2 \left (b x +a \right ) \sinh \left (b x +a \right )+2 \cosh \left (b x +a \right )\right )+3 a^{2} \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )-a^{3} \cosh \left (b x +a \right )}{4 b^{4}}+\frac {\left (3 b x +3 a \right )^{3} \cosh \left (3 b x +3 a \right )-3 \left (3 b x +3 a \right )^{2} \sinh \left (3 b x +3 a \right )+6 \left (3 b x +3 a \right ) \cosh \left (3 b x +3 a \right )-6 \sinh \left (3 b x +3 a \right )-9 a \left (\left (3 b x +3 a \right )^{2} \cosh \left (3 b x +3 a \right )-2 \left (3 b x +3 a \right ) \sinh \left (3 b x +3 a \right )+2 \cosh \left (3 b x +3 a \right )\right )+27 a^{2} \left (\left (3 b x +3 a \right ) \cosh \left (3 b x +3 a \right )-\sinh \left (3 b x +3 a \right )\right )-27 a^{3} \cosh \left (3 b x +3 a \right )}{324 b^{4}}\)	\(314\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*cosh(b*x+a)^2*sinh(b*x+a),x,method=_RETURNVERBOSE)

[Out]

1/4/b^4*((b*x+a)^3*cosh(b*x+a)-3*(b*x+a)^2*sinh(b*x+a)+6*(b*x+a)*cosh(b*x+a)-6*sinh(b*x+a)-3*a*((b*x+a)^2*cosh
(b*x+a)-2*(b*x+a)*sinh(b*x+a)+2*cosh(b*x+a))+3*a^2*((b*x+a)*cosh(b*x+a)-sinh(b*x+a))-a^3*cosh(b*x+a))+1/324/b^
4*((3*b*x+3*a)^3*cosh(3*b*x+3*a)-3*(3*b*x+3*a)^2*sinh(3*b*x+3*a)+6*(3*b*x+3*a)*cosh(3*b*x+3*a)-6*sinh(3*b*x+3*
a)-9*a*((3*b*x+3*a)^2*cosh(3*b*x+3*a)-2*(3*b*x+3*a)*sinh(3*b*x+3*a)+2*cosh(3*b*x+3*a))+27*a^2*((3*b*x+3*a)*cos
h(3*b*x+3*a)-sinh(3*b*x+3*a))-27*a^3*cosh(3*b*x+3*a))

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Maxima [A]
time = 0.30, size = 160, normalized size = 1.37 \begin {gather*} \frac {{\left (9 \, b^{3} x^{3} e^{\left (3 \, a\right )} - 9 \, b^{2} x^{2} e^{\left (3 \, a\right )} + 6 \, b x e^{\left (3 \, a\right )} - 2 \, e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )}}{216 \, b^{4}} + \frac {{\left (b^{3} x^{3} e^{a} - 3 \, b^{2} x^{2} e^{a} + 6 \, b x e^{a} - 6 \, e^{a}\right )} e^{\left (b x\right )}}{8 \, b^{4}} + \frac {{\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} e^{\left (-b x - a\right )}}{8 \, b^{4}} + \frac {{\left (9 \, b^{3} x^{3} + 9 \, b^{2} x^{2} + 6 \, b x + 2\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{216 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*cosh(b*x+a)^2*sinh(b*x+a),x, algorithm="maxima")

[Out]

1/216*(9*b^3*x^3*e^(3*a) - 9*b^2*x^2*e^(3*a) + 6*b*x*e^(3*a) - 2*e^(3*a))*e^(3*b*x)/b^4 + 1/8*(b^3*x^3*e^a - 3
*b^2*x^2*e^a + 6*b*x*e^a - 6*e^a)*e^(b*x)/b^4 + 1/8*(b^3*x^3 + 3*b^2*x^2 + 6*b*x + 6)*e^(-b*x - a)/b^4 + 1/216
*(9*b^3*x^3 + 9*b^2*x^2 + 6*b*x + 2)*e^(-3*b*x - 3*a)/b^4

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Fricas [A]
time = 0.39, size = 135, normalized size = 1.15 \begin {gather*} \frac {3 \, {\left (3 \, b^{3} x^{3} + 2 \, b x\right )} \cosh \left (b x + a\right )^{3} + 9 \, {\left (3 \, b^{3} x^{3} + 2 \, b x\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - {\left (9 \, b^{2} x^{2} + 2\right )} \sinh \left (b x + a\right )^{3} + 27 \, {\left (b^{3} x^{3} + 6 \, b x\right )} \cosh \left (b x + a\right ) - 3 \, {\left (27 \, b^{2} x^{2} + {\left (9 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )^{2} + 54\right )} \sinh \left (b x + a\right )}{108 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*cosh(b*x+a)^2*sinh(b*x+a),x, algorithm="fricas")

[Out]

1/108*(3*(3*b^3*x^3 + 2*b*x)*cosh(b*x + a)^3 + 9*(3*b^3*x^3 + 2*b*x)*cosh(b*x + a)*sinh(b*x + a)^2 - (9*b^2*x^
2 + 2)*sinh(b*x + a)^3 + 27*(b^3*x^3 + 6*b*x)*cosh(b*x + a) - 3*(27*b^2*x^2 + (9*b^2*x^2 + 2)*cosh(b*x + a)^2
+ 54)*sinh(b*x + a))/b^4

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Sympy [A]
time = 0.44, size = 146, normalized size = 1.25 \begin {gather*} \begin {cases} \frac {x^{3} \cosh ^{3}{\left (a + b x \right )}}{3 b} + \frac {2 x^{2} \sinh ^{3}{\left (a + b x \right )}}{3 b^{2}} - \frac {x^{2} \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{b^{2}} - \frac {4 x \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{3 b^{3}} + \frac {14 x \cosh ^{3}{\left (a + b x \right )}}{9 b^{3}} + \frac {40 \sinh ^{3}{\left (a + b x \right )}}{27 b^{4}} - \frac {14 \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{9 b^{4}} & \text {for}\: b \neq 0 \\\frac {x^{4} \sinh {\left (a \right )} \cosh ^{2}{\left (a \right )}}{4} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*cosh(b*x+a)**2*sinh(b*x+a),x)

[Out]

Piecewise((x**3*cosh(a + b*x)**3/(3*b) + 2*x**2*sinh(a + b*x)**3/(3*b**2) - x**2*sinh(a + b*x)*cosh(a + b*x)**
2/b**2 - 4*x*sinh(a + b*x)**2*cosh(a + b*x)/(3*b**3) + 14*x*cosh(a + b*x)**3/(9*b**3) + 40*sinh(a + b*x)**3/(2
7*b**4) - 14*sinh(a + b*x)*cosh(a + b*x)**2/(9*b**4), Ne(b, 0)), (x**4*sinh(a)*cosh(a)**2/4, True))

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Giac [A]
time = 0.39, size = 140, normalized size = 1.20 \begin {gather*} \frac {{\left (9 \, b^{3} x^{3} - 9 \, b^{2} x^{2} + 6 \, b x - 2\right )} e^{\left (3 \, b x + 3 \, a\right )}}{216 \, b^{4}} + \frac {{\left (b^{3} x^{3} - 3 \, b^{2} x^{2} + 6 \, b x - 6\right )} e^{\left (b x + a\right )}}{8 \, b^{4}} + \frac {{\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} e^{\left (-b x - a\right )}}{8 \, b^{4}} + \frac {{\left (9 \, b^{3} x^{3} + 9 \, b^{2} x^{2} + 6 \, b x + 2\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{216 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*cosh(b*x+a)^2*sinh(b*x+a),x, algorithm="giac")

[Out]

1/216*(9*b^3*x^3 - 9*b^2*x^2 + 6*b*x - 2)*e^(3*b*x + 3*a)/b^4 + 1/8*(b^3*x^3 - 3*b^2*x^2 + 6*b*x - 6)*e^(b*x +
a)/b^4 + 1/8*(b^3*x^3 + 3*b^2*x^2 + 6*b*x + 6)*e^(-b*x - a)/b^4 + 1/216*(9*b^3*x^3 + 9*b^2*x^2 + 6*b*x + 2)*e
^(-3*b*x - 3*a)/b^4

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Mupad [B]
time = 1.54, size = 108, normalized size = 0.92 \begin {gather*} \frac {\frac {2\,x\,{\mathrm {cosh}\left (a+b\,x\right )}^3}{9}+\frac {4\,x\,\mathrm {cosh}\left (a+b\,x\right )}{3}}{b^3}-\frac {\frac {2\,x^2\,\mathrm {sinh}\left (a+b\,x\right )}{3}+\frac {x^2\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )}{3}}{b^2}-\frac {40\,\mathrm {sinh}\left (a+b\,x\right )}{27\,b^4}-\frac {2\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )}{27\,b^4}+\frac {x^3\,{\mathrm {cosh}\left (a+b\,x\right )}^3}{3\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*cosh(a + b*x)^2*sinh(a + b*x),x)

[Out]

((4*x*cosh(a + b*x))/3 + (2*x*cosh(a + b*x)^3)/9)/b^3 - ((2*x^2*sinh(a + b*x))/3 + (x^2*cosh(a + b*x)^2*sinh(a
+ b*x))/3)/b^2 - (40*sinh(a + b*x))/(27*b^4) - (2*cosh(a + b*x)^2*sinh(a + b*x))/(27*b^4) + (x^3*cosh(a + b*x
)^3)/(3*b)

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