3.4.27 \(\int x^2 \cosh ^3(a+b x) \sinh ^3(a+b x) \, dx\) [327]

Optimal. Leaf size=105 \[ -\frac {3 \cosh (2 a+2 b x)}{128 b^3}-\frac {3 x^2 \cosh (2 a+2 b x)}{64 b}+\frac {\cosh (6 a+6 b x)}{3456 b^3}+\frac {x^2 \cosh (6 a+6 b x)}{192 b}+\frac {3 x \sinh (2 a+2 b x)}{64 b^2}-\frac {x \sinh (6 a+6 b x)}{576 b^2} \]

[Out]

-3/128*cosh(2*b*x+2*a)/b^3-3/64*x^2*cosh(2*b*x+2*a)/b+1/3456*cosh(6*b*x+6*a)/b^3+1/192*x^2*cosh(6*b*x+6*a)/b+3
/64*x*sinh(2*b*x+2*a)/b^2-1/576*x*sinh(6*b*x+6*a)/b^2

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Rubi [A]
time = 0.11, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {5556, 3377, 2718} \begin {gather*} -\frac {3 \cosh (2 a+2 b x)}{128 b^3}+\frac {\cosh (6 a+6 b x)}{3456 b^3}+\frac {3 x \sinh (2 a+2 b x)}{64 b^2}-\frac {x \sinh (6 a+6 b x)}{576 b^2}-\frac {3 x^2 \cosh (2 a+2 b x)}{64 b}+\frac {x^2 \cosh (6 a+6 b x)}{192 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*Cosh[2*a + 2*b*x])/(128*b^3) - (3*x^2*Cosh[2*a + 2*b*x])/(64*b) + Cosh[6*a + 6*b*x]/(3456*b^3) + (x^2*Cosh
[6*a + 6*b*x])/(192*b) + (3*x*Sinh[2*a + 2*b*x])/(64*b^2) - (x*Sinh[6*a + 6*b*x])/(576*b^2)

Rule 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[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 5556

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^2 \cosh ^3(a+b x) \sinh ^3(a+b x) \, dx &=\int \left (-\frac {3}{32} x^2 \sinh (2 a+2 b x)+\frac {1}{32} x^2 \sinh (6 a+6 b x)\right ) \, dx\\ &=\frac {1}{32} \int x^2 \sinh (6 a+6 b x) \, dx-\frac {3}{32} \int x^2 \sinh (2 a+2 b x) \, dx\\ &=-\frac {3 x^2 \cosh (2 a+2 b x)}{64 b}+\frac {x^2 \cosh (6 a+6 b x)}{192 b}-\frac {\int x \cosh (6 a+6 b x) \, dx}{96 b}+\frac {3 \int x \cosh (2 a+2 b x) \, dx}{32 b}\\ &=-\frac {3 x^2 \cosh (2 a+2 b x)}{64 b}+\frac {x^2 \cosh (6 a+6 b x)}{192 b}+\frac {3 x \sinh (2 a+2 b x)}{64 b^2}-\frac {x \sinh (6 a+6 b x)}{576 b^2}+\frac {\int \sinh (6 a+6 b x) \, dx}{576 b^2}-\frac {3 \int \sinh (2 a+2 b x) \, dx}{64 b^2}\\ &=-\frac {3 \cosh (2 a+2 b x)}{128 b^3}-\frac {3 x^2 \cosh (2 a+2 b x)}{64 b}+\frac {\cosh (6 a+6 b x)}{3456 b^3}+\frac {x^2 \cosh (6 a+6 b x)}{192 b}+\frac {3 x \sinh (2 a+2 b x)}{64 b^2}-\frac {x \sinh (6 a+6 b x)}{576 b^2}\\ \end {align*}

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Mathematica [A]
time = 0.14, size = 72, normalized size = 0.69 \begin {gather*} \frac {-81 \left (1+2 b^2 x^2\right ) \cosh (2 (a+b x))+\left (1+18 b^2 x^2\right ) \cosh (6 (a+b x))+6 b x (27 \sinh (2 (a+b x))-\sinh (6 (a+b x)))}{3456 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-81*(1 + 2*b^2*x^2)*Cosh[2*(a + b*x)] + (1 + 18*b^2*x^2)*Cosh[6*(a + b*x)] + 6*b*x*(27*Sinh[2*(a + b*x)] - Si
nh[6*(a + b*x)]))/(3456*b^3)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(207\) vs. \(2(93)=186\).
time = 1.20, size = 208, normalized size = 1.98

method	result	size
risch	\(\frac {\left (18 b^{2} x^{2}-6 b x +1\right ) {\mathrm e}^{6 b x +6 a}}{6912 b^{3}}-\frac {3 \left (2 b^{2} x^{2}-2 b x +1\right ) {\mathrm e}^{2 b x +2 a}}{256 b^{3}}-\frac {3 \left (2 b^{2} x^{2}+2 b x +1\right ) {\mathrm e}^{-2 b x -2 a}}{256 b^{3}}+\frac {\left (18 b^{2} x^{2}+6 b x +1\right ) {\mathrm e}^{-6 b x -6 a}}{6912 b^{3}}\)	\(114\)
default	\(-\frac {3 \left (\left (2 b x +2 a \right )^{2} \cosh \left (2 b x +2 a \right )-2 \left (2 b x +2 a \right ) \sinh \left (2 b x +2 a \right )+2 \cosh \left (2 b x +2 a \right )-4 a \left (\left (2 b x +2 a \right ) \cosh \left (2 b x +2 a \right )-\sinh \left (2 b x +2 a \right )\right )+4 a^{2} \cosh \left (2 b x +2 a \right )\right )}{256 b^{3}}+\frac {\left (6 b x +6 a \right )^{2} \cosh \left (6 b x +6 a \right )-2 \left (6 b x +6 a \right ) \sinh \left (6 b x +6 a \right )+2 \cosh \left (6 b x +6 a \right )-12 a \left (\left (6 b x +6 a \right ) \cosh \left (6 b x +6 a \right )-\sinh \left (6 b x +6 a \right )\right )+36 a^{2} \cosh \left (6 b x +6 a \right )}{6912 b^{3}}\)	\(208\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/256/b^3*((2*b*x+2*a)^2*cosh(2*b*x+2*a)-2*(2*b*x+2*a)*sinh(2*b*x+2*a)+2*cosh(2*b*x+2*a)-4*a*((2*b*x+2*a)*cos
h(2*b*x+2*a)-sinh(2*b*x+2*a))+4*a^2*cosh(2*b*x+2*a))+1/6912/b^3*((6*b*x+6*a)^2*cosh(6*b*x+6*a)-2*(6*b*x+6*a)*s
inh(6*b*x+6*a)+2*cosh(6*b*x+6*a)-12*a*((6*b*x+6*a)*cosh(6*b*x+6*a)-sinh(6*b*x+6*a))+36*a^2*cosh(6*b*x+6*a))

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Maxima [A]
time = 0.28, size = 127, normalized size = 1.21 \begin {gather*} \frac {{\left (18 \, b^{2} x^{2} e^{\left (6 \, a\right )} - 6 \, b x e^{\left (6 \, a\right )} + e^{\left (6 \, a\right )}\right )} e^{\left (6 \, b x\right )}}{6912 \, b^{3}} - \frac {3 \, {\left (2 \, b^{2} x^{2} e^{\left (2 \, a\right )} - 2 \, b x e^{\left (2 \, a\right )} + e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{256 \, b^{3}} - \frac {3 \, {\left (2 \, b^{2} x^{2} + 2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{256 \, b^{3}} + \frac {{\left (18 \, b^{2} x^{2} + 6 \, b x + 1\right )} e^{\left (-6 \, b x - 6 \, a\right )}}{6912 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6912*(18*b^2*x^2*e^(6*a) - 6*b*x*e^(6*a) + e^(6*a))*e^(6*b*x)/b^3 - 3/256*(2*b^2*x^2*e^(2*a) - 2*b*x*e^(2*a)
+ e^(2*a))*e^(2*b*x)/b^3 - 3/256*(2*b^2*x^2 + 2*b*x + 1)*e^(-2*b*x - 2*a)/b^3 + 1/6912*(18*b^2*x^2 + 6*b*x +
1)*e^(-6*b*x - 6*a)/b^3

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 202 vs. \(2 (93) = 186\).
time = 0.40, size = 202, normalized size = 1.92 \begin {gather*} -\frac {120 \, b x \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right )^{3} + 36 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} - {\left (18 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{6} - 15 \, {\left (18 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{4} - {\left (18 \, b^{2} x^{2} + 1\right )} \sinh \left (b x + a\right )^{6} + 81 \, {\left (2 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{2} - 3 \, {\left (5 \, {\left (18 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{4} - 54 \, b^{2} x^{2} - 27\right )} \sinh \left (b x + a\right )^{2} + 36 \, {\left (b x \cosh \left (b x + a\right )^{5} - 9 \, b x \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{3456 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3456*(120*b*x*cosh(b*x + a)^3*sinh(b*x + a)^3 + 36*b*x*cosh(b*x + a)*sinh(b*x + a)^5 - (18*b^2*x^2 + 1)*cos
h(b*x + a)^6 - 15*(18*b^2*x^2 + 1)*cosh(b*x + a)^2*sinh(b*x + a)^4 - (18*b^2*x^2 + 1)*sinh(b*x + a)^6 + 81*(2*
b^2*x^2 + 1)*cosh(b*x + a)^2 - 3*(5*(18*b^2*x^2 + 1)*cosh(b*x + a)^4 - 54*b^2*x^2 - 27)*sinh(b*x + a)^2 + 36*(
b*x*cosh(b*x + a)^5 - 9*b*x*cosh(b*x + a))*sinh(b*x + a))/b^3

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 212 vs. \(2 (102) = 204\).
time = 1.00, size = 212, normalized size = 2.02 \begin {gather*} \begin {cases} - \frac {x^{2} \sinh ^{6}{\left (a + b x \right )}}{24 b} + \frac {x^{2} \sinh ^{4}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{8 b} + \frac {x^{2} \sinh ^{2}{\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{8 b} - \frac {x^{2} \cosh ^{6}{\left (a + b x \right )}}{24 b} + \frac {x \sinh ^{5}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{12 b^{2}} - \frac {2 x \sinh ^{3}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{9 b^{2}} + \frac {x \sinh {\left (a + b x \right )} \cosh ^{5}{\left (a + b x \right )}}{12 b^{2}} - \frac {7 \sinh ^{6}{\left (a + b x \right )}}{216 b^{3}} + \frac {\sinh ^{4}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{18 b^{3}} - \frac {\cosh ^{6}{\left (a + b x \right )}}{72 b^{3}} & \text {for}\: b \neq 0 \\\frac {x^{3} \sinh ^{3}{\left (a \right )} \cosh ^{3}{\left (a \right )}}{3} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-x**2*sinh(a + b*x)**6/(24*b) + x**2*sinh(a + b*x)**4*cosh(a + b*x)**2/(8*b) + x**2*sinh(a + b*x)**
2*cosh(a + b*x)**4/(8*b) - x**2*cosh(a + b*x)**6/(24*b) + x*sinh(a + b*x)**5*cosh(a + b*x)/(12*b**2) - 2*x*sin
h(a + b*x)**3*cosh(a + b*x)**3/(9*b**2) + x*sinh(a + b*x)*cosh(a + b*x)**5/(12*b**2) - 7*sinh(a + b*x)**6/(216
*b**3) + sinh(a + b*x)**4*cosh(a + b*x)**2/(18*b**3) - cosh(a + b*x)**6/(72*b**3), Ne(b, 0)), (x**3*sinh(a)**3
*cosh(a)**3/3, True))

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Giac [A]
time = 0.39, size = 113, normalized size = 1.08 \begin {gather*} \frac {{\left (18 \, b^{2} x^{2} - 6 \, b x + 1\right )} e^{\left (6 \, b x + 6 \, a\right )}}{6912 \, b^{3}} - \frac {3 \, {\left (2 \, b^{2} x^{2} - 2 \, b x + 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{256 \, b^{3}} - \frac {3 \, {\left (2 \, b^{2} x^{2} + 2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{256 \, b^{3}} + \frac {{\left (18 \, b^{2} x^{2} + 6 \, b x + 1\right )} e^{\left (-6 \, b x - 6 \, a\right )}}{6912 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6912*(18*b^2*x^2 - 6*b*x + 1)*e^(6*b*x + 6*a)/b^3 - 3/256*(2*b^2*x^2 - 2*b*x + 1)*e^(2*b*x + 2*a)/b^3 - 3/25
6*(2*b^2*x^2 + 2*b*x + 1)*e^(-2*b*x - 2*a)/b^3 + 1/6912*(18*b^2*x^2 + 6*b*x + 1)*e^(-6*b*x - 6*a)/b^3

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Mupad [B]
time = 0.32, size = 89, normalized size = 0.85 \begin {gather*} -\frac {\frac {3\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{128}-\frac {\mathrm {cosh}\left (6\,a+6\,b\,x\right )}{3456}+b^2\,\left (\frac {3\,x^2\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{64}-\frac {x^2\,\mathrm {cosh}\left (6\,a+6\,b\,x\right )}{192}\right )-b\,\left (\frac {3\,x\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{64}-\frac {x\,\mathrm {sinh}\left (6\,a+6\,b\,x\right )}{576}\right )}{b^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((3*cosh(2*a + 2*b*x))/128 - cosh(6*a + 6*b*x)/3456 + b^2*((3*x^2*cosh(2*a + 2*b*x))/64 - (x^2*cosh(6*a + 6*b
*x))/192) - b*((3*x*sinh(2*a + 2*b*x))/64 - (x*sinh(6*a + 6*b*x))/576))/b^3

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