3.4.26 \(\int x^3 \cosh ^3(a+b x) \sinh ^3(a+b x) \, dx\) [326]

Optimal. Leaf size=143 \[ -\frac {9 x \cosh (2 a+2 b x)}{128 b^3}-\frac {3 x^3 \cosh (2 a+2 b x)}{64 b}+\frac {x \cosh (6 a+6 b x)}{1152 b^3}+\frac {x^3 \cosh (6 a+6 b x)}{192 b}+\frac {9 \sinh (2 a+2 b x)}{256 b^4}+\frac {9 x^2 \sinh (2 a+2 b x)}{128 b^2}-\frac {\sinh (6 a+6 b x)}{6912 b^4}-\frac {x^2 \sinh (6 a+6 b x)}{384 b^2} \]

[Out]

-9/128*x*cosh(2*b*x+2*a)/b^3-3/64*x^3*cosh(2*b*x+2*a)/b+1/1152*x*cosh(6*b*x+6*a)/b^3+1/192*x^3*cosh(6*b*x+6*a)
/b+9/256*sinh(2*b*x+2*a)/b^4+9/128*x^2*sinh(2*b*x+2*a)/b^2-1/6912*sinh(6*b*x+6*a)/b^4-1/384*x^2*sinh(6*b*x+6*a
)/b^2

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Rubi [A]
time = 0.15, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {5556, 3377, 2717} \begin {gather*} \frac {9 \sinh (2 a+2 b x)}{256 b^4}-\frac {\sinh (6 a+6 b x)}{6912 b^4}-\frac {9 x \cosh (2 a+2 b x)}{128 b^3}+\frac {x \cosh (6 a+6 b x)}{1152 b^3}+\frac {9 x^2 \sinh (2 a+2 b x)}{128 b^2}-\frac {x^2 \sinh (6 a+6 b x)}{384 b^2}-\frac {3 x^3 \cosh (2 a+2 b x)}{64 b}+\frac {x^3 \cosh (6 a+6 b x)}{192 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-9*x*Cosh[2*a + 2*b*x])/(128*b^3) - (3*x^3*Cosh[2*a + 2*b*x])/(64*b) + (x*Cosh[6*a + 6*b*x])/(1152*b^3) + (x^
3*Cosh[6*a + 6*b*x])/(192*b) + (9*Sinh[2*a + 2*b*x])/(256*b^4) + (9*x^2*Sinh[2*a + 2*b*x])/(128*b^2) - Sinh[6*
a + 6*b*x]/(6912*b^4) - (x^2*Sinh[6*a + 6*b*x])/(384*b^2)

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 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^3 \cosh ^3(a+b x) \sinh ^3(a+b x) \, dx &=\int \left (-\frac {3}{32} x^3 \sinh (2 a+2 b x)+\frac {1}{32} x^3 \sinh (6 a+6 b x)\right ) \, dx\\ &=\frac {1}{32} \int x^3 \sinh (6 a+6 b x) \, dx-\frac {3}{32} \int x^3 \sinh (2 a+2 b x) \, dx\\ &=-\frac {3 x^3 \cosh (2 a+2 b x)}{64 b}+\frac {x^3 \cosh (6 a+6 b x)}{192 b}-\frac {\int x^2 \cosh (6 a+6 b x) \, dx}{64 b}+\frac {9 \int x^2 \cosh (2 a+2 b x) \, dx}{64 b}\\ &=-\frac {3 x^3 \cosh (2 a+2 b x)}{64 b}+\frac {x^3 \cosh (6 a+6 b x)}{192 b}+\frac {9 x^2 \sinh (2 a+2 b x)}{128 b^2}-\frac {x^2 \sinh (6 a+6 b x)}{384 b^2}+\frac {\int x \sinh (6 a+6 b x) \, dx}{192 b^2}-\frac {9 \int x \sinh (2 a+2 b x) \, dx}{64 b^2}\\ &=-\frac {9 x \cosh (2 a+2 b x)}{128 b^3}-\frac {3 x^3 \cosh (2 a+2 b x)}{64 b}+\frac {x \cosh (6 a+6 b x)}{1152 b^3}+\frac {x^3 \cosh (6 a+6 b x)}{192 b}+\frac {9 x^2 \sinh (2 a+2 b x)}{128 b^2}-\frac {x^2 \sinh (6 a+6 b x)}{384 b^2}-\frac {\int \cosh (6 a+6 b x) \, dx}{1152 b^3}+\frac {9 \int \cosh (2 a+2 b x) \, dx}{128 b^3}\\ &=-\frac {9 x \cosh (2 a+2 b x)}{128 b^3}-\frac {3 x^3 \cosh (2 a+2 b x)}{64 b}+\frac {x \cosh (6 a+6 b x)}{1152 b^3}+\frac {x^3 \cosh (6 a+6 b x)}{192 b}+\frac {9 \sinh (2 a+2 b x)}{256 b^4}+\frac {9 x^2 \sinh (2 a+2 b x)}{128 b^2}-\frac {\sinh (6 a+6 b x)}{6912 b^4}-\frac {x^2 \sinh (6 a+6 b x)}{384 b^2}\\ \end {align*}

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Mathematica [A]
time = 0.59, size = 90, normalized size = 0.63 \begin {gather*} -\frac {81 b x \left (3+2 b^2 x^2\right ) \cosh (2 (a+b x))-3 \left (b x+6 b^3 x^3\right ) \cosh (6 (a+b x))+\left (-121-234 b^2 x^2+\left (1+18 b^2 x^2\right ) \cosh (4 (a+b x))\right ) \sinh (2 (a+b x))}{3456 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3456*(81*b*x*(3 + 2*b^2*x^2)*Cosh[2*(a + b*x)] - 3*(b*x + 6*b^3*x^3)*Cosh[6*(a + b*x)] + (-121 - 234*b^2*x^
2 + (1 + 18*b^2*x^2)*Cosh[4*(a + b*x)])*Sinh[2*(a + b*x)])/b^4

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(361\) vs. \(2(127)=254\).
time = 1.20, size = 362, normalized size = 2.53

method	result	size
risch	\(\frac {\left (36 b^{3} x^{3}-18 b^{2} x^{2}+6 b x -1\right ) {\mathrm e}^{6 b x +6 a}}{13824 b^{4}}-\frac {3 \left (4 b^{3} x^{3}-6 b^{2} x^{2}+6 b x -3\right ) {\mathrm e}^{2 b x +2 a}}{512 b^{4}}-\frac {3 \left (4 b^{3} x^{3}+6 b^{2} x^{2}+6 b x +3\right ) {\mathrm e}^{-2 b x -2 a}}{512 b^{4}}+\frac {\left (36 b^{3} x^{3}+18 b^{2} x^{2}+6 b x +1\right ) {\mathrm e}^{-6 b x -6 a}}{13824 b^{4}}\)	\(146\)
default	\(-\frac {3 \left (\left (2 b x +2 a \right )^{3} \cosh \left (2 b x +2 a \right )-3 \left (2 b x +2 a \right )^{2} \sinh \left (2 b x +2 a \right )+6 \left (2 b x +2 a \right ) \cosh \left (2 b x +2 a \right )-6 \sinh \left (2 b x +2 a \right )-6 a \left (\left (2 b x +2 a \right )^{2} \cosh \left (2 b x +2 a \right )-2 \left (2 b x +2 a \right ) \sinh \left (2 b x +2 a \right )+2 \cosh \left (2 b x +2 a \right )\right )+12 a^{2} \left (\left (2 b x +2 a \right ) \cosh \left (2 b x +2 a \right )-\sinh \left (2 b x +2 a \right )\right )-8 a^{3} \cosh \left (2 b x +2 a \right )\right )}{512 b^{4}}+\frac {\left (6 b x +6 a \right )^{3} \cosh \left (6 b x +6 a \right )-3 \left (6 b x +6 a \right )^{2} \sinh \left (6 b x +6 a \right )+6 \left (6 b x +6 a \right ) \cosh \left (6 b x +6 a \right )-6 \sinh \left (6 b x +6 a \right )-18 a \left (\left (6 b x +6 a \right )^{2} \cosh \left (6 b x +6 a \right )-2 \left (6 b x +6 a \right ) \sinh \left (6 b x +6 a \right )+2 \cosh \left (6 b x +6 a \right )\right )+108 a^{2} \left (\left (6 b x +6 a \right ) \cosh \left (6 b x +6 a \right )-\sinh \left (6 b x +6 a \right )\right )-216 a^{3} \cosh \left (6 b x +6 a \right )}{41472 b^{4}}\)	\(362\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/512/b^4*((2*b*x+2*a)^3*cosh(2*b*x+2*a)-3*(2*b*x+2*a)^2*sinh(2*b*x+2*a)+6*(2*b*x+2*a)*cosh(2*b*x+2*a)-6*sinh
(2*b*x+2*a)-6*a*((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))+12*a^2*((2*b*x
+2*a)*cosh(2*b*x+2*a)-sinh(2*b*x+2*a))-8*a^3*cosh(2*b*x+2*a))+1/41472/b^4*((6*b*x+6*a)^3*cosh(6*b*x+6*a)-3*(6*
b*x+6*a)^2*sinh(6*b*x+6*a)+6*(6*b*x+6*a)*cosh(6*b*x+6*a)-6*sinh(6*b*x+6*a)-18*a*((6*b*x+6*a)^2*cosh(6*b*x+6*a)
-2*(6*b*x+6*a)*sinh(6*b*x+6*a)+2*cosh(6*b*x+6*a))+108*a^2*((6*b*x+6*a)*cosh(6*b*x+6*a)-sinh(6*b*x+6*a))-216*a^
3*cosh(6*b*x+6*a))

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Maxima [A]
time = 0.27, size = 171, normalized size = 1.20 \begin {gather*} \frac {{\left (36 \, b^{3} x^{3} e^{\left (6 \, a\right )} - 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 )}}{13824 \, b^{4}} - \frac {3 \, {\left (4 \, b^{3} x^{3} e^{\left (2 \, a\right )} - 6 \, b^{2} x^{2} e^{\left (2 \, a\right )} + 6 \, b x e^{\left (2 \, a\right )} - 3 \, e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{512 \, b^{4}} - \frac {3 \, {\left (4 \, b^{3} x^{3} + 6 \, b^{2} x^{2} + 6 \, b x + 3\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{512 \, b^{4}} + \frac {{\left (36 \, b^{3} x^{3} + 18 \, b^{2} x^{2} + 6 \, b x + 1\right )} e^{\left (-6 \, b x - 6 \, a\right )}}{13824 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13824*(36*b^3*x^3*e^(6*a) - 18*b^2*x^2*e^(6*a) + 6*b*x*e^(6*a) - e^(6*a))*e^(6*b*x)/b^4 - 3/512*(4*b^3*x^3*e
^(2*a) - 6*b^2*x^2*e^(2*a) + 6*b*x*e^(2*a) - 3*e^(2*a))*e^(2*b*x)/b^4 - 3/512*(4*b^3*x^3 + 6*b^2*x^2 + 6*b*x +
3)*e^(-2*b*x - 2*a)/b^4 + 1/13824*(36*b^3*x^3 + 18*b^2*x^2 + 6*b*x + 1)*e^(-6*b*x - 6*a)/b^4

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Fricas [A]
time = 0.36, size = 248, normalized size = 1.73 \begin {gather*} \frac {3 \, {\left (6 \, b^{3} x^{3} + b x\right )} \cosh \left (b x + a\right )^{6} - 10 \, {\left (18 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right )^{3} + 45 \, {\left (6 \, b^{3} x^{3} + b x\right )} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{4} - 3 \, {\left (18 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + 3 \, {\left (6 \, b^{3} x^{3} + b x\right )} \sinh \left (b x + a\right )^{6} - 81 \, {\left (2 \, b^{3} x^{3} + 3 \, b x\right )} \cosh \left (b x + a\right )^{2} - 9 \, {\left (18 \, b^{3} x^{3} - 5 \, {\left (6 \, b^{3} x^{3} + b x\right )} \cosh \left (b x + a\right )^{4} + 27 \, b x\right )} \sinh \left (b x + a\right )^{2} - 3 \, {\left ({\left (18 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{5} - 81 \, {\left (2 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{3456 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3456*(3*(6*b^3*x^3 + b*x)*cosh(b*x + a)^6 - 10*(18*b^2*x^2 + 1)*cosh(b*x + a)^3*sinh(b*x + a)^3 + 45*(6*b^3*
x^3 + b*x)*cosh(b*x + a)^2*sinh(b*x + a)^4 - 3*(18*b^2*x^2 + 1)*cosh(b*x + a)*sinh(b*x + a)^5 + 3*(6*b^3*x^3 +
b*x)*sinh(b*x + a)^6 - 81*(2*b^3*x^3 + 3*b*x)*cosh(b*x + a)^2 - 9*(18*b^3*x^3 - 5*(6*b^3*x^3 + b*x)*cosh(b*x
+ a)^4 + 27*b*x)*sinh(b*x + a)^2 - 3*((18*b^2*x^2 + 1)*cosh(b*x + a)^5 - 81*(2*b^2*x^2 + 1)*cosh(b*x + a))*sin
h(b*x + a))/b^4

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 314 vs. \(2 (141) = 282\).
time = 1.40, size = 314, normalized size = 2.20 \begin {gather*} \begin {cases} - \frac {x^{3} \sinh ^{6}{\left (a + b x \right )}}{24 b} + \frac {x^{3} \sinh ^{4}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{8 b} + \frac {x^{3} \sinh ^{2}{\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{8 b} - \frac {x^{3} \cosh ^{6}{\left (a + b x \right )}}{24 b} + \frac {x^{2} \sinh ^{5}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{8 b^{2}} - \frac {x^{2} \sinh ^{3}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{3 b^{2}} + \frac {x^{2} \sinh {\left (a + b x \right )} \cosh ^{5}{\left (a + b x \right )}}{8 b^{2}} - \frac {5 x \sinh ^{6}{\left (a + b x \right )}}{72 b^{3}} + \frac {x \sinh ^{4}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{12 b^{3}} + \frac {x \sinh ^{2}{\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{12 b^{3}} - \frac {5 x \cosh ^{6}{\left (a + b x \right )}}{72 b^{3}} + \frac {5 \sinh ^{5}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{72 b^{4}} - \frac {31 \sinh ^{3}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{216 b^{4}} + \frac {5 \sinh {\left (a + b x \right )} \cosh ^{5}{\left (a + b x \right )}}{72 b^{4}} & \text {for}\: b \neq 0 \\\frac {x^{4} \sinh ^{3}{\left (a \right )} \cosh ^{3}{\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)**3*sinh(b*x+a)**3,x)

[Out]

Piecewise((-x**3*sinh(a + b*x)**6/(24*b) + x**3*sinh(a + b*x)**4*cosh(a + b*x)**2/(8*b) + x**3*sinh(a + b*x)**
2*cosh(a + b*x)**4/(8*b) - x**3*cosh(a + b*x)**6/(24*b) + x**2*sinh(a + b*x)**5*cosh(a + b*x)/(8*b**2) - x**2*
sinh(a + b*x)**3*cosh(a + b*x)**3/(3*b**2) + x**2*sinh(a + b*x)*cosh(a + b*x)**5/(8*b**2) - 5*x*sinh(a + b*x)*
*6/(72*b**3) + x*sinh(a + b*x)**4*cosh(a + b*x)**2/(12*b**3) + x*sinh(a + b*x)**2*cosh(a + b*x)**4/(12*b**3) -
5*x*cosh(a + b*x)**6/(72*b**3) + 5*sinh(a + b*x)**5*cosh(a + b*x)/(72*b**4) - 31*sinh(a + b*x)**3*cosh(a + b*
x)**3/(216*b**4) + 5*sinh(a + b*x)*cosh(a + b*x)**5/(72*b**4), Ne(b, 0)), (x**4*sinh(a)**3*cosh(a)**3/4, True)
)

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Giac [A]
time = 0.40, size = 145, normalized size = 1.01 \begin {gather*} \frac {{\left (36 \, b^{3} x^{3} - 18 \, b^{2} x^{2} + 6 \, b x - 1\right )} e^{\left (6 \, b x + 6 \, a\right )}}{13824 \, b^{4}} - \frac {3 \, {\left (4 \, b^{3} x^{3} - 6 \, b^{2} x^{2} + 6 \, b x - 3\right )} e^{\left (2 \, b x + 2 \, a\right )}}{512 \, b^{4}} - \frac {3 \, {\left (4 \, b^{3} x^{3} + 6 \, b^{2} x^{2} + 6 \, b x + 3\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{512 \, b^{4}} + \frac {{\left (36 \, b^{3} x^{3} + 18 \, b^{2} x^{2} + 6 \, b x + 1\right )} e^{\left (-6 \, b x - 6 \, a\right )}}{13824 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13824*(36*b^3*x^3 - 18*b^2*x^2 + 6*b*x - 1)*e^(6*b*x + 6*a)/b^4 - 3/512*(4*b^3*x^3 - 6*b^2*x^2 + 6*b*x - 3)*
e^(2*b*x + 2*a)/b^4 - 3/512*(4*b^3*x^3 + 6*b^2*x^2 + 6*b*x + 3)*e^(-2*b*x - 2*a)/b^4 + 1/13824*(36*b^3*x^3 + 1
8*b^2*x^2 + 6*b*x + 1)*e^(-6*b*x - 6*a)/b^4

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Mupad [B]
time = 1.72, size = 126, normalized size = 0.88 \begin {gather*} \frac {\frac {9\,x^2\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{128}-\frac {x^2\,\mathrm {sinh}\left (6\,a+6\,b\,x\right )}{384}}{b^2}-\frac {\frac {3\,x^3\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{64}-\frac {x^3\,\mathrm {cosh}\left (6\,a+6\,b\,x\right )}{192}}{b}-\frac {\frac {9\,x\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{128}-\frac {x\,\mathrm {cosh}\left (6\,a+6\,b\,x\right )}{1152}}{b^3}+\frac {9\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{256\,b^4}-\frac {\mathrm {sinh}\left (6\,a+6\,b\,x\right )}{6912\,b^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((9*x^2*sinh(2*a + 2*b*x))/128 - (x^2*sinh(6*a + 6*b*x))/384)/b^2 - ((3*x^3*cosh(2*a + 2*b*x))/64 - (x^3*cosh(
6*a + 6*b*x))/192)/b - ((9*x*cosh(2*a + 2*b*x))/128 - (x*cosh(6*a + 6*b*x))/1152)/b^3 + (9*sinh(2*a + 2*b*x))/
(256*b^4) - sinh(6*a + 6*b*x)/(6912*b^4)

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