3.3.99 \(\int x^3 \cosh ^3(a+b x) \sinh ^2(a+b x) \, dx\) [299]

Optimal. Leaf size=202 \[ \frac {3 \cosh (a+b x)}{4 b^4}+\frac {3 x^2 \cosh (a+b x)}{8 b^2}-\frac {\cosh (3 a+3 b x)}{216 b^4}-\frac {x^2 \cosh (3 a+3 b x)}{48 b^2}-\frac {3 \cosh (5 a+5 b x)}{5000 b^4}-\frac {3 x^2 \cosh (5 a+5 b x)}{400 b^2}-\frac {3 x \sinh (a+b x)}{4 b^3}-\frac {x^3 \sinh (a+b x)}{8 b}+\frac {x \sinh (3 a+3 b x)}{72 b^3}+\frac {x^3 \sinh (3 a+3 b x)}{48 b}+\frac {3 x \sinh (5 a+5 b x)}{1000 b^3}+\frac {x^3 \sinh (5 a+5 b x)}{80 b} \]

[Out]

3/4*cosh(b*x+a)/b^4+3/8*x^2*cosh(b*x+a)/b^2-1/216*cosh(3*b*x+3*a)/b^4-1/48*x^2*cosh(3*b*x+3*a)/b^2-3/5000*cosh
(5*b*x+5*a)/b^4-3/400*x^2*cosh(5*b*x+5*a)/b^2-3/4*x*sinh(b*x+a)/b^3-1/8*x^3*sinh(b*x+a)/b+1/72*x*sinh(3*b*x+3*
a)/b^3+1/48*x^3*sinh(3*b*x+3*a)/b+3/1000*x*sinh(5*b*x+5*a)/b^3+1/80*x^3*sinh(5*b*x+5*a)/b

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Rubi [A]
time = 0.20, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 14, 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 (a+b x)}{4 b^4}-\frac {\cosh (3 a+3 b x)}{216 b^4}-\frac {3 \cosh (5 a+5 b x)}{5000 b^4}-\frac {3 x \sinh (a+b x)}{4 b^3}+\frac {x \sinh (3 a+3 b x)}{72 b^3}+\frac {3 x \sinh (5 a+5 b x)}{1000 b^3}+\frac {3 x^2 \cosh (a+b x)}{8 b^2}-\frac {x^2 \cosh (3 a+3 b x)}{48 b^2}-\frac {3 x^2 \cosh (5 a+5 b x)}{400 b^2}-\frac {x^3 \sinh (a+b x)}{8 b}+\frac {x^3 \sinh (3 a+3 b x)}{48 b}+\frac {x^3 \sinh (5 a+5 b x)}{80 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*Cosh[a + b*x])/(4*b^4) + (3*x^2*Cosh[a + b*x])/(8*b^2) - Cosh[3*a + 3*b*x]/(216*b^4) - (x^2*Cosh[3*a + 3*b*
x])/(48*b^2) - (3*Cosh[5*a + 5*b*x])/(5000*b^4) - (3*x^2*Cosh[5*a + 5*b*x])/(400*b^2) - (3*x*Sinh[a + b*x])/(4
*b^3) - (x^3*Sinh[a + b*x])/(8*b) + (x*Sinh[3*a + 3*b*x])/(72*b^3) + (x^3*Sinh[3*a + 3*b*x])/(48*b) + (3*x*Sin
h[5*a + 5*b*x])/(1000*b^3) + (x^3*Sinh[5*a + 5*b*x])/(80*b)

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^3 \cosh ^3(a+b x) \sinh ^2(a+b x) \, dx &=\int \left (-\frac {1}{8} x^3 \cosh (a+b x)+\frac {1}{16} x^3 \cosh (3 a+3 b x)+\frac {1}{16} x^3 \cosh (5 a+5 b x)\right ) \, dx\\ &=\frac {1}{16} \int x^3 \cosh (3 a+3 b x) \, dx+\frac {1}{16} \int x^3 \cosh (5 a+5 b x) \, dx-\frac {1}{8} \int x^3 \cosh (a+b x) \, dx\\ &=-\frac {x^3 \sinh (a+b x)}{8 b}+\frac {x^3 \sinh (3 a+3 b x)}{48 b}+\frac {x^3 \sinh (5 a+5 b x)}{80 b}-\frac {3 \int x^2 \sinh (5 a+5 b x) \, dx}{80 b}-\frac {\int x^2 \sinh (3 a+3 b x) \, dx}{16 b}+\frac {3 \int x^2 \sinh (a+b x) \, dx}{8 b}\\ &=\frac {3 x^2 \cosh (a+b x)}{8 b^2}-\frac {x^2 \cosh (3 a+3 b x)}{48 b^2}-\frac {3 x^2 \cosh (5 a+5 b x)}{400 b^2}-\frac {x^3 \sinh (a+b x)}{8 b}+\frac {x^3 \sinh (3 a+3 b x)}{48 b}+\frac {x^3 \sinh (5 a+5 b x)}{80 b}+\frac {3 \int x \cosh (5 a+5 b x) \, dx}{200 b^2}+\frac {\int x \cosh (3 a+3 b x) \, dx}{24 b^2}-\frac {3 \int x \cosh (a+b x) \, dx}{4 b^2}\\ &=\frac {3 x^2 \cosh (a+b x)}{8 b^2}-\frac {x^2 \cosh (3 a+3 b x)}{48 b^2}-\frac {3 x^2 \cosh (5 a+5 b x)}{400 b^2}-\frac {3 x \sinh (a+b x)}{4 b^3}-\frac {x^3 \sinh (a+b x)}{8 b}+\frac {x \sinh (3 a+3 b x)}{72 b^3}+\frac {x^3 \sinh (3 a+3 b x)}{48 b}+\frac {3 x \sinh (5 a+5 b x)}{1000 b^3}+\frac {x^3 \sinh (5 a+5 b x)}{80 b}-\frac {3 \int \sinh (5 a+5 b x) \, dx}{1000 b^3}-\frac {\int \sinh (3 a+3 b x) \, dx}{72 b^3}+\frac {3 \int \sinh (a+b x) \, dx}{4 b^3}\\ &=\frac {3 \cosh (a+b x)}{4 b^4}+\frac {3 x^2 \cosh (a+b x)}{8 b^2}-\frac {\cosh (3 a+3 b x)}{216 b^4}-\frac {x^2 \cosh (3 a+3 b x)}{48 b^2}-\frac {3 \cosh (5 a+5 b x)}{5000 b^4}-\frac {3 x^2 \cosh (5 a+5 b x)}{400 b^2}-\frac {3 x \sinh (a+b x)}{4 b^3}-\frac {x^3 \sinh (a+b x)}{8 b}+\frac {x \sinh (3 a+3 b x)}{72 b^3}+\frac {x^3 \sinh (3 a+3 b x)}{48 b}+\frac {3 x \sinh (5 a+5 b x)}{1000 b^3}+\frac {x^3 \sinh (5 a+5 b x)}{80 b}\\ \end {align*}

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Mathematica [A]
time = 0.77, size = 125, normalized size = 0.62 \begin {gather*} \frac {101250 \left (2+b^2 x^2\right ) \cosh (a+b x)-625 \left (2+9 b^2 x^2\right ) \cosh (3 (a+b x))-81 \left (2+25 b^2 x^2\right ) \cosh (5 (a+b x))+30 b x \left (-6598-825 b^2 x^2+8 \left (38+75 b^2 x^2\right ) \cosh (2 (a+b x))+9 \left (6+25 b^2 x^2\right ) \cosh (4 (a+b x))\right ) \sinh (a+b x)}{270000 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(101250*(2 + b^2*x^2)*Cosh[a + b*x] - 625*(2 + 9*b^2*x^2)*Cosh[3*(a + b*x)] - 81*(2 + 25*b^2*x^2)*Cosh[5*(a +
b*x)] + 30*b*x*(-6598 - 825*b^2*x^2 + 8*(38 + 75*b^2*x^2)*Cosh[2*(a + b*x)] + 9*(6 + 25*b^2*x^2)*Cosh[4*(a + b
*x)])*Sinh[a + b*x])/(270000*b^4)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(493\) vs. \(2(178)=356\).
time = 2.05, size = 494, normalized size = 2.45

method	result	size
risch	\(\frac {\left (125 b^{3} x^{3}-75 b^{2} x^{2}+30 b x -6\right ) {\mathrm e}^{5 b x +5 a}}{20000 b^{4}}+\frac {\left (9 b^{3} x^{3}-9 b^{2} x^{2}+6 b x -2\right ) {\mathrm e}^{3 b x +3 a}}{864 b^{4}}-\frac {\left (b^{3} x^{3}-3 b^{2} x^{2}+6 b x -6\right ) {\mathrm e}^{b x +a}}{16 b^{4}}+\frac {\left (b^{3} x^{3}+3 b^{2} x^{2}+6 b x +6\right ) {\mathrm e}^{-b x -a}}{16 b^{4}}-\frac {\left (9 b^{3} x^{3}+9 b^{2} x^{2}+6 b x +2\right ) {\mathrm e}^{-3 b x -3 a}}{864 b^{4}}-\frac {\left (125 b^{3} x^{3}+75 b^{2} x^{2}+30 b x +6\right ) {\mathrm e}^{-5 b x -5 a}}{20000 b^{4}}\)	\(213\)
default	\(-\frac {\left (b x +a \right )^{3} \sinh \left (b x +a \right )-3 \left (b x +a \right )^{2} \cosh \left (b x +a \right )+6 \left (b x +a \right ) \sinh \left (b x +a \right )-6 \cosh \left (b x +a \right )-3 a \left (\left (b x +a \right )^{2} \sinh \left (b x +a \right )-2 \left (b x +a \right ) \cosh \left (b x +a \right )+2 \sinh \left (b x +a \right )\right )+3 a^{2} \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )-a^{3} \sinh \left (b x +a \right )}{8 b^{4}}+\frac {\left (3 b x +3 a \right )^{3} \sinh \left (3 b x +3 a \right )-3 \left (3 b x +3 a \right )^{2} \cosh \left (3 b x +3 a \right )+6 \left (3 b x +3 a \right ) \sinh \left (3 b x +3 a \right )-6 \cosh \left (3 b x +3 a \right )-9 a \left (\left (3 b x +3 a \right )^{2} \sinh \left (3 b x +3 a \right )-2 \left (3 b x +3 a \right ) \cosh \left (3 b x +3 a \right )+2 \sinh \left (3 b x +3 a \right )\right )+27 a^{2} \left (\left (3 b x +3 a \right ) \sinh \left (3 b x +3 a \right )-\cosh \left (3 b x +3 a \right )\right )-27 a^{3} \sinh \left (3 b x +3 a \right )}{1296 b^{4}}+\frac {\left (5 b x +5 a \right )^{3} \sinh \left (5 b x +5 a \right )-3 \left (5 b x +5 a \right )^{2} \cosh \left (5 b x +5 a \right )+6 \left (5 b x +5 a \right ) \sinh \left (5 b x +5 a \right )-6 \cosh \left (5 b x +5 a \right )-15 a \left (\left (5 b x +5 a \right )^{2} \sinh \left (5 b x +5 a \right )-2 \left (5 b x +5 a \right ) \cosh \left (5 b x +5 a \right )+2 \sinh \left (5 b x +5 a \right )\right )+75 a^{2} \left (\left (5 b x +5 a \right ) \sinh \left (5 b x +5 a \right )-\cosh \left (5 b x +5 a \right )\right )-125 a^{3} \sinh \left (5 b x +5 a \right )}{10000 b^{4}}\)	\(494\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8/b^4*((b*x+a)^3*sinh(b*x+a)-3*(b*x+a)^2*cosh(b*x+a)+6*(b*x+a)*sinh(b*x+a)-6*cosh(b*x+a)-3*a*((b*x+a)^2*sin
h(b*x+a)-2*(b*x+a)*cosh(b*x+a)+2*sinh(b*x+a))+3*a^2*((b*x+a)*sinh(b*x+a)-cosh(b*x+a))-a^3*sinh(b*x+a))+1/1296/
b^4*((3*b*x+3*a)^3*sinh(3*b*x+3*a)-3*(3*b*x+3*a)^2*cosh(3*b*x+3*a)+6*(3*b*x+3*a)*sinh(3*b*x+3*a)-6*cosh(3*b*x+
3*a)-9*a*((3*b*x+3*a)^2*sinh(3*b*x+3*a)-2*(3*b*x+3*a)*cosh(3*b*x+3*a)+2*sinh(3*b*x+3*a))+27*a^2*((3*b*x+3*a)*s
inh(3*b*x+3*a)-cosh(3*b*x+3*a))-27*a^3*sinh(3*b*x+3*a))+1/10000/b^4*((5*b*x+5*a)^3*sinh(5*b*x+5*a)-3*(5*b*x+5*
a)^2*cosh(5*b*x+5*a)+6*(5*b*x+5*a)*sinh(5*b*x+5*a)-6*cosh(5*b*x+5*a)-15*a*((5*b*x+5*a)^2*sinh(5*b*x+5*a)-2*(5*
b*x+5*a)*cosh(5*b*x+5*a)+2*sinh(5*b*x+5*a))+75*a^2*((5*b*x+5*a)*sinh(5*b*x+5*a)-cosh(5*b*x+5*a))-125*a^3*sinh(
5*b*x+5*a))

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Maxima [A]
time = 0.27, size = 245, normalized size = 1.21 \begin {gather*} \frac {{\left (125 \, b^{3} x^{3} e^{\left (5 \, a\right )} - 75 \, b^{2} x^{2} e^{\left (5 \, a\right )} + 30 \, b x e^{\left (5 \, a\right )} - 6 \, e^{\left (5 \, a\right )}\right )} e^{\left (5 \, b x\right )}}{20000 \, b^{4}} + \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 )}}{864 \, 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 )}}{16 \, b^{4}} + \frac {{\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} e^{\left (-b x - a\right )}}{16 \, 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 )}}{864 \, b^{4}} - \frac {{\left (125 \, b^{3} x^{3} + 75 \, b^{2} x^{2} + 30 \, b x + 6\right )} e^{\left (-5 \, b x - 5 \, a\right )}}{20000 \, 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)^2,x, algorithm="maxima")

[Out]

1/20000*(125*b^3*x^3*e^(5*a) - 75*b^2*x^2*e^(5*a) + 30*b*x*e^(5*a) - 6*e^(5*a))*e^(5*b*x)/b^4 + 1/864*(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 - 1/16*(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 + 1/16*(b^3*x^3 + 3*b^2*x^2 + 6*b*x + 6)*e^(-b*x - a)/b^4 - 1/864*(9*b^3*x^3
+ 9*b^2*x^2 + 6*b*x + 2)*e^(-3*b*x - 3*a)/b^4 - 1/20000*(125*b^3*x^3 + 75*b^2*x^2 + 30*b*x + 6)*e^(-5*b*x - 5*
a)/b^4

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Fricas [A]
time = 0.39, size = 274, normalized size = 1.36 \begin {gather*} -\frac {81 \, {\left (25 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )^{5} + 405 \, {\left (25 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{4} - 135 \, {\left (25 \, b^{3} x^{3} + 6 \, b x\right )} \sinh \left (b x + a\right )^{5} + 625 \, {\left (9 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )^{3} - 75 \, {\left (75 \, b^{3} x^{3} + 18 \, {\left (25 \, b^{3} x^{3} + 6 \, b x\right )} \cosh \left (b x + a\right )^{2} + 50 \, b x\right )} \sinh \left (b x + a\right )^{3} + 15 \, {\left (54 \, {\left (25 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )^{3} + 125 \, {\left (9 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} - 101250 \, {\left (b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right ) + 225 \, {\left (150 \, b^{3} x^{3} - 3 \, {\left (25 \, b^{3} x^{3} + 6 \, b x\right )} \cosh \left (b x + a\right )^{4} - 25 \, {\left (3 \, b^{3} x^{3} + 2 \, b x\right )} \cosh \left (b x + a\right )^{2} + 900 \, b x\right )} \sinh \left (b x + a\right )}{270000 \, 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)^2,x, algorithm="fricas")

[Out]

-1/270000*(81*(25*b^2*x^2 + 2)*cosh(b*x + a)^5 + 405*(25*b^2*x^2 + 2)*cosh(b*x + a)*sinh(b*x + a)^4 - 135*(25*
b^3*x^3 + 6*b*x)*sinh(b*x + a)^5 + 625*(9*b^2*x^2 + 2)*cosh(b*x + a)^3 - 75*(75*b^3*x^3 + 18*(25*b^3*x^3 + 6*b
*x)*cosh(b*x + a)^2 + 50*b*x)*sinh(b*x + a)^3 + 15*(54*(25*b^2*x^2 + 2)*cosh(b*x + a)^3 + 125*(9*b^2*x^2 + 2)*
cosh(b*x + a))*sinh(b*x + a)^2 - 101250*(b^2*x^2 + 2)*cosh(b*x + a) + 225*(150*b^3*x^3 - 3*(25*b^3*x^3 + 6*b*x
)*cosh(b*x + a)^4 - 25*(3*b^3*x^3 + 2*b*x)*cosh(b*x + a)^2 + 900*b*x)*sinh(b*x + a))/b^4

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Sympy [A]
time = 0.97, size = 253, normalized size = 1.25 \begin {gather*} \begin {cases} - \frac {2 x^{3} \sinh ^{5}{\left (a + b x \right )}}{15 b} + \frac {x^{3} \sinh ^{3}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{3 b} + \frac {2 x^{2} \sinh ^{4}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{5 b^{2}} - \frac {13 x^{2} \sinh ^{2}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{15 b^{2}} + \frac {26 x^{2} \cosh ^{5}{\left (a + b x \right )}}{75 b^{2}} - \frac {856 x \sinh ^{5}{\left (a + b x \right )}}{1125 b^{3}} + \frac {338 x \sinh ^{3}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{225 b^{3}} - \frac {52 x \sinh {\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{75 b^{3}} + \frac {856 \sinh ^{4}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{1125 b^{4}} - \frac {5114 \sinh ^{2}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{3375 b^{4}} + \frac {12568 \cosh ^{5}{\left (a + b x \right )}}{16875 b^{4}} & \text {for}\: b \neq 0 \\\frac {x^{4} \sinh ^{2}{\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)**2,x)

[Out]

Piecewise((-2*x**3*sinh(a + b*x)**5/(15*b) + x**3*sinh(a + b*x)**3*cosh(a + b*x)**2/(3*b) + 2*x**2*sinh(a + b*
x)**4*cosh(a + b*x)/(5*b**2) - 13*x**2*sinh(a + b*x)**2*cosh(a + b*x)**3/(15*b**2) + 26*x**2*cosh(a + b*x)**5/
(75*b**2) - 856*x*sinh(a + b*x)**5/(1125*b**3) + 338*x*sinh(a + b*x)**3*cosh(a + b*x)**2/(225*b**3) - 52*x*sin
h(a + b*x)*cosh(a + b*x)**4/(75*b**3) + 856*sinh(a + b*x)**4*cosh(a + b*x)/(1125*b**4) - 5114*sinh(a + b*x)**2
*cosh(a + b*x)**3/(3375*b**4) + 12568*cosh(a + b*x)**5/(16875*b**4), Ne(b, 0)), (x**4*sinh(a)**2*cosh(a)**3/4,
True))

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Giac [A]
time = 0.40, size = 212, normalized size = 1.05 \begin {gather*} \frac {{\left (125 \, b^{3} x^{3} - 75 \, b^{2} x^{2} + 30 \, b x - 6\right )} e^{\left (5 \, b x + 5 \, a\right )}}{20000 \, 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 )}}{864 \, b^{4}} - \frac {{\left (b^{3} x^{3} - 3 \, b^{2} x^{2} + 6 \, b x - 6\right )} e^{\left (b x + a\right )}}{16 \, b^{4}} + \frac {{\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} e^{\left (-b x - a\right )}}{16 \, 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 )}}{864 \, b^{4}} - \frac {{\left (125 \, b^{3} x^{3} + 75 \, b^{2} x^{2} + 30 \, b x + 6\right )} e^{\left (-5 \, b x - 5 \, a\right )}}{20000 \, 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)^2,x, algorithm="giac")

[Out]

1/20000*(125*b^3*x^3 - 75*b^2*x^2 + 30*b*x - 6)*e^(5*b*x + 5*a)/b^4 + 1/864*(9*b^3*x^3 - 9*b^2*x^2 + 6*b*x - 2
)*e^(3*b*x + 3*a)/b^4 - 1/16*(b^3*x^3 - 3*b^2*x^2 + 6*b*x - 6)*e^(b*x + a)/b^4 + 1/16*(b^3*x^3 + 3*b^2*x^2 + 6
*b*x + 6)*e^(-b*x - a)/b^4 - 1/864*(9*b^3*x^3 + 9*b^2*x^2 + 6*b*x + 2)*e^(-3*b*x - 3*a)/b^4 - 1/20000*(125*b^3
*x^3 + 75*b^2*x^2 + 30*b*x + 6)*e^(-5*b*x - 5*a)/b^4

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Mupad [B]
time = 0.50, size = 167, normalized size = 0.83 \begin {gather*} \frac {\frac {x\,\mathrm {sinh}\left (3\,a+3\,b\,x\right )}{72}-\frac {3\,x\,\mathrm {sinh}\left (a+b\,x\right )}{4}+\frac {3\,x\,\mathrm {sinh}\left (5\,a+5\,b\,x\right )}{1000}}{b^3}+\frac {\frac {x^3\,\mathrm {sinh}\left (3\,a+3\,b\,x\right )}{48}+\frac {x^3\,\mathrm {sinh}\left (5\,a+5\,b\,x\right )}{80}-\frac {x^3\,\mathrm {sinh}\left (a+b\,x\right )}{8}}{b}+\frac {3\,\mathrm {cosh}\left (a+b\,x\right )}{4\,b^4}-\frac {\mathrm {cosh}\left (3\,a+3\,b\,x\right )}{216\,b^4}-\frac {3\,\mathrm {cosh}\left (5\,a+5\,b\,x\right )}{5000\,b^4}-\frac {\frac {x^2\,\mathrm {cosh}\left (3\,a+3\,b\,x\right )}{48}-\frac {3\,x^2\,\mathrm {cosh}\left (a+b\,x\right )}{8}+\frac {3\,x^2\,\mathrm {cosh}\left (5\,a+5\,b\,x\right )}{400}}{b^2} \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)^2,x)

[Out]

((x*sinh(3*a + 3*b*x))/72 - (3*x*sinh(a + b*x))/4 + (3*x*sinh(5*a + 5*b*x))/1000)/b^3 + ((x^3*sinh(3*a + 3*b*x
))/48 + (x^3*sinh(5*a + 5*b*x))/80 - (x^3*sinh(a + b*x))/8)/b + (3*cosh(a + b*x))/(4*b^4) - cosh(3*a + 3*b*x)/
(216*b^4) - (3*cosh(5*a + 5*b*x))/(5000*b^4) - ((x^2*cosh(3*a + 3*b*x))/48 - (3*x^2*cosh(a + b*x))/8 + (3*x^2*
cosh(5*a + 5*b*x))/400)/b^2

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