3.16 \(\int (c+d x)^4 \cosh ^3(a+b x) \, dx\)

Optimal. Leaf size=225 \[ \frac{80 d^2 (c+d x)^2 \sinh (a+b x)}{9 b^3}-\frac{8 d^3 (c+d x) \cosh ^3(a+b x)}{27 b^4}-\frac{160 d^3 (c+d x) \cosh (a+b x)}{9 b^4}+\frac{4 d^2 (c+d x)^2 \sinh (a+b x) \cosh ^2(a+b x)}{9 b^3}-\frac{4 d (c+d x)^3 \cosh ^3(a+b x)}{9 b^2}-\frac{8 d (c+d x)^3 \cosh (a+b x)}{3 b^2}+\frac{8 d^4 \sinh ^3(a+b x)}{81 b^5}+\frac{488 d^4 \sinh (a+b x)}{27 b^5}+\frac{2 (c+d x)^4 \sinh (a+b x)}{3 b}+\frac{(c+d x)^4 \sinh (a+b x) \cosh ^2(a+b x)}{3 b} \]

[Out]

(-160*d^3*(c + d*x)*Cosh[a + b*x])/(9*b^4) - (8*d*(c + d*x)^3*Cosh[a + b*x])/(3*b^2) - (8*d^3*(c + d*x)*Cosh[a
+ b*x]^3)/(27*b^4) - (4*d*(c + d*x)^3*Cosh[a + b*x]^3)/(9*b^2) + (488*d^4*Sinh[a + b*x])/(27*b^5) + (80*d^2*(
c + d*x)^2*Sinh[a + b*x])/(9*b^3) + (2*(c + d*x)^4*Sinh[a + b*x])/(3*b) + (4*d^2*(c + d*x)^2*Cosh[a + b*x]^2*S
inh[a + b*x])/(9*b^3) + ((c + d*x)^4*Cosh[a + b*x]^2*Sinh[a + b*x])/(3*b) + (8*d^4*Sinh[a + b*x]^3)/(81*b^5)

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Rubi [A]  time = 0.281545, antiderivative size = 225, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3311, 3296, 2637, 2633} \[ \frac{80 d^2 (c+d x)^2 \sinh (a+b x)}{9 b^3}-\frac{8 d^3 (c+d x) \cosh ^3(a+b x)}{27 b^4}-\frac{160 d^3 (c+d x) \cosh (a+b x)}{9 b^4}+\frac{4 d^2 (c+d x)^2 \sinh (a+b x) \cosh ^2(a+b x)}{9 b^3}-\frac{4 d (c+d x)^3 \cosh ^3(a+b x)}{9 b^2}-\frac{8 d (c+d x)^3 \cosh (a+b x)}{3 b^2}+\frac{8 d^4 \sinh ^3(a+b x)}{81 b^5}+\frac{488 d^4 \sinh (a+b x)}{27 b^5}+\frac{2 (c+d x)^4 \sinh (a+b x)}{3 b}+\frac{(c+d x)^4 \sinh (a+b x) \cosh ^2(a+b x)}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^4*Cosh[a + b*x]^3,x]

[Out]

(-160*d^3*(c + d*x)*Cosh[a + b*x])/(9*b^4) - (8*d*(c + d*x)^3*Cosh[a + b*x])/(3*b^2) - (8*d^3*(c + d*x)*Cosh[a
+ b*x]^3)/(27*b^4) - (4*d*(c + d*x)^3*Cosh[a + b*x]^3)/(9*b^2) + (488*d^4*Sinh[a + b*x])/(27*b^5) + (80*d^2*(
c + d*x)^2*Sinh[a + b*x])/(9*b^3) + (2*(c + d*x)^4*Sinh[a + b*x])/(3*b) + (4*d^2*(c + d*x)^2*Cosh[a + b*x]^2*S
inh[a + b*x])/(9*b^3) + ((c + d*x)^4*Cosh[a + b*x]^2*Sinh[a + b*x])/(3*b) + (8*d^4*Sinh[a + b*x]^3)/(81*b^5)

Rule 3311

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 3296

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 2637

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

Rule 2633

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]

Rubi steps

\begin{align*} \int (c+d x)^4 \cosh ^3(a+b x) \, dx &=-\frac{4 d (c+d x)^3 \cosh ^3(a+b x)}{9 b^2}+\frac{(c+d x)^4 \cosh ^2(a+b x) \sinh (a+b x)}{3 b}+\frac{2}{3} \int (c+d x)^4 \cosh (a+b x) \, dx+\frac{\left (4 d^2\right ) \int (c+d x)^2 \cosh ^3(a+b x) \, dx}{3 b^2}\\ &=-\frac{8 d^3 (c+d x) \cosh ^3(a+b x)}{27 b^4}-\frac{4 d (c+d x)^3 \cosh ^3(a+b x)}{9 b^2}+\frac{2 (c+d x)^4 \sinh (a+b x)}{3 b}+\frac{4 d^2 (c+d x)^2 \cosh ^2(a+b x) \sinh (a+b x)}{9 b^3}+\frac{(c+d x)^4 \cosh ^2(a+b x) \sinh (a+b x)}{3 b}-\frac{(8 d) \int (c+d x)^3 \sinh (a+b x) \, dx}{3 b}+\frac{\left (8 d^2\right ) \int (c+d x)^2 \cosh (a+b x) \, dx}{9 b^2}+\frac{\left (8 d^4\right ) \int \cosh ^3(a+b x) \, dx}{27 b^4}\\ &=-\frac{8 d (c+d x)^3 \cosh (a+b x)}{3 b^2}-\frac{8 d^3 (c+d x) \cosh ^3(a+b x)}{27 b^4}-\frac{4 d (c+d x)^3 \cosh ^3(a+b x)}{9 b^2}+\frac{8 d^2 (c+d x)^2 \sinh (a+b x)}{9 b^3}+\frac{2 (c+d x)^4 \sinh (a+b x)}{3 b}+\frac{4 d^2 (c+d x)^2 \cosh ^2(a+b x) \sinh (a+b x)}{9 b^3}+\frac{(c+d x)^4 \cosh ^2(a+b x) \sinh (a+b x)}{3 b}+\frac{\left (8 d^2\right ) \int (c+d x)^2 \cosh (a+b x) \, dx}{b^2}-\frac{\left (16 d^3\right ) \int (c+d x) \sinh (a+b x) \, dx}{9 b^3}+\frac{\left (8 i d^4\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-i \sinh (a+b x)\right )}{27 b^5}\\ &=-\frac{16 d^3 (c+d x) \cosh (a+b x)}{9 b^4}-\frac{8 d (c+d x)^3 \cosh (a+b x)}{3 b^2}-\frac{8 d^3 (c+d x) \cosh ^3(a+b x)}{27 b^4}-\frac{4 d (c+d x)^3 \cosh ^3(a+b x)}{9 b^2}+\frac{8 d^4 \sinh (a+b x)}{27 b^5}+\frac{80 d^2 (c+d x)^2 \sinh (a+b x)}{9 b^3}+\frac{2 (c+d x)^4 \sinh (a+b x)}{3 b}+\frac{4 d^2 (c+d x)^2 \cosh ^2(a+b x) \sinh (a+b x)}{9 b^3}+\frac{(c+d x)^4 \cosh ^2(a+b x) \sinh (a+b x)}{3 b}+\frac{8 d^4 \sinh ^3(a+b x)}{81 b^5}-\frac{\left (16 d^3\right ) \int (c+d x) \sinh (a+b x) \, dx}{b^3}+\frac{\left (16 d^4\right ) \int \cosh (a+b x) \, dx}{9 b^4}\\ &=-\frac{160 d^3 (c+d x) \cosh (a+b x)}{9 b^4}-\frac{8 d (c+d x)^3 \cosh (a+b x)}{3 b^2}-\frac{8 d^3 (c+d x) \cosh ^3(a+b x)}{27 b^4}-\frac{4 d (c+d x)^3 \cosh ^3(a+b x)}{9 b^2}+\frac{56 d^4 \sinh (a+b x)}{27 b^5}+\frac{80 d^2 (c+d x)^2 \sinh (a+b x)}{9 b^3}+\frac{2 (c+d x)^4 \sinh (a+b x)}{3 b}+\frac{4 d^2 (c+d x)^2 \cosh ^2(a+b x) \sinh (a+b x)}{9 b^3}+\frac{(c+d x)^4 \cosh ^2(a+b x) \sinh (a+b x)}{3 b}+\frac{8 d^4 \sinh ^3(a+b x)}{81 b^5}+\frac{\left (16 d^4\right ) \int \cosh (a+b x) \, dx}{b^4}\\ &=-\frac{160 d^3 (c+d x) \cosh (a+b x)}{9 b^4}-\frac{8 d (c+d x)^3 \cosh (a+b x)}{3 b^2}-\frac{8 d^3 (c+d x) \cosh ^3(a+b x)}{27 b^4}-\frac{4 d (c+d x)^3 \cosh ^3(a+b x)}{9 b^2}+\frac{488 d^4 \sinh (a+b x)}{27 b^5}+\frac{80 d^2 (c+d x)^2 \sinh (a+b x)}{9 b^3}+\frac{2 (c+d x)^4 \sinh (a+b x)}{3 b}+\frac{4 d^2 (c+d x)^2 \cosh ^2(a+b x) \sinh (a+b x)}{9 b^3}+\frac{(c+d x)^4 \cosh ^2(a+b x) \sinh (a+b x)}{3 b}+\frac{8 d^4 \sinh ^3(a+b x)}{81 b^5}\\ \end{align*}

Mathematica [A]  time = 0.934196, size = 385, normalized size = 1.71 \[ \frac{1458 b^4 c^2 d^2 x^2 \sinh (a+b x)+162 b^4 c^2 d^2 x^2 \sinh (3 (a+b x))+2916 b^2 c^2 d^2 \sinh (a+b x)+36 b^2 c^2 d^2 \sinh (3 (a+b x))+972 b^4 c^3 d x \sinh (a+b x)+108 b^4 c^3 d x \sinh (3 (a+b x))+243 b^4 c^4 \sinh (a+b x)+27 b^4 c^4 \sinh (3 (a+b x))+972 b^4 c d^3 x^3 \sinh (a+b x)+108 b^4 c d^3 x^3 \sinh (3 (a+b x))+5832 b^2 c d^3 x \sinh (a+b x)+72 b^2 c d^3 x \sinh (3 (a+b x))-972 b d (c+d x) \cosh (a+b x) \left (b^2 (c+d x)^2+6 d^2\right )-12 b d (c+d x) \cosh (3 (a+b x)) \left (3 b^2 (c+d x)^2+2 d^2\right )+243 b^4 d^4 x^4 \sinh (a+b x)+27 b^4 d^4 x^4 \sinh (3 (a+b x))+2916 b^2 d^4 x^2 \sinh (a+b x)+36 b^2 d^4 x^2 \sinh (3 (a+b x))+5832 d^4 \sinh (a+b x)+8 d^4 \sinh (3 (a+b x))}{324 b^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^4*Cosh[a + b*x]^3,x]

[Out]

(-972*b*d*(c + d*x)*(6*d^2 + b^2*(c + d*x)^2)*Cosh[a + b*x] - 12*b*d*(c + d*x)*(2*d^2 + 3*b^2*(c + d*x)^2)*Cos
h[3*(a + b*x)] + 243*b^4*c^4*Sinh[a + b*x] + 2916*b^2*c^2*d^2*Sinh[a + b*x] + 5832*d^4*Sinh[a + b*x] + 972*b^4
*c^3*d*x*Sinh[a + b*x] + 5832*b^2*c*d^3*x*Sinh[a + b*x] + 1458*b^4*c^2*d^2*x^2*Sinh[a + b*x] + 2916*b^2*d^4*x^
2*Sinh[a + b*x] + 972*b^4*c*d^3*x^3*Sinh[a + b*x] + 243*b^4*d^4*x^4*Sinh[a + b*x] + 27*b^4*c^4*Sinh[3*(a + b*x
)] + 36*b^2*c^2*d^2*Sinh[3*(a + b*x)] + 8*d^4*Sinh[3*(a + b*x)] + 108*b^4*c^3*d*x*Sinh[3*(a + b*x)] + 72*b^2*c
*d^3*x*Sinh[3*(a + b*x)] + 162*b^4*c^2*d^2*x^2*Sinh[3*(a + b*x)] + 36*b^2*d^4*x^2*Sinh[3*(a + b*x)] + 108*b^4*
c*d^3*x^3*Sinh[3*(a + b*x)] + 27*b^4*d^4*x^4*Sinh[3*(a + b*x)])/(324*b^5)

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Maple [B]  time = 0.013, size = 1217, normalized size = 5.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^4*cosh(b*x+a)^3,x)

[Out]

1/b*(1/b^4*d^4*(2/3*(b*x+a)^4*sinh(b*x+a)+1/3*(b*x+a)^4*sinh(b*x+a)*cosh(b*x+a)^2-28/9*(b*x+a)^3*cosh(b*x+a)+8
0/9*(b*x+a)^2*sinh(b*x+a)-488/27*(b*x+a)*cosh(b*x+a)+1456/81*sinh(b*x+a)-4/9*(b*x+a)^3*cosh(b*x+a)*sinh(b*x+a)
^2+4/9*(b*x+a)^2*sinh(b*x+a)*cosh(b*x+a)^2-8/27*(b*x+a)*sinh(b*x+a)^2*cosh(b*x+a)+8/81*sinh(b*x+a)*cosh(b*x+a)
^2)-4/b^4*d^4*a*(2/3*(b*x+a)^3*sinh(b*x+a)+1/3*(b*x+a)^3*sinh(b*x+a)*cosh(b*x+a)^2-7/3*(b*x+a)^2*cosh(b*x+a)+4
0/9*(b*x+a)*sinh(b*x+a)-122/27*cosh(b*x+a)-1/3*(b*x+a)^2*sinh(b*x+a)^2*cosh(b*x+a)+2/9*(b*x+a)*sinh(b*x+a)*cos
h(b*x+a)^2-2/27*sinh(b*x+a)^2*cosh(b*x+a))+6/b^4*d^4*a^2*(1/3*(b*x+a)^2*sinh(b*x+a)*cosh(b*x+a)^2+2/3*(b*x+a)^
2*sinh(b*x+a)-2/9*(b*x+a)*sinh(b*x+a)^2*cosh(b*x+a)-14/9*(b*x+a)*cosh(b*x+a)+2/27*sinh(b*x+a)*cosh(b*x+a)^2+40
/27*sinh(b*x+a))-4/b^4*d^4*a^3*(2/3*(b*x+a)*sinh(b*x+a)+1/3*(b*x+a)*sinh(b*x+a)*cosh(b*x+a)^2-7/9*cosh(b*x+a)-
1/9*sinh(b*x+a)^2*cosh(b*x+a))+1/b^4*d^4*a^4*(2/3+1/3*cosh(b*x+a)^2)*sinh(b*x+a)+4/b^3*c*d^3*(2/3*(b*x+a)^3*si
nh(b*x+a)+1/3*(b*x+a)^3*sinh(b*x+a)*cosh(b*x+a)^2-7/3*(b*x+a)^2*cosh(b*x+a)+40/9*(b*x+a)*sinh(b*x+a)-122/27*co
sh(b*x+a)-1/3*(b*x+a)^2*sinh(b*x+a)^2*cosh(b*x+a)+2/9*(b*x+a)*sinh(b*x+a)*cosh(b*x+a)^2-2/27*sinh(b*x+a)^2*cos
h(b*x+a))-12/b^3*c*d^3*a*(1/3*(b*x+a)^2*sinh(b*x+a)*cosh(b*x+a)^2+2/3*(b*x+a)^2*sinh(b*x+a)-2/9*(b*x+a)*sinh(b
*x+a)^2*cosh(b*x+a)-14/9*(b*x+a)*cosh(b*x+a)+2/27*sinh(b*x+a)*cosh(b*x+a)^2+40/27*sinh(b*x+a))+12/b^3*c*d^3*a^
2*(2/3*(b*x+a)*sinh(b*x+a)+1/3*(b*x+a)*sinh(b*x+a)*cosh(b*x+a)^2-7/9*cosh(b*x+a)-1/9*sinh(b*x+a)^2*cosh(b*x+a)
)-4/b^3*c*d^3*a^3*(2/3+1/3*cosh(b*x+a)^2)*sinh(b*x+a)+6/b^2*c^2*d^2*(1/3*(b*x+a)^2*sinh(b*x+a)*cosh(b*x+a)^2+2
/3*(b*x+a)^2*sinh(b*x+a)-2/9*(b*x+a)*sinh(b*x+a)^2*cosh(b*x+a)-14/9*(b*x+a)*cosh(b*x+a)+2/27*sinh(b*x+a)*cosh(
b*x+a)^2+40/27*sinh(b*x+a))-12/b^2*c^2*d^2*a*(2/3*(b*x+a)*sinh(b*x+a)+1/3*(b*x+a)*sinh(b*x+a)*cosh(b*x+a)^2-7/
9*cosh(b*x+a)-1/9*sinh(b*x+a)^2*cosh(b*x+a))+6/b^2*c^2*d^2*a^2*(2/3+1/3*cosh(b*x+a)^2)*sinh(b*x+a)+4/b*c^3*d*(
2/3*(b*x+a)*sinh(b*x+a)+1/3*(b*x+a)*sinh(b*x+a)*cosh(b*x+a)^2-7/9*cosh(b*x+a)-1/9*sinh(b*x+a)^2*cosh(b*x+a))-4
/b*c^3*d*a*(2/3+1/3*cosh(b*x+a)^2)*sinh(b*x+a)+c^4*(2/3+1/3*cosh(b*x+a)^2)*sinh(b*x+a))

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Maxima [B]  time = 1.16462, size = 869, normalized size = 3.86 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^4*cosh(b*x+a)^3,x, algorithm="maxima")

[Out]

1/18*c^3*d*((3*b*x*e^(3*a) - e^(3*a))*e^(3*b*x)/b^2 + 27*(b*x*e^a - e^a)*e^(b*x)/b^2 - 27*(b*x + 1)*e^(-b*x -
a)/b^2 - (3*b*x + 1)*e^(-3*b*x - 3*a)/b^2) + 1/24*c^4*(e^(3*b*x + 3*a)/b + 9*e^(b*x + a)/b - 9*e^(-b*x - a)/b
- e^(-3*b*x - 3*a)/b) + 1/36*c^2*d^2*((9*b^2*x^2*e^(3*a) - 6*b*x*e^(3*a) + 2*e^(3*a))*e^(3*b*x)/b^3 + 81*(b^2*
x^2*e^a - 2*b*x*e^a + 2*e^a)*e^(b*x)/b^3 - 81*(b^2*x^2 + 2*b*x + 2)*e^(-b*x - a)/b^3 - (9*b^2*x^2 + 6*b*x + 2)
*e^(-3*b*x - 3*a)/b^3) + 1/54*c*d^3*((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 + 81*(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 - 81*(b^3*x^3 + 3*b^2*x^2 + 6*b*x
+ 6)*e^(-b*x - a)/b^4 - (9*b^3*x^3 + 9*b^2*x^2 + 6*b*x + 2)*e^(-3*b*x - 3*a)/b^4) + 1/648*d^4*((27*b^4*x^4*e^(
3*a) - 36*b^3*x^3*e^(3*a) + 36*b^2*x^2*e^(3*a) - 24*b*x*e^(3*a) + 8*e^(3*a))*e^(3*b*x)/b^5 + 243*(b^4*x^4*e^a
- 4*b^3*x^3*e^a + 12*b^2*x^2*e^a - 24*b*x*e^a + 24*e^a)*e^(b*x)/b^5 - 243*(b^4*x^4 + 4*b^3*x^3 + 12*b^2*x^2 +
24*b*x + 24)*e^(-b*x - a)/b^5 - (27*b^4*x^4 + 36*b^3*x^3 + 36*b^2*x^2 + 24*b*x + 8)*e^(-3*b*x - 3*a)/b^5)

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Fricas [B]  time = 2.04164, size = 1149, normalized size = 5.11 \begin{align*} -\frac{12 \,{\left (3 \, b^{3} d^{4} x^{3} + 9 \, b^{3} c d^{3} x^{2} + 3 \, b^{3} c^{3} d + 2 \, b c d^{3} +{\left (9 \, b^{3} c^{2} d^{2} + 2 \, b d^{4}\right )} x\right )} \cosh \left (b x + a\right )^{3} + 36 \,{\left (3 \, b^{3} d^{4} x^{3} + 9 \, b^{3} c d^{3} x^{2} + 3 \, b^{3} c^{3} d + 2 \, b c d^{3} +{\left (9 \, b^{3} c^{2} d^{2} + 2 \, b d^{4}\right )} x\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} -{\left (27 \, b^{4} d^{4} x^{4} + 108 \, b^{4} c d^{3} x^{3} + 27 \, b^{4} c^{4} + 36 \, b^{2} c^{2} d^{2} + 8 \, d^{4} + 18 \,{\left (9 \, b^{4} c^{2} d^{2} + 2 \, b^{2} d^{4}\right )} x^{2} + 36 \,{\left (3 \, b^{4} c^{3} d + 2 \, b^{2} c d^{3}\right )} x\right )} \sinh \left (b x + a\right )^{3} + 972 \,{\left (b^{3} d^{4} x^{3} + 3 \, b^{3} c d^{3} x^{2} + b^{3} c^{3} d + 6 \, b c d^{3} + 3 \,{\left (b^{3} c^{2} d^{2} + 2 \, b d^{4}\right )} x\right )} \cosh \left (b x + a\right ) - 3 \,{\left (81 \, b^{4} d^{4} x^{4} + 324 \, b^{4} c d^{3} x^{3} + 81 \, b^{4} c^{4} + 972 \, b^{2} c^{2} d^{2} + 1944 \, d^{4} + 486 \,{\left (b^{4} c^{2} d^{2} + 2 \, b^{2} d^{4}\right )} x^{2} +{\left (27 \, b^{4} d^{4} x^{4} + 108 \, b^{4} c d^{3} x^{3} + 27 \, b^{4} c^{4} + 36 \, b^{2} c^{2} d^{2} + 8 \, d^{4} + 18 \,{\left (9 \, b^{4} c^{2} d^{2} + 2 \, b^{2} d^{4}\right )} x^{2} + 36 \,{\left (3 \, b^{4} c^{3} d + 2 \, b^{2} c d^{3}\right )} x\right )} \cosh \left (b x + a\right )^{2} + 324 \,{\left (b^{4} c^{3} d + 6 \, b^{2} c d^{3}\right )} x\right )} \sinh \left (b x + a\right )}{324 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^4*cosh(b*x+a)^3,x, algorithm="fricas")

[Out]

-1/324*(12*(3*b^3*d^4*x^3 + 9*b^3*c*d^3*x^2 + 3*b^3*c^3*d + 2*b*c*d^3 + (9*b^3*c^2*d^2 + 2*b*d^4)*x)*cosh(b*x
+ a)^3 + 36*(3*b^3*d^4*x^3 + 9*b^3*c*d^3*x^2 + 3*b^3*c^3*d + 2*b*c*d^3 + (9*b^3*c^2*d^2 + 2*b*d^4)*x)*cosh(b*x
+ a)*sinh(b*x + a)^2 - (27*b^4*d^4*x^4 + 108*b^4*c*d^3*x^3 + 27*b^4*c^4 + 36*b^2*c^2*d^2 + 8*d^4 + 18*(9*b^4*
c^2*d^2 + 2*b^2*d^4)*x^2 + 36*(3*b^4*c^3*d + 2*b^2*c*d^3)*x)*sinh(b*x + a)^3 + 972*(b^3*d^4*x^3 + 3*b^3*c*d^3*
x^2 + b^3*c^3*d + 6*b*c*d^3 + 3*(b^3*c^2*d^2 + 2*b*d^4)*x)*cosh(b*x + a) - 3*(81*b^4*d^4*x^4 + 324*b^4*c*d^3*x
^3 + 81*b^4*c^4 + 972*b^2*c^2*d^2 + 1944*d^4 + 486*(b^4*c^2*d^2 + 2*b^2*d^4)*x^2 + (27*b^4*d^4*x^4 + 108*b^4*c
*d^3*x^3 + 27*b^4*c^4 + 36*b^2*c^2*d^2 + 8*d^4 + 18*(9*b^4*c^2*d^2 + 2*b^2*d^4)*x^2 + 36*(3*b^4*c^3*d + 2*b^2*
c*d^3)*x)*cosh(b*x + a)^2 + 324*(b^4*c^3*d + 6*b^2*c*d^3)*x)*sinh(b*x + a))/b^5

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Sympy [A]  time = 11.2935, size = 772, normalized size = 3.43 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**4*cosh(b*x+a)**3,x)

[Out]

Piecewise((-2*c**4*sinh(a + b*x)**3/(3*b) + c**4*sinh(a + b*x)*cosh(a + b*x)**2/b - 8*c**3*d*x*sinh(a + b*x)**
3/(3*b) + 4*c**3*d*x*sinh(a + b*x)*cosh(a + b*x)**2/b - 4*c**2*d**2*x**2*sinh(a + b*x)**3/b + 6*c**2*d**2*x**2
*sinh(a + b*x)*cosh(a + b*x)**2/b - 8*c*d**3*x**3*sinh(a + b*x)**3/(3*b) + 4*c*d**3*x**3*sinh(a + b*x)*cosh(a
+ b*x)**2/b - 2*d**4*x**4*sinh(a + b*x)**3/(3*b) + d**4*x**4*sinh(a + b*x)*cosh(a + b*x)**2/b + 8*c**3*d*sinh(
a + b*x)**2*cosh(a + b*x)/(3*b**2) - 28*c**3*d*cosh(a + b*x)**3/(9*b**2) + 8*c**2*d**2*x*sinh(a + b*x)**2*cosh
(a + b*x)/b**2 - 28*c**2*d**2*x*cosh(a + b*x)**3/(3*b**2) + 8*c*d**3*x**2*sinh(a + b*x)**2*cosh(a + b*x)/b**2
- 28*c*d**3*x**2*cosh(a + b*x)**3/(3*b**2) + 8*d**4*x**3*sinh(a + b*x)**2*cosh(a + b*x)/(3*b**2) - 28*d**4*x**
3*cosh(a + b*x)**3/(9*b**2) - 80*c**2*d**2*sinh(a + b*x)**3/(9*b**3) + 28*c**2*d**2*sinh(a + b*x)*cosh(a + b*x
)**2/(3*b**3) - 160*c*d**3*x*sinh(a + b*x)**3/(9*b**3) + 56*c*d**3*x*sinh(a + b*x)*cosh(a + b*x)**2/(3*b**3) -
80*d**4*x**2*sinh(a + b*x)**3/(9*b**3) + 28*d**4*x**2*sinh(a + b*x)*cosh(a + b*x)**2/(3*b**3) + 160*c*d**3*si
nh(a + b*x)**2*cosh(a + b*x)/(9*b**4) - 488*c*d**3*cosh(a + b*x)**3/(27*b**4) + 160*d**4*x*sinh(a + b*x)**2*co
sh(a + b*x)/(9*b**4) - 488*d**4*x*cosh(a + b*x)**3/(27*b**4) - 1456*d**4*sinh(a + b*x)**3/(81*b**5) + 488*d**4
*sinh(a + b*x)*cosh(a + b*x)**2/(27*b**5), Ne(b, 0)), ((c**4*x + 2*c**3*d*x**2 + 2*c**2*d**2*x**3 + c*d**3*x**
4 + d**4*x**5/5)*cosh(a)**3, True))

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Giac [B]  time = 1.39823, size = 883, normalized size = 3.92 \begin{align*} \frac{{\left (27 \, b^{4} d^{4} x^{4} + 108 \, b^{4} c d^{3} x^{3} + 162 \, b^{4} c^{2} d^{2} x^{2} - 36 \, b^{3} d^{4} x^{3} + 108 \, b^{4} c^{3} d x - 108 \, b^{3} c d^{3} x^{2} + 27 \, b^{4} c^{4} - 108 \, b^{3} c^{2} d^{2} x + 36 \, b^{2} d^{4} x^{2} - 36 \, b^{3} c^{3} d + 72 \, b^{2} c d^{3} x + 36 \, b^{2} c^{2} d^{2} - 24 \, b d^{4} x - 24 \, b c d^{3} + 8 \, d^{4}\right )} e^{\left (3 \, b x + 3 \, a\right )}}{648 \, b^{5}} + \frac{3 \,{\left (b^{4} d^{4} x^{4} + 4 \, b^{4} c d^{3} x^{3} + 6 \, b^{4} c^{2} d^{2} x^{2} - 4 \, b^{3} d^{4} x^{3} + 4 \, b^{4} c^{3} d x - 12 \, b^{3} c d^{3} x^{2} + b^{4} c^{4} - 12 \, b^{3} c^{2} d^{2} x + 12 \, b^{2} d^{4} x^{2} - 4 \, b^{3} c^{3} d + 24 \, b^{2} c d^{3} x + 12 \, b^{2} c^{2} d^{2} - 24 \, b d^{4} x - 24 \, b c d^{3} + 24 \, d^{4}\right )} e^{\left (b x + a\right )}}{8 \, b^{5}} - \frac{3 \,{\left (b^{4} d^{4} x^{4} + 4 \, b^{4} c d^{3} x^{3} + 6 \, b^{4} c^{2} d^{2} x^{2} + 4 \, b^{3} d^{4} x^{3} + 4 \, b^{4} c^{3} d x + 12 \, b^{3} c d^{3} x^{2} + b^{4} c^{4} + 12 \, b^{3} c^{2} d^{2} x + 12 \, b^{2} d^{4} x^{2} + 4 \, b^{3} c^{3} d + 24 \, b^{2} c d^{3} x + 12 \, b^{2} c^{2} d^{2} + 24 \, b d^{4} x + 24 \, b c d^{3} + 24 \, d^{4}\right )} e^{\left (-b x - a\right )}}{8 \, b^{5}} - \frac{{\left (27 \, b^{4} d^{4} x^{4} + 108 \, b^{4} c d^{3} x^{3} + 162 \, b^{4} c^{2} d^{2} x^{2} + 36 \, b^{3} d^{4} x^{3} + 108 \, b^{4} c^{3} d x + 108 \, b^{3} c d^{3} x^{2} + 27 \, b^{4} c^{4} + 108 \, b^{3} c^{2} d^{2} x + 36 \, b^{2} d^{4} x^{2} + 36 \, b^{3} c^{3} d + 72 \, b^{2} c d^{3} x + 36 \, b^{2} c^{2} d^{2} + 24 \, b d^{4} x + 24 \, b c d^{3} + 8 \, d^{4}\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{648 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^4*cosh(b*x+a)^3,x, algorithm="giac")

[Out]

1/648*(27*b^4*d^4*x^4 + 108*b^4*c*d^3*x^3 + 162*b^4*c^2*d^2*x^2 - 36*b^3*d^4*x^3 + 108*b^4*c^3*d*x - 108*b^3*c
*d^3*x^2 + 27*b^4*c^4 - 108*b^3*c^2*d^2*x + 36*b^2*d^4*x^2 - 36*b^3*c^3*d + 72*b^2*c*d^3*x + 36*b^2*c^2*d^2 -
24*b*d^4*x - 24*b*c*d^3 + 8*d^4)*e^(3*b*x + 3*a)/b^5 + 3/8*(b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + 6*b^4*c^2*d^2*x^2
- 4*b^3*d^4*x^3 + 4*b^4*c^3*d*x - 12*b^3*c*d^3*x^2 + b^4*c^4 - 12*b^3*c^2*d^2*x + 12*b^2*d^4*x^2 - 4*b^3*c^3*d
+ 24*b^2*c*d^3*x + 12*b^2*c^2*d^2 - 24*b*d^4*x - 24*b*c*d^3 + 24*d^4)*e^(b*x + a)/b^5 - 3/8*(b^4*d^4*x^4 + 4*
b^4*c*d^3*x^3 + 6*b^4*c^2*d^2*x^2 + 4*b^3*d^4*x^3 + 4*b^4*c^3*d*x + 12*b^3*c*d^3*x^2 + b^4*c^4 + 12*b^3*c^2*d^
2*x + 12*b^2*d^4*x^2 + 4*b^3*c^3*d + 24*b^2*c*d^3*x + 12*b^2*c^2*d^2 + 24*b*d^4*x + 24*b*c*d^3 + 24*d^4)*e^(-b
*x - a)/b^5 - 1/648*(27*b^4*d^4*x^4 + 108*b^4*c*d^3*x^3 + 162*b^4*c^2*d^2*x^2 + 36*b^3*d^4*x^3 + 108*b^4*c^3*d
*x + 108*b^3*c*d^3*x^2 + 27*b^4*c^4 + 108*b^3*c^2*d^2*x + 36*b^2*d^4*x^2 + 36*b^3*c^3*d + 72*b^2*c*d^3*x + 36*
b^2*c^2*d^2 + 24*b*d^4*x + 24*b*c*d^3 + 8*d^4)*e^(-3*b*x - 3*a)/b^5