3.87 \(\int x (a+b x^3)^2 \cosh (c+d x) \, dx\)

Optimal. Leaf size=234 \[ -\frac{a^2 \cosh (c+d x)}{d^2}+\frac{a^2 x \sinh (c+d x)}{d}+\frac{24 a b x^2 \sinh (c+d x)}{d^3}-\frac{8 a b x^3 \cosh (c+d x)}{d^2}+\frac{48 a b \sinh (c+d x)}{d^5}-\frac{48 a b x \cosh (c+d x)}{d^4}+\frac{2 a b x^4 \sinh (c+d x)}{d}+\frac{42 b^2 x^5 \sinh (c+d x)}{d^3}+\frac{840 b^2 x^3 \sinh (c+d x)}{d^5}-\frac{7 b^2 x^6 \cosh (c+d x)}{d^2}-\frac{210 b^2 x^4 \cosh (c+d x)}{d^4}-\frac{2520 b^2 x^2 \cosh (c+d x)}{d^6}+\frac{5040 b^2 x \sinh (c+d x)}{d^7}-\frac{5040 b^2 \cosh (c+d x)}{d^8}+\frac{b^2 x^7 \sinh (c+d x)}{d} \]

[Out]

(-5040*b^2*Cosh[c + d*x])/d^8 - (a^2*Cosh[c + d*x])/d^2 - (48*a*b*x*Cosh[c + d*x])/d^4 - (2520*b^2*x^2*Cosh[c
+ d*x])/d^6 - (8*a*b*x^3*Cosh[c + d*x])/d^2 - (210*b^2*x^4*Cosh[c + d*x])/d^4 - (7*b^2*x^6*Cosh[c + d*x])/d^2
+ (48*a*b*Sinh[c + d*x])/d^5 + (5040*b^2*x*Sinh[c + d*x])/d^7 + (a^2*x*Sinh[c + d*x])/d + (24*a*b*x^2*Sinh[c +
d*x])/d^3 + (840*b^2*x^3*Sinh[c + d*x])/d^5 + (2*a*b*x^4*Sinh[c + d*x])/d + (42*b^2*x^5*Sinh[c + d*x])/d^3 +
(b^2*x^7*Sinh[c + d*x])/d

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Rubi [A]  time = 0.39526, antiderivative size = 234, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {5287, 3296, 2638, 2637} \[ -\frac{a^2 \cosh (c+d x)}{d^2}+\frac{a^2 x \sinh (c+d x)}{d}+\frac{24 a b x^2 \sinh (c+d x)}{d^3}-\frac{8 a b x^3 \cosh (c+d x)}{d^2}+\frac{48 a b \sinh (c+d x)}{d^5}-\frac{48 a b x \cosh (c+d x)}{d^4}+\frac{2 a b x^4 \sinh (c+d x)}{d}+\frac{42 b^2 x^5 \sinh (c+d x)}{d^3}+\frac{840 b^2 x^3 \sinh (c+d x)}{d^5}-\frac{7 b^2 x^6 \cosh (c+d x)}{d^2}-\frac{210 b^2 x^4 \cosh (c+d x)}{d^4}-\frac{2520 b^2 x^2 \cosh (c+d x)}{d^6}+\frac{5040 b^2 x \sinh (c+d x)}{d^7}-\frac{5040 b^2 \cosh (c+d x)}{d^8}+\frac{b^2 x^7 \sinh (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x^3)^2*Cosh[c + d*x],x]

[Out]

(-5040*b^2*Cosh[c + d*x])/d^8 - (a^2*Cosh[c + d*x])/d^2 - (48*a*b*x*Cosh[c + d*x])/d^4 - (2520*b^2*x^2*Cosh[c
+ d*x])/d^6 - (8*a*b*x^3*Cosh[c + d*x])/d^2 - (210*b^2*x^4*Cosh[c + d*x])/d^4 - (7*b^2*x^6*Cosh[c + d*x])/d^2
+ (48*a*b*Sinh[c + d*x])/d^5 + (5040*b^2*x*Sinh[c + d*x])/d^7 + (a^2*x*Sinh[c + d*x])/d + (24*a*b*x^2*Sinh[c +
d*x])/d^3 + (840*b^2*x^3*Sinh[c + d*x])/d^5 + (2*a*b*x^4*Sinh[c + d*x])/d + (42*b^2*x^5*Sinh[c + d*x])/d^3 +
(b^2*x^7*Sinh[c + d*x])/d

Rule 5287

Int[Cosh[(c_.) + (d_.)*(x_)]*((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[Cosh[c + d*x], (e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]

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 2638

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

Rule 2637

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

Rubi steps

\begin{align*} \int x \left (a+b x^3\right )^2 \cosh (c+d x) \, dx &=\int \left (a^2 x \cosh (c+d x)+2 a b x^4 \cosh (c+d x)+b^2 x^7 \cosh (c+d x)\right ) \, dx\\ &=a^2 \int x \cosh (c+d x) \, dx+(2 a b) \int x^4 \cosh (c+d x) \, dx+b^2 \int x^7 \cosh (c+d x) \, dx\\ &=\frac{a^2 x \sinh (c+d x)}{d}+\frac{2 a b x^4 \sinh (c+d x)}{d}+\frac{b^2 x^7 \sinh (c+d x)}{d}-\frac{a^2 \int \sinh (c+d x) \, dx}{d}-\frac{(8 a b) \int x^3 \sinh (c+d x) \, dx}{d}-\frac{\left (7 b^2\right ) \int x^6 \sinh (c+d x) \, dx}{d}\\ &=-\frac{a^2 \cosh (c+d x)}{d^2}-\frac{8 a b x^3 \cosh (c+d x)}{d^2}-\frac{7 b^2 x^6 \cosh (c+d x)}{d^2}+\frac{a^2 x \sinh (c+d x)}{d}+\frac{2 a b x^4 \sinh (c+d x)}{d}+\frac{b^2 x^7 \sinh (c+d x)}{d}+\frac{(24 a b) \int x^2 \cosh (c+d x) \, dx}{d^2}+\frac{\left (42 b^2\right ) \int x^5 \cosh (c+d x) \, dx}{d^2}\\ &=-\frac{a^2 \cosh (c+d x)}{d^2}-\frac{8 a b x^3 \cosh (c+d x)}{d^2}-\frac{7 b^2 x^6 \cosh (c+d x)}{d^2}+\frac{a^2 x \sinh (c+d x)}{d}+\frac{24 a b x^2 \sinh (c+d x)}{d^3}+\frac{2 a b x^4 \sinh (c+d x)}{d}+\frac{42 b^2 x^5 \sinh (c+d x)}{d^3}+\frac{b^2 x^7 \sinh (c+d x)}{d}-\frac{(48 a b) \int x \sinh (c+d x) \, dx}{d^3}-\frac{\left (210 b^2\right ) \int x^4 \sinh (c+d x) \, dx}{d^3}\\ &=-\frac{a^2 \cosh (c+d x)}{d^2}-\frac{48 a b x \cosh (c+d x)}{d^4}-\frac{8 a b x^3 \cosh (c+d x)}{d^2}-\frac{210 b^2 x^4 \cosh (c+d x)}{d^4}-\frac{7 b^2 x^6 \cosh (c+d x)}{d^2}+\frac{a^2 x \sinh (c+d x)}{d}+\frac{24 a b x^2 \sinh (c+d x)}{d^3}+\frac{2 a b x^4 \sinh (c+d x)}{d}+\frac{42 b^2 x^5 \sinh (c+d x)}{d^3}+\frac{b^2 x^7 \sinh (c+d x)}{d}+\frac{(48 a b) \int \cosh (c+d x) \, dx}{d^4}+\frac{\left (840 b^2\right ) \int x^3 \cosh (c+d x) \, dx}{d^4}\\ &=-\frac{a^2 \cosh (c+d x)}{d^2}-\frac{48 a b x \cosh (c+d x)}{d^4}-\frac{8 a b x^3 \cosh (c+d x)}{d^2}-\frac{210 b^2 x^4 \cosh (c+d x)}{d^4}-\frac{7 b^2 x^6 \cosh (c+d x)}{d^2}+\frac{48 a b \sinh (c+d x)}{d^5}+\frac{a^2 x \sinh (c+d x)}{d}+\frac{24 a b x^2 \sinh (c+d x)}{d^3}+\frac{840 b^2 x^3 \sinh (c+d x)}{d^5}+\frac{2 a b x^4 \sinh (c+d x)}{d}+\frac{42 b^2 x^5 \sinh (c+d x)}{d^3}+\frac{b^2 x^7 \sinh (c+d x)}{d}-\frac{\left (2520 b^2\right ) \int x^2 \sinh (c+d x) \, dx}{d^5}\\ &=-\frac{a^2 \cosh (c+d x)}{d^2}-\frac{48 a b x \cosh (c+d x)}{d^4}-\frac{2520 b^2 x^2 \cosh (c+d x)}{d^6}-\frac{8 a b x^3 \cosh (c+d x)}{d^2}-\frac{210 b^2 x^4 \cosh (c+d x)}{d^4}-\frac{7 b^2 x^6 \cosh (c+d x)}{d^2}+\frac{48 a b \sinh (c+d x)}{d^5}+\frac{a^2 x \sinh (c+d x)}{d}+\frac{24 a b x^2 \sinh (c+d x)}{d^3}+\frac{840 b^2 x^3 \sinh (c+d x)}{d^5}+\frac{2 a b x^4 \sinh (c+d x)}{d}+\frac{42 b^2 x^5 \sinh (c+d x)}{d^3}+\frac{b^2 x^7 \sinh (c+d x)}{d}+\frac{\left (5040 b^2\right ) \int x \cosh (c+d x) \, dx}{d^6}\\ &=-\frac{a^2 \cosh (c+d x)}{d^2}-\frac{48 a b x \cosh (c+d x)}{d^4}-\frac{2520 b^2 x^2 \cosh (c+d x)}{d^6}-\frac{8 a b x^3 \cosh (c+d x)}{d^2}-\frac{210 b^2 x^4 \cosh (c+d x)}{d^4}-\frac{7 b^2 x^6 \cosh (c+d x)}{d^2}+\frac{48 a b \sinh (c+d x)}{d^5}+\frac{5040 b^2 x \sinh (c+d x)}{d^7}+\frac{a^2 x \sinh (c+d x)}{d}+\frac{24 a b x^2 \sinh (c+d x)}{d^3}+\frac{840 b^2 x^3 \sinh (c+d x)}{d^5}+\frac{2 a b x^4 \sinh (c+d x)}{d}+\frac{42 b^2 x^5 \sinh (c+d x)}{d^3}+\frac{b^2 x^7 \sinh (c+d x)}{d}-\frac{\left (5040 b^2\right ) \int \sinh (c+d x) \, dx}{d^7}\\ &=-\frac{5040 b^2 \cosh (c+d x)}{d^8}-\frac{a^2 \cosh (c+d x)}{d^2}-\frac{48 a b x \cosh (c+d x)}{d^4}-\frac{2520 b^2 x^2 \cosh (c+d x)}{d^6}-\frac{8 a b x^3 \cosh (c+d x)}{d^2}-\frac{210 b^2 x^4 \cosh (c+d x)}{d^4}-\frac{7 b^2 x^6 \cosh (c+d x)}{d^2}+\frac{48 a b \sinh (c+d x)}{d^5}+\frac{5040 b^2 x \sinh (c+d x)}{d^7}+\frac{a^2 x \sinh (c+d x)}{d}+\frac{24 a b x^2 \sinh (c+d x)}{d^3}+\frac{840 b^2 x^3 \sinh (c+d x)}{d^5}+\frac{2 a b x^4 \sinh (c+d x)}{d}+\frac{42 b^2 x^5 \sinh (c+d x)}{d^3}+\frac{b^2 x^7 \sinh (c+d x)}{d}\\ \end{align*}

Mathematica [A]  time = 0.291489, size = 139, normalized size = 0.59 \[ \frac{d \left (a^2 d^6 x+2 a b d^2 \left (d^4 x^4+12 d^2 x^2+24\right )+b^2 x \left (d^6 x^6+42 d^4 x^4+840 d^2 x^2+5040\right )\right ) \sinh (c+d x)-\left (a^2 d^6+8 a b d^4 x \left (d^2 x^2+6\right )+7 b^2 \left (d^6 x^6+30 d^4 x^4+360 d^2 x^2+720\right )\right ) \cosh (c+d x)}{d^8} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x^3)^2*Cosh[c + d*x],x]

[Out]

(-((a^2*d^6 + 8*a*b*d^4*x*(6 + d^2*x^2) + 7*b^2*(720 + 360*d^2*x^2 + 30*d^4*x^4 + d^6*x^6))*Cosh[c + d*x]) + d
*(a^2*d^6*x + 2*a*b*d^2*(24 + 12*d^2*x^2 + d^4*x^4) + b^2*x*(5040 + 840*d^2*x^2 + 42*d^4*x^4 + d^6*x^6))*Sinh[
c + d*x])/d^8

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Maple [B]  time = 0.008, size = 818, normalized size = 3.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d^2*(a^2*((d*x+c)*sinh(d*x+c)-cosh(d*x+c))+1/d^6*b^2*((d*x+c)^7*sinh(d*x+c)-7*(d*x+c)^6*cosh(d*x+c)+42*(d*x+
c)^5*sinh(d*x+c)-210*(d*x+c)^4*cosh(d*x+c)+840*(d*x+c)^3*sinh(d*x+c)-2520*(d*x+c)^2*cosh(d*x+c)+5040*(d*x+c)*s
inh(d*x+c)-5040*cosh(d*x+c))+7/d^6*b^2*c^6*((d*x+c)*sinh(d*x+c)-cosh(d*x+c))+12/d^3*b*c^2*a*((d*x+c)^2*sinh(d*
x+c)-2*(d*x+c)*cosh(d*x+c)+2*sinh(d*x+c))-8/d^3*b*c*a*((d*x+c)^3*sinh(d*x+c)-3*(d*x+c)^2*cosh(d*x+c)+6*(d*x+c)
*sinh(d*x+c)-6*cosh(d*x+c))-21/d^6*b^2*c^5*((d*x+c)^2*sinh(d*x+c)-2*(d*x+c)*cosh(d*x+c)+2*sinh(d*x+c))-8/d^3*b
*c^3*a*((d*x+c)*sinh(d*x+c)-cosh(d*x+c))+35/d^6*b^2*c^4*((d*x+c)^3*sinh(d*x+c)-3*(d*x+c)^2*cosh(d*x+c)+6*(d*x+
c)*sinh(d*x+c)-6*cosh(d*x+c))-35/d^6*b^2*c^3*((d*x+c)^4*sinh(d*x+c)-4*(d*x+c)^3*cosh(d*x+c)+12*(d*x+c)^2*sinh(
d*x+c)-24*(d*x+c)*cosh(d*x+c)+24*sinh(d*x+c))+2/d^3*b*a*((d*x+c)^4*sinh(d*x+c)-4*(d*x+c)^3*cosh(d*x+c)+12*(d*x
+c)^2*sinh(d*x+c)-24*(d*x+c)*cosh(d*x+c)+24*sinh(d*x+c))-7/d^6*b^2*c*((d*x+c)^6*sinh(d*x+c)-6*(d*x+c)^5*cosh(d
*x+c)+30*(d*x+c)^4*sinh(d*x+c)-120*(d*x+c)^3*cosh(d*x+c)+360*(d*x+c)^2*sinh(d*x+c)-720*(d*x+c)*cosh(d*x+c)+720
*sinh(d*x+c))+21/d^6*b^2*c^2*((d*x+c)^5*sinh(d*x+c)-5*(d*x+c)^4*cosh(d*x+c)+20*(d*x+c)^3*sinh(d*x+c)-60*(d*x+c
)^2*cosh(d*x+c)+120*(d*x+c)*sinh(d*x+c)-120*cosh(d*x+c))-c*a^2*sinh(d*x+c)-1/d^6*b^2*c^7*sinh(d*x+c)+2/d^3*b*c
^4*a*sinh(d*x+c))

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Maxima [A]  time = 1.07065, size = 517, normalized size = 2.21 \begin{align*} -\frac{1}{80} \, d{\left (\frac{20 \,{\left (d^{2} x^{2} e^{c} - 2 \, d x e^{c} + 2 \, e^{c}\right )} a^{2} e^{\left (d x\right )}}{d^{3}} + \frac{20 \,{\left (d^{2} x^{2} + 2 \, d x + 2\right )} a^{2} e^{\left (-d x - c\right )}}{d^{3}} + \frac{16 \,{\left (d^{5} x^{5} e^{c} - 5 \, d^{4} x^{4} e^{c} + 20 \, d^{3} x^{3} e^{c} - 60 \, d^{2} x^{2} e^{c} + 120 \, d x e^{c} - 120 \, e^{c}\right )} a b e^{\left (d x\right )}}{d^{6}} + \frac{16 \,{\left (d^{5} x^{5} + 5 \, d^{4} x^{4} + 20 \, d^{3} x^{3} + 60 \, d^{2} x^{2} + 120 \, d x + 120\right )} a b e^{\left (-d x - c\right )}}{d^{6}} + \frac{5 \,{\left (d^{8} x^{8} e^{c} - 8 \, d^{7} x^{7} e^{c} + 56 \, d^{6} x^{6} e^{c} - 336 \, d^{5} x^{5} e^{c} + 1680 \, d^{4} x^{4} e^{c} - 6720 \, d^{3} x^{3} e^{c} + 20160 \, d^{2} x^{2} e^{c} - 40320 \, d x e^{c} + 40320 \, e^{c}\right )} b^{2} e^{\left (d x\right )}}{d^{9}} + \frac{5 \,{\left (d^{8} x^{8} + 8 \, d^{7} x^{7} + 56 \, d^{6} x^{6} + 336 \, d^{5} x^{5} + 1680 \, d^{4} x^{4} + 6720 \, d^{3} x^{3} + 20160 \, d^{2} x^{2} + 40320 \, d x + 40320\right )} b^{2} e^{\left (-d x - c\right )}}{d^{9}}\right )} + \frac{1}{40} \,{\left (5 \, b^{2} x^{8} + 16 \, a b x^{5} + 20 \, a^{2} x^{2}\right )} \cosh \left (d x + c\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/80*d*(20*(d^2*x^2*e^c - 2*d*x*e^c + 2*e^c)*a^2*e^(d*x)/d^3 + 20*(d^2*x^2 + 2*d*x + 2)*a^2*e^(-d*x - c)/d^3
+ 16*(d^5*x^5*e^c - 5*d^4*x^4*e^c + 20*d^3*x^3*e^c - 60*d^2*x^2*e^c + 120*d*x*e^c - 120*e^c)*a*b*e^(d*x)/d^6 +
16*(d^5*x^5 + 5*d^4*x^4 + 20*d^3*x^3 + 60*d^2*x^2 + 120*d*x + 120)*a*b*e^(-d*x - c)/d^6 + 5*(d^8*x^8*e^c - 8*
d^7*x^7*e^c + 56*d^6*x^6*e^c - 336*d^5*x^5*e^c + 1680*d^4*x^4*e^c - 6720*d^3*x^3*e^c + 20160*d^2*x^2*e^c - 403
20*d*x*e^c + 40320*e^c)*b^2*e^(d*x)/d^9 + 5*(d^8*x^8 + 8*d^7*x^7 + 56*d^6*x^6 + 336*d^5*x^5 + 1680*d^4*x^4 + 6
720*d^3*x^3 + 20160*d^2*x^2 + 40320*d*x + 40320)*b^2*e^(-d*x - c)/d^9) + 1/40*(5*b^2*x^8 + 16*a*b*x^5 + 20*a^2
*x^2)*cosh(d*x + c)

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Fricas [A]  time = 1.75902, size = 358, normalized size = 1.53 \begin{align*} -\frac{{\left (7 \, b^{2} d^{6} x^{6} + 8 \, a b d^{6} x^{3} + 210 \, b^{2} d^{4} x^{4} + a^{2} d^{6} + 48 \, a b d^{4} x + 2520 \, b^{2} d^{2} x^{2} + 5040 \, b^{2}\right )} \cosh \left (d x + c\right ) -{\left (b^{2} d^{7} x^{7} + 2 \, a b d^{7} x^{4} + 42 \, b^{2} d^{5} x^{5} + 24 \, a b d^{5} x^{2} + 840 \, b^{2} d^{3} x^{3} + 48 \, a b d^{3} +{\left (a^{2} d^{7} + 5040 \, b^{2} d\right )} x\right )} \sinh \left (d x + c\right )}{d^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((7*b^2*d^6*x^6 + 8*a*b*d^6*x^3 + 210*b^2*d^4*x^4 + a^2*d^6 + 48*a*b*d^4*x + 2520*b^2*d^2*x^2 + 5040*b^2)*cos
h(d*x + c) - (b^2*d^7*x^7 + 2*a*b*d^7*x^4 + 42*b^2*d^5*x^5 + 24*a*b*d^5*x^2 + 840*b^2*d^3*x^3 + 48*a*b*d^3 + (
a^2*d^7 + 5040*b^2*d)*x)*sinh(d*x + c))/d^8

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Sympy [A]  time = 15.3885, size = 284, normalized size = 1.21 \begin{align*} \begin{cases} \frac{a^{2} x \sinh{\left (c + d x \right )}}{d} - \frac{a^{2} \cosh{\left (c + d x \right )}}{d^{2}} + \frac{2 a b x^{4} \sinh{\left (c + d x \right )}}{d} - \frac{8 a b x^{3} \cosh{\left (c + d x \right )}}{d^{2}} + \frac{24 a b x^{2} \sinh{\left (c + d x \right )}}{d^{3}} - \frac{48 a b x \cosh{\left (c + d x \right )}}{d^{4}} + \frac{48 a b \sinh{\left (c + d x \right )}}{d^{5}} + \frac{b^{2} x^{7} \sinh{\left (c + d x \right )}}{d} - \frac{7 b^{2} x^{6} \cosh{\left (c + d x \right )}}{d^{2}} + \frac{42 b^{2} x^{5} \sinh{\left (c + d x \right )}}{d^{3}} - \frac{210 b^{2} x^{4} \cosh{\left (c + d x \right )}}{d^{4}} + \frac{840 b^{2} x^{3} \sinh{\left (c + d x \right )}}{d^{5}} - \frac{2520 b^{2} x^{2} \cosh{\left (c + d x \right )}}{d^{6}} + \frac{5040 b^{2} x \sinh{\left (c + d x \right )}}{d^{7}} - \frac{5040 b^{2} \cosh{\left (c + d x \right )}}{d^{8}} & \text{for}\: d \neq 0 \\\left (\frac{a^{2} x^{2}}{2} + \frac{2 a b x^{5}}{5} + \frac{b^{2} x^{8}}{8}\right ) \cosh{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*x*sinh(c + d*x)/d - a**2*cosh(c + d*x)/d**2 + 2*a*b*x**4*sinh(c + d*x)/d - 8*a*b*x**3*cosh(c +
d*x)/d**2 + 24*a*b*x**2*sinh(c + d*x)/d**3 - 48*a*b*x*cosh(c + d*x)/d**4 + 48*a*b*sinh(c + d*x)/d**5 + b**2*x
**7*sinh(c + d*x)/d - 7*b**2*x**6*cosh(c + d*x)/d**2 + 42*b**2*x**5*sinh(c + d*x)/d**3 - 210*b**2*x**4*cosh(c
+ d*x)/d**4 + 840*b**2*x**3*sinh(c + d*x)/d**5 - 2520*b**2*x**2*cosh(c + d*x)/d**6 + 5040*b**2*x*sinh(c + d*x)
/d**7 - 5040*b**2*cosh(c + d*x)/d**8, Ne(d, 0)), ((a**2*x**2/2 + 2*a*b*x**5/5 + b**2*x**8/8)*cosh(c), True))

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Giac [A]  time = 1.23338, size = 409, normalized size = 1.75 \begin{align*} \frac{{\left (b^{2} d^{7} x^{7} - 7 \, b^{2} d^{6} x^{6} + 2 \, a b d^{7} x^{4} + 42 \, b^{2} d^{5} x^{5} - 8 \, a b d^{6} x^{3} + a^{2} d^{7} x - 210 \, b^{2} d^{4} x^{4} + 24 \, a b d^{5} x^{2} - a^{2} d^{6} + 840 \, b^{2} d^{3} x^{3} - 48 \, a b d^{4} x - 2520 \, b^{2} d^{2} x^{2} + 48 \, a b d^{3} + 5040 \, b^{2} d x - 5040 \, b^{2}\right )} e^{\left (d x + c\right )}}{2 \, d^{8}} - \frac{{\left (b^{2} d^{7} x^{7} + 7 \, b^{2} d^{6} x^{6} + 2 \, a b d^{7} x^{4} + 42 \, b^{2} d^{5} x^{5} + 8 \, a b d^{6} x^{3} + a^{2} d^{7} x + 210 \, b^{2} d^{4} x^{4} + 24 \, a b d^{5} x^{2} + a^{2} d^{6} + 840 \, b^{2} d^{3} x^{3} + 48 \, a b d^{4} x + 2520 \, b^{2} d^{2} x^{2} + 48 \, a b d^{3} + 5040 \, b^{2} d x + 5040 \, b^{2}\right )} e^{\left (-d x - c\right )}}{2 \, d^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b^2*d^7*x^7 - 7*b^2*d^6*x^6 + 2*a*b*d^7*x^4 + 42*b^2*d^5*x^5 - 8*a*b*d^6*x^3 + a^2*d^7*x - 210*b^2*d^4*x^
4 + 24*a*b*d^5*x^2 - a^2*d^6 + 840*b^2*d^3*x^3 - 48*a*b*d^4*x - 2520*b^2*d^2*x^2 + 48*a*b*d^3 + 5040*b^2*d*x -
5040*b^2)*e^(d*x + c)/d^8 - 1/2*(b^2*d^7*x^7 + 7*b^2*d^6*x^6 + 2*a*b*d^7*x^4 + 42*b^2*d^5*x^5 + 8*a*b*d^6*x^3
+ a^2*d^7*x + 210*b^2*d^4*x^4 + 24*a*b*d^5*x^2 + a^2*d^6 + 840*b^2*d^3*x^3 + 48*a*b*d^4*x + 2520*b^2*d^2*x^2
+ 48*a*b*d^3 + 5040*b^2*d*x + 5040*b^2)*e^(-d*x - c)/d^8