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

Optimal. Leaf size=186 \[ \frac{a^2 \sinh (c+d x)}{d}-\frac{6 a b x^2 \cosh (c+d x)}{d^2}+\frac{12 a b x \sinh (c+d x)}{d^3}-\frac{12 a b \cosh (c+d x)}{d^4}+\frac{2 a b x^3 \sinh (c+d x)}{d}+\frac{30 b^2 x^4 \sinh (c+d x)}{d^3}+\frac{360 b^2 x^2 \sinh (c+d x)}{d^5}-\frac{6 b^2 x^5 \cosh (c+d x)}{d^2}-\frac{120 b^2 x^3 \cosh (c+d x)}{d^4}+\frac{720 b^2 \sinh (c+d x)}{d^7}-\frac{720 b^2 x \cosh (c+d x)}{d^6}+\frac{b^2 x^6 \sinh (c+d x)}{d} \]

[Out]

(-12*a*b*Cosh[c + d*x])/d^4 - (720*b^2*x*Cosh[c + d*x])/d^6 - (6*a*b*x^2*Cosh[c + d*x])/d^2 - (120*b^2*x^3*Cos
h[c + d*x])/d^4 - (6*b^2*x^5*Cosh[c + d*x])/d^2 + (720*b^2*Sinh[c + d*x])/d^7 + (a^2*Sinh[c + d*x])/d + (12*a*
b*x*Sinh[c + d*x])/d^3 + (360*b^2*x^2*Sinh[c + d*x])/d^5 + (2*a*b*x^3*Sinh[c + d*x])/d + (30*b^2*x^4*Sinh[c +
d*x])/d^3 + (b^2*x^6*Sinh[c + d*x])/d

________________________________________________________________________________________

Rubi [A]  time = 0.287106, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5277, 2637, 3296, 2638} \[ \frac{a^2 \sinh (c+d x)}{d}-\frac{6 a b x^2 \cosh (c+d x)}{d^2}+\frac{12 a b x \sinh (c+d x)}{d^3}-\frac{12 a b \cosh (c+d x)}{d^4}+\frac{2 a b x^3 \sinh (c+d x)}{d}+\frac{30 b^2 x^4 \sinh (c+d x)}{d^3}+\frac{360 b^2 x^2 \sinh (c+d x)}{d^5}-\frac{6 b^2 x^5 \cosh (c+d x)}{d^2}-\frac{120 b^2 x^3 \cosh (c+d x)}{d^4}+\frac{720 b^2 \sinh (c+d x)}{d^7}-\frac{720 b^2 x \cosh (c+d x)}{d^6}+\frac{b^2 x^6 \sinh (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^2*Cosh[c + d*x],x]

[Out]

(-12*a*b*Cosh[c + d*x])/d^4 - (720*b^2*x*Cosh[c + d*x])/d^6 - (6*a*b*x^2*Cosh[c + d*x])/d^2 - (120*b^2*x^3*Cos
h[c + d*x])/d^4 - (6*b^2*x^5*Cosh[c + d*x])/d^2 + (720*b^2*Sinh[c + d*x])/d^7 + (a^2*Sinh[c + d*x])/d + (12*a*
b*x*Sinh[c + d*x])/d^3 + (360*b^2*x^2*Sinh[c + d*x])/d^5 + (2*a*b*x^3*Sinh[c + d*x])/d + (30*b^2*x^4*Sinh[c +
d*x])/d^3 + (b^2*x^6*Sinh[c + d*x])/d

Rule 5277

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

Rule 2637

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

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]

Rubi steps

\begin{align*} \int \left (a+b x^3\right )^2 \cosh (c+d x) \, dx &=\int \left (a^2 \cosh (c+d x)+2 a b x^3 \cosh (c+d x)+b^2 x^6 \cosh (c+d x)\right ) \, dx\\ &=a^2 \int \cosh (c+d x) \, dx+(2 a b) \int x^3 \cosh (c+d x) \, dx+b^2 \int x^6 \cosh (c+d x) \, dx\\ &=\frac{a^2 \sinh (c+d x)}{d}+\frac{2 a b x^3 \sinh (c+d x)}{d}+\frac{b^2 x^6 \sinh (c+d x)}{d}-\frac{(6 a b) \int x^2 \sinh (c+d x) \, dx}{d}-\frac{\left (6 b^2\right ) \int x^5 \sinh (c+d x) \, dx}{d}\\ &=-\frac{6 a b x^2 \cosh (c+d x)}{d^2}-\frac{6 b^2 x^5 \cosh (c+d x)}{d^2}+\frac{a^2 \sinh (c+d x)}{d}+\frac{2 a b x^3 \sinh (c+d x)}{d}+\frac{b^2 x^6 \sinh (c+d x)}{d}+\frac{(12 a b) \int x \cosh (c+d x) \, dx}{d^2}+\frac{\left (30 b^2\right ) \int x^4 \cosh (c+d x) \, dx}{d^2}\\ &=-\frac{6 a b x^2 \cosh (c+d x)}{d^2}-\frac{6 b^2 x^5 \cosh (c+d x)}{d^2}+\frac{a^2 \sinh (c+d x)}{d}+\frac{12 a b x \sinh (c+d x)}{d^3}+\frac{2 a b x^3 \sinh (c+d x)}{d}+\frac{30 b^2 x^4 \sinh (c+d x)}{d^3}+\frac{b^2 x^6 \sinh (c+d x)}{d}-\frac{(12 a b) \int \sinh (c+d x) \, dx}{d^3}-\frac{\left (120 b^2\right ) \int x^3 \sinh (c+d x) \, dx}{d^3}\\ &=-\frac{12 a b \cosh (c+d x)}{d^4}-\frac{6 a b x^2 \cosh (c+d x)}{d^2}-\frac{120 b^2 x^3 \cosh (c+d x)}{d^4}-\frac{6 b^2 x^5 \cosh (c+d x)}{d^2}+\frac{a^2 \sinh (c+d x)}{d}+\frac{12 a b x \sinh (c+d x)}{d^3}+\frac{2 a b x^3 \sinh (c+d x)}{d}+\frac{30 b^2 x^4 \sinh (c+d x)}{d^3}+\frac{b^2 x^6 \sinh (c+d x)}{d}+\frac{\left (360 b^2\right ) \int x^2 \cosh (c+d x) \, dx}{d^4}\\ &=-\frac{12 a b \cosh (c+d x)}{d^4}-\frac{6 a b x^2 \cosh (c+d x)}{d^2}-\frac{120 b^2 x^3 \cosh (c+d x)}{d^4}-\frac{6 b^2 x^5 \cosh (c+d x)}{d^2}+\frac{a^2 \sinh (c+d x)}{d}+\frac{12 a b x \sinh (c+d x)}{d^3}+\frac{360 b^2 x^2 \sinh (c+d x)}{d^5}+\frac{2 a b x^3 \sinh (c+d x)}{d}+\frac{30 b^2 x^4 \sinh (c+d x)}{d^3}+\frac{b^2 x^6 \sinh (c+d x)}{d}-\frac{\left (720 b^2\right ) \int x \sinh (c+d x) \, dx}{d^5}\\ &=-\frac{12 a b \cosh (c+d x)}{d^4}-\frac{720 b^2 x \cosh (c+d x)}{d^6}-\frac{6 a b x^2 \cosh (c+d x)}{d^2}-\frac{120 b^2 x^3 \cosh (c+d x)}{d^4}-\frac{6 b^2 x^5 \cosh (c+d x)}{d^2}+\frac{a^2 \sinh (c+d x)}{d}+\frac{12 a b x \sinh (c+d x)}{d^3}+\frac{360 b^2 x^2 \sinh (c+d x)}{d^5}+\frac{2 a b x^3 \sinh (c+d x)}{d}+\frac{30 b^2 x^4 \sinh (c+d x)}{d^3}+\frac{b^2 x^6 \sinh (c+d x)}{d}+\frac{\left (720 b^2\right ) \int \cosh (c+d x) \, dx}{d^6}\\ &=-\frac{12 a b \cosh (c+d x)}{d^4}-\frac{720 b^2 x \cosh (c+d x)}{d^6}-\frac{6 a b x^2 \cosh (c+d x)}{d^2}-\frac{120 b^2 x^3 \cosh (c+d x)}{d^4}-\frac{6 b^2 x^5 \cosh (c+d x)}{d^2}+\frac{720 b^2 \sinh (c+d x)}{d^7}+\frac{a^2 \sinh (c+d x)}{d}+\frac{12 a b x \sinh (c+d x)}{d^3}+\frac{360 b^2 x^2 \sinh (c+d x)}{d^5}+\frac{2 a b x^3 \sinh (c+d x)}{d}+\frac{30 b^2 x^4 \sinh (c+d x)}{d^3}+\frac{b^2 x^6 \sinh (c+d x)}{d}\\ \end{align*}

Mathematica [A]  time = 0.219807, size = 111, normalized size = 0.6 \[ \frac{\left (a^2 d^6+2 a b d^4 x \left (d^2 x^2+6\right )+b^2 \left (d^6 x^6+30 d^4 x^4+360 d^2 x^2+720\right )\right ) \sinh (c+d x)-6 b d \left (a d^2 \left (d^2 x^2+2\right )+b x \left (d^4 x^4+20 d^2 x^2+120\right )\right ) \cosh (c+d x)}{d^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*b*d*(a*d^2*(2 + d^2*x^2) + b*x*(120 + 20*d^2*x^2 + d^4*x^4))*Cosh[c + d*x] + (a^2*d^6 + 2*a*b*d^4*x*(6 + d
^2*x^2) + b^2*(720 + 360*d^2*x^2 + 30*d^4*x^4 + d^6*x^6))*Sinh[c + d*x])/d^7

________________________________________________________________________________________

Maple [B]  time = 0.009, size = 592, normalized size = 3.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(-6/d^6*b^2*c*((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))+15/d^6*b^2*c^2*((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))-20/d^6*b^2*c^3*((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))+2/d^3*b*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))+a^2*sinh(d*x+c)+1/d^6*b^2*((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))+1/d^6*b^2*c^6*sinh(d*x+c)+6/d^3*b*c^2*a*((d*x+c)*sinh(d*x+c)-cosh(d*x+c))-6/d^3*b*c*a*((d*x+c)^2*sinh(d*
x+c)-2*(d*x+c)*cosh(d*x+c)+2*sinh(d*x+c))-2/d^3*b*c^3*a*sinh(d*x+c)-6/d^6*b^2*c^5*((d*x+c)*sinh(d*x+c)-cosh(d*
x+c))+15/d^6*b^2*c^4*((d*x+c)^2*sinh(d*x+c)-2*(d*x+c)*cosh(d*x+c)+2*sinh(d*x+c)))

________________________________________________________________________________________

Maxima [A]  time = 1.04638, size = 328, normalized size = 1.76 \begin{align*} \frac{a^{2} e^{\left (d x + c\right )}}{2 \, d} - \frac{a^{2} e^{\left (-d x - c\right )}}{2 \, d} + \frac{{\left (d^{3} x^{3} e^{c} - 3 \, d^{2} x^{2} e^{c} + 6 \, d x e^{c} - 6 \, e^{c}\right )} a b e^{\left (d x\right )}}{d^{4}} - \frac{{\left (d^{3} x^{3} + 3 \, d^{2} x^{2} + 6 \, d x + 6\right )} a b e^{\left (-d x - c\right )}}{d^{4}} + \frac{{\left (d^{6} x^{6} e^{c} - 6 \, d^{5} x^{5} e^{c} + 30 \, d^{4} x^{4} e^{c} - 120 \, d^{3} x^{3} e^{c} + 360 \, d^{2} x^{2} e^{c} - 720 \, d x e^{c} + 720 \, e^{c}\right )} b^{2} e^{\left (d x\right )}}{2 \, d^{7}} - \frac{{\left (d^{6} x^{6} + 6 \, d^{5} x^{5} + 30 \, d^{4} x^{4} + 120 \, d^{3} x^{3} + 360 \, d^{2} x^{2} + 720 \, d x + 720\right )} b^{2} e^{\left (-d x - c\right )}}{2 \, d^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a^2*e^(d*x + c)/d - 1/2*a^2*e^(-d*x - c)/d + (d^3*x^3*e^c - 3*d^2*x^2*e^c + 6*d*x*e^c - 6*e^c)*a*b*e^(d*x)
/d^4 - (d^3*x^3 + 3*d^2*x^2 + 6*d*x + 6)*a*b*e^(-d*x - c)/d^4 + 1/2*(d^6*x^6*e^c - 6*d^5*x^5*e^c + 30*d^4*x^4*
e^c - 120*d^3*x^3*e^c + 360*d^2*x^2*e^c - 720*d*x*e^c + 720*e^c)*b^2*e^(d*x)/d^7 - 1/2*(d^6*x^6 + 6*d^5*x^5 +
30*d^4*x^4 + 120*d^3*x^3 + 360*d^2*x^2 + 720*d*x + 720)*b^2*e^(-d*x - c)/d^7

________________________________________________________________________________________

Fricas [A]  time = 1.7411, size = 285, normalized size = 1.53 \begin{align*} -\frac{6 \,{\left (b^{2} d^{5} x^{5} + a b d^{5} x^{2} + 20 \, b^{2} d^{3} x^{3} + 2 \, a b d^{3} + 120 \, b^{2} d x\right )} \cosh \left (d x + c\right ) -{\left (b^{2} d^{6} x^{6} + 2 \, a b d^{6} x^{3} + 30 \, b^{2} d^{4} x^{4} + a^{2} d^{6} + 12 \, a b d^{4} x + 360 \, b^{2} d^{2} x^{2} + 720 \, b^{2}\right )} \sinh \left (d x + c\right )}{d^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(6*(b^2*d^5*x^5 + a*b*d^5*x^2 + 20*b^2*d^3*x^3 + 2*a*b*d^3 + 120*b^2*d*x)*cosh(d*x + c) - (b^2*d^6*x^6 + 2*a*
b*d^6*x^3 + 30*b^2*d^4*x^4 + a^2*d^6 + 12*a*b*d^4*x + 360*b^2*d^2*x^2 + 720*b^2)*sinh(d*x + c))/d^7

________________________________________________________________________________________

Sympy [A]  time = 9.36333, size = 226, normalized size = 1.22 \begin{align*} \begin{cases} \frac{a^{2} \sinh{\left (c + d x \right )}}{d} + \frac{2 a b x^{3} \sinh{\left (c + d x \right )}}{d} - \frac{6 a b x^{2} \cosh{\left (c + d x \right )}}{d^{2}} + \frac{12 a b x \sinh{\left (c + d x \right )}}{d^{3}} - \frac{12 a b \cosh{\left (c + d x \right )}}{d^{4}} + \frac{b^{2} x^{6} \sinh{\left (c + d x \right )}}{d} - \frac{6 b^{2} x^{5} \cosh{\left (c + d x \right )}}{d^{2}} + \frac{30 b^{2} x^{4} \sinh{\left (c + d x \right )}}{d^{3}} - \frac{120 b^{2} x^{3} \cosh{\left (c + d x \right )}}{d^{4}} + \frac{360 b^{2} x^{2} \sinh{\left (c + d x \right )}}{d^{5}} - \frac{720 b^{2} x \cosh{\left (c + d x \right )}}{d^{6}} + \frac{720 b^{2} \sinh{\left (c + d x \right )}}{d^{7}} & \text{for}\: d \neq 0 \\\left (a^{2} x + \frac{a b x^{4}}{2} + \frac{b^{2} x^{7}}{7}\right ) \cosh{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*sinh(c + d*x)/d + 2*a*b*x**3*sinh(c + d*x)/d - 6*a*b*x**2*cosh(c + d*x)/d**2 + 12*a*b*x*sinh(c
+ d*x)/d**3 - 12*a*b*cosh(c + d*x)/d**4 + b**2*x**6*sinh(c + d*x)/d - 6*b**2*x**5*cosh(c + d*x)/d**2 + 30*b**
2*x**4*sinh(c + d*x)/d**3 - 120*b**2*x**3*cosh(c + d*x)/d**4 + 360*b**2*x**2*sinh(c + d*x)/d**5 - 720*b**2*x*c
osh(c + d*x)/d**6 + 720*b**2*sinh(c + d*x)/d**7, Ne(d, 0)), ((a**2*x + a*b*x**4/2 + b**2*x**7/7)*cosh(c), True
))

________________________________________________________________________________________

Giac [A]  time = 1.38075, size = 329, normalized size = 1.77 \begin{align*} \frac{{\left (b^{2} d^{6} x^{6} - 6 \, b^{2} d^{5} x^{5} + 2 \, a b d^{6} x^{3} + 30 \, b^{2} d^{4} x^{4} - 6 \, a b d^{5} x^{2} + a^{2} d^{6} - 120 \, b^{2} d^{3} x^{3} + 12 \, a b d^{4} x + 360 \, b^{2} d^{2} x^{2} - 12 \, a b d^{3} - 720 \, b^{2} d x + 720 \, b^{2}\right )} e^{\left (d x + c\right )}}{2 \, d^{7}} - \frac{{\left (b^{2} d^{6} x^{6} + 6 \, b^{2} d^{5} x^{5} + 2 \, a b d^{6} x^{3} + 30 \, b^{2} d^{4} x^{4} + 6 \, a b d^{5} x^{2} + a^{2} d^{6} + 120 \, b^{2} d^{3} x^{3} + 12 \, a b d^{4} x + 360 \, b^{2} d^{2} x^{2} + 12 \, a b d^{3} + 720 \, b^{2} d x + 720 \, b^{2}\right )} e^{\left (-d x - c\right )}}{2 \, d^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b^2*d^6*x^6 - 6*b^2*d^5*x^5 + 2*a*b*d^6*x^3 + 30*b^2*d^4*x^4 - 6*a*b*d^5*x^2 + a^2*d^6 - 120*b^2*d^3*x^3
+ 12*a*b*d^4*x + 360*b^2*d^2*x^2 - 12*a*b*d^3 - 720*b^2*d*x + 720*b^2)*e^(d*x + c)/d^7 - 1/2*(b^2*d^6*x^6 + 6*
b^2*d^5*x^5 + 2*a*b*d^6*x^3 + 30*b^2*d^4*x^4 + 6*a*b*d^5*x^2 + a^2*d^6 + 120*b^2*d^3*x^3 + 12*a*b*d^4*x + 360*
b^2*d^2*x^2 + 12*a*b*d^3 + 720*b^2*d*x + 720*b^2)*e^(-d*x - c)/d^7