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

Optimal. Leaf size=66 \[ \frac{a \sinh (c+d x)}{d}-\frac{3 b x^2 \cosh (c+d x)}{d^2}+\frac{6 b x \sinh (c+d x)}{d^3}-\frac{6 b \cosh (c+d x)}{d^4}+\frac{b x^3 \sinh (c+d x)}{d} \]

[Out]

(-6*b*Cosh[c + d*x])/d^4 - (3*b*x^2*Cosh[c + d*x])/d^2 + (a*Sinh[c + d*x])/d + (6*b*x*Sinh[c + d*x])/d^3 + (b*
x^3*Sinh[c + d*x])/d

________________________________________________________________________________________

Rubi [A]  time = 0.103047, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {5277, 2637, 3296, 2638} \[ \frac{a \sinh (c+d x)}{d}-\frac{3 b x^2 \cosh (c+d x)}{d^2}+\frac{6 b x \sinh (c+d x)}{d^3}-\frac{6 b \cosh (c+d x)}{d^4}+\frac{b x^3 \sinh (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)*Cosh[c + d*x],x]

[Out]

(-6*b*Cosh[c + d*x])/d^4 - (3*b*x^2*Cosh[c + d*x])/d^2 + (a*Sinh[c + d*x])/d + (6*b*x*Sinh[c + d*x])/d^3 + (b*
x^3*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 ) \cosh (c+d x) \, dx &=\int \left (a \cosh (c+d x)+b x^3 \cosh (c+d x)\right ) \, dx\\ &=a \int \cosh (c+d x) \, dx+b \int x^3 \cosh (c+d x) \, dx\\ &=\frac{a \sinh (c+d x)}{d}+\frac{b x^3 \sinh (c+d x)}{d}-\frac{(3 b) \int x^2 \sinh (c+d x) \, dx}{d}\\ &=-\frac{3 b x^2 \cosh (c+d x)}{d^2}+\frac{a \sinh (c+d x)}{d}+\frac{b x^3 \sinh (c+d x)}{d}+\frac{(6 b) \int x \cosh (c+d x) \, dx}{d^2}\\ &=-\frac{3 b x^2 \cosh (c+d x)}{d^2}+\frac{a \sinh (c+d x)}{d}+\frac{6 b x \sinh (c+d x)}{d^3}+\frac{b x^3 \sinh (c+d x)}{d}-\frac{(6 b) \int \sinh (c+d x) \, dx}{d^3}\\ &=-\frac{6 b \cosh (c+d x)}{d^4}-\frac{3 b x^2 \cosh (c+d x)}{d^2}+\frac{a \sinh (c+d x)}{d}+\frac{6 b x \sinh (c+d x)}{d^3}+\frac{b x^3 \sinh (c+d x)}{d}\\ \end{align*}

Mathematica [A]  time = 0.0808538, size = 49, normalized size = 0.74 \[ \frac{d \left (a d^2+b x \left (d^2 x^2+6\right )\right ) \sinh (c+d x)-3 b \left (d^2 x^2+2\right ) \cosh (c+d x)}{d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*b*(2 + d^2*x^2)*Cosh[c + d*x] + d*(a*d^2 + b*x*(6 + d^2*x^2))*Sinh[c + d*x])/d^4

________________________________________________________________________________________

Maple [B]  time = 0.007, size = 158, normalized size = 2.4 \begin{align*}{\frac{1}{d} \left ({\frac{b \left ( \left ( dx+c \right ) ^{3}\sinh \left ( dx+c \right ) -3\, \left ( dx+c \right ) ^{2}\cosh \left ( dx+c \right ) +6\, \left ( dx+c \right ) \sinh \left ( dx+c \right ) -6\,\cosh \left ( dx+c \right ) \right ) }{{d}^{3}}}-3\,{\frac{cb \left ( \left ( dx+c \right ) ^{2}\sinh \left ( dx+c \right ) -2\, \left ( dx+c \right ) \cosh \left ( dx+c \right ) +2\,\sinh \left ( dx+c \right ) \right ) }{{d}^{3}}}+3\,{\frac{b{c}^{2} \left ( \left ( dx+c \right ) \sinh \left ( dx+c \right ) -\cosh \left ( dx+c \right ) \right ) }{{d}^{3}}}-{\frac{b{c}^{3}\sinh \left ( dx+c \right ) }{{d}^{3}}}+a\sinh \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(1/d^3*b*((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))-3/d^3*b*c*((d
*x+c)^2*sinh(d*x+c)-2*(d*x+c)*cosh(d*x+c)+2*sinh(d*x+c))+3/d^3*b*c^2*((d*x+c)*sinh(d*x+c)-cosh(d*x+c))-1/d^3*b
*c^3*sinh(d*x+c)+a*sinh(d*x+c))

________________________________________________________________________________________

Maxima [A]  time = 1.02864, size = 140, normalized size = 2.12 \begin{align*} \frac{a e^{\left (d x + c\right )}}{2 \, d} - \frac{a 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 )} b e^{\left (d x\right )}}{2 \, d^{4}} - \frac{{\left (d^{3} x^{3} + 3 \, d^{2} x^{2} + 6 \, d x + 6\right )} b e^{\left (-d x - c\right )}}{2 \, d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a*e^(d*x + c)/d - 1/2*a*e^(-d*x - c)/d + 1/2*(d^3*x^3*e^c - 3*d^2*x^2*e^c + 6*d*x*e^c - 6*e^c)*b*e^(d*x)/d
^4 - 1/2*(d^3*x^3 + 3*d^2*x^2 + 6*d*x + 6)*b*e^(-d*x - c)/d^4

________________________________________________________________________________________

Fricas [A]  time = 1.81258, size = 119, normalized size = 1.8 \begin{align*} -\frac{3 \,{\left (b d^{2} x^{2} + 2 \, b\right )} \cosh \left (d x + c\right ) -{\left (b d^{3} x^{3} + a d^{3} + 6 \, b d x\right )} \sinh \left (d x + c\right )}{d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 1.40829, size = 82, normalized size = 1.24 \begin{align*} \begin{cases} \frac{a \sinh{\left (c + d x \right )}}{d} + \frac{b x^{3} \sinh{\left (c + d x \right )}}{d} - \frac{3 b x^{2} \cosh{\left (c + d x \right )}}{d^{2}} + \frac{6 b x \sinh{\left (c + d x \right )}}{d^{3}} - \frac{6 b \cosh{\left (c + d x \right )}}{d^{4}} & \text{for}\: d \neq 0 \\\left (a x + \frac{b x^{4}}{4}\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)*cosh(d*x+c),x)

[Out]

Piecewise((a*sinh(c + d*x)/d + b*x**3*sinh(c + d*x)/d - 3*b*x**2*cosh(c + d*x)/d**2 + 6*b*x*sinh(c + d*x)/d**3
- 6*b*cosh(c + d*x)/d**4, Ne(d, 0)), ((a*x + b*x**4/4)*cosh(c), True))

________________________________________________________________________________________

Giac [A]  time = 1.22369, size = 119, normalized size = 1.8 \begin{align*} \frac{{\left (b d^{3} x^{3} - 3 \, b d^{2} x^{2} + a d^{3} + 6 \, b d x - 6 \, b\right )} e^{\left (d x + c\right )}}{2 \, d^{4}} - \frac{{\left (b d^{3} x^{3} + 3 \, b d^{2} x^{2} + a d^{3} + 6 \, b d x + 6 \, b\right )} e^{\left (-d x - c\right )}}{2 \, d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b*d^3*x^3 - 3*b*d^2*x^2 + a*d^3 + 6*b*d*x - 6*b)*e^(d*x + c)/d^4 - 1/2*(b*d^3*x^3 + 3*b*d^2*x^2 + a*d^3 +
6*b*d*x + 6*b)*e^(-d*x - c)/d^4