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

Optimal. Leaf size=51 \[ \frac{a \sinh (c+d x)}{d}+\frac{2 b \sinh (c+d x)}{d^3}-\frac{2 b x \cosh (c+d x)}{d^2}+\frac{b x^2 \sinh (c+d x)}{d} \]

[Out]

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

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Rubi [A]  time = 0.0667648, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {5277, 2637, 3296} \[ \frac{a \sinh (c+d x)}{d}+\frac{2 b \sinh (c+d x)}{d^3}-\frac{2 b x \cosh (c+d x)}{d^2}+\frac{b x^2 \sinh (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)*Cosh[c + d*x],x]

[Out]

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

Rubi steps

\begin{align*} \int \left (a+b x^2\right ) \cosh (c+d x) \, dx &=\int \left (a \cosh (c+d x)+b x^2 \cosh (c+d x)\right ) \, dx\\ &=a \int \cosh (c+d x) \, dx+b \int x^2 \cosh (c+d x) \, dx\\ &=\frac{a \sinh (c+d x)}{d}+\frac{b x^2 \sinh (c+d x)}{d}-\frac{(2 b) \int x \sinh (c+d x) \, dx}{d}\\ &=-\frac{2 b x \cosh (c+d x)}{d^2}+\frac{a \sinh (c+d x)}{d}+\frac{b x^2 \sinh (c+d x)}{d}+\frac{(2 b) \int \cosh (c+d x) \, dx}{d^2}\\ &=-\frac{2 b x \cosh (c+d x)}{d^2}+\frac{2 b \sinh (c+d x)}{d^3}+\frac{a \sinh (c+d x)}{d}+\frac{b x^2 \sinh (c+d x)}{d}\\ \end{align*}

Mathematica [A]  time = 0.0667749, size = 40, normalized size = 0.78 \[ \frac{\left (a d^2+b \left (d^2 x^2+2\right )\right ) \sinh (c+d x)-2 b d x \cosh (c+d x)}{d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.008, size = 97, normalized size = 1.9 \begin{align*}{\frac{1}{d} \left ({\frac{b \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}^{2}}}-2\,{\frac{cb \left ( \left ( dx+c \right ) \sinh \left ( dx+c \right ) -\cosh \left ( dx+c \right ) \right ) }{{d}^{2}}}+{\frac{b{c}^{2}\sinh \left ( dx+c \right ) }{{d}^{2}}}+a\sinh \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.01897, size = 116, normalized size = 2.27 \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^{2} x^{2} e^{c} - 2 \, d x e^{c} + 2 \, e^{c}\right )} b e^{\left (d x\right )}}{2 \, d^{3}} - \frac{{\left (d^{2} x^{2} + 2 \, d x + 2\right )} b e^{\left (-d x - c\right )}}{2 \, d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+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^2*x^2*e^c - 2*d*x*e^c + 2*e^c)*b*e^(d*x)/d^3 - 1/2*(d^2*x^
2 + 2*d*x + 2)*b*e^(-d*x - c)/d^3

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Fricas [A]  time = 1.9949, size = 97, normalized size = 1.9 \begin{align*} -\frac{2 \, b d x \cosh \left (d x + c\right ) -{\left (b d^{2} x^{2} + a d^{2} + 2 \, b\right )} \sinh \left (d x + c\right )}{d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.674157, size = 65, normalized size = 1.27 \begin{align*} \begin{cases} \frac{a \sinh{\left (c + d x \right )}}{d} + \frac{b x^{2} \sinh{\left (c + d x \right )}}{d} - \frac{2 b x \cosh{\left (c + d x \right )}}{d^{2}} + \frac{2 b \sinh{\left (c + d x \right )}}{d^{3}} & \text{for}\: d \neq 0 \\\left (a x + \frac{b x^{3}}{3}\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**2+a)*cosh(d*x+c),x)

[Out]

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

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Giac [A]  time = 1.15273, size = 95, normalized size = 1.86 \begin{align*} \frac{{\left (b d^{2} x^{2} + a d^{2} - 2 \, b d x + 2 \, b\right )} e^{\left (d x + c\right )}}{2 \, d^{3}} - \frac{{\left (b d^{2} x^{2} + a d^{2} + 2 \, b d x + 2 \, b\right )} e^{\left (-d x - c\right )}}{2 \, d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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