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

Optimal. Leaf size=49 \[ -\frac{2 b (a+b x) \cosh (c+d x)}{d^2}+\frac{(a+b x)^2 \sinh (c+d x)}{d}+\frac{2 b^2 \sinh (c+d x)}{d^3} \]

[Out]

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

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Rubi [A]  time = 0.0479988, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3296, 2637} \[ -\frac{2 b (a+b x) \cosh (c+d x)}{d^2}+\frac{(a+b x)^2 \sinh (c+d x)}{d}+\frac{2 b^2 \sinh (c+d x)}{d^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2*Cosh[c + d*x],x]

[Out]

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

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]

Rubi steps

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

Mathematica [A]  time = 0.142182, size = 56, normalized size = 1.14 \[ \frac{\left (a^2 d^2+2 a b d^2 x+b^2 \left (d^2 x^2+2\right )\right ) \sinh (c+d x)-2 b d (a+b x) \cosh (c+d x)}{d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.007, size = 147, normalized size = 3. \begin{align*}{\frac{1}{d} \left ({\frac{{b}^{2} \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{c{b}^{2} \left ( \left ( dx+c \right ) \sinh \left ( dx+c \right ) -\cosh \left ( dx+c \right ) \right ) }{{d}^{2}}}+2\,{\frac{ab \left ( \left ( dx+c \right ) \sinh \left ( dx+c \right ) -\cosh \left ( dx+c \right ) \right ) }{d}}+{\frac{{c}^{2}{b}^{2}\sinh \left ( dx+c \right ) }{{d}^{2}}}-2\,{\frac{cba\sinh \left ( dx+c \right ) }{d}}+{a}^{2}\sinh \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.20621, size = 182, normalized size = 3.71 \begin{align*} \frac{a^{2} e^{\left (d x + c\right )}}{2 \, d} + \frac{{\left (d x e^{c} - e^{c}\right )} a b e^{\left (d x\right )}}{d^{2}} - \frac{{\left (d x + 1\right )} a b e^{\left (-d x - c\right )}}{d^{2}} - \frac{a^{2} 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^{2} e^{\left (d x\right )}}{2 \, d^{3}} - \frac{{\left (d^{2} x^{2} + 2 \, d x + 2\right )} b^{2} 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+a)^2*cosh(d*x+c),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.94877, size = 140, normalized size = 2.86 \begin{align*} -\frac{2 \,{\left (b^{2} d x + a b d\right )} \cosh \left (d x + c\right ) -{\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2} + 2 \, b^{2}\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+a)^2*cosh(d*x+c),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.0641, size = 112, normalized size = 2.29 \begin{align*} \begin{cases} \frac{a^{2} \sinh{\left (c + d x \right )}}{d} + \frac{2 a b x \sinh{\left (c + d x \right )}}{d} - \frac{2 a b \cosh{\left (c + d x \right )}}{d^{2}} + \frac{b^{2} x^{2} \sinh{\left (c + d x \right )}}{d} - \frac{2 b^{2} x \cosh{\left (c + d x \right )}}{d^{2}} + \frac{2 b^{2} \sinh{\left (c + d x \right )}}{d^{3}} & \text{for}\: d \neq 0 \\\left (a^{2} x + a b x^{2} + \frac{b^{2} 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+a)**2*cosh(d*x+c),x)

[Out]

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

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Giac [B]  time = 1.21508, size = 151, normalized size = 3.08 \begin{align*} \frac{{\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2} - 2 \, b^{2} d x - 2 \, a b d + 2 \, b^{2}\right )} e^{\left (d x + c\right )}}{2 \, d^{3}} - \frac{{\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2} + 2 \, b^{2} d x + 2 \, a b d + 2 \, b^{2}\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+a)^2*cosh(d*x+c),x, algorithm="giac")

[Out]

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