3.3 \(\int (c+d x)^2 \sinh (a+b x) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0496901, 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, 2638} \[ -\frac{2 d (c+d x) \sinh (a+b x)}{b^2}+\frac{2 d^2 \cosh (a+b x)}{b^3}+\frac{(c+d x)^2 \cosh (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^2*Sinh[a + b*x],x]

[Out]

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

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

Mathematica [A]  time = 0.157426, size = 44, normalized size = 0.9 \[ \frac{\cosh (a+b x) \left (b^2 (c+d x)^2+2 d^2\right )-2 b d (c+d x) \sinh (a+b x)}{b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.006, size = 147, normalized size = 3. \begin{align*}{\frac{1}{b} \left ({\frac{{d}^{2} \left ( \left ( bx+a \right ) ^{2}\cosh \left ( bx+a \right ) -2\, \left ( bx+a \right ) \sinh \left ( bx+a \right ) +2\,\cosh \left ( bx+a \right ) \right ) }{{b}^{2}}}-2\,{\frac{{d}^{2}a \left ( \left ( bx+a \right ) \cosh \left ( bx+a \right ) -\sinh \left ( bx+a \right ) \right ) }{{b}^{2}}}+2\,{\frac{dc \left ( \left ( bx+a \right ) \cosh \left ( bx+a \right ) -\sinh \left ( bx+a \right ) \right ) }{b}}+{\frac{{d}^{2}{a}^{2}\cosh \left ( bx+a \right ) }{{b}^{2}}}-2\,{\frac{dac\cosh \left ( bx+a \right ) }{b}}+{c}^{2}\cosh \left ( bx+a \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.1176, size = 181, normalized size = 3.69 \begin{align*} \frac{c^{2} e^{\left (b x + a\right )}}{2 \, b} + \frac{{\left (b x e^{a} - e^{a}\right )} c d e^{\left (b x\right )}}{b^{2}} + \frac{c^{2} e^{\left (-b x - a\right )}}{2 \, b} + \frac{{\left (b x + 1\right )} c d e^{\left (-b x - a\right )}}{b^{2}} + \frac{{\left (b^{2} x^{2} e^{a} - 2 \, b x e^{a} + 2 \, e^{a}\right )} d^{2} e^{\left (b x\right )}}{2 \, b^{3}} + \frac{{\left (b^{2} x^{2} + 2 \, b x + 2\right )} d^{2} e^{\left (-b x - a\right )}}{2 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.56058, size = 139, normalized size = 2.84 \begin{align*} \frac{{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} + 2 \, d^{2}\right )} \cosh \left (b x + a\right ) - 2 \,{\left (b d^{2} x + b c d\right )} \sinh \left (b x + a\right )}{b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.721562, size = 112, normalized size = 2.29 \begin{align*} \begin{cases} \frac{c^{2} \cosh{\left (a + b x \right )}}{b} + \frac{2 c d x \cosh{\left (a + b x \right )}}{b} + \frac{d^{2} x^{2} \cosh{\left (a + b x \right )}}{b} - \frac{2 c d \sinh{\left (a + b x \right )}}{b^{2}} - \frac{2 d^{2} x \sinh{\left (a + b x \right )}}{b^{2}} + \frac{2 d^{2} \cosh{\left (a + b x \right )}}{b^{3}} & \text{for}\: b \neq 0 \\\left (c^{2} x + c d x^{2} + \frac{d^{2} x^{3}}{3}\right ) \sinh{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.16676, size = 151, normalized size = 3.08 \begin{align*} \frac{{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} - 2 \, b d^{2} x - 2 \, b c d + 2 \, d^{2}\right )} e^{\left (b x + a\right )}}{2 \, b^{3}} + \frac{{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} + 2 \, b d^{2} x + 2 \, b c d + 2 \, d^{2}\right )} e^{\left (-b x - a\right )}}{2 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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