3.100 \(\int (c+d x)^2 (a+a \cosh (e+f x)) \, dx\)

Optimal. Leaf size=67 \[ -\frac{2 a d (c+d x) \cosh (e+f x)}{f^2}+\frac{a (c+d x)^2 \sinh (e+f x)}{f}+\frac{a (c+d x)^3}{3 d}+\frac{2 a d^2 \sinh (e+f x)}{f^3} \]

[Out]

(a*(c + d*x)^3)/(3*d) - (2*a*d*(c + d*x)*Cosh[e + f*x])/f^2 + (2*a*d^2*Sinh[e + f*x])/f^3 + (a*(c + d*x)^2*Sin
h[e + f*x])/f

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Rubi [A]  time = 0.0906686, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3317, 3296, 2637} \[ -\frac{2 a d (c+d x) \cosh (e+f x)}{f^2}+\frac{a (c+d x)^2 \sinh (e+f x)}{f}+\frac{a (c+d x)^3}{3 d}+\frac{2 a d^2 \sinh (e+f x)}{f^3} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^2*(a + a*Cosh[e + f*x]),x]

[Out]

(a*(c + d*x)^3)/(3*d) - (2*a*d*(c + d*x)*Cosh[e + f*x])/f^2 + (2*a*d^2*Sinh[e + f*x])/f^3 + (a*(c + d*x)^2*Sin
h[e + f*x])/f

Rule 3317

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, (a + b*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 0] && (EqQ[n, 1] ||
IGtQ[m, 0] || NeQ[a^2 - b^2, 0])

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

Mathematica [A]  time = 0.314141, size = 80, normalized size = 1.19 \[ a \left (\frac{\left (c^2 f^2+2 c d f^2 x+d^2 \left (f^2 x^2+2\right )\right ) \sinh (e+f x)}{f^3}+c^2 x-\frac{2 d (c+d x) \cosh (e+f x)}{f^2}+c d x^2+\frac{d^2 x^3}{3}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.012, size = 240, normalized size = 3.6 \begin{align*}{\frac{1}{f} \left ({\frac{a{d}^{2} \left ( fx+e \right ) ^{3}}{3\,{f}^{2}}}+{\frac{a{d}^{2} \left ( \left ( fx+e \right ) ^{2}\sinh \left ( fx+e \right ) -2\, \left ( fx+e \right ) \cosh \left ( fx+e \right ) +2\,\sinh \left ( fx+e \right ) \right ) }{{f}^{2}}}-{\frac{{d}^{2}ea \left ( fx+e \right ) ^{2}}{{f}^{2}}}-2\,{\frac{{d}^{2}ea \left ( \left ( fx+e \right ) \sinh \left ( fx+e \right ) -\cosh \left ( fx+e \right ) \right ) }{{f}^{2}}}+{\frac{cda \left ( fx+e \right ) ^{2}}{f}}+2\,{\frac{cda \left ( \left ( fx+e \right ) \sinh \left ( fx+e \right ) -\cosh \left ( fx+e \right ) \right ) }{f}}+{\frac{{d}^{2}{e}^{2}a \left ( fx+e \right ) }{{f}^{2}}}+{\frac{{d}^{2}{e}^{2}a\sinh \left ( fx+e \right ) }{{f}^{2}}}-2\,{\frac{deca \left ( fx+e \right ) }{f}}-2\,{\frac{deca\sinh \left ( fx+e \right ) }{f}}+{c}^{2}a \left ( fx+e \right ) +{c}^{2}a\sinh \left ( fx+e \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.09106, size = 190, normalized size = 2.84 \begin{align*} \frac{1}{3} \, a d^{2} x^{3} + a c d x^{2} + a c^{2} x + a c d{\left (\frac{{\left (f x e^{e} - e^{e}\right )} e^{\left (f x\right )}}{f^{2}} - \frac{{\left (f x + 1\right )} e^{\left (-f x - e\right )}}{f^{2}}\right )} + \frac{1}{2} \, a d^{2}{\left (\frac{{\left (f^{2} x^{2} e^{e} - 2 \, f x e^{e} + 2 \, e^{e}\right )} e^{\left (f x\right )}}{f^{3}} - \frac{{\left (f^{2} x^{2} + 2 \, f x + 2\right )} e^{\left (-f x - e\right )}}{f^{3}}\right )} + \frac{a c^{2} \sinh \left (f x + e\right )}{f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.9719, size = 231, normalized size = 3.45 \begin{align*} \frac{a d^{2} f^{3} x^{3} + 3 \, a c d f^{3} x^{2} + 3 \, a c^{2} f^{3} x - 6 \,{\left (a d^{2} f x + a c d f\right )} \cosh \left (f x + e\right ) + 3 \,{\left (a d^{2} f^{2} x^{2} + 2 \, a c d f^{2} x + a c^{2} f^{2} + 2 \, a d^{2}\right )} \sinh \left (f x + e\right )}{3 \, f^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(a*d^2*f^3*x^3 + 3*a*c*d*f^3*x^2 + 3*a*c^2*f^3*x - 6*(a*d^2*f*x + a*c*d*f)*cosh(f*x + e) + 3*(a*d^2*f^2*x^
2 + 2*a*c*d*f^2*x + a*c^2*f^2 + 2*a*d^2)*sinh(f*x + e))/f^3

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Sympy [A]  time = 0.945921, size = 151, normalized size = 2.25 \begin{align*} \begin{cases} a c^{2} x + \frac{a c^{2} \sinh{\left (e + f x \right )}}{f} + a c d x^{2} + \frac{2 a c d x \sinh{\left (e + f x \right )}}{f} - \frac{2 a c d \cosh{\left (e + f x \right )}}{f^{2}} + \frac{a d^{2} x^{3}}{3} + \frac{a d^{2} x^{2} \sinh{\left (e + f x \right )}}{f} - \frac{2 a d^{2} x \cosh{\left (e + f x \right )}}{f^{2}} + \frac{2 a d^{2} \sinh{\left (e + f x \right )}}{f^{3}} & \text{for}\: f \neq 0 \\\left (a \cosh{\left (e \right )} + a\right ) \left (c^{2} x + c d x^{2} + \frac{d^{2} x^{3}}{3}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**2*(a+a*cosh(f*x+e)),x)

[Out]

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

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Giac [B]  time = 1.29276, size = 200, normalized size = 2.99 \begin{align*} \frac{1}{3} \, a d^{2} x^{3} + a c d x^{2} + a c^{2} x + \frac{{\left (a d^{2} f^{2} x^{2} + 2 \, a c d f^{2} x + a c^{2} f^{2} - 2 \, a d^{2} f x - 2 \, a c d f + 2 \, a d^{2}\right )} e^{\left (f x + e\right )}}{2 \, f^{3}} - \frac{{\left (a d^{2} f^{2} x^{2} + 2 \, a c d f^{2} x + a c^{2} f^{2} + 2 \, a d^{2} f x + 2 \, a c d f + 2 \, a d^{2}\right )} e^{\left (-f x - e\right )}}{2 \, f^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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