3.121 \(\int x^3 \sqrt{a+a \cosh (c+d x)} \, dx\)

Optimal. Leaf size=110 \[ -\frac{12 x^2 \sqrt{a \cosh (c+d x)+a}}{d^2}-\frac{96 \sqrt{a \cosh (c+d x)+a}}{d^4}+\frac{48 x \tanh \left (\frac{c}{2}+\frac{d x}{2}\right ) \sqrt{a \cosh (c+d x)+a}}{d^3}+\frac{2 x^3 \tanh \left (\frac{c}{2}+\frac{d x}{2}\right ) \sqrt{a \cosh (c+d x)+a}}{d} \]

[Out]

(-96*Sqrt[a + a*Cosh[c + d*x]])/d^4 - (12*x^2*Sqrt[a + a*Cosh[c + d*x]])/d^2 + (48*x*Sqrt[a + a*Cosh[c + d*x]]
*Tanh[c/2 + (d*x)/2])/d^3 + (2*x^3*Sqrt[a + a*Cosh[c + d*x]]*Tanh[c/2 + (d*x)/2])/d

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Rubi [A]  time = 0.147767, antiderivative size = 110, 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 = {3319, 3296, 2638} \[ -\frac{12 x^2 \sqrt{a \cosh (c+d x)+a}}{d^2}-\frac{96 \sqrt{a \cosh (c+d x)+a}}{d^4}+\frac{48 x \tanh \left (\frac{c}{2}+\frac{d x}{2}\right ) \sqrt{a \cosh (c+d x)+a}}{d^3}+\frac{2 x^3 \tanh \left (\frac{c}{2}+\frac{d x}{2}\right ) \sqrt{a \cosh (c+d x)+a}}{d} \]

Antiderivative was successfully verified.

[In]

Int[x^3*Sqrt[a + a*Cosh[c + d*x]],x]

[Out]

(-96*Sqrt[a + a*Cosh[c + d*x]])/d^4 - (12*x^2*Sqrt[a + a*Cosh[c + d*x]])/d^2 + (48*x*Sqrt[a + a*Cosh[c + d*x]]
*Tanh[c/2 + (d*x)/2])/d^3 + (2*x^3*Sqrt[a + a*Cosh[c + d*x]]*Tanh[c/2 + (d*x)/2])/d

Rule 3319

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[((2*a)^IntPart[n
]*(a + b*Sin[e + f*x])^FracPart[n])/Sin[e/2 + (a*Pi)/(4*b) + (f*x)/2]^(2*FracPart[n]), Int[(c + d*x)^m*Sin[e/2
+ (a*Pi)/(4*b) + (f*x)/2]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[n
+ 1/2] && (GtQ[n, 0] || IGtQ[m, 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 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int x^3 \sqrt{a+a \cosh (c+d x)} \, dx &=\left (\sqrt{a+a \cosh (c+d x)} \csc \left (\frac{1}{2} \left (i c+\frac{\pi }{2}\right )+\frac{\pi }{4}+\frac{i d x}{2}\right )\right ) \int x^3 \sin \left (\frac{1}{2} \left (i c+\frac{\pi }{2}\right )+\frac{\pi }{4}+\frac{i d x}{2}\right ) \, dx\\ &=\frac{2 x^3 \sqrt{a+a \cosh (c+d x)} \tanh \left (\frac{c}{2}+\frac{d x}{2}\right )}{d}-\frac{\left (6 \sqrt{a+a \cosh (c+d x)} \csc \left (\frac{1}{2} \left (i c+\frac{\pi }{2}\right )+\frac{\pi }{4}+\frac{i d x}{2}\right )\right ) \int x^2 \sinh \left (\frac{c}{2}+\frac{d x}{2}\right ) \, dx}{d}\\ &=-\frac{12 x^2 \sqrt{a+a \cosh (c+d x)}}{d^2}+\frac{2 x^3 \sqrt{a+a \cosh (c+d x)} \tanh \left (\frac{c}{2}+\frac{d x}{2}\right )}{d}+\frac{\left (24 \sqrt{a+a \cosh (c+d x)} \csc \left (\frac{1}{2} \left (i c+\frac{\pi }{2}\right )+\frac{\pi }{4}+\frac{i d x}{2}\right )\right ) \int x \cosh \left (\frac{c}{2}+\frac{d x}{2}\right ) \, dx}{d^2}\\ &=-\frac{12 x^2 \sqrt{a+a \cosh (c+d x)}}{d^2}+\frac{48 x \sqrt{a+a \cosh (c+d x)} \tanh \left (\frac{c}{2}+\frac{d x}{2}\right )}{d^3}+\frac{2 x^3 \sqrt{a+a \cosh (c+d x)} \tanh \left (\frac{c}{2}+\frac{d x}{2}\right )}{d}-\frac{\left (48 \sqrt{a+a \cosh (c+d x)} \csc \left (\frac{1}{2} \left (i c+\frac{\pi }{2}\right )+\frac{\pi }{4}+\frac{i d x}{2}\right )\right ) \int \sinh \left (\frac{c}{2}+\frac{d x}{2}\right ) \, dx}{d^3}\\ &=-\frac{96 \sqrt{a+a \cosh (c+d x)}}{d^4}-\frac{12 x^2 \sqrt{a+a \cosh (c+d x)}}{d^2}+\frac{48 x \sqrt{a+a \cosh (c+d x)} \tanh \left (\frac{c}{2}+\frac{d x}{2}\right )}{d^3}+\frac{2 x^3 \sqrt{a+a \cosh (c+d x)} \tanh \left (\frac{c}{2}+\frac{d x}{2}\right )}{d}\\ \end{align*}

Mathematica [A]  time = 0.197331, size = 53, normalized size = 0.48 \[ \frac{2 \left (d x \left (d^2 x^2+24\right ) \tanh \left (\frac{1}{2} (c+d x)\right )-6 \left (d^2 x^2+8\right )\right ) \sqrt{a (\cosh (c+d x)+1)}}{d^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*Sqrt[a + a*Cosh[c + d*x]],x]

[Out]

(2*Sqrt[a*(1 + Cosh[c + d*x])]*(-6*(8 + d^2*x^2) + d*x*(24 + d^2*x^2)*Tanh[(c + d*x)/2]))/d^4

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Maple [A]  time = 0.106, size = 108, normalized size = 1. \begin{align*}{\frac{\sqrt{2} \left ({d}^{3}{x}^{3}{{\rm e}^{dx+c}}-{d}^{3}{x}^{3}-6\,{d}^{2}{x}^{2}{{\rm e}^{dx+c}}-6\,{d}^{2}{x}^{2}+24\,dx{{\rm e}^{dx+c}}-24\,dx-48\,{{\rm e}^{dx+c}}-48 \right ) }{ \left ({{\rm e}^{dx+c}}+1 \right ){d}^{4}}\sqrt{a \left ({{\rm e}^{dx+c}}+1 \right ) ^{2}{{\rm e}^{-dx-c}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a+a*cosh(d*x+c))^(1/2),x)

[Out]

2^(1/2)*(a*(exp(d*x+c)+1)^2*exp(-d*x-c))^(1/2)/(exp(d*x+c)+1)*(d^3*x^3*exp(d*x+c)-d^3*x^3-6*d^2*x^2*exp(d*x+c)
-6*d^2*x^2+24*d*x*exp(d*x+c)-24*d*x-48*exp(d*x+c)-48)/d^4

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Maxima [A]  time = 1.7096, size = 162, normalized size = 1.47 \begin{align*} -\frac{{\left (\sqrt{2} \sqrt{a} d^{3} x^{3} + 6 \, \sqrt{2} \sqrt{a} d^{2} x^{2} + 24 \, \sqrt{2} \sqrt{a} d x -{\left (\sqrt{2} \sqrt{a} d^{3} x^{3} e^{c} - 6 \, \sqrt{2} \sqrt{a} d^{2} x^{2} e^{c} + 24 \, \sqrt{2} \sqrt{a} d x e^{c} - 48 \, \sqrt{2} \sqrt{a} e^{c}\right )} e^{\left (d x\right )} + 48 \, \sqrt{2} \sqrt{a}\right )} e^{\left (-\frac{1}{2} \, d x - \frac{1}{2} \, c\right )}}{d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+a*cosh(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

-(sqrt(2)*sqrt(a)*d^3*x^3 + 6*sqrt(2)*sqrt(a)*d^2*x^2 + 24*sqrt(2)*sqrt(a)*d*x - (sqrt(2)*sqrt(a)*d^3*x^3*e^c
- 6*sqrt(2)*sqrt(a)*d^2*x^2*e^c + 24*sqrt(2)*sqrt(a)*d*x*e^c - 48*sqrt(2)*sqrt(a)*e^c)*e^(d*x) + 48*sqrt(2)*sq
rt(a))*e^(-1/2*d*x - 1/2*c)/d^4

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+a*cosh(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \sqrt{a \left (\cosh{\left (c + d x \right )} + 1\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(a+a*cosh(d*x+c))**(1/2),x)

[Out]

Integral(x**3*sqrt(a*(cosh(c + d*x) + 1)), x)

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Giac [A]  time = 1.30485, size = 198, normalized size = 1.8 \begin{align*} \frac{\sqrt{2}{\left (\sqrt{a} d^{3} x^{3} e^{\left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )} - \sqrt{a} d^{3} x^{3} e^{\left (-\frac{1}{2} \, d x - \frac{1}{2} \, c\right )} - 6 \, \sqrt{a} d^{2} x^{2} e^{\left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )} - 6 \, \sqrt{a} d^{2} x^{2} e^{\left (-\frac{1}{2} \, d x - \frac{1}{2} \, c\right )} + 24 \, \sqrt{a} d x e^{\left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )} - 24 \, \sqrt{a} d x e^{\left (-\frac{1}{2} \, d x - \frac{1}{2} \, c\right )} - 48 \, \sqrt{a} e^{\left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )} - 48 \, \sqrt{a} e^{\left (-\frac{1}{2} \, d x - \frac{1}{2} \, c\right )}\right )}}{d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+a*cosh(d*x+c))^(1/2),x, algorithm="giac")

[Out]

sqrt(2)*(sqrt(a)*d^3*x^3*e^(1/2*d*x + 1/2*c) - sqrt(a)*d^3*x^3*e^(-1/2*d*x - 1/2*c) - 6*sqrt(a)*d^2*x^2*e^(1/2
*d*x + 1/2*c) - 6*sqrt(a)*d^2*x^2*e^(-1/2*d*x - 1/2*c) + 24*sqrt(a)*d*x*e^(1/2*d*x + 1/2*c) - 24*sqrt(a)*d*x*e
^(-1/2*d*x - 1/2*c) - 48*sqrt(a)*e^(1/2*d*x + 1/2*c) - 48*sqrt(a)*e^(-1/2*d*x - 1/2*c))/d^4