\(\int x^3 \sqrt {a+a \cosh (c+d x)} \, dx\) [121]

Optimal result

Integrand size = 18, antiderivative size = 110 \[ \int x^3 \sqrt {a+a \cosh (c+d x)} \, dx=-\frac {96 \sqrt {a+a \cosh (c+d x)}}{d^4}-\frac {12 x^2 \sqrt {a+a \cosh (c+d x)}}{d^2}+\frac {48 x \sqrt {a+a \cosh (c+d x)} \tanh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d^3}+\frac {2 x^3 \sqrt {a+a \cosh (c+d x)} \tanh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d} \]

[Out]

-96*(a+a*cosh(d*x+c))^(1/2)/d^4-12*x^2*(a+a*cosh(d*x+c))^(1/2)/d^2+48*x*(a+a*cosh(d*x+c))^(1/2)*tanh(1/2*d*x+1
/2*c)/d^3+2*x^3*(a+a*cosh(d*x+c))^(1/2)*tanh(1/2*d*x+1/2*c)/d

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3400, 3377, 2718} \[ \int x^3 \sqrt {a+a \cosh (c+d x)} \, dx=-\frac {96 \sqrt {a \cosh (c+d x)+a}}{d^4}+\frac {48 x \tanh \left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a}}{d^3}-\frac {12 x^2 \sqrt {a \cosh (c+d x)+a}}{d^2}+\frac {2 x^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a}}{d} \]

[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 2718

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

Rule 3377

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 3400

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])

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 0.23 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.48 \[ \int x^3 \sqrt {a+a \cosh (c+d x)} \, dx=\frac {2 \sqrt {a (1+\cosh (c+d x))} \left (-6 \left (8+d^2 x^2\right )+d x \left (24+d^2 x^2\right ) \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{d^4} \]

[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

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.98

method	result	size
risch	\(\frac {\sqrt {2}\, \sqrt {a \left ({\mathrm e}^{d x +c}+1\right )^{2} {\mathrm e}^{-d x -c}}\, \left (d^{3} x^{3} {\mathrm e}^{d x +c}-d^{3} x^{3}-6 d^{2} x^{2} {\mathrm e}^{d x +c}-6 x^{2} d^{2}+24 d x \,{\mathrm e}^{d x +c}-24 d x -48 \,{\mathrm e}^{d x +c}-48\right )}{\left ({\mathrm e}^{d x +c}+1\right ) d^{4}}\)	\(108\)

[In]

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

[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*x^2*d^2+24*d*x*exp(d*x+c)-24*d*x-48*exp(d*x+c)-48)/d^4

Fricas [F(-2)]

Exception generated. \[ \int x^3 \sqrt {a+a \cosh (c+d x)} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

Sympy [F]

\[ \int x^3 \sqrt {a+a \cosh (c+d x)} \, dx=\int x^{3} \sqrt {a \left (\cosh {\left (c + d x \right )} + 1\right )}\, dx \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.09 \[ \int x^3 \sqrt {a+a \cosh (c+d x)} \, dx=-\frac {{\left (\sqrt {2} \sqrt {a} d^{3} x^{3} + 6 \, \sqrt {2} \sqrt {a} d^{2} x^{2} + 24 \, \sqrt {2} \sqrt {a} d x - {\left (\sqrt {2} \sqrt {a} d^{3} x^{3} e^{c} - 6 \, \sqrt {2} \sqrt {a} d^{2} x^{2} e^{c} + 24 \, \sqrt {2} \sqrt {a} d x e^{c} - 48 \, \sqrt {2} \sqrt {a} e^{c}\right )} e^{\left (d x\right )} + 48 \, \sqrt {2} \sqrt {a}\right )} e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )}}{d^{4}} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.34 \[ \int x^3 \sqrt {a+a \cosh (c+d x)} \, dx=\frac {\sqrt {2} {\left (\sqrt {a} d^{3} x^{3} e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} - \sqrt {a} d^{3} x^{3} e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )} - 6 \, \sqrt {a} d^{2} x^{2} e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} - 6 \, \sqrt {a} d^{2} x^{2} e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )} + 24 \, \sqrt {a} d x e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} - 24 \, \sqrt {a} d x e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )} - 48 \, \sqrt {a} e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} - 48 \, \sqrt {a} e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )}\right )}}{d^{4}} \]

[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

Mupad [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.06 \[ \int x^3 \sqrt {a+a \cosh (c+d x)} \, dx=-\frac {\sqrt {a+a\,\left (\frac {{\mathrm {e}}^{c+d\,x}}{2}+\frac {{\mathrm {e}}^{-c-d\,x}}{2}\right )}\,\left (\frac {96\,{\mathrm {e}}^{c+d\,x}}{d^4}+\frac {48\,x}{d^3}+\frac {96}{d^4}+\frac {2\,x^3}{d}+\frac {12\,x^2}{d^2}-\frac {2\,x^3\,{\mathrm {e}}^{c+d\,x}}{d}+\frac {12\,x^2\,{\mathrm {e}}^{c+d\,x}}{d^2}-\frac {48\,x\,{\mathrm {e}}^{c+d\,x}}{d^3}\right )}{{\mathrm {e}}^{c+d\,x}+1} \]

[In]

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

[Out]

-((a + a*(exp(c + d*x)/2 + exp(- c - d*x)/2))^(1/2)*((96*exp(c + d*x))/d^4 + (48*x)/d^3 + 96/d^4 + (2*x^3)/d +
(12*x^2)/d^2 - (2*x^3*exp(c + d*x))/d + (12*x^2*exp(c + d*x))/d^2 - (48*x*exp(c + d*x))/d^3))/(exp(c + d*x) +
1)