Integrand size = 14, antiderivative size = 68 \[ \int x^3 \sqrt {a+a \cosh (x)} \, dx=-96 \sqrt {a+a \cosh (x)}-12 x^2 \sqrt {a+a \cosh (x)}+48 x \sqrt {a+a \cosh (x)} \tanh \left (\frac {x}{2}\right )+2 x^3 \sqrt {a+a \cosh (x)} \tanh \left (\frac {x}{2}\right ) \] Output:
-96*(a+a*cosh(x))^(1/2)-12*x^2*(a+a*cosh(x))^(1/2)+48*x*(a+a*cosh(x))^(1/2 )*tanh(1/2*x)+2*x^3*(a+a*cosh(x))^(1/2)*tanh(1/2*x)
Time = 0.08 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.49 \[ \int x^3 \sqrt {a+a \cosh (x)} \, dx=2 \sqrt {a (1+\cosh (x))} \left (-6 \left (8+x^2\right )+x \left (24+x^2\right ) \tanh \left (\frac {x}{2}\right )\right ) \] Input:
Integrate[x^3*Sqrt[a + a*Cosh[x]],x]
Output:
2*Sqrt[a*(1 + Cosh[x])]*(-6*(8 + x^2) + x*(24 + x^2)*Tanh[x/2])
Result contains complex when optimal does not.
Time = 0.52 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.01, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 3800, 3042, 3777, 26, 3042, 26, 3777, 3042, 3777, 26, 3042, 26, 3118}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x^3 \sqrt {a \cosh (x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int x^3 \sqrt {a+a \sin \left (\frac {\pi }{2}+i x\right )}dx\) |
\(\Big \downarrow \) 3800 |
\(\displaystyle \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \int x^3 \cosh \left (\frac {x}{2}\right )dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \int x^3 \sin \left (\frac {i x}{2}+\frac {\pi }{2}\right )dx\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (2 x^3 \sinh \left (\frac {x}{2}\right )-6 i \int -i x^2 \sinh \left (\frac {x}{2}\right )dx\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (2 x^3 \sinh \left (\frac {x}{2}\right )-6 \int x^2 \sinh \left (\frac {x}{2}\right )dx\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (2 x^3 \sinh \left (\frac {x}{2}\right )-6 \int -i x^2 \sin \left (\frac {i x}{2}\right )dx\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (2 x^3 \sinh \left (\frac {x}{2}\right )+6 i \int x^2 \sin \left (\frac {i x}{2}\right )dx\right )\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (2 x^3 \sinh \left (\frac {x}{2}\right )+6 i \left (2 i x^2 \cosh \left (\frac {x}{2}\right )-4 i \int x \cosh \left (\frac {x}{2}\right )dx\right )\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (2 x^3 \sinh \left (\frac {x}{2}\right )+6 i \left (2 i x^2 \cosh \left (\frac {x}{2}\right )-4 i \int x \sin \left (\frac {i x}{2}+\frac {\pi }{2}\right )dx\right )\right )\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (2 x^3 \sinh \left (\frac {x}{2}\right )+6 i \left (2 i x^2 \cosh \left (\frac {x}{2}\right )-4 i \left (2 x \sinh \left (\frac {x}{2}\right )-2 i \int -i \sinh \left (\frac {x}{2}\right )dx\right )\right )\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (2 x^3 \sinh \left (\frac {x}{2}\right )+6 i \left (2 i x^2 \cosh \left (\frac {x}{2}\right )-4 i \left (2 x \sinh \left (\frac {x}{2}\right )-2 \int \sinh \left (\frac {x}{2}\right )dx\right )\right )\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (2 x^3 \sinh \left (\frac {x}{2}\right )+6 i \left (2 i x^2 \cosh \left (\frac {x}{2}\right )-4 i \left (2 x \sinh \left (\frac {x}{2}\right )-2 \int -i \sin \left (\frac {i x}{2}\right )dx\right )\right )\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (2 x^3 \sinh \left (\frac {x}{2}\right )+6 i \left (2 i x^2 \cosh \left (\frac {x}{2}\right )-4 i \left (2 x \sinh \left (\frac {x}{2}\right )+2 i \int \sin \left (\frac {i x}{2}\right )dx\right )\right )\right )\) |
\(\Big \downarrow \) 3118 |
\(\displaystyle \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (2 x^3 \sinh \left (\frac {x}{2}\right )+6 i \left (2 i x^2 \cosh \left (\frac {x}{2}\right )-4 i \left (2 x \sinh \left (\frac {x}{2}\right )-4 \cosh \left (\frac {x}{2}\right )\right )\right )\right )\) |
Input:
Int[x^3*Sqrt[a + a*Cosh[x]],x]
Output:
Sqrt[a + a*Cosh[x]]*Sech[x/2]*(2*x^3*Sinh[x/2] + (6*I)*((2*I)*x^2*Cosh[x/2 ] - (4*I)*(-4*Cosh[x/2] + 2*x*Sinh[x/2])))
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( -(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*C os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(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])
Time = 0.23 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.91
method | result | size |
risch | \(\frac {\sqrt {2}\, \sqrt {a \left ({\mathrm e}^{x}+1\right )^{2} {\mathrm e}^{-x}}\, \left (x^{3} {\mathrm e}^{x}-x^{3}-6 x^{2} {\mathrm e}^{x}-6 x^{2}+24 x \,{\mathrm e}^{x}-24 x -48 \,{\mathrm e}^{x}-48\right )}{{\mathrm e}^{x}+1}\) | \(62\) |
Input:
int(x^3*(a+cosh(x)*a)^(1/2),x,method=_RETURNVERBOSE)
Output:
2^(1/2)*(a*(exp(x)+1)^2*exp(-x))^(1/2)/(exp(x)+1)*(x^3*exp(x)-x^3-6*x^2*ex p(x)-6*x^2+24*x*exp(x)-24*x-48*exp(x)-48)
Exception generated. \[ \int x^3 \sqrt {a+a \cosh (x)} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^3*(a+a*cosh(x))^(1/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (has polynomial part)
\[ \int x^3 \sqrt {a+a \cosh (x)} \, dx=\int x^{3} \sqrt {a \left (\cosh {\left (x \right )} + 1\right )}\, dx \] Input:
integrate(x**3*(a+a*cosh(x))**(1/2),x)
Output:
Integral(x**3*sqrt(a*(cosh(x) + 1)), x)
Time = 0.12 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.29 \[ \int x^3 \sqrt {a+a \cosh (x)} \, dx=-{\left (\sqrt {2} \sqrt {a} x^{3} + 6 \, \sqrt {2} \sqrt {a} x^{2} + 24 \, \sqrt {2} \sqrt {a} x - {\left (\sqrt {2} \sqrt {a} x^{3} - 6 \, \sqrt {2} \sqrt {a} x^{2} + 24 \, \sqrt {2} \sqrt {a} x - 48 \, \sqrt {2} \sqrt {a}\right )} e^{x} + 48 \, \sqrt {2} \sqrt {a}\right )} e^{\left (-\frac {1}{2} \, x\right )} \] Input:
integrate(x^3*(a+a*cosh(x))^(1/2),x, algorithm="maxima")
Output:
-(sqrt(2)*sqrt(a)*x^3 + 6*sqrt(2)*sqrt(a)*x^2 + 24*sqrt(2)*sqrt(a)*x - (sq rt(2)*sqrt(a)*x^3 - 6*sqrt(2)*sqrt(a)*x^2 + 24*sqrt(2)*sqrt(a)*x - 48*sqrt (2)*sqrt(a))*e^x + 48*sqrt(2)*sqrt(a))*e^(-1/2*x)
\[ \int x^3 \sqrt {a+a \cosh (x)} \, dx=\int { \sqrt {a \cosh \left (x\right ) + a} x^{3} \,d x } \] Input:
integrate(x^3*(a+a*cosh(x))^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(a*cosh(x) + a)*x^3, x)
Time = 1.87 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.93 \[ \int x^3 \sqrt {a+a \cosh (x)} \, dx=-\frac {\sqrt {a+a\,\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}\,\left (48\,x+96\,{\mathrm {e}}^x+12\,x^2\,{\mathrm {e}}^x-2\,x^3\,{\mathrm {e}}^x-48\,x\,{\mathrm {e}}^x+12\,x^2+2\,x^3+96\right )}{{\mathrm {e}}^x+1} \] Input:
int(x^3*(a + a*cosh(x))^(1/2),x)
Output:
-((a + a*(exp(-x)/2 + exp(x)/2))^(1/2)*(48*x + 96*exp(x) + 12*x^2*exp(x) - 2*x^3*exp(x) - 48*x*exp(x) + 12*x^2 + 2*x^3 + 96))/(exp(x) + 1)
\[ \int x^3 \sqrt {a+a \cosh (x)} \, dx=\sqrt {a}\, \left (\int \sqrt {\cosh \left (x \right )+1}\, x^{3}d x \right ) \] Input:
int(x^3*(a+a*cosh(x))^(1/2),x)
Output:
sqrt(a)*int(sqrt(cosh(x) + 1)*x**3,x)