Integrand size = 18, antiderivative size = 88 \[ \int x^2 \sqrt {a+a \cosh (c+d x)} \, dx=-\frac {8 x \sqrt {a+a \cosh (c+d x)}}{d^2}+\frac {16 \sqrt {a+a \cosh (c+d x)} \tanh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d^3}+\frac {2 x^2 \sqrt {a+a \cosh (c+d x)} \tanh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d} \] Output:
-8*x*(a+a*cosh(d*x+c))^(1/2)/d^2+16*(a+a*cosh(d*x+c))^(1/2)*tanh(1/2*d*x+1 /2*c)/d^3+2*x^2*(a+a*cosh(d*x+c))^(1/2)*tanh(1/2*d*x+1/2*c)/d
Time = 0.15 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.50 \[ \int x^2 \sqrt {a+a \cosh (c+d x)} \, dx=\frac {2 \sqrt {a (1+\cosh (c+d x))} \left (-4 d x+\left (8+d^2 x^2\right ) \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{d^3} \] Input:
Integrate[x^2*Sqrt[a + a*Cosh[c + d*x]],x]
Output:
(2*Sqrt[a*(1 + Cosh[c + d*x])]*(-4*d*x + (8 + d^2*x^2)*Tanh[(c + d*x)/2])) /d^3
Result contains complex when optimal does not.
Time = 0.52 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.12, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {3042, 3800, 3042, 3777, 26, 3042, 26, 3777, 3042, 3117}
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.
\(\displaystyle \int x^2 \sqrt {a \cosh (c+d x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int x^2 \sqrt {a+a \sin \left (i c+i d x+\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 3800 |
\(\displaystyle \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a} \int x^2 \cosh \left (\frac {c}{2}+\frac {d x}{2}\right )dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a} \int x^2 \sin \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{2}\right )dx\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a} \left (\frac {2 x^2 \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}-\frac {4 i \int -i x \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )dx}{d}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a} \left (\frac {2 x^2 \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}-\frac {4 \int x \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )dx}{d}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a} \left (\frac {2 x^2 \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}-\frac {4 \int -i x \sin \left (\frac {i c}{2}+\frac {i d x}{2}\right )dx}{d}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a} \left (\frac {2 x^2 \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}+\frac {4 i \int x \sin \left (\frac {i c}{2}+\frac {i d x}{2}\right )dx}{d}\right )\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a} \left (\frac {2 x^2 \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}+\frac {4 i \left (\frac {2 i x \cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}-\frac {2 i \int \cosh \left (\frac {c}{2}+\frac {d x}{2}\right )dx}{d}\right )}{d}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a} \left (\frac {2 x^2 \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}+\frac {4 i \left (\frac {2 i x \cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}-\frac {2 i \int \sin \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{2}\right )dx}{d}\right )}{d}\right )\) |
\(\Big \downarrow \) 3117 |
\(\displaystyle \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a} \left (\frac {2 x^2 \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}+\frac {4 i \left (\frac {2 i x \cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}-\frac {4 i \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d^2}\right )}{d}\right )\) |
Input:
Int[x^2*Sqrt[a + a*Cosh[c + d*x]],x]
Output:
Sqrt[a + a*Cosh[c + d*x]]*Sech[c/2 + (d*x)/2]*((2*x^2*Sinh[c/2 + (d*x)/2]) /d + ((4*I)*(((2*I)*x*Cosh[c/2 + (d*x)/2])/d - ((4*I)*Sinh[c/2 + (d*x)/2]) /d^2))/d)
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[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
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 = 0.42 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.98
method | result | size |
risch | \(\frac {\sqrt {2}\, \sqrt {a \left ({\mathrm e}^{d x +c}+1\right )^{2} {\mathrm e}^{-d x -c}}\, \left (d^{2} x^{2} {\mathrm e}^{d x +c}-x^{2} d^{2}-4 d x \,{\mathrm e}^{d x +c}-4 d x +8 \,{\mathrm e}^{d x +c}-8\right )}{\left ({\mathrm e}^{d x +c}+1\right ) d^{3}}\) | \(86\) |
Input:
int(x^2*(a+a*cosh(d*x+c))^(1/2),x,method=_RETURNVERBOSE)
Output:
2^(1/2)*(a*(exp(d*x+c)+1)^2*exp(-d*x-c))^(1/2)/(exp(d*x+c)+1)*(d^2*x^2*exp (d*x+c)-x^2*d^2-4*d*x*exp(d*x+c)-4*d*x+8*exp(d*x+c)-8)/d^3
Exception generated. \[ \int x^2 \sqrt {a+a \cosh (c+d x)} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^2*(a+a*cosh(d*x+c))^(1/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (has polynomial part)
\[ \int x^2 \sqrt {a+a \cosh (c+d x)} \, dx=\int x^{2} \sqrt {a \left (\cosh {\left (c + d x \right )} + 1\right )}\, dx \] Input:
integrate(x**2*(a+a*cosh(d*x+c))**(1/2),x)
Output:
Integral(x**2*sqrt(a*(cosh(c + d*x) + 1)), x)
Time = 0.16 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.02 \[ \int x^2 \sqrt {a+a \cosh (c+d x)} \, dx=-\frac {{\left (\sqrt {2} \sqrt {a} d^{2} x^{2} + 4 \, \sqrt {2} \sqrt {a} d x - {\left (\sqrt {2} \sqrt {a} d^{2} x^{2} e^{c} - 4 \, \sqrt {2} \sqrt {a} d x e^{c} + 8 \, \sqrt {2} \sqrt {a} e^{c}\right )} e^{\left (d x\right )} + 8 \, \sqrt {2} \sqrt {a}\right )} e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )}}{d^{3}} \] Input:
integrate(x^2*(a+a*cosh(d*x+c))^(1/2),x, algorithm="maxima")
Output:
-(sqrt(2)*sqrt(a)*d^2*x^2 + 4*sqrt(2)*sqrt(a)*d*x - (sqrt(2)*sqrt(a)*d^2*x ^2*e^c - 4*sqrt(2)*sqrt(a)*d*x*e^c + 8*sqrt(2)*sqrt(a)*e^c)*e^(d*x) + 8*sq rt(2)*sqrt(a))*e^(-1/2*d*x - 1/2*c)/d^3
Time = 0.11 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.22 \[ \int x^2 \sqrt {a+a \cosh (c+d x)} \, dx=\frac {\sqrt {2} {\left (\sqrt {a} d^{2} x^{2} e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} - \sqrt {a} d^{2} x^{2} e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )} - 4 \, \sqrt {a} d x e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} - 4 \, \sqrt {a} d x e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )} + 8 \, \sqrt {a} e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} - 8 \, \sqrt {a} e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )}\right )}}{d^{3}} \] Input:
integrate(x^2*(a+a*cosh(d*x+c))^(1/2),x, algorithm="giac")
Output:
sqrt(2)*(sqrt(a)*d^2*x^2*e^(1/2*d*x + 1/2*c) - sqrt(a)*d^2*x^2*e^(-1/2*d*x - 1/2*c) - 4*sqrt(a)*d*x*e^(1/2*d*x + 1/2*c) - 4*sqrt(a)*d*x*e^(-1/2*d*x - 1/2*c) + 8*sqrt(a)*e^(1/2*d*x + 1/2*c) - 8*sqrt(a)*e^(-1/2*d*x - 1/2*c)) /d^3
Time = 1.89 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.08 \[ \int x^2 \sqrt {a+a \cosh (c+d x)} \, dx=-\frac {\sqrt {a+a\,\left (\frac {{\mathrm {e}}^{c+d\,x}}{2}+\frac {{\mathrm {e}}^{-c-d\,x}}{2}\right )}\,\left (\frac {8\,x}{d^2}-\frac {16\,{\mathrm {e}}^{c+d\,x}}{d^3}+\frac {16}{d^3}+\frac {2\,x^2}{d}-\frac {2\,x^2\,{\mathrm {e}}^{c+d\,x}}{d}+\frac {8\,x\,{\mathrm {e}}^{c+d\,x}}{d^2}\right )}{{\mathrm {e}}^{c+d\,x}+1} \] Input:
int(x^2*(a + a*cosh(c + d*x))^(1/2),x)
Output:
-((a + a*(exp(c + d*x)/2 + exp(- c - d*x)/2))^(1/2)*((8*x)/d^2 - (16*exp(c + d*x))/d^3 + 16/d^3 + (2*x^2)/d - (2*x^2*exp(c + d*x))/d + (8*x*exp(c + d*x))/d^2))/(exp(c + d*x) + 1)
\[ \int x^2 \sqrt {a+a \cosh (c+d x)} \, dx=\sqrt {a}\, \left (\int \sqrt {\cosh \left (d x +c \right )+1}\, x^{2}d x \right ) \] Input:
int(x^2*(a+a*cosh(d*x+c))^(1/2),x)
Output:
sqrt(a)*int(sqrt(cosh(c + d*x) + 1)*x**2,x)