Integrand size = 17, antiderivative size = 268 \[ \int x^2 \cosh ^2\left (a+b x+c x^2\right ) \, dx=\frac {x^3}{6}+\frac {b^2 e^{-2 a+\frac {b^2}{2 c}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{32 c^{5/2}}+\frac {e^{-2 a+\frac {b^2}{2 c}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{32 c^{3/2}}+\frac {b^2 e^{2 a-\frac {b^2}{2 c}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{32 c^{5/2}}-\frac {e^{2 a-\frac {b^2}{2 c}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{32 c^{3/2}}-\frac {b \sinh \left (2 a+2 b x+2 c x^2\right )}{16 c^2}+\frac {x \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c} \] Output:
1/6*x^3+1/64*b^2*exp(-2*a+1/2*b^2/c)*2^(1/2)*Pi^(1/2)*erf(1/2*(2*c*x+b)*2^ (1/2)/c^(1/2))/c^(5/2)+1/64*exp(-2*a+1/2*b^2/c)*2^(1/2)*Pi^(1/2)*erf(1/2*( 2*c*x+b)*2^(1/2)/c^(1/2))/c^(3/2)+1/64*b^2*exp(2*a-1/2*b^2/c)*2^(1/2)*Pi^( 1/2)*erfi(1/2*(2*c*x+b)*2^(1/2)/c^(1/2))/c^(5/2)-1/64*exp(2*a-1/2*b^2/c)*2 ^(1/2)*Pi^(1/2)*erfi(1/2*(2*c*x+b)*2^(1/2)/c^(1/2))/c^(3/2)-1/16*b*sinh(2* c*x^2+2*b*x+2*a)/c^2+1/8*x*sinh(2*c*x^2+2*b*x+2*a)/c
Time = 0.50 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.66 \[ \int x^2 \cosh ^2\left (a+b x+c x^2\right ) \, dx=\frac {3 \left (b^2+c\right ) \sqrt {2 \pi } \text {erf}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right ) \left (\cosh \left (2 a-\frac {b^2}{2 c}\right )-\sinh \left (2 a-\frac {b^2}{2 c}\right )\right )+3 \left (b^2-c\right ) \sqrt {2 \pi } \text {erfi}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right ) \left (\cosh \left (2 a-\frac {b^2}{2 c}\right )+\sinh \left (2 a-\frac {b^2}{2 c}\right )\right )+4 \sqrt {c} \left (8 c^2 x^3-3 (b-2 c x) \sinh (2 (a+x (b+c x)))\right )}{192 c^{5/2}} \] Input:
Integrate[x^2*Cosh[a + b*x + c*x^2]^2,x]
Output:
(3*(b^2 + c)*Sqrt[2*Pi]*Erf[(b + 2*c*x)/(Sqrt[2]*Sqrt[c])]*(Cosh[2*a - b^2 /(2*c)] - Sinh[2*a - b^2/(2*c)]) + 3*(b^2 - c)*Sqrt[2*Pi]*Erfi[(b + 2*c*x) /(Sqrt[2]*Sqrt[c])]*(Cosh[2*a - b^2/(2*c)] + Sinh[2*a - b^2/(2*c)]) + 4*Sq rt[c]*(8*c^2*x^3 - 3*(b - 2*c*x)*Sinh[2*(a + x*(b + c*x))]))/(192*c^(5/2))
Time = 0.54 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {5918, 2009}
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.
| \(\displaystyle \int x^2 \cosh ^2\left (a+b x+c x^2\right ) \, dx\) |
| \(\Big \downarrow \) 5918 |
| \(\displaystyle \int \left (\frac {1}{2} x^2 \cosh \left (2 a+2 b x+2 c x^2\right )+\frac {x^2}{2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {\frac {\pi }{2}} e^{\frac {b^2}{2 c}-2 a} \text {erf}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{32 c^{3/2}}+\frac {\sqrt {\frac {\pi }{2}} b^2 e^{\frac {b^2}{2 c}-2 a} \text {erf}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{32 c^{5/2}}-\frac {\sqrt {\frac {\pi }{2}} e^{2 a-\frac {b^2}{2 c}} \text {erfi}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{32 c^{3/2}}+\frac {\sqrt {\frac {\pi }{2}} b^2 e^{2 a-\frac {b^2}{2 c}} \text {erfi}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{32 c^{5/2}}-\frac {b \sinh \left (2 a+2 b x+2 c x^2\right )}{16 c^2}+\frac {x \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}+\frac {x^3}{6}\) |
Input:
Int[x^2*Cosh[a + b*x + c*x^2]^2,x]
Output:
x^3/6 + (b^2*E^(-2*a + b^2/(2*c))*Sqrt[Pi/2]*Erf[(b + 2*c*x)/(Sqrt[2]*Sqrt [c])])/(32*c^(5/2)) + (E^(-2*a + b^2/(2*c))*Sqrt[Pi/2]*Erf[(b + 2*c*x)/(Sq rt[2]*Sqrt[c])])/(32*c^(3/2)) + (b^2*E^(2*a - b^2/(2*c))*Sqrt[Pi/2]*Erfi[( b + 2*c*x)/(Sqrt[2]*Sqrt[c])])/(32*c^(5/2)) - (E^(2*a - b^2/(2*c))*Sqrt[Pi /2]*Erfi[(b + 2*c*x)/(Sqrt[2]*Sqrt[c])])/(32*c^(3/2)) - (b*Sinh[2*a + 2*b* x + 2*c*x^2])/(16*c^2) + (x*Sinh[2*a + 2*b*x + 2*c*x^2])/(8*c)
Int[Cosh[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]^(n_)*((d_.) + (e_.)*(x_))^(m_.) , x_Symbol] :> Int[ExpandTrigReduce[(d + e*x)^m, Cosh[a + b*x + c*x^2]^n, x ], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 1]
Time = 1.30 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.05
| method | result | size |
| risch | \(\frac {x^{3}}{6}-\frac {x \,{\mathrm e}^{-2 c \,x^{2}-2 b x -2 a}}{16 c}+\frac {b \,{\mathrm e}^{-2 c \,x^{2}-2 b x -2 a}}{32 c^{2}}+\frac {b^{2} \sqrt {\pi }\, {\mathrm e}^{-\frac {4 a c -b^{2}}{2 c}} \sqrt {2}\, \operatorname {erf}\left (\sqrt {2}\, \sqrt {c}\, x +\frac {b \sqrt {2}}{2 \sqrt {c}}\right )}{64 c^{\frac {5}{2}}}+\frac {\sqrt {\pi }\, {\mathrm e}^{-\frac {4 a c -b^{2}}{2 c}} \sqrt {2}\, \operatorname {erf}\left (\sqrt {2}\, \sqrt {c}\, x +\frac {b \sqrt {2}}{2 \sqrt {c}}\right )}{64 c^{\frac {3}{2}}}+\frac {x \,{\mathrm e}^{2 c \,x^{2}+2 b x +2 a}}{16 c}-\frac {b \,{\mathrm e}^{2 c \,x^{2}+2 b x +2 a}}{32 c^{2}}-\frac {b^{2} \sqrt {\pi }\, {\mathrm e}^{\frac {4 a c -b^{2}}{2 c}} \operatorname {erf}\left (-\sqrt {-2 c}\, x +\frac {b}{\sqrt {-2 c}}\right )}{32 c^{2} \sqrt {-2 c}}+\frac {\sqrt {\pi }\, {\mathrm e}^{\frac {4 a c -b^{2}}{2 c}} \operatorname {erf}\left (-\sqrt {-2 c}\, x +\frac {b}{\sqrt {-2 c}}\right )}{32 c \sqrt {-2 c}}\) | \(281\) |
Input:
int(x^2*cosh(c*x^2+b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
1/6*x^3-1/16/c*x*exp(-2*c*x^2-2*b*x-2*a)+1/32*b/c^2*exp(-2*c*x^2-2*b*x-2*a )+1/64*b^2/c^(5/2)*Pi^(1/2)*exp(-1/2*(4*a*c-b^2)/c)*2^(1/2)*erf(2^(1/2)*c^ (1/2)*x+1/2*b*2^(1/2)/c^(1/2))+1/64/c^(3/2)*Pi^(1/2)*exp(-1/2*(4*a*c-b^2)/ c)*2^(1/2)*erf(2^(1/2)*c^(1/2)*x+1/2*b*2^(1/2)/c^(1/2))+1/16/c*x*exp(2*c*x ^2+2*b*x+2*a)-1/32*b/c^2*exp(2*c*x^2+2*b*x+2*a)-1/32*b^2/c^2*Pi^(1/2)*exp( 1/2*(4*a*c-b^2)/c)/(-2*c)^(1/2)*erf(-(-2*c)^(1/2)*x+b/(-2*c)^(1/2))+1/32/c *Pi^(1/2)*exp(1/2*(4*a*c-b^2)/c)/(-2*c)^(1/2)*erf(-(-2*c)^(1/2)*x+b/(-2*c) ^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 766 vs. \(2 (210) = 420\).
Time = 0.10 (sec) , antiderivative size = 766, normalized size of antiderivative = 2.86 \[ \int x^2 \cosh ^2\left (a+b x+c x^2\right ) \, dx =\text {Too large to display} \] Input:
integrate(x^2*cosh(c*x^2+b*x+a)^2,x, algorithm="fricas")
Output:
1/192*(32*c^3*x^3*cosh(c*x^2 + b*x + a)^2 + 6*(2*c^2*x - b*c)*cosh(c*x^2 + b*x + a)^4 + 24*(2*c^2*x - b*c)*cosh(c*x^2 + b*x + a)*sinh(c*x^2 + b*x + a)^3 + 6*(2*c^2*x - b*c)*sinh(c*x^2 + b*x + a)^4 - 3*sqrt(2)*sqrt(pi)*((b^ 2 - c)*cosh(c*x^2 + b*x + a)^2*cosh(-1/2*(b^2 - 4*a*c)/c) + (b^2 - c)*cosh (c*x^2 + b*x + a)^2*sinh(-1/2*(b^2 - 4*a*c)/c) + ((b^2 - c)*cosh(-1/2*(b^2 - 4*a*c)/c) + (b^2 - c)*sinh(-1/2*(b^2 - 4*a*c)/c))*sinh(c*x^2 + b*x + a) ^2 + 2*((b^2 - c)*cosh(c*x^2 + b*x + a)*cosh(-1/2*(b^2 - 4*a*c)/c) + (b^2 - c)*cosh(c*x^2 + b*x + a)*sinh(-1/2*(b^2 - 4*a*c)/c))*sinh(c*x^2 + b*x + a))*sqrt(-c)*erf(1/2*sqrt(2)*(2*c*x + b)*sqrt(-c)/c) + 3*sqrt(2)*sqrt(pi)* ((b^2 + c)*cosh(c*x^2 + b*x + a)^2*cosh(-1/2*(b^2 - 4*a*c)/c) - (b^2 + c)* cosh(c*x^2 + b*x + a)^2*sinh(-1/2*(b^2 - 4*a*c)/c) + ((b^2 + c)*cosh(-1/2* (b^2 - 4*a*c)/c) - (b^2 + c)*sinh(-1/2*(b^2 - 4*a*c)/c))*sinh(c*x^2 + b*x + a)^2 + 2*((b^2 + c)*cosh(c*x^2 + b*x + a)*cosh(-1/2*(b^2 - 4*a*c)/c) - ( b^2 + c)*cosh(c*x^2 + b*x + a)*sinh(-1/2*(b^2 - 4*a*c)/c))*sinh(c*x^2 + b* x + a))*sqrt(c)*erf(1/2*sqrt(2)*(2*c*x + b)/sqrt(c)) - 12*c^2*x + 4*(8*c^3 *x^3 + 9*(2*c^2*x - b*c)*cosh(c*x^2 + b*x + a)^2)*sinh(c*x^2 + b*x + a)^2 + 6*b*c + 8*(8*c^3*x^3*cosh(c*x^2 + b*x + a) + 3*(2*c^2*x - b*c)*cosh(c*x^ 2 + b*x + a)^3)*sinh(c*x^2 + b*x + a))/(c^3*cosh(c*x^2 + b*x + a)^2 + 2*c^ 3*cosh(c*x^2 + b*x + a)*sinh(c*x^2 + b*x + a) + c^3*sinh(c*x^2 + b*x + a)^ 2)
\[ \int x^2 \cosh ^2\left (a+b x+c x^2\right ) \, dx=\int x^{2} \cosh ^{2}{\left (a + b x + c x^{2} \right )}\, dx \] Input:
integrate(x**2*cosh(c*x**2+b*x+a)**2,x)
Output:
Integral(x**2*cosh(a + b*x + c*x**2)**2, x)
Time = 0.17 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.10 \[ \int x^2 \cosh ^2\left (a+b x+c x^2\right ) \, dx=\frac {1}{6} \, x^{3} + \frac {\sqrt {2} {\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b^{2} {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (2 \, c x + b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt {-\frac {{\left (2 \, c x + b\right )}^{2}}{c}} c^{\frac {5}{2}}} - \frac {2 \, \sqrt {2} b e^{\left (\frac {{\left (2 \, c x + b\right )}^{2}}{2 \, c}\right )}}{c^{\frac {3}{2}}} - \frac {2 \, {\left (2 \, c x + b\right )}^{3} \Gamma \left (\frac {3}{2}, -\frac {{\left (2 \, c x + b\right )}^{2}}{2 \, c}\right )}{\left (-\frac {{\left (2 \, c x + b\right )}^{2}}{c}\right )^{\frac {3}{2}} c^{\frac {5}{2}}}\right )} e^{\left (2 \, a - \frac {b^{2}}{2 \, c}\right )}}{64 \, \sqrt {c}} - \frac {\sqrt {2} {\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b^{2} {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {\frac {{\left (2 \, c x + b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt {\frac {{\left (2 \, c x + b\right )}^{2}}{c}} \left (-c\right )^{\frac {5}{2}}} + \frac {2 \, \sqrt {2} b c e^{\left (-\frac {{\left (2 \, c x + b\right )}^{2}}{2 \, c}\right )}}{\left (-c\right )^{\frac {5}{2}}} - \frac {2 \, {\left (2 \, c x + b\right )}^{3} \Gamma \left (\frac {3}{2}, \frac {{\left (2 \, c x + b\right )}^{2}}{2 \, c}\right )}{\left (\frac {{\left (2 \, c x + b\right )}^{2}}{c}\right )^{\frac {3}{2}} \left (-c\right )^{\frac {5}{2}}}\right )} e^{\left (-2 \, a + \frac {b^{2}}{2 \, c}\right )}}{64 \, \sqrt {-c}} \] Input:
integrate(x^2*cosh(c*x^2+b*x+a)^2,x, algorithm="maxima")
Output:
1/6*x^3 + 1/64*sqrt(2)*(sqrt(pi)*(2*c*x + b)*b^2*(erf(sqrt(1/2)*sqrt(-(2*c *x + b)^2/c)) - 1)/(sqrt(-(2*c*x + b)^2/c)*c^(5/2)) - 2*sqrt(2)*b*e^(1/2*( 2*c*x + b)^2/c)/c^(3/2) - 2*(2*c*x + b)^3*gamma(3/2, -1/2*(2*c*x + b)^2/c) /((-(2*c*x + b)^2/c)^(3/2)*c^(5/2)))*e^(2*a - 1/2*b^2/c)/sqrt(c) - 1/64*sq rt(2)*(sqrt(pi)*(2*c*x + b)*b^2*(erf(sqrt(1/2)*sqrt((2*c*x + b)^2/c)) - 1) /(sqrt((2*c*x + b)^2/c)*(-c)^(5/2)) + 2*sqrt(2)*b*c*e^(-1/2*(2*c*x + b)^2/ c)/(-c)^(5/2) - 2*(2*c*x + b)^3*gamma(3/2, 1/2*(2*c*x + b)^2/c)/(((2*c*x + b)^2/c)^(3/2)*(-c)^(5/2)))*e^(-2*a + 1/2*b^2/c)/sqrt(-c)
Time = 0.12 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.68 \[ \int x^2 \cosh ^2\left (a+b x+c x^2\right ) \, dx=\frac {1}{6} \, x^{3} - \frac {\frac {\sqrt {2} \sqrt {\pi } {\left (b^{2} + c\right )} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} \sqrt {c} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (\frac {b^{2} - 4 \, a c}{2 \, c}\right )}}{\sqrt {c}} + 2 \, {\left (c {\left (2 \, x + \frac {b}{c}\right )} - 2 \, b\right )} e^{\left (-2 \, c x^{2} - 2 \, b x - 2 \, a\right )}}{64 \, c^{2}} - \frac {\frac {\sqrt {2} \sqrt {\pi } {\left (b^{2} - c\right )} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} \sqrt {-c} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right )}}{\sqrt {-c}} - 2 \, {\left (c {\left (2 \, x + \frac {b}{c}\right )} - 2 \, b\right )} e^{\left (2 \, c x^{2} + 2 \, b x + 2 \, a\right )}}{64 \, c^{2}} \] Input:
integrate(x^2*cosh(c*x^2+b*x+a)^2,x, algorithm="giac")
Output:
1/6*x^3 - 1/64*(sqrt(2)*sqrt(pi)*(b^2 + c)*erf(-1/2*sqrt(2)*sqrt(c)*(2*x + b/c))*e^(1/2*(b^2 - 4*a*c)/c)/sqrt(c) + 2*(c*(2*x + b/c) - 2*b)*e^(-2*c*x ^2 - 2*b*x - 2*a))/c^2 - 1/64*(sqrt(2)*sqrt(pi)*(b^2 - c)*erf(-1/2*sqrt(2) *sqrt(-c)*(2*x + b/c))*e^(-1/2*(b^2 - 4*a*c)/c)/sqrt(-c) - 2*(c*(2*x + b/c ) - 2*b)*e^(2*c*x^2 + 2*b*x + 2*a))/c^2
Timed out. \[ \int x^2 \cosh ^2\left (a+b x+c x^2\right ) \, dx=\int x^2\,{\mathrm {cosh}\left (c\,x^2+b\,x+a\right )}^2 \,d x \] Input:
int(x^2*cosh(a + b*x + c*x^2)^2,x)
Output:
int(x^2*cosh(a + b*x + c*x^2)^2, x)
\[ \int x^2 \cosh ^2\left (a+b x+c x^2\right ) \, dx=\frac {-3 \sqrt {\pi }\, e^{2 c \,x^{2}+2 b x +4 a} \mathrm {erf}\left (\frac {2 c i x +b i}{\sqrt {c}\, \sqrt {2}}\right ) b^{2} i +3 \sqrt {\pi }\, e^{2 c \,x^{2}+2 b x +4 a} \mathrm {erf}\left (\frac {2 c i x +b i}{\sqrt {c}\, \sqrt {2}}\right ) c i -3 e^{\frac {8 c^{2} x^{2}+8 b c x +8 a c +b^{2}}{2 c}} \sqrt {c}\, \sqrt {2}\, b +6 e^{\frac {8 c^{2} x^{2}+8 b c x +8 a c +b^{2}}{2 c}} \sqrt {c}\, \sqrt {2}\, c x +16 e^{\frac {4 c^{2} x^{2}+4 b c x +4 a c +b^{2}}{2 c}} \sqrt {c}\, \sqrt {2}\, c^{2} x^{3}+6 e^{\frac {4 c^{2} x^{2}+4 b c x +b^{2}}{2 c}} \sqrt {c}\, \sqrt {2}\, \left (\int \frac {1}{e^{2 c \,x^{2}+2 b x}}d x \right ) b^{2}+6 e^{\frac {4 c^{2} x^{2}+4 b c x +b^{2}}{2 c}} \sqrt {c}\, \sqrt {2}\, \left (\int \frac {1}{e^{2 c \,x^{2}+2 b x}}d x \right ) c +3 e^{\frac {b^{2}}{2 c}} \sqrt {c}\, \sqrt {2}\, b -6 e^{\frac {b^{2}}{2 c}} \sqrt {c}\, \sqrt {2}\, c x}{96 e^{\frac {4 c^{2} x^{2}+4 b c x +4 a c +b^{2}}{2 c}} \sqrt {c}\, \sqrt {2}\, c^{2}} \] Input:
int(x^2*cosh(c*x^2+b*x+a)^2,x)
Output:
( - 3*sqrt(pi)*e**(4*a + 2*b*x + 2*c*x**2)*erf((b*i + 2*c*i*x)/(sqrt(c)*sq rt(2)))*b**2*i + 3*sqrt(pi)*e**(4*a + 2*b*x + 2*c*x**2)*erf((b*i + 2*c*i*x )/(sqrt(c)*sqrt(2)))*c*i - 3*e**((8*a*c + b**2 + 8*b*c*x + 8*c**2*x**2)/(2 *c))*sqrt(c)*sqrt(2)*b + 6*e**((8*a*c + b**2 + 8*b*c*x + 8*c**2*x**2)/(2*c ))*sqrt(c)*sqrt(2)*c*x + 16*e**((4*a*c + b**2 + 4*b*c*x + 4*c**2*x**2)/(2* c))*sqrt(c)*sqrt(2)*c**2*x**3 + 6*e**((b**2 + 4*b*c*x + 4*c**2*x**2)/(2*c) )*sqrt(c)*sqrt(2)*int(1/e**(2*b*x + 2*c*x**2),x)*b**2 + 6*e**((b**2 + 4*b* c*x + 4*c**2*x**2)/(2*c))*sqrt(c)*sqrt(2)*int(1/e**(2*b*x + 2*c*x**2),x)*c + 3*e**(b**2/(2*c))*sqrt(c)*sqrt(2)*b - 6*e**(b**2/(2*c))*sqrt(c)*sqrt(2) *c*x)/(96*e**((4*a*c + b**2 + 4*b*c*x + 4*c**2*x**2)/(2*c))*sqrt(c)*sqrt(2 )*c**2)