Integrand size = 13, antiderivative size = 111 \[ \int x \sinh \left (a+b x+c x^2\right ) \, dx=\frac {\cosh \left (a+b x+c x^2\right )}{2 c}+\frac {b e^{-a+\frac {b^2}{4 c}} \sqrt {\pi } \text {erf}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{8 c^{3/2}}-\frac {b e^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{8 c^{3/2}} \]
1/2*cosh(c*x^2+b*x+a)/c+1/8*b*exp(-a+1/4*b^2/c)*erf(1/2*(2*c*x+b)/c^(1/2)) *Pi^(1/2)/c^(3/2)-1/8*b*exp(a-1/4*b^2/c)*erfi(1/2*(2*c*x+b)/c^(1/2))*Pi^(1 /2)/c^(3/2)
Time = 0.13 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.17 \[ \int x \sinh \left (a+b x+c x^2\right ) \, dx=\frac {4 \sqrt {c} \cosh (a+x (b+c x))+b \sqrt {\pi } \text {erf}\left (\frac {b+2 c x}{2 \sqrt {c}}\right ) \left (\cosh \left (a-\frac {b^2}{4 c}\right )-\sinh \left (a-\frac {b^2}{4 c}\right )\right )-b \sqrt {\pi } \text {erfi}\left (\frac {b+2 c x}{2 \sqrt {c}}\right ) \left (\cosh \left (a-\frac {b^2}{4 c}\right )+\sinh \left (a-\frac {b^2}{4 c}\right )\right )}{8 c^{3/2}} \]
(4*Sqrt[c]*Cosh[a + x*(b + c*x)] + b*Sqrt[Pi]*Erf[(b + 2*c*x)/(2*Sqrt[c])] *(Cosh[a - b^2/(4*c)] - Sinh[a - b^2/(4*c)]) - b*Sqrt[Pi]*Erfi[(b + 2*c*x) /(2*Sqrt[c])]*(Cosh[a - b^2/(4*c)] + Sinh[a - b^2/(4*c)]))/(8*c^(3/2))
Time = 0.36 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {5905, 5897, 2664, 2633, 2634}
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.
| \(\displaystyle \int x \sinh \left (a+b x+c x^2\right ) \, dx\) |
| \(\Big \downarrow \) 5905 |
| \(\displaystyle \frac {\cosh \left (a+b x+c x^2\right )}{2 c}-\frac {b \int \sinh \left (c x^2+b x+a\right )dx}{2 c}\) |
| \(\Big \downarrow \) 5897 |
| \(\displaystyle \frac {\cosh \left (a+b x+c x^2\right )}{2 c}-\frac {b \left (\frac {1}{2} \int e^{c x^2+b x+a}dx-\frac {1}{2} \int e^{-c x^2-b x-a}dx\right )}{2 c}\) |
| \(\Big \downarrow \) 2664 |
| \(\displaystyle \frac {\cosh \left (a+b x+c x^2\right )}{2 c}-\frac {b \left (\frac {1}{2} e^{a-\frac {b^2}{4 c}} \int e^{\frac {(b+2 c x)^2}{4 c}}dx-\frac {1}{2} e^{\frac {b^2}{4 c}-a} \int e^{-\frac {(b+2 c x)^2}{4 c}}dx\right )}{2 c}\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle \frac {\cosh \left (a+b x+c x^2\right )}{2 c}-\frac {b \left (\frac {\sqrt {\pi } e^{a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}}-\frac {1}{2} e^{\frac {b^2}{4 c}-a} \int e^{-\frac {(b+2 c x)^2}{4 c}}dx\right )}{2 c}\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle \frac {\cosh \left (a+b x+c x^2\right )}{2 c}-\frac {b \left (\frac {\sqrt {\pi } e^{a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}}-\frac {\sqrt {\pi } e^{\frac {b^2}{4 c}-a} \text {erf}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}}\right )}{2 c}\) |
Cosh[a + b*x + c*x^2]/(2*c) - (b*(-1/4*(E^(-a + b^2/(4*c))*Sqrt[Pi]*Erf[(b + 2*c*x)/(2*Sqrt[c])])/Sqrt[c] + (E^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[(b + 2* c*x)/(2*Sqrt[c])])/(4*Sqrt[c])))/(2*c)
3.1.2.3.1 Defintions of rubi rules used
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ F, a, b, c, d}, x] && PosQ[b]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F], 2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; Fr eeQ[{F, a, b, c, d}, x] && NegQ[b]
Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[F^(a - b^2/ (4*c)) Int[F^((b + 2*c*x)^2/(4*c)), x], x] /; FreeQ[{F, a, b, c}, x]
Int[Sinh[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[1/2 Int[E^ (a + b*x + c*x^2), x], x] - Simp[1/2 Int[E^(-a - b*x - c*x^2), x], x] /; FreeQ[{a, b, c}, x]
Int[((d_.) + (e_.)*(x_))*Sinh[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[e*(Cosh[a + b*x + c*x^2]/(2*c)), x] - Simp[(b*e - 2*c*d)/(2*c) I nt[Sinh[a + b*x + c*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b*e - 2*c*d, 0]
Time = 0.36 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.12
| method | result | size |
| risch | \(\frac {{\mathrm e}^{-c \,x^{2}-b x -a}}{4 c}+\frac {b \sqrt {\pi }\, {\mathrm e}^{-\frac {4 a c -b^{2}}{4 c}} \operatorname {erf}\left (\sqrt {c}\, x +\frac {b}{2 \sqrt {c}}\right )}{8 c^{\frac {3}{2}}}+\frac {{\mathrm e}^{c \,x^{2}+b x +a}}{4 c}+\frac {b \sqrt {\pi }\, {\mathrm e}^{\frac {4 a c -b^{2}}{4 c}} \operatorname {erf}\left (-\sqrt {-c}\, x +\frac {b}{2 \sqrt {-c}}\right )}{8 c \sqrt {-c}}\) | \(124\) |
1/4/c*exp(-c*x^2-b*x-a)+1/8*b/c^(3/2)*Pi^(1/2)*exp(-1/4*(4*a*c-b^2)/c)*erf (c^(1/2)*x+1/2*b/c^(1/2))+1/4/c*exp(c*x^2+b*x+a)+1/8*b/c*Pi^(1/2)*exp(1/4* (4*a*c-b^2)/c)/(-c)^(1/2)*erf(-(-c)^(1/2)*x+1/2*b/(-c)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 343 vs. \(2 (83) = 166\).
Time = 0.26 (sec) , antiderivative size = 343, normalized size of antiderivative = 3.09 \[ \int x \sinh \left (a+b x+c x^2\right ) \, dx=\frac {2 \, c \cosh \left (c x^{2} + b x + a\right )^{2} + \sqrt {\pi } {\left (b \cosh \left (c x^{2} + b x + a\right ) \cosh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) + b \cosh \left (c x^{2} + b x + a\right ) \sinh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) + {\left (b \cosh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) + b \sinh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right )\right )} \sinh \left (c x^{2} + b x + a\right )\right )} \sqrt {-c} \operatorname {erf}\left (\frac {{\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, c}\right ) + \sqrt {\pi } {\left (b \cosh \left (c x^{2} + b x + a\right ) \cosh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) - b \cosh \left (c x^{2} + b x + a\right ) \sinh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) + {\left (b \cosh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) - b \sinh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right )\right )} \sinh \left (c x^{2} + b x + a\right )\right )} \sqrt {c} \operatorname {erf}\left (\frac {2 \, c x + b}{2 \, \sqrt {c}}\right ) + 4 \, c \cosh \left (c x^{2} + b x + a\right ) \sinh \left (c x^{2} + b x + a\right ) + 2 \, c \sinh \left (c x^{2} + b x + a\right )^{2} + 2 \, c}{8 \, {\left (c^{2} \cosh \left (c x^{2} + b x + a\right ) + c^{2} \sinh \left (c x^{2} + b x + a\right )\right )}} \]
1/8*(2*c*cosh(c*x^2 + b*x + a)^2 + sqrt(pi)*(b*cosh(c*x^2 + b*x + a)*cosh( -1/4*(b^2 - 4*a*c)/c) + b*cosh(c*x^2 + b*x + a)*sinh(-1/4*(b^2 - 4*a*c)/c) + (b*cosh(-1/4*(b^2 - 4*a*c)/c) + b*sinh(-1/4*(b^2 - 4*a*c)/c))*sinh(c*x^ 2 + b*x + a))*sqrt(-c)*erf(1/2*(2*c*x + b)*sqrt(-c)/c) + sqrt(pi)*(b*cosh( c*x^2 + b*x + a)*cosh(-1/4*(b^2 - 4*a*c)/c) - b*cosh(c*x^2 + b*x + a)*sinh (-1/4*(b^2 - 4*a*c)/c) + (b*cosh(-1/4*(b^2 - 4*a*c)/c) - b*sinh(-1/4*(b^2 - 4*a*c)/c))*sinh(c*x^2 + b*x + a))*sqrt(c)*erf(1/2*(2*c*x + b)/sqrt(c)) + 4*c*cosh(c*x^2 + b*x + a)*sinh(c*x^2 + b*x + a) + 2*c*sinh(c*x^2 + b*x + a)^2 + 2*c)/(c^2*cosh(c*x^2 + b*x + a) + c^2*sinh(c*x^2 + b*x + a))
\[ \int x \sinh \left (a+b x+c x^2\right ) \, dx=\int x \sinh {\left (a + b x + c x^{2} \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 611 vs. \(2 (83) = 166\).
Time = 0.39 (sec) , antiderivative size = 611, normalized size of antiderivative = 5.50 \[ \int x \sinh \left (a+b x+c x^2\right ) \, dx=\frac {1}{2} \, x^{2} \sinh \left (c x^{2} + b x + a\right ) - \frac {{\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b^{2} {\left (\operatorname {erf}\left (\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 {4 \, b e^{\left (\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}}{c^{\frac {3}{2}}} - \frac {4 \, {\left (2 \, c x + b\right )}^{3} \Gamma \left (\frac {3}{2}, -\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}{\left (-\frac {{\left (2 \, c x + b\right )}^{2}}{c}\right )^{\frac {3}{2}} c^{\frac {5}{2}}}\right )} b e^{\left (a - \frac {b^{2}}{4 \, c}\right )}}{32 \, \sqrt {c}} + \frac {1}{32} \, {\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b^{3} {\left (\operatorname {erf}\left (\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 {7}{2}}} - \frac {6 \, b^{2} e^{\left (\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}}{c^{\frac {5}{2}}} - \frac {12 \, {\left (2 \, c x + b\right )}^{3} b \Gamma \left (\frac {3}{2}, -\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}{\left (-\frac {{\left (2 \, c x + b\right )}^{2}}{c}\right )^{\frac {3}{2}} c^{\frac {7}{2}}} + \frac {8 \, \Gamma \left (2, -\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}{c^{\frac {3}{2}}}\right )} \sqrt {c} e^{\left (a - \frac {b^{2}}{4 \, c}\right )} + \frac {{\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b^{2} {\left (\operatorname {erf}\left (\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 {4 \, b c e^{\left (-\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}}{\left (-c\right )^{\frac {5}{2}}} - \frac {4 \, {\left (2 \, c x + b\right )}^{3} \Gamma \left (\frac {3}{2}, \frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}{\left (\frac {{\left (2 \, c x + b\right )}^{2}}{c}\right )^{\frac {3}{2}} \left (-c\right )^{\frac {5}{2}}}\right )} b e^{\left (-a + \frac {b^{2}}{4 \, c}\right )}}{32 \, \sqrt {-c}} + \frac {{\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b^{3} {\left (\operatorname {erf}\left (\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 {7}{2}}} + \frac {6 \, b^{2} c e^{\left (-\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}}{\left (-c\right )^{\frac {7}{2}}} - \frac {12 \, {\left (2 \, c x + b\right )}^{3} b \Gamma \left (\frac {3}{2}, \frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}{\left (\frac {{\left (2 \, c x + b\right )}^{2}}{c}\right )^{\frac {3}{2}} \left (-c\right )^{\frac {7}{2}}} + \frac {8 \, c^{2} \Gamma \left (2, \frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}{\left (-c\right )^{\frac {7}{2}}}\right )} c e^{\left (-a + \frac {b^{2}}{4 \, c}\right )}}{32 \, \sqrt {-c}} \]
1/2*x^2*sinh(c*x^2 + b*x + a) - 1/32*(sqrt(pi)*(2*c*x + b)*b^2*(erf(1/2*sq rt(-(2*c*x + b)^2/c)) - 1)/(sqrt(-(2*c*x + b)^2/c)*c^(5/2)) - 4*b*e^(1/4*( 2*c*x + b)^2/c)/c^(3/2) - 4*(2*c*x + b)^3*gamma(3/2, -1/4*(2*c*x + b)^2/c) /((-(2*c*x + b)^2/c)^(3/2)*c^(5/2)))*b*e^(a - 1/4*b^2/c)/sqrt(c) + 1/32*(s qrt(pi)*(2*c*x + b)*b^3*(erf(1/2*sqrt(-(2*c*x + b)^2/c)) - 1)/(sqrt(-(2*c* x + b)^2/c)*c^(7/2)) - 6*b^2*e^(1/4*(2*c*x + b)^2/c)/c^(5/2) - 12*(2*c*x + b)^3*b*gamma(3/2, -1/4*(2*c*x + b)^2/c)/((-(2*c*x + b)^2/c)^(3/2)*c^(7/2) ) + 8*gamma(2, -1/4*(2*c*x + b)^2/c)/c^(3/2))*sqrt(c)*e^(a - 1/4*b^2/c) + 1/32*(sqrt(pi)*(2*c*x + b)*b^2*(erf(1/2*sqrt((2*c*x + b)^2/c)) - 1)/(sqrt( (2*c*x + b)^2/c)*(-c)^(5/2)) + 4*b*c*e^(-1/4*(2*c*x + b)^2/c)/(-c)^(5/2) - 4*(2*c*x + b)^3*gamma(3/2, 1/4*(2*c*x + b)^2/c)/(((2*c*x + b)^2/c)^(3/2)* (-c)^(5/2)))*b*e^(-a + 1/4*b^2/c)/sqrt(-c) + 1/32*(sqrt(pi)*(2*c*x + b)*b^ 3*(erf(1/2*sqrt((2*c*x + b)^2/c)) - 1)/(sqrt((2*c*x + b)^2/c)*(-c)^(7/2)) + 6*b^2*c*e^(-1/4*(2*c*x + b)^2/c)/(-c)^(7/2) - 12*(2*c*x + b)^3*b*gamma(3 /2, 1/4*(2*c*x + b)^2/c)/(((2*c*x + b)^2/c)^(3/2)*(-c)^(7/2)) + 8*c^2*gamm a(2, 1/4*(2*c*x + b)^2/c)/(-c)^(7/2))*c*e^(-a + 1/4*b^2/c)/sqrt(-c)
Time = 0.30 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.09 \[ \int x \sinh \left (a+b x+c x^2\right ) \, dx=-\frac {\frac {\sqrt {\pi } b \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {c} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (\frac {b^{2} - 4 \, a c}{4 \, c}\right )}}{\sqrt {c}} - 2 \, e^{\left (-c x^{2} - b x - a\right )}}{8 \, c} + \frac {\frac {\sqrt {\pi } b \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right )}}{\sqrt {-c}} + 2 \, e^{\left (c x^{2} + b x + a\right )}}{8 \, c} \]
-1/8*(sqrt(pi)*b*erf(-1/2*sqrt(c)*(2*x + b/c))*e^(1/4*(b^2 - 4*a*c)/c)/sqr t(c) - 2*e^(-c*x^2 - b*x - a))/c + 1/8*(sqrt(pi)*b*erf(-1/2*sqrt(-c)*(2*x + b/c))*e^(-1/4*(b^2 - 4*a*c)/c)/sqrt(-c) + 2*e^(c*x^2 + b*x + a))/c
Timed out. \[ \int x \sinh \left (a+b x+c x^2\right ) \, dx=\int x\,\mathrm {sinh}\left (c\,x^2+b\,x+a\right ) \,d x \]