Integrand size = 19, antiderivative size = 160 \[ \int (d+e x) \sinh ^2\left (a+b x+c x^2\right ) \, dx=-\frac {(d+e x)^2}{4 e}+\frac {(2 c d-b e) e^{-2 a+\frac {b^2}{2 c}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{16 c^{3/2}}+\frac {(2 c d-b e) e^{2 a-\frac {b^2}{2 c}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{16 c^{3/2}}+\frac {e \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c} \]
-1/4*(e*x+d)^2/e+1/8*e*sinh(2*c*x^2+2*b*x+2*a)/c+1/32*(-b*e+2*c*d)*exp(-2* a+1/2*b^2/c)*erf(1/2*(2*c*x+b)*2^(1/2)/c^(1/2))*2^(1/2)*Pi^(1/2)/c^(3/2)+1 /32*(-b*e+2*c*d)*exp(2*a-1/2*b^2/c)*erfi(1/2*(2*c*x+b)*2^(1/2)/c^(1/2))*2^ (1/2)*Pi^(1/2)/c^(3/2)
Time = 0.47 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.11 \[ \int (d+e x) \sinh ^2\left (a+b x+c x^2\right ) \, dx=\frac {(2 c d-b e) \sqrt {2 \pi } \text {erf}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right ) \left (\cosh \left (2 a-\frac {b^2}{2 c}\right )-\sinh \left (2 a-\frac {b^2}{2 c}\right )\right )+(2 c d-b e) \sqrt {2 \pi } \text {erfi}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right ) \left (\cosh \left (2 a-\frac {b^2}{2 c}\right )+\sinh \left (2 a-\frac {b^2}{2 c}\right )\right )+4 \sqrt {c} (-2 c x (2 d+e x)+e \sinh (2 (a+x (b+c x))))}{32 c^{3/2}} \]
((2*c*d - b*e)*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)]) + (2*c*d - b*e)*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*Sqrt[c]*(-2*c*x*(2*d + e*x) + e*Sinh[2*(a + x*(b + c*x))]))/(32*c^(3/2))
Time = 0.35 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {5917, 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.
| \(\displaystyle \int (d+e x) \sinh ^2\left (a+b x+c x^2\right ) \, dx\) |
| \(\Big \downarrow \) 5917 |
| \(\displaystyle \int \left (\frac {1}{2} (d+e x) \cosh \left (2 a+2 b x+2 c x^2\right )+\frac {1}{2} (-d-e x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {\frac {\pi }{2}} e^{\frac {b^2}{2 c}-2 a} (2 c d-b e) \text {erf}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{16 c^{3/2}}+\frac {\sqrt {\frac {\pi }{2}} e^{2 a-\frac {b^2}{2 c}} (2 c d-b e) \text {erfi}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{16 c^{3/2}}+\frac {e \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}-\frac {(d+e x)^2}{4 e}\) |
-1/4*(d + e*x)^2/e + ((2*c*d - b*e)*E^(-2*a + b^2/(2*c))*Sqrt[Pi/2]*Erf[(b + 2*c*x)/(Sqrt[2]*Sqrt[c])])/(16*c^(3/2)) + ((2*c*d - b*e)*E^(2*a - b^2/( 2*c))*Sqrt[Pi/2]*Erfi[(b + 2*c*x)/(Sqrt[2]*Sqrt[c])])/(16*c^(3/2)) + (e*Si nh[2*a + 2*b*x + 2*c*x^2])/(8*c)
3.1.32.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]^(n_) , x_Symbol] :> Int[ExpandTrigReduce[(d + e*x)^m, Sinh[a + b*x + c*x^2]^n, x ], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 1]
Time = 0.72 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.44
| method | result | size |
| risch | \(-\frac {e \,x^{2}}{4}-\frac {d x}{2}+\frac {\operatorname {erf}\left (\sqrt {2}\, \sqrt {c}\, x +\frac {b \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}\, \sqrt {\pi }\, d \,{\mathrm e}^{-\frac {4 a c -b^{2}}{2 c}}}{16 \sqrt {c}}-\frac {{\mathrm e}^{-2 a} e \,{\mathrm e}^{-2 x \left (c x +b \right )}}{16 c}-\frac {{\mathrm e}^{-2 a} e b \sqrt {\pi }\, {\mathrm e}^{\frac {b^{2}}{2 c}} \sqrt {2}\, \operatorname {erf}\left (\sqrt {2}\, \sqrt {c}\, x +\frac {b \sqrt {2}}{2 \sqrt {c}}\right )}{32 c^{\frac {3}{2}}}-\frac {\operatorname {erf}\left (-\sqrt {-2 c}\, x +\frac {b}{\sqrt {-2 c}}\right ) \sqrt {\pi }\, d \,{\mathrm e}^{\frac {4 a c -b^{2}}{2 c}}}{8 \sqrt {-2 c}}+\frac {{\mathrm e}^{2 a} e \,{\mathrm e}^{2 x \left (c x +b \right )}}{16 c}+\frac {{\mathrm e}^{2 a} e b \sqrt {\pi }\, {\mathrm e}^{-\frac {b^{2}}{2 c}} \operatorname {erf}\left (-\sqrt {-2 c}\, x +\frac {b}{\sqrt {-2 c}}\right )}{16 c \sqrt {-2 c}}\) | \(231\) |
-1/4*e*x^2-1/2*d*x+1/16*erf(2^(1/2)*c^(1/2)*x+1/2*b*2^(1/2)/c^(1/2))/c^(1/ 2)*2^(1/2)*Pi^(1/2)*d*exp(-1/2*(4*a*c-b^2)/c)-1/16*exp(-2*a)*e/c*exp(-2*x* (c*x+b))-1/32*exp(-2*a)*e*b/c^(3/2)*Pi^(1/2)*exp(1/2*b^2/c)*2^(1/2)*erf(2^ (1/2)*c^(1/2)*x+1/2*b*2^(1/2)/c^(1/2))-1/8*erf(-(-2*c)^(1/2)*x+b/(-2*c)^(1 /2))/(-2*c)^(1/2)*Pi^(1/2)*d*exp(1/2*(4*a*c-b^2)/c)+1/16*exp(2*a)*e/c*exp( 2*x*(c*x+b))+1/16*exp(2*a)*e*b/c*Pi^(1/2)*exp(-1/2*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. 777 vs. \(2 (130) = 260\).
Time = 0.27 (sec) , antiderivative size = 777, normalized size of antiderivative = 4.86 \[ \int (d+e x) \sinh ^2\left (a+b x+c x^2\right ) \, dx=\frac {2 \, c e \cosh \left (c x^{2} + b x + a\right )^{4} + 8 \, c e \cosh \left (c x^{2} + b x + a\right ) \sinh \left (c x^{2} + b x + a\right )^{3} + 2 \, c e \sinh \left (c x^{2} + b x + a\right )^{4} - \sqrt {2} \sqrt {\pi } {\left ({\left (2 \, c d - b e\right )} \cosh \left (c x^{2} + b x + a\right )^{2} \cosh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) + {\left (2 \, c d - b e\right )} \cosh \left (c x^{2} + b x + a\right )^{2} \sinh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) + {\left ({\left (2 \, c d - b e\right )} \cosh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) + {\left (2 \, c d - b e\right )} \sinh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right )\right )} \sinh \left (c x^{2} + b x + a\right )^{2} + 2 \, {\left ({\left (2 \, c d - b e\right )} \cosh \left (c x^{2} + b x + a\right ) \cosh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) + {\left (2 \, c d - b e\right )} \cosh \left (c x^{2} + b x + a\right ) \sinh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right )\right )} \sinh \left (c x^{2} + b x + a\right )\right )} \sqrt {-c} \operatorname {erf}\left (\frac {\sqrt {2} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, c}\right ) + \sqrt {2} \sqrt {\pi } {\left ({\left (2 \, c d - b e\right )} \cosh \left (c x^{2} + b x + a\right )^{2} \cosh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) - {\left (2 \, c d - b e\right )} \cosh \left (c x^{2} + b x + a\right )^{2} \sinh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) + {\left ({\left (2 \, c d - b e\right )} \cosh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) - {\left (2 \, c d - b e\right )} \sinh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right )\right )} \sinh \left (c x^{2} + b x + a\right )^{2} + 2 \, {\left ({\left (2 \, c d - b e\right )} \cosh \left (c x^{2} + b x + a\right ) \cosh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) - {\left (2 \, c d - b e\right )} \cosh \left (c x^{2} + b x + a\right ) \sinh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right )\right )} \sinh \left (c x^{2} + b x + a\right )\right )} \sqrt {c} \operatorname {erf}\left (\frac {\sqrt {2} {\left (2 \, c x + b\right )}}{2 \, \sqrt {c}}\right ) - 8 \, {\left (c^{2} e x^{2} + 2 \, c^{2} d x\right )} \cosh \left (c x^{2} + b x + a\right )^{2} - 4 \, {\left (2 \, c^{2} e x^{2} + 4 \, c^{2} d x - 3 \, c e \cosh \left (c x^{2} + b x + a\right )^{2}\right )} \sinh \left (c x^{2} + b x + a\right )^{2} - 2 \, c e + 8 \, {\left (c e \cosh \left (c x^{2} + b x + a\right )^{3} - 2 \, {\left (c^{2} e x^{2} + 2 \, c^{2} d x\right )} \cosh \left (c x^{2} + b x + a\right )\right )} \sinh \left (c x^{2} + b x + a\right )}{32 \, {\left (c^{2} \cosh \left (c x^{2} + b x + a\right )^{2} + 2 \, c^{2} \cosh \left (c x^{2} + b x + a\right ) \sinh \left (c x^{2} + b x + a\right ) + c^{2} \sinh \left (c x^{2} + b x + a\right )^{2}\right )}} \]
1/32*(2*c*e*cosh(c*x^2 + b*x + a)^4 + 8*c*e*cosh(c*x^2 + b*x + a)*sinh(c*x ^2 + b*x + a)^3 + 2*c*e*sinh(c*x^2 + b*x + a)^4 - sqrt(2)*sqrt(pi)*((2*c*d - b*e)*cosh(c*x^2 + b*x + a)^2*cosh(-1/2*(b^2 - 4*a*c)/c) + (2*c*d - b*e) *cosh(c*x^2 + b*x + a)^2*sinh(-1/2*(b^2 - 4*a*c)/c) + ((2*c*d - b*e)*cosh( -1/2*(b^2 - 4*a*c)/c) + (2*c*d - b*e)*sinh(-1/2*(b^2 - 4*a*c)/c))*sinh(c*x ^2 + b*x + a)^2 + 2*((2*c*d - b*e)*cosh(c*x^2 + b*x + a)*cosh(-1/2*(b^2 - 4*a*c)/c) + (2*c*d - b*e)*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) + sqrt(2)*sqrt(pi)*((2*c*d - b*e)*cosh(c*x^2 + b*x + a)^2*cosh(-1/2*(b^2 - 4*a*c)/c) - (2*c*d - b*e)*cosh(c*x^2 + b*x + a)^2*sinh(-1/2*(b^2 - 4*a*c) /c) + ((2*c*d - b*e)*cosh(-1/2*(b^2 - 4*a*c)/c) - (2*c*d - b*e)*sinh(-1/2* (b^2 - 4*a*c)/c))*sinh(c*x^2 + b*x + a)^2 + 2*((2*c*d - b*e)*cosh(c*x^2 + b*x + a)*cosh(-1/2*(b^2 - 4*a*c)/c) - (2*c*d - b*e)*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)) - 8*(c^2*e*x^2 + 2*c^2*d*x)*cosh(c*x^2 + b*x + a)^2 - 4*(2*c^2*e*x^2 + 4*c^2*d*x - 3*c*e*cosh(c*x^2 + b*x + a)^2)*sinh(c*x^2 + b*x + a)^2 - 2*c*e + 8*(c*e*cosh(c*x^2 + b*x + a)^3 - 2*(c^2*e*x^2 + 2*c^ 2*d*x)*cosh(c*x^2 + b*x + a))*sinh(c*x^2 + b*x + a))/(c^2*cosh(c*x^2 + b*x + a)^2 + 2*c^2*cosh(c*x^2 + b*x + a)*sinh(c*x^2 + b*x + a) + c^2*sinh(c*x ^2 + b*x + a)^2)
\[ \int (d+e x) \sinh ^2\left (a+b x+c x^2\right ) \, dx=\int \left (d + e x\right ) \sinh ^{2}{\left (a + b x + c x^{2} \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 299 vs. \(2 (130) = 260\).
Time = 0.37 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.87 \[ \int (d+e x) \sinh ^2\left (a+b x+c x^2\right ) \, dx=\frac {1}{16} \, {\left (\frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\sqrt {2} \sqrt {-c} x - \frac {\sqrt {2} b}{2 \, \sqrt {-c}}\right ) e^{\left (2 \, a - \frac {b^{2}}{2 \, c}\right )}}{\sqrt {-c}} + \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\sqrt {2} \sqrt {c} x + \frac {\sqrt {2} b}{2 \, \sqrt {c}}\right ) e^{\left (-2 \, a + \frac {b^{2}}{2 \, c}\right )}}{\sqrt {c}} - 8 \, x\right )} d - \frac {1}{32} \, {\left (8 \, x^{2} + \frac {\sqrt {2} {\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (2 \, c x + b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt {-\frac {{\left (2 \, c x + b\right )}^{2}}{c}} c^{\frac {3}{2}}} - \frac {\sqrt {2} e^{\left (\frac {{\left (2 \, c x + b\right )}^{2}}{2 \, c}\right )}}{\sqrt {c}}\right )} e^{\left (2 \, a - \frac {b^{2}}{2 \, c}\right )}}{\sqrt {c}} + \frac {\sqrt {2} {\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {\frac {{\left (2 \, c x + b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt {\frac {{\left (2 \, c x + b\right )}^{2}}{c}} \left (-c\right )^{\frac {3}{2}}} + \frac {\sqrt {2} c e^{\left (-\frac {{\left (2 \, c x + b\right )}^{2}}{2 \, c}\right )}}{\left (-c\right )^{\frac {3}{2}}}\right )} e^{\left (-2 \, a + \frac {b^{2}}{2 \, c}\right )}}{\sqrt {-c}}\right )} e \]
1/16*(sqrt(2)*sqrt(pi)*erf(sqrt(2)*sqrt(-c)*x - 1/2*sqrt(2)*b/sqrt(-c))*e^ (2*a - 1/2*b^2/c)/sqrt(-c) + sqrt(2)*sqrt(pi)*erf(sqrt(2)*sqrt(c)*x + 1/2* sqrt(2)*b/sqrt(c))*e^(-2*a + 1/2*b^2/c)/sqrt(c) - 8*x)*d - 1/32*(8*x^2 + s qrt(2)*(sqrt(pi)*(2*c*x + b)*b*(erf(sqrt(1/2)*sqrt(-(2*c*x + b)^2/c)) - 1) /(sqrt(-(2*c*x + b)^2/c)*c^(3/2)) - sqrt(2)*e^(1/2*(2*c*x + b)^2/c)/sqrt(c ))*e^(2*a - 1/2*b^2/c)/sqrt(c) + sqrt(2)*(sqrt(pi)*(2*c*x + b)*b*(erf(sqrt (1/2)*sqrt((2*c*x + b)^2/c)) - 1)/(sqrt((2*c*x + b)^2/c)*(-c)^(3/2)) + sqr t(2)*c*e^(-1/2*(2*c*x + b)^2/c)/(-c)^(3/2))*e^(-2*a + 1/2*b^2/c)/sqrt(-c)) *e
Time = 0.29 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.03 \[ \int (d+e x) \sinh ^2\left (a+b x+c x^2\right ) \, dx=-\frac {1}{4} \, e x^{2} - \frac {1}{2} \, d x - \frac {\frac {\sqrt {2} \sqrt {\pi } {\left (2 \, c d - b e\right )} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} \sqrt {c} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (\frac {b^{2} - 4 \, a c}{2 \, c}\right )}}{\sqrt {c}} + 2 \, e e^{\left (-2 \, c x^{2} - 2 \, b x - 2 \, a\right )}}{32 \, c} - \frac {\frac {\sqrt {2} \sqrt {\pi } {\left (2 \, c d - b e\right )} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} \sqrt {-c} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right )}}{\sqrt {-c}} - 2 \, e e^{\left (2 \, c x^{2} + 2 \, b x + 2 \, a\right )}}{32 \, c} \]
-1/4*e*x^2 - 1/2*d*x - 1/32*(sqrt(2)*sqrt(pi)*(2*c*d - b*e)*erf(-1/2*sqrt( 2)*sqrt(c)*(2*x + b/c))*e^(1/2*(b^2 - 4*a*c)/c)/sqrt(c) + 2*e*e^(-2*c*x^2 - 2*b*x - 2*a))/c - 1/32*(sqrt(2)*sqrt(pi)*(2*c*d - b*e)*erf(-1/2*sqrt(2)* sqrt(-c)*(2*x + b/c))*e^(-1/2*(b^2 - 4*a*c)/c)/sqrt(-c) - 2*e*e^(2*c*x^2 + 2*b*x + 2*a))/c
Timed out. \[ \int (d+e x) \sinh ^2\left (a+b x+c x^2\right ) \, dx=\int {\mathrm {sinh}\left (c\,x^2+b\,x+a\right )}^2\,\left (d+e\,x\right ) \,d x \]