Integrand size = 21, antiderivative size = 311 \[ \int (d+e x)^2 \sinh ^2\left (a+b x+c x^2\right ) \, dx=-\frac {(d+e x)^3}{6 e}+\frac {e^2 e^{-2 a+\frac {b^2}{2 c}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{32 c^{3/2}}+\frac {(2 c d-b e)^2 e^{-2 a+\frac {b^2}{2 c}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{32 c^{5/2}}-\frac {e^2 e^{2 a-\frac {b^2}{2 c}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{32 c^{3/2}}+\frac {(2 c d-b e)^2 e^{2 a-\frac {b^2}{2 c}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{32 c^{5/2}}+\frac {e (2 c d-b e) \sinh \left (2 a+2 b x+2 c x^2\right )}{16 c^2}+\frac {e (d+e x) \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c} \] Output:
-1/6*(e*x+d)^3/e+1/64*e^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^(3/2)+1/64*(-b*e+2*c*d)^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*e^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^(3/2)+1 /64*(-b*e+2*c*d)^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)+1/16*e*(-b*e+2*c*d)*sinh(2*c*x^2+2*b*x+2*a)/c^2+1 /8*e*(e*x+d)*sinh(2*c*x^2+2*b*x+2*a)/c
Time = 0.89 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.77 \[ \int (d+e x)^2 \sinh ^2\left (a+b x+c x^2\right ) \, dx=\frac {3 \left (4 c^2 d^2+b^2 e^2+c e (-4 b d+e)\right ) \sqrt {2 \pi } \text {erf}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right ) \left (\cosh \left (2 a-\frac {b^2}{2 c}\right )-\sinh \left (2 a-\frac {b^2}{2 c}\right )\right )+3 \left (4 c^2 d^2+b^2 e^2-c e (4 b d+e)\right ) \sqrt {2 \pi } \text {erfi}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right ) \left (\cosh \left (2 a-\frac {b^2}{2 c}\right )+\sinh \left (2 a-\frac {b^2}{2 c}\right )\right )-4 \sqrt {c} \left (8 c^2 x \left (3 d^2+3 d e x+e^2 x^2\right )-3 e (4 c d-b e+2 c e x) \sinh (2 (a+x (b+c x)))\right )}{192 c^{5/2}} \] Input:
Integrate[(d + e*x)^2*Sinh[a + b*x + c*x^2]^2,x]
Output:
(3*(4*c^2*d^2 + b^2*e^2 + c*e*(-4*b*d + e))*Sqrt[2*Pi]*Erf[(b + 2*c*x)/(Sq rt[2]*Sqrt[c])]*(Cosh[2*a - b^2/(2*c)] - Sinh[2*a - b^2/(2*c)]) + 3*(4*c^2 *d^2 + b^2*e^2 - c*e*(4*b*d + e))*Sqrt[2*Pi]*Erfi[(b + 2*c*x)/(Sqrt[2]*Sqr t[c])]*(Cosh[2*a - b^2/(2*c)] + Sinh[2*a - b^2/(2*c)]) - 4*Sqrt[c]*(8*c^2* x*(3*d^2 + 3*d*e*x + e^2*x^2) - 3*e*(4*c*d - b*e + 2*c*e*x)*Sinh[2*(a + x* (b + c*x))]))/(192*c^(5/2))
Time = 0.62 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, 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.
| \(\displaystyle \int (d+e x)^2 \sinh ^2\left (a+b x+c x^2\right ) \, dx\) |
| \(\Big \downarrow \) 5917 |
| \(\displaystyle \int \left (\frac {1}{2} (d+e x)^2 \cosh \left (2 a+2 b x+2 c x^2\right )-\frac {1}{2} (d+e x)^2\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {\frac {\pi }{2}} e^{\frac {b^2}{2 c}-2 a} (2 c d-b e)^2 \text {erf}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{32 c^{5/2}}+\frac {\sqrt {\frac {\pi }{2}} e^{2 a-\frac {b^2}{2 c}} (2 c d-b e)^2 \text {erfi}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{32 c^{5/2}}+\frac {\sqrt {\frac {\pi }{2}} e^2 e^{\frac {b^2}{2 c}-2 a} \text {erf}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{32 c^{3/2}}-\frac {\sqrt {\frac {\pi }{2}} e^2 e^{2 a-\frac {b^2}{2 c}} \text {erfi}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{32 c^{3/2}}+\frac {e (2 c d-b e) \sinh \left (2 a+2 b x+2 c x^2\right )}{16 c^2}+\frac {e (d+e x) \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}-\frac {(d+e x)^3}{6 e}\) |
Input:
Int[(d + e*x)^2*Sinh[a + b*x + c*x^2]^2,x]
Output:
-1/6*(d + e*x)^3/e + (e^2*E^(-2*a + b^2/(2*c))*Sqrt[Pi/2]*Erf[(b + 2*c*x)/ (Sqrt[2]*Sqrt[c])])/(32*c^(3/2)) + ((2*c*d - b*e)^2*E^(-2*a + b^2/(2*c))*S qrt[Pi/2]*Erf[(b + 2*c*x)/(Sqrt[2]*Sqrt[c])])/(32*c^(5/2)) - (e^2*E^(2*a - b^2/(2*c))*Sqrt[Pi/2]*Erfi[(b + 2*c*x)/(Sqrt[2]*Sqrt[c])])/(32*c^(3/2)) + ((2*c*d - b*e)^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*c*d - b*e)*Sinh[2*a + 2*b*x + 2*c*x^2])/( 16*c^2) + (e*(d + e*x)*Sinh[2*a + 2*b*x + 2*c*x^2])/(8*c)
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]
Leaf count of result is larger than twice the leaf count of optimal. \(527\) vs. \(2(253)=506\).
Time = 0.88 (sec) , antiderivative size = 528, normalized size of antiderivative = 1.70
| method | result | size |
| risch | \(-\frac {d^{2} x}{2}-\frac {e^{2} x^{3}}{6}+\frac {\operatorname {erf}\left (\sqrt {2}\, \sqrt {c}\, x +\frac {b \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}\, \sqrt {\pi }\, d^{2} {\mathrm e}^{-\frac {4 a c -b^{2}}{2 c}}}{16 \sqrt {c}}-\frac {{\mathrm e}^{-2 a} e^{2} x \,{\mathrm e}^{-2 x \left (c x +b \right )}}{16 c}+\frac {{\mathrm e}^{-2 a} e^{2} b \,{\mathrm e}^{-2 x \left (c x +b \right )}}{32 c^{2}}+\frac {{\mathrm e}^{-2 a} e^{2} b^{2} \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 )}{64 c^{\frac {5}{2}}}+\frac {{\mathrm e}^{-2 a} e^{2} \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 )}{64 c^{\frac {3}{2}}}-\frac {{\mathrm e}^{-2 a} d e \,{\mathrm e}^{-2 x \left (c x +b \right )}}{8 c}-\frac {{\mathrm e}^{-2 a} d 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 )}{16 c^{\frac {3}{2}}}-\frac {\operatorname {erf}\left (-\sqrt {-2 c}\, x +\frac {b}{\sqrt {-2 c}}\right ) \sqrt {\pi }\, d^{2} {\mathrm e}^{\frac {4 a c -b^{2}}{2 c}}}{8 \sqrt {-2 c}}+\frac {{\mathrm e}^{2 a} e^{2} x \,{\mathrm e}^{2 x \left (c x +b \right )}}{16 c}-\frac {{\mathrm e}^{2 a} e^{2} b \,{\mathrm e}^{2 x \left (c x +b \right )}}{32 c^{2}}-\frac {{\mathrm e}^{2 a} e^{2} b^{2} \sqrt {\pi }\, {\mathrm e}^{-\frac {b^{2}}{2 c}} \operatorname {erf}\left (-\sqrt {-2 c}\, x +\frac {b}{\sqrt {-2 c}}\right )}{32 c^{2} \sqrt {-2 c}}+\frac {{\mathrm e}^{2 a} e^{2} \sqrt {\pi }\, {\mathrm e}^{-\frac {b^{2}}{2 c}} \operatorname {erf}\left (-\sqrt {-2 c}\, x +\frac {b}{\sqrt {-2 c}}\right )}{32 c \sqrt {-2 c}}+\frac {{\mathrm e}^{2 a} d e \,{\mathrm e}^{2 x \left (c x +b \right )}}{8 c}+\frac {{\mathrm e}^{2 a} d e b \sqrt {\pi }\, {\mathrm e}^{-\frac {b^{2}}{2 c}} \operatorname {erf}\left (-\sqrt {-2 c}\, x +\frac {b}{\sqrt {-2 c}}\right )}{8 c \sqrt {-2 c}}-\frac {d e \,x^{2}}{2}\) | \(528\) |
Input:
int((e*x+d)^2*sinh(c*x^2+b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
-1/2*d^2*x-1/6*e^2*x^3+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^2*exp(-1/2*(4*a*c-b^2)/c)-1/16*exp(-2*a)*e^2/c*x *exp(-2*x*(c*x+b))+1/32*exp(-2*a)*e^2/c^2*b*exp(-2*x*(c*x+b))+1/64*exp(-2* a)*e^2/c^(5/2)*b^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))+1/64*exp(-2*a)*e^2/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*exp(-2*a)*d*e/c*ex p(-2*x*(c*x+b))-1/16*exp(-2*a)*d*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^2*exp(1/2*(4*a*c-b^2)/c)+1/16*exp(2* a)*e^2/c*x*exp(2*x*(c*x+b))-1/32*exp(2*a)*e^2/c^2*b*exp(2*x*(c*x+b))-1/32* exp(2*a)*e^2/c^2*b^2*Pi^(1/2)*exp(-1/2*b^2/c)/(-2*c)^(1/2)*erf(-(-2*c)^(1/ 2)*x+b/(-2*c)^(1/2))+1/32*exp(2*a)*e^2/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))+1/8*exp(2*a)*d*e/c*exp(2*x*(c*x+b ))+1/8*exp(2*a)*d*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))-1/2*d*e*x^2
Leaf count of result is larger than twice the leaf count of optimal. 1142 vs. \(2 (253) = 506\).
Time = 0.10 (sec) , antiderivative size = 1142, normalized size of antiderivative = 3.67 \[ \int (d+e x)^2 \sinh ^2\left (a+b x+c x^2\right ) \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)^2*sinh(c*x^2+b*x+a)^2,x, algorithm="fricas")
Output:
-1/192*(12*c^2*e^2*x - 6*(2*c^2*e^2*x + 4*c^2*d*e - b*c*e^2)*cosh(c*x^2 + b*x + a)^4 - 24*(2*c^2*e^2*x + 4*c^2*d*e - b*c*e^2)*cosh(c*x^2 + b*x + a)* sinh(c*x^2 + b*x + a)^3 - 6*(2*c^2*e^2*x + 4*c^2*d*e - b*c*e^2)*sinh(c*x^2 + b*x + a)^4 + 24*c^2*d*e - 6*b*c*e^2 + 3*sqrt(2)*sqrt(pi)*((4*c^2*d^2 - 4*b*c*d*e + (b^2 - c)*e^2)*cosh(c*x^2 + b*x + a)^2*cosh(-1/2*(b^2 - 4*a*c) /c) + (4*c^2*d^2 - 4*b*c*d*e + (b^2 - c)*e^2)*cosh(c*x^2 + b*x + a)^2*sinh (-1/2*(b^2 - 4*a*c)/c) + ((4*c^2*d^2 - 4*b*c*d*e + (b^2 - c)*e^2)*cosh(-1/ 2*(b^2 - 4*a*c)/c) + (4*c^2*d^2 - 4*b*c*d*e + (b^2 - c)*e^2)*sinh(-1/2*(b^ 2 - 4*a*c)/c))*sinh(c*x^2 + b*x + a)^2 + 2*((4*c^2*d^2 - 4*b*c*d*e + (b^2 - c)*e^2)*cosh(c*x^2 + b*x + a)*cosh(-1/2*(b^2 - 4*a*c)/c) + (4*c^2*d^2 - 4*b*c*d*e + (b^2 - c)*e^2)*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)*((4*c^2*d^2 - 4*b*c*d*e + (b^2 + c)*e^2)*cosh(c*x^2 + b*x + a)^2*cosh(-1/2*(b^2 - 4*a*c)/c) - (4*c^2*d^2 - 4*b*c*d*e + (b^2 + c)*e^2)*cosh(c*x^2 + b*x + a)^2*sinh(-1/2*(b^2 - 4*a*c)/c) + ((4*c^2*d^2 - 4*b*c*d*e + (b^2 + c)*e^2)*cosh(-1/2*(b^2 - 4*a*c)/c) - (4*c^2*d^2 - 4*b* c*d*e + (b^2 + c)*e^2)*sinh(-1/2*(b^2 - 4*a*c)/c))*sinh(c*x^2 + b*x + a)^2 + 2*((4*c^2*d^2 - 4*b*c*d*e + (b^2 + c)*e^2)*cosh(c*x^2 + b*x + a)*cosh(- 1/2*(b^2 - 4*a*c)/c) - (4*c^2*d^2 - 4*b*c*d*e + (b^2 + c)*e^2)*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)*e...
\[ \int (d+e x)^2 \sinh ^2\left (a+b x+c x^2\right ) \, dx=\int \left (d + e x\right )^{2} \sinh ^{2}{\left (a + b x + c x^{2} \right )}\, dx \] Input:
integrate((e*x+d)**2*sinh(c*x**2+b*x+a)**2,x)
Output:
Integral((d + e*x)**2*sinh(a + b*x + c*x**2)**2, x)
Leaf count of result is larger than twice the leaf count of optimal. 601 vs. \(2 (253) = 506\).
Time = 0.32 (sec) , antiderivative size = 601, normalized size of antiderivative = 1.93 \[ \int (d+e x)^2 \sinh ^2\left (a+b x+c x^2\right ) \, dx =\text {Too large to display} \] Input:
integrate((e*x+d)^2*sinh(c*x^2+b*x+a)^2,x, algorithm="maxima")
Output:
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^2 - 1/16*(8*x^2 + 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)) - 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(sq rt(1/2)*sqrt((2*c*x + b)^2/c)) - 1)/(sqrt((2*c*x + b)^2/c)*(-c)^(3/2)) + s qrt(2)*c*e^(-1/2*(2*c*x + b)^2/c)/(-c)^(3/2))*e^(-2*a + 1/2*b^2/c)/sqrt(-c ))*d*e - 1/192*(32*x^3 - 3*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) + 3*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*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))*e^2
Time = 0.14 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.85 \[ \int (d+e x)^2 \sinh ^2\left (a+b x+c x^2\right ) \, dx=-\frac {1}{6} \, e^{2} x^{3} - \frac {1}{2} \, d e x^{2} - \frac {1}{2} \, d^{2} x - \frac {\frac {\sqrt {2} \sqrt {\pi } {\left (4 \, c^{2} d^{2} - 4 \, b c d e + b^{2} e^{2} + c e^{2}\right )} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} \sqrt {c} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (\frac {b^{2} - 4 \, a c}{2 \, c}\right )}}{\sqrt {c}} + 2 \, {\left (c e^{2} {\left (2 \, x + \frac {b}{c}\right )} + 4 \, c d e - 2 \, b e^{2}\right )} e^{\left (-2 \, c x^{2} - 2 \, b x - 2 \, a\right )}}{64 \, c^{2}} - \frac {\frac {\sqrt {2} \sqrt {\pi } {\left (4 \, c^{2} d^{2} - 4 \, b c d e + b^{2} e^{2} - c e^{2}\right )} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} \sqrt {-c} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right )}}{\sqrt {-c}} - 2 \, {\left (c e^{2} {\left (2 \, x + \frac {b}{c}\right )} + 4 \, c d e - 2 \, b e^{2}\right )} e^{\left (2 \, c x^{2} + 2 \, b x + 2 \, a\right )}}{64 \, c^{2}} \] Input:
integrate((e*x+d)^2*sinh(c*x^2+b*x+a)^2,x, algorithm="giac")
Output:
-1/6*e^2*x^3 - 1/2*d*e*x^2 - 1/2*d^2*x - 1/64*(sqrt(2)*sqrt(pi)*(4*c^2*d^2 - 4*b*c*d*e + b^2*e^2 + c*e^2)*erf(-1/2*sqrt(2)*sqrt(c)*(2*x + b/c))*e^(1 /2*(b^2 - 4*a*c)/c)/sqrt(c) + 2*(c*e^2*(2*x + b/c) + 4*c*d*e - 2*b*e^2)*e^ (-2*c*x^2 - 2*b*x - 2*a))/c^2 - 1/64*(sqrt(2)*sqrt(pi)*(4*c^2*d^2 - 4*b*c* d*e + b^2*e^2 - c*e^2)*erf(-1/2*sqrt(2)*sqrt(-c)*(2*x + b/c))*e^(-1/2*(b^2 - 4*a*c)/c)/sqrt(-c) - 2*(c*e^2*(2*x + b/c) + 4*c*d*e - 2*b*e^2)*e^(2*c*x ^2 + 2*b*x + 2*a))/c^2
Timed out. \[ \int (d+e x)^2 \sinh ^2\left (a+b x+c x^2\right ) \, dx=\int {\mathrm {sinh}\left (c\,x^2+b\,x+a\right )}^2\,{\left (d+e\,x\right )}^2 \,d x \] Input:
int(sinh(a + b*x + c*x^2)^2*(d + e*x)^2,x)
Output:
int(sinh(a + b*x + c*x^2)^2*(d + e*x)^2, x)
\[ \int (d+e x)^2 \sinh ^2\left (a+b x+c x^2\right ) \, dx=\frac {-3 \sqrt {\pi }\, e^{2 c \,x^{2}+2 b x +4 a} \mathrm {erf}\left (\frac {2 c i x +b i}{\sqrt {c}\, \sqrt {2}}\right ) b^{2} e^{2} i +12 \sqrt {\pi }\, e^{2 c \,x^{2}+2 b x +4 a} \mathrm {erf}\left (\frac {2 c i x +b i}{\sqrt {c}\, \sqrt {2}}\right ) b c d e i -12 \sqrt {\pi }\, e^{2 c \,x^{2}+2 b x +4 a} \mathrm {erf}\left (\frac {2 c i x +b i}{\sqrt {c}\, \sqrt {2}}\right ) c^{2} d^{2} i +3 \sqrt {\pi }\, e^{2 c \,x^{2}+2 b x +4 a} \mathrm {erf}\left (\frac {2 c i x +b i}{\sqrt {c}\, \sqrt {2}}\right ) c \,e^{2} i -3 e^{\frac {8 c^{2} x^{2}+8 b c x +8 a c +b^{2}}{2 c}} \sqrt {c}\, \sqrt {2}\, b \,e^{2}+12 e^{\frac {8 c^{2} x^{2}+8 b c x +8 a c +b^{2}}{2 c}} \sqrt {c}\, \sqrt {2}\, c d e +6 e^{\frac {8 c^{2} x^{2}+8 b c x +8 a c +b^{2}}{2 c}} \sqrt {c}\, \sqrt {2}\, c \,e^{2} x -48 e^{\frac {4 c^{2} x^{2}+4 b c x +4 a c +b^{2}}{2 c}} \sqrt {c}\, \sqrt {2}\, c^{2} d^{2} x -48 e^{\frac {4 c^{2} x^{2}+4 b c x +4 a c +b^{2}}{2 c}} \sqrt {c}\, \sqrt {2}\, c^{2} d e \,x^{2}-16 e^{\frac {4 c^{2} x^{2}+4 b c x +4 a c +b^{2}}{2 c}} \sqrt {c}\, \sqrt {2}\, c^{2} e^{2} x^{3}+6 e^{\frac {4 c^{2} x^{2}+4 b c x +b^{2}}{2 c}} \sqrt {c}\, \sqrt {2}\, \left (\int \frac {1}{e^{2 c \,x^{2}+2 b x}}d x \right ) b^{2} e^{2}-24 e^{\frac {4 c^{2} x^{2}+4 b c x +b^{2}}{2 c}} \sqrt {c}\, \sqrt {2}\, \left (\int \frac {1}{e^{2 c \,x^{2}+2 b x}}d x \right ) b c d e +24 e^{\frac {4 c^{2} x^{2}+4 b c x +b^{2}}{2 c}} \sqrt {c}\, \sqrt {2}\, \left (\int \frac {1}{e^{2 c \,x^{2}+2 b x}}d x \right ) c^{2} d^{2}+6 e^{\frac {4 c^{2} x^{2}+4 b c x +b^{2}}{2 c}} \sqrt {c}\, \sqrt {2}\, \left (\int \frac {1}{e^{2 c \,x^{2}+2 b x}}d x \right ) c \,e^{2}+3 e^{\frac {b^{2}}{2 c}} \sqrt {c}\, \sqrt {2}\, b \,e^{2}-12 e^{\frac {b^{2}}{2 c}} \sqrt {c}\, \sqrt {2}\, c d e -6 e^{\frac {b^{2}}{2 c}} \sqrt {c}\, \sqrt {2}\, c \,e^{2} x}{96 e^{\frac {4 c^{2} x^{2}+4 b c x +4 a c +b^{2}}{2 c}} \sqrt {c}\, \sqrt {2}\, c^{2}} \] Input:
int((e*x+d)^2*sinh(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*e**2*i + 12*sqrt(pi)*e**(4*a + 2*b*x + 2*c*x**2)*erf((b*i + 2 *c*i*x)/(sqrt(c)*sqrt(2)))*b*c*d*e*i - 12*sqrt(pi)*e**(4*a + 2*b*x + 2*c*x **2)*erf((b*i + 2*c*i*x)/(sqrt(c)*sqrt(2)))*c**2*d**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*e**2*i - 3 *e**((8*a*c + b**2 + 8*b*c*x + 8*c**2*x**2)/(2*c))*sqrt(c)*sqrt(2)*b*e**2 + 12*e**((8*a*c + b**2 + 8*b*c*x + 8*c**2*x**2)/(2*c))*sqrt(c)*sqrt(2)*c*d *e + 6*e**((8*a*c + b**2 + 8*b*c*x + 8*c**2*x**2)/(2*c))*sqrt(c)*sqrt(2)*c *e**2*x - 48*e**((4*a*c + b**2 + 4*b*c*x + 4*c**2*x**2)/(2*c))*sqrt(c)*sqr t(2)*c**2*d**2*x - 48*e**((4*a*c + b**2 + 4*b*c*x + 4*c**2*x**2)/(2*c))*sq rt(c)*sqrt(2)*c**2*d*e*x**2 - 16*e**((4*a*c + b**2 + 4*b*c*x + 4*c**2*x**2 )/(2*c))*sqrt(c)*sqrt(2)*c**2*e**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*e**2 - 24* 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*c*d*e + 24*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**2*d**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* e**2 + 3*e**(b**2/(2*c))*sqrt(c)*sqrt(2)*b*e**2 - 12*e**(b**2/(2*c))*sqrt( c)*sqrt(2)*c*d*e - 6*e**(b**2/(2*c))*sqrt(c)*sqrt(2)*c*e**2*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)