\(\int (d+e x)^2 \sinh ^2(a+b x+c x^2) \, dx\) [31]

Optimal result

Integrand size = 21, antiderivative size = 311 \[ \int (d+e x)^2 \sinh ^2\left (a+b x+c x^2\right ) \, dx=-\frac {(d+e x)^3}{6 e}+\frac {e^2 e^{-2 a+\frac {b^2}{2 c}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{32 c^{3/2}}+\frac {(2 c d-b e)^2 e^{-2 a+\frac {b^2}{2 c}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{32 c^{5/2}}-\frac {e^2 e^{2 a-\frac {b^2}{2 c}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{32 c^{3/2}}+\frac {(2 c d-b e)^2 e^{2 a-\frac {b^2}{2 c}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{32 c^{5/2}}+\frac {e (2 c d-b e) \sinh \left (2 a+2 b x+2 c x^2\right )}{16 c^2}+\frac {e (d+e x) \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c} \]

[Out]

-1/6*(e*x+d)^3/e+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+1/64*
e^2*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)+1/64*(-b*e+2*c*d)^2*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^(5/2)-1/64*e^2*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)+1/64*(-b*e+2*c*d)^2*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^(5/2)

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {5502, 5495, 5491, 5483, 2266, 2235, 2236, 5482} \[ \int (d+e x)^2 \sinh ^2\left (a+b x+c x^2\right ) \, dx=\frac {\sqrt {\frac {\pi }{2}} e^{\frac {b^2}{2 c}-2 a} (2 c d-b e)^2 \text {erf}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{32 c^{5/2}}+\frac {\sqrt {\frac {\pi }{2}} e^{2 a-\frac {b^2}{2 c}} (2 c d-b e)^2 \text {erfi}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{32 c^{5/2}}+\frac {\sqrt {\frac {\pi }{2}} e^2 e^{\frac {b^2}{2 c}-2 a} \text {erf}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{32 c^{3/2}}-\frac {\sqrt {\frac {\pi }{2}} e^2 e^{2 a-\frac {b^2}{2 c}} \text {erfi}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{32 c^{3/2}}+\frac {e (2 c d-b e) \sinh \left (2 a+2 b x+2 c x^2\right )}{16 c^2}+\frac {e (d+e x) \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}-\frac {(d+e x)^3}{6 e} \]

[In]

Int[(d + e*x)^2*Sinh[a + b*x + c*x^2]^2,x]

[Out]

-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))*Sqrt[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)

Rule 2235

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]

Rule 2236

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] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 2266

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[F^(a - b^2/(4*c)), Int[F^((b + 2*c*x)^2/(4*c))
, x], x] /; FreeQ[{F, a, b, c}, x]

Rule 5482

Int[Sinh[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[1/2, Int[E^(a + b*x + c*x^2), x], x] - Dist[1/2
, Int[E^(-a - b*x - c*x^2), x], x] /; FreeQ[{a, b, c}, x]

Rule 5483

Int[Cosh[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[1/2, Int[E^(a + b*x + c*x^2), x], x] + Dist[1/2
, Int[E^(-a - b*x - c*x^2), x], x] /; FreeQ[{a, b, c}, x]

Rule 5491

Int[Cosh[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]*((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[e*(Sinh[a + b*x + c*x^2]/(
2*c)), x] - Dist[(b*e - 2*c*d)/(2*c), Int[Cosh[a + b*x + c*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b*
e - 2*c*d, 0]

Rule 5495

Int[Cosh[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]*((d_.) + (e_.)*(x_))^(m_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*(
Sinh[a + b*x + c*x^2]/(2*c)), x] + (-Dist[(b*e - 2*c*d)/(2*c), Int[(d + e*x)^(m - 1)*Cosh[a + b*x + c*x^2], x]
, x] - Dist[e^2*((m - 1)/(2*c)), Int[(d + e*x)^(m - 2)*Sinh[a + b*x + c*x^2], x], x]) /; FreeQ[{a, b, c, d, e}
, x] && GtQ[m, 1] && NeQ[b*e - 2*c*d, 0]

Rule 5502

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]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {1}{2} (d+e x)^2+\frac {1}{2} (d+e x)^2 \cosh \left (2 a+2 b x+2 c x^2\right )\right ) \, dx \\ & = -\frac {(d+e x)^3}{6 e}+\frac {1}{2} \int (d+e x)^2 \cosh \left (2 a+2 b x+2 c x^2\right ) \, dx \\ & = -\frac {(d+e x)^3}{6 e}+\frac {e (d+e x) \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}-\frac {e^2 \int \sinh \left (2 a+2 b x+2 c x^2\right ) \, dx}{8 c}+\frac {(2 c d-b e) \int (d+e x) \cosh \left (2 a+2 b x+2 c x^2\right ) \, dx}{4 c} \\ & = -\frac {(d+e x)^3}{6 e}+\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 {e^2 \int e^{-2 a-2 b x-2 c x^2} \, dx}{16 c}-\frac {e^2 \int e^{2 a+2 b x+2 c x^2} \, dx}{16 c}+\frac {(2 c d-b e)^2 \int \cosh \left (2 a+2 b x+2 c x^2\right ) \, dx}{8 c^2} \\ & = -\frac {(d+e x)^3}{6 e}+\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 {(2 c d-b e)^2 \int e^{-2 a-2 b x-2 c x^2} \, dx}{16 c^2}+\frac {(2 c d-b e)^2 \int e^{2 a+2 b x+2 c x^2} \, dx}{16 c^2}-\frac {\left (e^2 e^{2 a-\frac {b^2}{2 c}}\right ) \int e^{\frac {(2 b+4 c x)^2}{8 c}} \, dx}{16 c}+\frac {\left (e^2 e^{-2 a+\frac {b^2}{2 c}}\right ) \int e^{-\frac {(-2 b-4 c x)^2}{8 c}} \, dx}{16 c} \\ & = -\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 {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 {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 {\left ((2 c d-b e)^2 e^{2 a-\frac {b^2}{2 c}}\right ) \int e^{\frac {(2 b+4 c x)^2}{8 c}} \, dx}{16 c^2}+\frac {\left ((2 c d-b e)^2 e^{-2 a+\frac {b^2}{2 c}}\right ) \int e^{-\frac {(-2 b-4 c x)^2}{8 c}} \, dx}{16 c^2} \\ & = -\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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.95 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.77 \[ \int (d+e x)^2 \sinh ^2\left (a+b x+c x^2\right ) \, dx=\frac {3 \left (4 c^2 d^2+b^2 e^2+c e (-4 b d+e)\right ) \sqrt {2 \pi } \text {erf}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right ) \left (\cosh \left (2 a-\frac {b^2}{2 c}\right )-\sinh \left (2 a-\frac {b^2}{2 c}\right )\right )+3 \left (4 c^2 d^2+b^2 e^2-c e (4 b d+e)\right ) \sqrt {2 \pi } \text {erfi}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right ) \left (\cosh \left (2 a-\frac {b^2}{2 c}\right )+\sinh \left (2 a-\frac {b^2}{2 c}\right )\right )-4 \sqrt {c} \left (8 c^2 x \left (3 d^2+3 d e x+e^2 x^2\right )-3 e (4 c d-b e+2 c e x) \sinh (2 (a+x (b+c x)))\right )}{192 c^{5/2}} \]

[In]

Integrate[(d + e*x)^2*Sinh[a + b*x + c*x^2]^2,x]

[Out]

(3*(4*c^2*d^2 + b^2*e^2 + c*e*(-4*b*d + 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)]) + 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]*S
qrt[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))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(527\) vs. \(2(253)=506\).

Time = 0.90 (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}\)	\(528\)

[In]

int((e*x+d)^2*sinh(c*x^2+b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

-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*b/c^2*exp(-2*x*(c*x+b))+1/64*exp(-
2*a)*e^2*b^2/c^(5/2)*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*exp(-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*b/c^2*exp(2*x*(c*x+b))-1/32*exp(2*a)*e^2*b^2/c^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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1142 vs. \(2 (253) = 506\).

Time = 0.27 (sec) , antiderivative size = 1142, normalized size of antiderivative = 3.67 \[ \int (d+e x)^2 \sinh ^2\left (a+b x+c x^2\right ) \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)^2*sinh(c*x^2+b*x+a)^2,x, algorithm="fricas")

[Out]

-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)*cos
h(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)*sqr
t(-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)*sin
h(-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)) + 32*(c^3*e^2*x^3 + 3*c^3*d
*e*x^2 + 3*c^3*d^2*x)*cosh(c*x^2 + b*x + a)^2 + 4*(8*c^3*e^2*x^3 + 24*c^3*d*e*x^2 + 24*c^3*d^2*x - 9*(2*c^2*e^
2*x + 4*c^2*d*e - b*c*e^2)*cosh(c*x^2 + b*x + a)^2)*sinh(c*x^2 + b*x + a)^2 - 8*(3*(2*c^2*e^2*x + 4*c^2*d*e -
b*c*e^2)*cosh(c*x^2 + b*x + a)^3 - 8*(c^3*e^2*x^3 + 3*c^3*d*e*x^2 + 3*c^3*d^2*x)*cosh(c*x^2 + b*x + a))*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)

Sympy [F]

\[ \int (d+e x)^2 \sinh ^2\left (a+b x+c x^2\right ) \, dx=\int \left (d + e x\right )^{2} \sinh ^{2}{\left (a + b x + c x^{2} \right )}\, dx \]

[In]

integrate((e*x+d)**2*sinh(c*x**2+b*x+a)**2,x)

[Out]

Integral((d + e*x)**2*sinh(a + b*x + c*x**2)**2, x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 601 vs. \(2 (253) = 506\).

Time = 0.46 (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=\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^{2} - \frac {1}{16} \, {\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 )} d e - \frac {1}{192} \, {\left (32 \, x^{3} - \frac {3 \, \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 )}}{\sqrt {c}} + \frac {3 \, \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 )}}{\sqrt {-c}}\right )} e^{2} \]

[In]

integrate((e*x+d)^2*sinh(c*x^2+b*x+a)^2,x, algorithm="maxima")

[Out]

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*(er
f(sqrt(1/2)*sqrt((2*c*x + b)^2/c)) - 1)/(sqrt((2*c*x + b)^2/c)*(-c)^(3/2)) + sqrt(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(sq
rt(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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.85 \[ \int (d+e x)^2 \sinh ^2\left (a+b x+c x^2\right ) \, dx=-\frac {1}{6} \, e^{2} x^{3} - \frac {1}{2} \, d e x^{2} - \frac {1}{2} \, d^{2} x - \frac {\frac {\sqrt {2} \sqrt {\pi } {\left (4 \, c^{2} d^{2} - 4 \, b c d e + b^{2} e^{2} + c e^{2}\right )} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} \sqrt {c} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (\frac {b^{2} - 4 \, a c}{2 \, c}\right )}}{\sqrt {c}} + 2 \, {\left (c e^{2} {\left (2 \, x + \frac {b}{c}\right )} + 4 \, c d e - 2 \, b e^{2}\right )} e^{\left (-2 \, c x^{2} - 2 \, b x - 2 \, a\right )}}{64 \, c^{2}} - \frac {\frac {\sqrt {2} \sqrt {\pi } {\left (4 \, c^{2} d^{2} - 4 \, b c d e + b^{2} e^{2} - c e^{2}\right )} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} \sqrt {-c} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right )}}{\sqrt {-c}} - 2 \, {\left (c e^{2} {\left (2 \, x + \frac {b}{c}\right )} + 4 \, c d e - 2 \, b e^{2}\right )} e^{\left (2 \, c x^{2} + 2 \, b x + 2 \, a\right )}}{64 \, c^{2}} \]

[In]

integrate((e*x+d)^2*sinh(c*x^2+b*x+a)^2,x, algorithm="giac")

[Out]

-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*sq
rt(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

Mupad [F(-1)]

Timed out. \[ \int (d+e x)^2 \sinh ^2\left (a+b x+c x^2\right ) \, dx=\int {\mathrm {sinh}\left (c\,x^2+b\,x+a\right )}^2\,{\left (d+e\,x\right )}^2 \,d x \]

[In]

int(sinh(a + b*x + c*x^2)^2*(d + e*x)^2,x)

[Out]

int(sinh(a + b*x + c*x^2)^2*(d + e*x)^2, x)