\(\int (d+e x) \cosh ^2(a+b x+c x^2) \, dx\) [32]

Optimal result

Integrand size = 19, antiderivative size = 160 \[ \int (d+e x) \cosh ^2\left (a+b x+c x^2\right ) \, dx=\frac {(d+e x)^2}{4 e}+\frac {(2 c d-b e) e^{-2 a+\frac {b^2}{2 c}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{16 c^{3/2}}+\frac {(2 c d-b e) e^{2 a-\frac {b^2}{2 c}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{16 c^{3/2}}+\frac {e \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c} \] Output:

1/4*(e*x+d)^2/e+1/32*(-b*e+2*c*d)*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)+1/32*(-b*e+2*c*d)*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)+1/8*e*sinh 
(2*c*x^2+2*b*x+2*a)/c
 

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.11 \[ \int (d+e x) \cosh ^2\left (a+b x+c x^2\right ) \, dx=\frac {(2 c d-b e) \sqrt {2 \pi } \text {erf}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right ) \left (\cosh \left (2 a-\frac {b^2}{2 c}\right )-\sinh \left (2 a-\frac {b^2}{2 c}\right )\right )+(2 c d-b e) \sqrt {2 \pi } \text {erfi}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right ) \left (\cosh \left (2 a-\frac {b^2}{2 c}\right )+\sinh \left (2 a-\frac {b^2}{2 c}\right )\right )+4 \sqrt {c} (2 c x (2 d+e x)+e \sinh (2 (a+x (b+c x))))}{32 c^{3/2}} \] Input:

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

Output:

((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))
 

Rubi [A] (verified)

Time = 0.38 (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 = {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.

\(\displaystyle \int (d+e x) \cosh ^2\left (a+b x+c x^2\right ) \, dx\)

\(\Big \downarrow \) 5918

\(\displaystyle \int \left (\frac {1}{2} (d+e x) \cosh \left (2 a+2 b x+2 c x^2\right )+\frac {1}{2} (d+e x)\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {\frac {\pi }{2}} e^{\frac {b^2}{2 c}-2 a} (2 c d-b e) \text {erf}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{16 c^{3/2}}+\frac {\sqrt {\frac {\pi }{2}} e^{2 a-\frac {b^2}{2 c}} (2 c d-b e) \text {erfi}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{16 c^{3/2}}+\frac {e \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}+\frac {(d+e x)^2}{4 e}\)

Input:

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

Output:

(d + e*x)^2/(4*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*Sin 
h[2*a + 2*b*x + 2*c*x^2])/(8*c)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5918

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]
 

Maple [A] (verified)

Time = 1.39 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.44

method	result	size
risch	\(\frac {e \,x^{2}}{4}+\frac {d x}{2}+\frac {\operatorname {erf}\left (\sqrt {2}\, \sqrt {c}\, x +\frac {b \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}\, \sqrt {\pi }\, d \,{\mathrm e}^{-\frac {4 a c -b^{2}}{2 c}}}{16 \sqrt {c}}-\frac {{\mathrm e}^{-2 a} e \,{\mathrm e}^{-2 x \left (c x +b \right )}}{16 c}-\frac {{\mathrm e}^{-2 a} e b \sqrt {\pi }\, {\mathrm e}^{\frac {b^{2}}{2 c}} \sqrt {2}\, \operatorname {erf}\left (\sqrt {2}\, \sqrt {c}\, x +\frac {b \sqrt {2}}{2 \sqrt {c}}\right )}{32 c^{\frac {3}{2}}}-\frac {\operatorname {erf}\left (-\sqrt {-2 c}\, x +\frac {b}{\sqrt {-2 c}}\right ) \sqrt {\pi }\, d \,{\mathrm e}^{\frac {4 a c -b^{2}}{2 c}}}{8 \sqrt {-2 c}}+\frac {{\mathrm e}^{2 a} e \,{\mathrm e}^{2 x \left (c x +b \right )}}{16 c}+\frac {{\mathrm e}^{2 a} e b \sqrt {\pi }\, {\mathrm e}^{-\frac {b^{2}}{2 c}} \operatorname {erf}\left (-\sqrt {-2 c}\, x +\frac {b}{\sqrt {-2 c}}\right )}{16 c \sqrt {-2 c}}\)	\(231\)

Input:

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

Output:

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))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 777 vs. \(2 (130) = 260\).

Time = 0.11 (sec) , antiderivative size = 777, normalized size of antiderivative = 4.86 \[ \int (d+e x) \cosh ^2\left (a+b x+c x^2\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

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)
 

Sympy [F]

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

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

Output:

Integral((d + e*x)*cosh(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. 301 vs. \(2 (130) = 260\).

Time = 0.24 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.88 \[ \int (d+e x) \cosh ^2\left (a+b x+c x^2\right ) \, dx=\frac {1}{16} \, {\left (\frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\sqrt {2} \sqrt {-c} x - \frac {\sqrt {2} b}{2 \, \sqrt {-c}}\right ) e^{\left (2 \, a - \frac {b^{2}}{2 \, c}\right )}}{\sqrt {-c}} + \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\sqrt {2} \sqrt {c} x + \frac {\sqrt {2} b}{2 \, \sqrt {c}}\right ) e^{\left (-2 \, a + \frac {b^{2}}{2 \, c}\right )}}{\sqrt {c}} + 8 \, x\right )} d + \frac {1}{32} \, {\left (8 \, x^{2} - \frac {\sqrt {2} {\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (2 \, c x + b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt {-\frac {{\left (2 \, c x + b\right )}^{2}}{c}} c^{\frac {3}{2}}} - \frac {\sqrt {2} e^{\left (\frac {{\left (2 \, c x + b\right )}^{2}}{2 \, c}\right )}}{\sqrt {c}}\right )} e^{\left (2 \, a - \frac {b^{2}}{2 \, c}\right )}}{\sqrt {c}} - \frac {\sqrt {2} {\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {\frac {{\left (2 \, c x + b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt {\frac {{\left (2 \, c x + b\right )}^{2}}{c}} \left (-c\right )^{\frac {3}{2}}} + \frac {\sqrt {2} c e^{\left (-\frac {{\left (2 \, c x + b\right )}^{2}}{2 \, c}\right )}}{\left (-c\right )^{\frac {3}{2}}}\right )} e^{\left (-2 \, a + \frac {b^{2}}{2 \, c}\right )}}{\sqrt {-c}}\right )} e \] Input:

integrate((e*x+d)*cosh(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 + 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
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.03 \[ \int (d+e x) \cosh ^2\left (a+b x+c x^2\right ) \, dx=\frac {1}{4} \, e x^{2} + \frac {1}{2} \, d x - \frac {\frac {\sqrt {2} \sqrt {\pi } {\left (2 \, c d - b e\right )} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} \sqrt {c} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (\frac {b^{2} - 4 \, a c}{2 \, c}\right )}}{\sqrt {c}} + 2 \, e e^{\left (-2 \, c x^{2} - 2 \, b x - 2 \, a\right )}}{32 \, c} - \frac {\frac {\sqrt {2} \sqrt {\pi } {\left (2 \, c d - b e\right )} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} \sqrt {-c} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right )}}{\sqrt {-c}} - 2 \, e e^{\left (2 \, c x^{2} + 2 \, b x + 2 \, a\right )}}{32 \, c} \] Input:

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

Output:

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)*s 
qrt(-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
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (d+e x) \cosh ^2\left (a+b x+c x^2\right ) \, dx=\frac {\sqrt {\pi }\, e^{2 c \,x^{2}+2 b x +4 a} \mathrm {erf}\left (\frac {2 c i x +b i}{\sqrt {c}\, \sqrt {2}}\right ) b e i -2 \sqrt {\pi }\, e^{2 c \,x^{2}+2 b x +4 a} \mathrm {erf}\left (\frac {2 c i x +b i}{\sqrt {c}\, \sqrt {2}}\right ) c d i +e^{\frac {8 c^{2} x^{2}+8 b c x +8 a c +b^{2}}{2 c}} \sqrt {c}\, \sqrt {2}\, e +8 e^{\frac {4 c^{2} x^{2}+4 b c x +4 a c +b^{2}}{2 c}} \sqrt {c}\, \sqrt {2}\, c d x +4 e^{\frac {4 c^{2} x^{2}+4 b c x +4 a c +b^{2}}{2 c}} \sqrt {c}\, \sqrt {2}\, c e \,x^{2}-2 e^{\frac {4 c^{2} x^{2}+4 b c x +b^{2}}{2 c}} \sqrt {c}\, \sqrt {2}\, \left (\int \frac {1}{e^{2 c \,x^{2}+2 b x}}d x \right ) b e +4 e^{\frac {4 c^{2} x^{2}+4 b c x +b^{2}}{2 c}} \sqrt {c}\, \sqrt {2}\, \left (\int \frac {1}{e^{2 c \,x^{2}+2 b x}}d x \right ) c d -e^{\frac {b^{2}}{2 c}} \sqrt {c}\, \sqrt {2}\, e}{16 e^{\frac {4 c^{2} x^{2}+4 b c x +4 a c +b^{2}}{2 c}} \sqrt {c}\, \sqrt {2}\, c} \] Input:

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

Output:

(sqrt(pi)*e**(4*a + 2*b*x + 2*c*x**2)*erf((b*i + 2*c*i*x)/(sqrt(c)*sqrt(2) 
))*b*e*i - 2*sqrt(pi)*e**(4*a + 2*b*x + 2*c*x**2)*erf((b*i + 2*c*i*x)/(sqr 
t(c)*sqrt(2)))*c*d*i + e**((8*a*c + b**2 + 8*b*c*x + 8*c**2*x**2)/(2*c))*s 
qrt(c)*sqrt(2)*e + 8*e**((4*a*c + b**2 + 4*b*c*x + 4*c**2*x**2)/(2*c))*sqr 
t(c)*sqrt(2)*c*d*x + 4*e**((4*a*c + b**2 + 4*b*c*x + 4*c**2*x**2)/(2*c))*s 
qrt(c)*sqrt(2)*c*e*x**2 - 2*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*e + 4*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*d - e* 
*(b**2/(2*c))*sqrt(c)*sqrt(2)*e)/(16*e**((4*a*c + b**2 + 4*b*c*x + 4*c**2* 
x**2)/(2*c))*sqrt(c)*sqrt(2)*c)