\(\int f^{a+c x^2} \cosh ^2(d+e x+f x^2) \, dx\) [319]

Optimal result

Integrand size = 23, antiderivative size = 183 \[ \int f^{a+c x^2} \cosh ^2\left (d+e x+f x^2\right ) \, dx=\frac {f^a \sqrt {\pi } \text {erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )}{4 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{-2 d+\frac {e^2}{2 f-c \log (f)}} f^a \sqrt {\pi } \text {erf}\left (\frac {e+x (2 f-c \log (f))}{\sqrt {2 f-c \log (f)}}\right )}{8 \sqrt {2 f-c \log (f)}}+\frac {e^{2 d-\frac {e^2}{2 f+c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {e+x (2 f+c \log (f))}{\sqrt {2 f+c \log (f)}}\right )}{8 \sqrt {2 f+c \log (f)}} \] Output:

1/4*f^a*Pi^(1/2)*erfi(c^(1/2)*x*ln(f)^(1/2))/c^(1/2)/ln(f)^(1/2)+1/8*exp(- 
2*d+e^2/(2*f-c*ln(f)))*f^a*Pi^(1/2)*erf((e+x*(2*f-c*ln(f)))/(2*f-c*ln(f))^ 
(1/2))/(2*f-c*ln(f))^(1/2)+1/8*exp(2*d-e^2/(2*f+c*ln(f)))*f^a*Pi^(1/2)*erf 
i((e+x*(2*f+c*ln(f)))/(2*f+c*ln(f))^(1/2))/(2*f+c*ln(f))^(1/2)
 

Mathematica [A] (verified)

Time = 0.94 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.41 \[ \int f^{a+c x^2} \cosh ^2\left (d+e x+f x^2\right ) \, dx=\frac {e^{\frac {e^2}{2 f-c \log (f)}} f^a \sqrt {\pi } \left (-2 e^{\frac {e^2}{-2 f+c \log (f)}} \text {erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right ) \left (4 f^2-c^2 \log ^2(f)\right )-\sqrt {c} \sqrt {\log (f)} \left (\text {erf}\left (\frac {e+2 f x-c x \log (f)}{\sqrt {2 f-c \log (f)}}\right ) \sqrt {2 f-c \log (f)} (2 f+c \log (f)) (\cosh (2 d)-\sinh (2 d))+e^{\frac {4 e^2 f}{-4 f^2+c^2 \log ^2(f)}} \text {erfi}\left (\frac {e+2 f x+c x \log (f)}{\sqrt {2 f+c \log (f)}}\right ) (2 f-c \log (f)) \sqrt {2 f+c \log (f)} (\cosh (2 d)+\sinh (2 d))\right )\right )}{8 \sqrt {c} \sqrt {\log (f)} \left (-4 f^2+c^2 \log ^2(f)\right )} \] Input:

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

Output:

(E^(e^2/(2*f - c*Log[f]))*f^a*Sqrt[Pi]*(-2*E^(e^2/(-2*f + c*Log[f]))*Erfi[ 
Sqrt[c]*x*Sqrt[Log[f]]]*(4*f^2 - c^2*Log[f]^2) - Sqrt[c]*Sqrt[Log[f]]*(Erf 
[(e + 2*f*x - c*x*Log[f])/Sqrt[2*f - c*Log[f]]]*Sqrt[2*f - c*Log[f]]*(2*f 
+ c*Log[f])*(Cosh[2*d] - Sinh[2*d]) + E^((4*e^2*f)/(-4*f^2 + c^2*Log[f]^2) 
)*Erfi[(e + 2*f*x + c*x*Log[f])/Sqrt[2*f + c*Log[f]]]*(2*f - c*Log[f])*Sqr 
t[2*f + c*Log[f]]*(Cosh[2*d] + Sinh[2*d]))))/(8*Sqrt[c]*Sqrt[Log[f]]*(-4*f 
^2 + c^2*Log[f]^2))
 

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {6039, 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.

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

\(\Big \downarrow \) 6039

\(\displaystyle \int \left (\frac {1}{4} f^{a+c x^2} e^{-2 d-2 e x-2 f x^2}+\frac {1}{4} f^{a+c x^2} e^{2 d+2 e x+2 f x^2}+\frac {1}{2} f^{a+c x^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {\pi } f^a e^{\frac {e^2}{2 f-c \log (f)}-2 d} \text {erf}\left (\frac {x (2 f-c \log (f))+e}{\sqrt {2 f-c \log (f)}}\right )}{8 \sqrt {2 f-c \log (f)}}+\frac {\sqrt {\pi } f^a e^{2 d-\frac {e^2}{c \log (f)+2 f}} \text {erfi}\left (\frac {x (c \log (f)+2 f)+e}{\sqrt {c \log (f)+2 f}}\right )}{8 \sqrt {c \log (f)+2 f}}+\frac {\sqrt {\pi } f^a \text {erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )}{4 \sqrt {c} \sqrt {\log (f)}}\)

Input:

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

Output:

(f^a*Sqrt[Pi]*Erfi[Sqrt[c]*x*Sqrt[Log[f]]])/(4*Sqrt[c]*Sqrt[Log[f]]) + (E^ 
(-2*d + e^2/(2*f - c*Log[f]))*f^a*Sqrt[Pi]*Erf[(e + x*(2*f - c*Log[f]))/Sq 
rt[2*f - c*Log[f]]])/(8*Sqrt[2*f - c*Log[f]]) + (E^(2*d - e^2/(2*f + c*Log 
[f]))*f^a*Sqrt[Pi]*Erfi[(e + x*(2*f + c*Log[f]))/Sqrt[2*f + c*Log[f]]])/(8 
*Sqrt[2*f + c*Log[f]])
 

Defintions of rubi rules used

rule 2009

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

rule 6039

Int[Cosh[v_]^(n_.)*(F_)^(u_), x_Symbol] :> Int[ExpandTrigToExp[F^u, Cosh[v] 
^n, x], x] /; FreeQ[F, x] && (LinearQ[u, x] || PolyQ[u, x, 2]) && (LinearQ[ 
v, x] || PolyQ[v, x, 2]) && IGtQ[n, 0]
 

Maple [A] (verified)

Time = 1.61 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.97

method	result	size
risch	\(\frac {\operatorname {erf}\left (x \sqrt {2 f -c \ln \left (f \right )}+\frac {e}{\sqrt {2 f -c \ln \left (f \right )}}\right ) \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {2 d \ln \left (f \right ) c -4 d f +e^{2}}{c \ln \left (f \right )-2 f}}}{8 \sqrt {2 f -c \ln \left (f \right )}}-\frac {\operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )-2 f}\, x +\frac {e}{\sqrt {-c \ln \left (f \right )-2 f}}\right ) \sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {2 d \ln \left (f \right ) c +4 d f -e^{2}}{2 f +c \ln \left (f \right )}}}{8 \sqrt {-c \ln \left (f \right )-2 f}}+\frac {f^{a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-c \ln \left (f \right )}\, x \right )}{4 \sqrt {-c \ln \left (f \right )}}\)	\(177\)

Input:

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

Output:

1/8*erf(x*(2*f-c*ln(f))^(1/2)+e/(2*f-c*ln(f))^(1/2))/(2*f-c*ln(f))^(1/2)*P 
i^(1/2)*f^a*exp(-(2*d*ln(f)*c-4*d*f+e^2)/(c*ln(f)-2*f))-1/8*erf(-(-c*ln(f) 
-2*f)^(1/2)*x+e/(-c*ln(f)-2*f)^(1/2))/(-c*ln(f)-2*f)^(1/2)*Pi^(1/2)*f^a*ex 
p((2*d*ln(f)*c+4*d*f-e^2)/(2*f+c*ln(f)))+1/4*f^a*Pi^(1/2)/(-c*ln(f))^(1/2) 
*erf((-c*ln(f))^(1/2)*x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 420 vs. \(2 (155) = 310\).

Time = 0.10 (sec) , antiderivative size = 420, normalized size of antiderivative = 2.30 \[ \int f^{a+c x^2} \cosh ^2\left (d+e x+f x^2\right ) \, dx=-\frac {2 \, {\left (\sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} - 4 \, f^{2}\right )} \cosh \left (a \log \left (f\right )\right ) + \sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} - 4 \, f^{2}\right )} \sinh \left (a \log \left (f\right )\right )\right )} \sqrt {-c \log \left (f\right )} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x\right ) + {\left (\sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} + 2 \, c f \log \left (f\right )\right )} \cosh \left (\frac {a c \log \left (f\right )^{2} - e^{2} + 4 \, d f - 2 \, {\left (c d + a f\right )} \log \left (f\right )}{c \log \left (f\right ) - 2 \, f}\right ) + \sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} + 2 \, c f \log \left (f\right )\right )} \sinh \left (\frac {a c \log \left (f\right )^{2} - e^{2} + 4 \, d f - 2 \, {\left (c d + a f\right )} \log \left (f\right )}{c \log \left (f\right ) - 2 \, f}\right )\right )} \sqrt {-c \log \left (f\right ) + 2 \, f} \operatorname {erf}\left (\frac {{\left (c x \log \left (f\right ) - 2 \, f x - e\right )} \sqrt {-c \log \left (f\right ) + 2 \, f}}{c \log \left (f\right ) - 2 \, f}\right ) + {\left (\sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} - 2 \, c f \log \left (f\right )\right )} \cosh \left (\frac {a c \log \left (f\right )^{2} - e^{2} + 4 \, d f + 2 \, {\left (c d + a f\right )} \log \left (f\right )}{c \log \left (f\right ) + 2 \, f}\right ) + \sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} - 2 \, c f \log \left (f\right )\right )} \sinh \left (\frac {a c \log \left (f\right )^{2} - e^{2} + 4 \, d f + 2 \, {\left (c d + a f\right )} \log \left (f\right )}{c \log \left (f\right ) + 2 \, f}\right )\right )} \sqrt {-c \log \left (f\right ) - 2 \, f} \operatorname {erf}\left (\frac {{\left (c x \log \left (f\right ) + 2 \, f x + e\right )} \sqrt {-c \log \left (f\right ) - 2 \, f}}{c \log \left (f\right ) + 2 \, f}\right )}{8 \, {\left (c^{3} \log \left (f\right )^{3} - 4 \, c f^{2} \log \left (f\right )\right )}} \] Input:

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

Output:

-1/8*(2*(sqrt(pi)*(c^2*log(f)^2 - 4*f^2)*cosh(a*log(f)) + sqrt(pi)*(c^2*lo 
g(f)^2 - 4*f^2)*sinh(a*log(f)))*sqrt(-c*log(f))*erf(sqrt(-c*log(f))*x) + ( 
sqrt(pi)*(c^2*log(f)^2 + 2*c*f*log(f))*cosh((a*c*log(f)^2 - e^2 + 4*d*f - 
2*(c*d + a*f)*log(f))/(c*log(f) - 2*f)) + sqrt(pi)*(c^2*log(f)^2 + 2*c*f*l 
og(f))*sinh((a*c*log(f)^2 - e^2 + 4*d*f - 2*(c*d + a*f)*log(f))/(c*log(f) 
- 2*f)))*sqrt(-c*log(f) + 2*f)*erf((c*x*log(f) - 2*f*x - e)*sqrt(-c*log(f) 
 + 2*f)/(c*log(f) - 2*f)) + (sqrt(pi)*(c^2*log(f)^2 - 2*c*f*log(f))*cosh(( 
a*c*log(f)^2 - e^2 + 4*d*f + 2*(c*d + a*f)*log(f))/(c*log(f) + 2*f)) + sqr 
t(pi)*(c^2*log(f)^2 - 2*c*f*log(f))*sinh((a*c*log(f)^2 - e^2 + 4*d*f + 2*( 
c*d + a*f)*log(f))/(c*log(f) + 2*f)))*sqrt(-c*log(f) - 2*f)*erf((c*x*log(f 
) + 2*f*x + e)*sqrt(-c*log(f) - 2*f)/(c*log(f) + 2*f)))/(c^3*log(f)^3 - 4* 
c*f^2*log(f))
 

Sympy [F]

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

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

Output:

Integral(f**(a + c*x**2)*cosh(d + e*x + f*x**2)**2, x)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.88 \[ \int f^{a+c x^2} \cosh ^2\left (d+e x+f x^2\right ) \, dx=\frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) - 2 \, f} x - \frac {e}{\sqrt {-c \log \left (f\right ) - 2 \, f}}\right ) e^{\left (2 \, d - \frac {e^{2}}{c \log \left (f\right ) + 2 \, f}\right )}}{8 \, \sqrt {-c \log \left (f\right ) - 2 \, f}} + \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) + 2 \, f} x + \frac {e}{\sqrt {-c \log \left (f\right ) + 2 \, f}}\right ) e^{\left (-2 \, d - \frac {e^{2}}{c \log \left (f\right ) - 2 \, f}\right )}}{8 \, \sqrt {-c \log \left (f\right ) + 2 \, f}} + \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x\right )}{4 \, \sqrt {-c \log \left (f\right )}} \] Input:

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

Output:

1/8*sqrt(pi)*f^a*erf(sqrt(-c*log(f) - 2*f)*x - e/sqrt(-c*log(f) - 2*f))*e^ 
(2*d - e^2/(c*log(f) + 2*f))/sqrt(-c*log(f) - 2*f) + 1/8*sqrt(pi)*f^a*erf( 
sqrt(-c*log(f) + 2*f)*x + e/sqrt(-c*log(f) + 2*f))*e^(-2*d - e^2/(c*log(f) 
 - 2*f))/sqrt(-c*log(f) + 2*f) + 1/4*sqrt(pi)*f^a*erf(sqrt(-c*log(f))*x)/s 
qrt(-c*log(f))
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.08 \[ \int f^{a+c x^2} \cosh ^2\left (d+e x+f x^2\right ) \, dx=-\frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (-\sqrt {-c \log \left (f\right )} x\right )}{4 \, \sqrt {-c \log \left (f\right )}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-c \log \left (f\right ) - 2 \, f} {\left (x + \frac {e}{c \log \left (f\right ) + 2 \, f}\right )}\right ) e^{\left (\frac {a c \log \left (f\right )^{2} + 2 \, c d \log \left (f\right ) + 2 \, a f \log \left (f\right ) - e^{2} + 4 \, d f}{c \log \left (f\right ) + 2 \, f}\right )}}{8 \, \sqrt {-c \log \left (f\right ) - 2 \, f}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-c \log \left (f\right ) + 2 \, f} {\left (x - \frac {e}{c \log \left (f\right ) - 2 \, f}\right )}\right ) e^{\left (\frac {a c \log \left (f\right )^{2} - 2 \, c d \log \left (f\right ) - 2 \, a f \log \left (f\right ) - e^{2} + 4 \, d f}{c \log \left (f\right ) - 2 \, f}\right )}}{8 \, \sqrt {-c \log \left (f\right ) + 2 \, f}} \] Input:

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

Output:

-1/4*sqrt(pi)*f^a*erf(-sqrt(-c*log(f))*x)/sqrt(-c*log(f)) - 1/8*sqrt(pi)*e 
rf(-sqrt(-c*log(f) - 2*f)*(x + e/(c*log(f) + 2*f)))*e^((a*c*log(f)^2 + 2*c 
*d*log(f) + 2*a*f*log(f) - e^2 + 4*d*f)/(c*log(f) + 2*f))/sqrt(-c*log(f) - 
 2*f) - 1/8*sqrt(pi)*erf(-sqrt(-c*log(f) + 2*f)*(x - e/(c*log(f) - 2*f)))* 
e^((a*c*log(f)^2 - 2*c*d*log(f) - 2*a*f*log(f) - e^2 + 4*d*f)/(c*log(f) - 
2*f))/sqrt(-c*log(f) + 2*f)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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