\(\int f^{a+c x^2} \cosh ^2(d+e x) \, dx\) [313]

Optimal result

Integrand size = 18, antiderivative size = 161 \[ \int f^{a+c x^2} \cosh ^2(d+e x) \, dx=\frac {f^a \sqrt {\pi } \text {erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )}{4 \sqrt {c} \sqrt {\log (f)}}-\frac {e^{-2 d-\frac {e^2}{c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {e-c x \log (f)}{\sqrt {c} \sqrt {\log (f)}}\right )}{8 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{2 d-\frac {e^2}{c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {e+c x \log (f)}{\sqrt {c} \sqrt {\log (f)}}\right )}{8 \sqrt {c} \sqrt {\log (f)}} \] 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/c/ln(f))*f^a*Pi^(1/2)*erfi((e-c*x*ln(f))/c^(1/2)/ln(f)^(1/2))/c^(1 
/2)/ln(f)^(1/2)+1/8*exp(2*d-e^2/c/ln(f))*f^a*Pi^(1/2)*erfi((e+c*x*ln(f))/c 
^(1/2)/ln(f)^(1/2))/c^(1/2)/ln(f)^(1/2)
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.81 \[ \int f^{a+c x^2} \cosh ^2(d+e x) \, dx=\frac {e^{-\frac {e^2}{c \log (f)}} f^a \sqrt {\pi } \left (2 e^{\frac {e^2}{c \log (f)}} \text {erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )+\text {erfi}\left (\frac {-e+c x \log (f)}{\sqrt {c} \sqrt {\log (f)}}\right ) (\cosh (2 d)-\sinh (2 d))+\text {erfi}\left (\frac {e+c x \log (f)}{\sqrt {c} \sqrt {\log (f)}}\right ) (\cosh (2 d)+\sinh (2 d))\right )}{8 \sqrt {c} \sqrt {\log (f)}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, 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.

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

\(\Big \downarrow \) 6039

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

\(\Big \downarrow \) 2009

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

Input:

Int[f^(a + c*x^2)*Cosh[d + e*x]^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/(c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(e - c*x*Log[f])/(Sqrt[c]*Sqrt[L 
og[f]])])/(8*Sqrt[c]*Sqrt[Log[f]]) + (E^(2*d - e^2/(c*Log[f]))*f^a*Sqrt[Pi 
]*Erfi[(e + c*x*Log[f])/(Sqrt[c]*Sqrt[Log[f]])])/(8*Sqrt[c]*Sqrt[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 = 1.00 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.86

method	result	size
risch	\(\frac {\operatorname {erf}\left (\sqrt {-c \ln \left (f \right )}\, x +\frac {e}{\sqrt {-c \ln \left (f \right )}}\right ) \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {2 d \ln \left (f \right ) c +e^{2}}{\ln \left (f \right ) c}}}{8 \sqrt {-c \ln \left (f \right )}}-\frac {\operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {e}{\sqrt {-c \ln \left (f \right )}}\right ) \sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {2 d \ln \left (f \right ) c -e^{2}}{\ln \left (f \right ) c}}}{8 \sqrt {-c \ln \left (f \right )}}+\frac {f^{a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-c \ln \left (f \right )}\, x \right )}{4 \sqrt {-c \ln \left (f \right )}}\)	\(139\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.50 \[ \int f^{a+c x^2} \cosh ^2(d+e x) \, dx=-\frac {2 \, \sqrt {-c \log \left (f\right )} {\left (\sqrt {\pi } \cosh \left (a \log \left (f\right )\right ) + \sqrt {\pi } \sinh \left (a \log \left (f\right )\right )\right )} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x\right ) + \sqrt {-c \log \left (f\right )} {\left (\sqrt {\pi } \cosh \left (\frac {a c \log \left (f\right )^{2} + 2 \, c d \log \left (f\right ) - e^{2}}{c \log \left (f\right )}\right ) + \sqrt {\pi } \sinh \left (\frac {a c \log \left (f\right )^{2} + 2 \, c d \log \left (f\right ) - e^{2}}{c \log \left (f\right )}\right )\right )} \operatorname {erf}\left (\frac {{\left (c x \log \left (f\right ) + e\right )} \sqrt {-c \log \left (f\right )}}{c \log \left (f\right )}\right ) + \sqrt {-c \log \left (f\right )} {\left (\sqrt {\pi } \cosh \left (\frac {a c \log \left (f\right )^{2} - 2 \, c d \log \left (f\right ) - e^{2}}{c \log \left (f\right )}\right ) + \sqrt {\pi } \sinh \left (\frac {a c \log \left (f\right )^{2} - 2 \, c d \log \left (f\right ) - e^{2}}{c \log \left (f\right )}\right )\right )} \operatorname {erf}\left (\frac {{\left (c x \log \left (f\right ) - e\right )} \sqrt {-c \log \left (f\right )}}{c \log \left (f\right )}\right )}{8 \, c \log \left (f\right )} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.81 \[ \int f^{a+c x^2} \cosh ^2(d+e x) \, dx=\frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x - \frac {e}{\sqrt {-c \log \left (f\right )}}\right ) e^{\left (2 \, d - \frac {e^{2}}{c \log \left (f\right )}\right )}}{8 \, \sqrt {-c \log \left (f\right )}} + \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x + \frac {e}{\sqrt {-c \log \left (f\right )}}\right ) e^{\left (-2 \, d - \frac {e^{2}}{c \log \left (f\right )}\right )}}{8 \, \sqrt {-c \log \left (f\right )}} + \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x\right )}{4 \, \sqrt {-c \log \left (f\right )}} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.93 \[ \int f^{a+c x^2} \cosh ^2(d+e x) \, dx=-\frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (-\sqrt {-c \log \left (f\right )} x\right )}{4 \, \sqrt {-c \log \left (f\right )}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-c \log \left (f\right )} {\left (x + \frac {e}{c \log \left (f\right )}\right )}\right ) e^{\left (\frac {a c \log \left (f\right )^{2} + 2 \, c d \log \left (f\right ) - e^{2}}{c \log \left (f\right )}\right )}}{8 \, \sqrt {-c \log \left (f\right )}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-c \log \left (f\right )} {\left (x - \frac {e}{c \log \left (f\right )}\right )}\right ) e^{\left (\frac {a c \log \left (f\right )^{2} - 2 \, c d \log \left (f\right ) - e^{2}}{c \log \left (f\right )}\right )}}{8 \, \sqrt {-c \log \left (f\right )}} \] Input:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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