\(\int f^{a+b x+c x^2} \sinh (d+e x) \, dx\) [357]

Optimal result

Integrand size = 19, antiderivative size = 153 \[ \int f^{a+b x+c x^2} \sinh (d+e x) \, dx=\frac {e^{-d-\frac {(e-b \log (f))^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {e-b \log (f)-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{4 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{d-\frac {(e+b \log (f))^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {e+b \log (f)+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{4 \sqrt {c} \sqrt {\log (f)}} \] Output:

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

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.88 \[ \int f^{a+b x+c x^2} \sinh (d+e x) \, dx=\frac {e^{-\frac {e (e+2 b \log (f))}{4 c \log (f)}} f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \left (-e^{\frac {b e}{c}} \text {erfi}\left (\frac {-e+(b+2 c x) \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right ) (\cosh (d)-\sinh (d))+\text {erfi}\left (\frac {e+(b+2 c x) \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right ) (\cosh (d)+\sinh (d))\right )}{4 \sqrt {c} \sqrt {\log (f)}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 153, 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 = {6038, 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.

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

\(\Big \downarrow \) 6038

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(E^(-d - (e - b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(e - b*Log[f] - 
2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(4*Sqrt[c]*Sqrt[Log[f]]) + (E^(d 
- (e + b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(e + b*Log[f] + 2*c*x*L 
og[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(4*Sqrt[c]*Sqrt[Log[f]])
 

Defintions of rubi rules used

rule 2009

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

rule 6038

Int[(F_)^(u_)*Sinh[v_]^(n_.), x_Symbol] :> Int[ExpandTrigToExp[F^u, Sinh[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 = 0.14 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.06

method	result	size
risch	\(-\frac {\operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {e +b \ln \left (f \right )}{2 \sqrt {-c \ln \left (f \right )}}\right ) \sqrt {\pi }\, f^{a} f^{-\frac {b^{2}}{4 c}} {\mathrm e}^{-\frac {2 \ln \left (f \right ) b e -4 d \ln \left (f \right ) c +e^{2}}{4 \ln \left (f \right ) c}}}{4 \sqrt {-c \ln \left (f \right )}}+\frac {\operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {b \ln \left (f \right )-e}{2 \sqrt {-c \ln \left (f \right )}}\right ) \sqrt {\pi }\, f^{a} f^{-\frac {b^{2}}{4 c}} {\mathrm e}^{\frac {2 \ln \left (f \right ) b e -4 d \ln \left (f \right ) c -e^{2}}{4 \ln \left (f \right ) c}}}{4 \sqrt {-c \ln \left (f \right )}}\)	\(162\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 263 vs. \(2 (121) = 242\).

Time = 0.10 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.72 \[ \int f^{a+b x+c x^2} \sinh (d+e x) \, dx=-\frac {\sqrt {-c \log \left (f\right )} {\left (\sqrt {\pi } \cosh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} + e^{2} - 2 \, {\left (2 \, c d - b e\right )} \log \left (f\right )}{4 \, c \log \left (f\right )}\right ) + \sqrt {\pi } \sinh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} + e^{2} - 2 \, {\left (2 \, c d - b e\right )} \log \left (f\right )}{4 \, c \log \left (f\right )}\right )\right )} \operatorname {erf}\left (\frac {{\left ({\left (2 \, c x + b\right )} \log \left (f\right ) + e\right )} \sqrt {-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) - \sqrt {-c \log \left (f\right )} {\left (\sqrt {\pi } \cosh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} + e^{2} + 2 \, {\left (2 \, c d - b e\right )} \log \left (f\right )}{4 \, c \log \left (f\right )}\right ) + \sqrt {\pi } \sinh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} + e^{2} + 2 \, {\left (2 \, c d - b e\right )} \log \left (f\right )}{4 \, c \log \left (f\right )}\right )\right )} \operatorname {erf}\left (\frac {{\left ({\left (2 \, c x + b\right )} \log \left (f\right ) - e\right )} \sqrt {-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right )}{4 \, c \log \left (f\right )} \] Input:

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

Output:

-1/4*(sqrt(-c*log(f))*(sqrt(pi)*cosh(-1/4*((b^2 - 4*a*c)*log(f)^2 + e^2 - 
2*(2*c*d - b*e)*log(f))/(c*log(f))) + sqrt(pi)*sinh(-1/4*((b^2 - 4*a*c)*lo 
g(f)^2 + e^2 - 2*(2*c*d - b*e)*log(f))/(c*log(f))))*erf(1/2*((2*c*x + b)*l 
og(f) + e)*sqrt(-c*log(f))/(c*log(f))) - sqrt(-c*log(f))*(sqrt(pi)*cosh(-1 
/4*((b^2 - 4*a*c)*log(f)^2 + e^2 + 2*(2*c*d - b*e)*log(f))/(c*log(f))) + s 
qrt(pi)*sinh(-1/4*((b^2 - 4*a*c)*log(f)^2 + e^2 + 2*(2*c*d - b*e)*log(f))/ 
(c*log(f))))*erf(1/2*((2*c*x + b)*log(f) - e)*sqrt(-c*log(f))/(c*log(f)))) 
/(c*log(f))
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.84 \[ \int f^{a+b x+c x^2} \sinh (d+e x) \, dx=\frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x - \frac {b \log \left (f\right ) + e}{2 \, \sqrt {-c \log \left (f\right )}}\right ) e^{\left (d - \frac {{\left (b \log \left (f\right ) + e\right )}^{2}}{4 \, c \log \left (f\right )}\right )}}{4 \, \sqrt {-c \log \left (f\right )}} - \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x - \frac {b \log \left (f\right ) - e}{2 \, \sqrt {-c \log \left (f\right )}}\right ) e^{\left (-d - \frac {{\left (b \log \left (f\right ) - e\right )}^{2}}{4 \, c \log \left (f\right )}\right )}}{4 \, \sqrt {-c \log \left (f\right )}} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.09 \[ \int f^{a+b x+c x^2} \sinh (d+e x) \, dx=\frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right )} {\left (2 \, x + \frac {b \log \left (f\right ) - e}{c \log \left (f\right )}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2} - 4 \, a c \log \left (f\right )^{2} + 4 \, c d \log \left (f\right ) - 2 \, b e \log \left (f\right ) + e^{2}}{4 \, c \log \left (f\right )}\right )}}{4 \, \sqrt {-c \log \left (f\right )}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right )} {\left (2 \, x + \frac {b \log \left (f\right ) + e}{c \log \left (f\right )}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2} - 4 \, a c \log \left (f\right )^{2} - 4 \, c d \log \left (f\right ) + 2 \, b e \log \left (f\right ) + e^{2}}{4 \, c \log \left (f\right )}\right )}}{4 \, \sqrt {-c \log \left (f\right )}} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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