\(\int f^{a+b x} \sinh ^3(d+e x+f x^2) \, dx\) [347]

Optimal result

Integrand size = 21, antiderivative size = 257 \[ \int f^{a+b x} \sinh ^3\left (d+e x+f x^2\right ) \, dx=\frac {3}{16} e^{-d+\frac {(e-b \log (f))^2}{4 f}} f^{-\frac {1}{2}+a} \sqrt {\pi } \text {erf}\left (\frac {e+2 f x-b \log (f)}{2 \sqrt {f}}\right )-\frac {1}{16} e^{-3 d+\frac {(3 e-b \log (f))^2}{12 f}} f^{-\frac {1}{2}+a} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {3 e+6 f x-b \log (f)}{2 \sqrt {3} \sqrt {f}}\right )-\frac {3}{16} e^{d-\frac {(e+b \log (f))^2}{4 f}} f^{-\frac {1}{2}+a} \sqrt {\pi } \text {erfi}\left (\frac {e+2 f x+b \log (f)}{2 \sqrt {f}}\right )+\frac {1}{16} e^{3 d-\frac {(3 e+b \log (f))^2}{12 f}} f^{-\frac {1}{2}+a} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {3 e+6 f x+b \log (f)}{2 \sqrt {3} \sqrt {f}}\right ) \] Output:

3/16*exp(-d+1/4*(e-b*ln(f))^2/f)*f^(-1/2+a)*Pi^(1/2)*erf(1/2*(e+2*f*x-b*ln 
(f))/f^(1/2))-1/48*exp(-3*d+1/12*(3*e-b*ln(f))^2/f)*f^(-1/2+a)*3^(1/2)*Pi^ 
(1/2)*erf(1/6*(3*e+6*f*x-b*ln(f))*3^(1/2)/f^(1/2))-3/16*exp(d-1/4*(e+b*ln( 
f))^2/f)*f^(-1/2+a)*Pi^(1/2)*erfi(1/2*(e+2*f*x+b*ln(f))/f^(1/2))+1/48*exp( 
3*d-1/12*(3*e+b*ln(f))^2/f)*f^(-1/2+a)*3^(1/2)*Pi^(1/2)*erfi(1/6*(3*e+6*f* 
x+b*ln(f))*3^(1/2)/f^(1/2))
 

Mathematica [A] (verified)

Time = 0.53 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.38 \[ \int f^{a+b x} \sinh ^3\left (d+e x+f x^2\right ) \, dx=\frac {1}{16} e^{-\frac {3 e^2+b^2 \log ^2(f)}{4 f}} f^{a-\frac {b e+f}{2 f}} \sqrt {\frac {\pi }{3}} \left (-3 \sqrt {3} e^{\frac {e^2}{2 f}} \cosh (d) \text {erfi}\left (\frac {e+2 f x+b \log (f)}{2 \sqrt {f}}\right )+e^{\frac {b^2 \log ^2(f)}{6 f}} \cosh (3 d) \text {erfi}\left (\frac {3 e+6 f x+b \log (f)}{2 \sqrt {3} \sqrt {f}}\right )+3 \sqrt {3} e^{\frac {2 e^2+b^2 \log ^2(f)}{2 f}} \text {erf}\left (\frac {e+2 f x-b \log (f)}{2 \sqrt {f}}\right ) (\cosh (d)-\sinh (d))-3 \sqrt {3} e^{\frac {e^2}{2 f}} \text {erfi}\left (\frac {e+2 f x+b \log (f)}{2 \sqrt {f}}\right ) \sinh (d)-e^{\frac {9 e^2+2 b^2 \log ^2(f)}{6 f}} \text {erf}\left (\frac {3 e+6 f x-b \log (f)}{2 \sqrt {3} \sqrt {f}}\right ) (\cosh (3 d)-\sinh (3 d))+e^{\frac {b^2 \log ^2(f)}{6 f}} \text {erfi}\left (\frac {3 e+6 f x+b \log (f)}{2 \sqrt {3} \sqrt {f}}\right ) \sinh (3 d)\right ) \] Input:

Integrate[f^(a + b*x)*Sinh[d + e*x + f*x^2]^3,x]
 

Output:

(f^(a - (b*e + f)/(2*f))*Sqrt[Pi/3]*(-3*Sqrt[3]*E^(e^2/(2*f))*Cosh[d]*Erfi 
[(e + 2*f*x + b*Log[f])/(2*Sqrt[f])] + E^((b^2*Log[f]^2)/(6*f))*Cosh[3*d]* 
Erfi[(3*e + 6*f*x + b*Log[f])/(2*Sqrt[3]*Sqrt[f])] + 3*Sqrt[3]*E^((2*e^2 + 
 b^2*Log[f]^2)/(2*f))*Erf[(e + 2*f*x - b*Log[f])/(2*Sqrt[f])]*(Cosh[d] - S 
inh[d]) - 3*Sqrt[3]*E^(e^2/(2*f))*Erfi[(e + 2*f*x + b*Log[f])/(2*Sqrt[f])] 
*Sinh[d] - E^((9*e^2 + 2*b^2*Log[f]^2)/(6*f))*Erf[(3*e + 6*f*x - b*Log[f]) 
/(2*Sqrt[3]*Sqrt[f])]*(Cosh[3*d] - Sinh[3*d]) + E^((b^2*Log[f]^2)/(6*f))*E 
rfi[(3*e + 6*f*x + b*Log[f])/(2*Sqrt[3]*Sqrt[f])]*Sinh[3*d]))/(16*E^((3*e^ 
2 + b^2*Log[f]^2)/(4*f)))
 

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, 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.

\(\displaystyle \int f^{a+b x} \sinh ^3\left (d+e x+f x^2\right ) \, dx\)

\(\Big \downarrow \) 6038

\(\displaystyle \int \left (\frac {3}{8} f^{a+b x} \exp \left (-3 \left (d+e x+f x^2\right )+2 d+2 e x+2 f x^2\right )-\frac {3}{8} f^{a+b x} \exp \left (-3 \left (d+e x+f x^2\right )+4 d+4 e x+4 f x^2\right )+\frac {1}{8} f^{a+b x} \exp \left (-3 \left (d+e x+f x^2\right )+6 d+6 e x+6 f x^2\right )-\frac {1}{8} f^{a+b x} e^{-3 \left (d+e x+f x^2\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3}{16} \sqrt {\pi } f^{a-\frac {1}{2}} e^{\frac {(e-b \log (f))^2}{4 f}-d} \text {erf}\left (\frac {-b \log (f)+e+2 f x}{2 \sqrt {f}}\right )-\frac {1}{16} \sqrt {\frac {\pi }{3}} f^{a-\frac {1}{2}} e^{\frac {(3 e-b \log (f))^2}{12 f}-3 d} \text {erf}\left (\frac {-b \log (f)+3 e+6 f x}{2 \sqrt {3} \sqrt {f}}\right )-\frac {3}{16} \sqrt {\pi } f^{a-\frac {1}{2}} e^{d-\frac {(b \log (f)+e)^2}{4 f}} \text {erfi}\left (\frac {b \log (f)+e+2 f x}{2 \sqrt {f}}\right )+\frac {1}{16} \sqrt {\frac {\pi }{3}} f^{a-\frac {1}{2}} e^{3 d-\frac {(b \log (f)+3 e)^2}{12 f}} \text {erfi}\left (\frac {b \log (f)+3 e+6 f x}{2 \sqrt {3} \sqrt {f}}\right )\)

Input:

Int[f^(a + b*x)*Sinh[d + e*x + f*x^2]^3,x]
 

Output:

(3*E^(-d + (e - b*Log[f])^2/(4*f))*f^(-1/2 + a)*Sqrt[Pi]*Erf[(e + 2*f*x - 
b*Log[f])/(2*Sqrt[f])])/16 - (E^(-3*d + (3*e - b*Log[f])^2/(12*f))*f^(-1/2 
 + a)*Sqrt[Pi/3]*Erf[(3*e + 6*f*x - b*Log[f])/(2*Sqrt[3]*Sqrt[f])])/16 - ( 
3*E^(d - (e + b*Log[f])^2/(4*f))*f^(-1/2 + a)*Sqrt[Pi]*Erfi[(e + 2*f*x + b 
*Log[f])/(2*Sqrt[f])])/16 + (E^(3*d - (3*e + b*Log[f])^2/(12*f))*f^(-1/2 + 
 a)*Sqrt[Pi/3]*Erfi[(3*e + 6*f*x + b*Log[f])/(2*Sqrt[3]*Sqrt[f])])/16
 

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 = 1.69 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.03

method	result	size
risch	\(-\frac {\operatorname {erf}\left (-\sqrt {-3 f}\, x +\frac {3 e +b \ln \left (f \right )}{2 \sqrt {-3 f}}\right ) \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {b^{2} \ln \left (f \right )^{2}+6 \ln \left (f \right ) b e -36 d f +9 e^{2}}{12 f}}}{16 \sqrt {-3 f}}+\frac {\operatorname {erf}\left (-\sqrt {3}\, \sqrt {f}\, x +\frac {\left (b \ln \left (f \right )-3 e \right ) \sqrt {3}}{6 \sqrt {f}}\right ) \sqrt {3}\, \sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {b^{2} \ln \left (f \right )^{2}-6 \ln \left (f \right ) b e -36 d f +9 e^{2}}{12 f}}}{48 \sqrt {f}}-\frac {3 \,\operatorname {erf}\left (-\sqrt {f}\, x +\frac {b \ln \left (f \right )-e}{2 \sqrt {f}}\right ) \sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {b^{2} \ln \left (f \right )^{2}-2 \ln \left (f \right ) b e -4 d f +e^{2}}{4 f}}}{16 \sqrt {f}}+\frac {3 \,\operatorname {erf}\left (-\sqrt {-f}\, x +\frac {e +b \ln \left (f \right )}{2 \sqrt {-f}}\right ) \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {b^{2} \ln \left (f \right )^{2}+2 \ln \left (f \right ) b e -4 d f +e^{2}}{4 f}}}{16 \sqrt {-f}}\)	\(265\)

Input:

int(f^(b*x+a)*sinh(f*x^2+e*x+d)^3,x,method=_RETURNVERBOSE)
 

Output:

-1/16*erf(-(-3*f)^(1/2)*x+1/2*(3*e+b*ln(f))/(-3*f)^(1/2))/(-3*f)^(1/2)*Pi^ 
(1/2)*f^a*exp(-1/12*(b^2*ln(f)^2+6*ln(f)*b*e-36*d*f+9*e^2)/f)+1/48*erf(-3^ 
(1/2)*f^(1/2)*x+1/6*(b*ln(f)-3*e)*3^(1/2)/f^(1/2))/f^(1/2)*3^(1/2)*Pi^(1/2 
)*f^a*exp(1/12*(b^2*ln(f)^2-6*ln(f)*b*e-36*d*f+9*e^2)/f)-3/16*erf(-f^(1/2) 
*x+1/2*(b*ln(f)-e)/f^(1/2))/f^(1/2)*Pi^(1/2)*f^a*exp(1/4*(b^2*ln(f)^2-2*ln 
(f)*b*e-4*d*f+e^2)/f)+3/16*erf(-(-f)^(1/2)*x+1/2*(e+b*ln(f))/(-f)^(1/2))/( 
-f)^(1/2)*Pi^(1/2)*f^a*exp(-1/4*(b^2*ln(f)^2+2*ln(f)*b*e-4*d*f+e^2)/f)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 541 vs. \(2 (199) = 398\).

Time = 0.10 (sec) , antiderivative size = 541, normalized size of antiderivative = 2.11 \[ \int f^{a+b x} \sinh ^3\left (d+e x+f x^2\right ) \, dx =\text {Too large to display} \] Input:

integrate(f^(b*x+a)*sinh(f*x^2+e*x+d)^3,x, algorithm="fricas")
 

Output:

-1/48*(sqrt(3)*sqrt(pi)*sqrt(-f)*cosh(1/12*(b^2*log(f)^2 + 9*e^2 - 36*d*f 
+ 6*(b*e - 2*a*f)*log(f))/f)*erf(1/6*sqrt(3)*(6*f*x + b*log(f) + 3*e)*sqrt 
(-f)/f) - sqrt(3)*sqrt(pi)*sqrt(f)*cosh(1/12*(b^2*log(f)^2 + 9*e^2 - 36*d* 
f - 6*(b*e - 2*a*f)*log(f))/f)*erf(-1/6*sqrt(3)*(6*f*x - b*log(f) + 3*e)/s 
qrt(f)) - sqrt(3)*sqrt(pi)*sqrt(-f)*erf(1/6*sqrt(3)*(6*f*x + b*log(f) + 3* 
e)*sqrt(-f)/f)*sinh(1/12*(b^2*log(f)^2 + 9*e^2 - 36*d*f + 6*(b*e - 2*a*f)* 
log(f))/f) - sqrt(3)*sqrt(pi)*sqrt(f)*erf(-1/6*sqrt(3)*(6*f*x - b*log(f) + 
 3*e)/sqrt(f))*sinh(1/12*(b^2*log(f)^2 + 9*e^2 - 36*d*f - 6*(b*e - 2*a*f)* 
log(f))/f) - 9*sqrt(pi)*sqrt(-f)*cosh(1/4*(b^2*log(f)^2 + e^2 - 4*d*f + 2* 
(b*e - 2*a*f)*log(f))/f)*erf(1/2*(2*f*x + b*log(f) + e)*sqrt(-f)/f) + 9*sq 
rt(pi)*sqrt(f)*cosh(1/4*(b^2*log(f)^2 + e^2 - 4*d*f - 2*(b*e - 2*a*f)*log( 
f))/f)*erf(-1/2*(2*f*x - b*log(f) + e)/sqrt(f)) + 9*sqrt(pi)*sqrt(-f)*erf( 
1/2*(2*f*x + b*log(f) + e)*sqrt(-f)/f)*sinh(1/4*(b^2*log(f)^2 + e^2 - 4*d* 
f + 2*(b*e - 2*a*f)*log(f))/f) + 9*sqrt(pi)*sqrt(f)*erf(-1/2*(2*f*x - b*lo 
g(f) + e)/sqrt(f))*sinh(1/4*(b^2*log(f)^2 + e^2 - 4*d*f - 2*(b*e - 2*a*f)* 
log(f))/f))/f
 

Sympy [F]

\[ \int f^{a+b x} \sinh ^3\left (d+e x+f x^2\right ) \, dx=\int f^{a + b x} \sinh ^{3}{\left (d + e x + f x^{2} \right )}\, dx \] Input:

integrate(f**(b*x+a)*sinh(f*x**2+e*x+d)**3,x)
 

Output:

Integral(f**(a + b*x)*sinh(d + e*x + f*x**2)**3, x)
 

Maxima [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.89 \[ \int f^{a+b x} \sinh ^3\left (d+e x+f x^2\right ) \, dx=\frac {\sqrt {3} \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {3} \sqrt {-f} x - \frac {\sqrt {3} {\left (b \log \left (f\right ) + 3 \, e\right )}}{6 \, \sqrt {-f}}\right ) e^{\left (3 \, d - \frac {{\left (b \log \left (f\right ) + 3 \, e\right )}^{2}}{12 \, f}\right )}}{48 \, \sqrt {-f}} + \frac {3}{16} \, \sqrt {\pi } f^{a - \frac {1}{2}} \operatorname {erf}\left (\sqrt {f} x - \frac {b \log \left (f\right ) - e}{2 \, \sqrt {f}}\right ) e^{\left (-d + \frac {{\left (b \log \left (f\right ) - e\right )}^{2}}{4 \, f}\right )} - \frac {\sqrt {3} \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {3} \sqrt {f} x - \frac {\sqrt {3} {\left (b \log \left (f\right ) - 3 \, e\right )}}{6 \, \sqrt {f}}\right ) e^{\left (-3 \, d + \frac {{\left (b \log \left (f\right ) - 3 \, e\right )}^{2}}{12 \, f}\right )}}{48 \, \sqrt {f}} - \frac {3 \, \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-f} x - \frac {b \log \left (f\right ) + e}{2 \, \sqrt {-f}}\right ) e^{\left (d - \frac {{\left (b \log \left (f\right ) + e\right )}^{2}}{4 \, f}\right )}}{16 \, \sqrt {-f}} \] Input:

integrate(f^(b*x+a)*sinh(f*x^2+e*x+d)^3,x, algorithm="maxima")
 

Output:

1/48*sqrt(3)*sqrt(pi)*f^a*erf(sqrt(3)*sqrt(-f)*x - 1/6*sqrt(3)*(b*log(f) + 
 3*e)/sqrt(-f))*e^(3*d - 1/12*(b*log(f) + 3*e)^2/f)/sqrt(-f) + 3/16*sqrt(p 
i)*f^(a - 1/2)*erf(sqrt(f)*x - 1/2*(b*log(f) - e)/sqrt(f))*e^(-d + 1/4*(b* 
log(f) - e)^2/f) - 1/48*sqrt(3)*sqrt(pi)*f^a*erf(sqrt(3)*sqrt(f)*x - 1/6*s 
qrt(3)*(b*log(f) - 3*e)/sqrt(f))*e^(-3*d + 1/12*(b*log(f) - 3*e)^2/f)/sqrt 
(f) - 3/16*sqrt(pi)*f^a*erf(sqrt(-f)*x - 1/2*(b*log(f) + e)/sqrt(-f))*e^(d 
 - 1/4*(b*log(f) + e)^2/f)/sqrt(-f)
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.09 \[ \int f^{a+b x} \sinh ^3\left (d+e x+f x^2\right ) \, dx=-\frac {\sqrt {3} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{6} \, \sqrt {3} \sqrt {-f} {\left (6 \, x + \frac {b \log \left (f\right ) + 3 \, e}{f}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2} + 6 \, b e \log \left (f\right ) - 12 \, a f \log \left (f\right ) + 9 \, e^{2} - 36 \, d f}{12 \, f}\right )}}{48 \, \sqrt {-f}} + \frac {\sqrt {3} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{6} \, \sqrt {3} \sqrt {f} {\left (6 \, x - \frac {b \log \left (f\right ) - 3 \, e}{f}\right )}\right ) e^{\left (\frac {b^{2} \log \left (f\right )^{2} - 6 \, b e \log \left (f\right ) + 12 \, a f \log \left (f\right ) + 9 \, e^{2} - 36 \, d f}{12 \, f}\right )}}{48 \, \sqrt {f}} + \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-f} {\left (2 \, x + \frac {b \log \left (f\right ) + e}{f}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2} + 2 \, b e \log \left (f\right ) - 4 \, a f \log \left (f\right ) + e^{2} - 4 \, d f}{4 \, f}\right )}}{16 \, \sqrt {-f}} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {f} {\left (2 \, x - \frac {b \log \left (f\right ) - e}{f}\right )}\right ) e^{\left (\frac {b^{2} \log \left (f\right )^{2} - 2 \, b e \log \left (f\right ) + 4 \, a f \log \left (f\right ) + e^{2} - 4 \, d f}{4 \, f}\right )}}{16 \, \sqrt {f}} \] Input:

integrate(f^(b*x+a)*sinh(f*x^2+e*x+d)^3,x, algorithm="giac")
 

Output:

-1/48*sqrt(3)*sqrt(pi)*erf(-1/6*sqrt(3)*sqrt(-f)*(6*x + (b*log(f) + 3*e)/f 
))*e^(-1/12*(b^2*log(f)^2 + 6*b*e*log(f) - 12*a*f*log(f) + 9*e^2 - 36*d*f) 
/f)/sqrt(-f) + 1/48*sqrt(3)*sqrt(pi)*erf(-1/6*sqrt(3)*sqrt(f)*(6*x - (b*lo 
g(f) - 3*e)/f))*e^(1/12*(b^2*log(f)^2 - 6*b*e*log(f) + 12*a*f*log(f) + 9*e 
^2 - 36*d*f)/f)/sqrt(f) + 3/16*sqrt(pi)*erf(-1/2*sqrt(-f)*(2*x + (b*log(f) 
 + e)/f))*e^(-1/4*(b^2*log(f)^2 + 2*b*e*log(f) - 4*a*f*log(f) + e^2 - 4*d* 
f)/f)/sqrt(-f) - 3/16*sqrt(pi)*erf(-1/2*sqrt(f)*(2*x - (b*log(f) - e)/f))* 
e^(1/4*(b^2*log(f)^2 - 2*b*e*log(f) + 4*a*f*log(f) + e^2 - 4*d*f)/f)/sqrt( 
f)
 

Mupad [F(-1)]

Timed out. \[ \int f^{a+b x} \sinh ^3\left (d+e x+f x^2\right ) \, dx=\int f^{a+b\,x}\,{\mathrm {sinh}\left (f\,x^2+e\,x+d\right )}^3 \,d x \] Input:

int(f^(a + b*x)*sinh(d + e*x + f*x^2)^3,x)
 

Output:

int(f^(a + b*x)*sinh(d + e*x + f*x^2)^3, x)
 

Reduce [F]

\[ \int f^{a+b x} \sinh ^3\left (d+e x+f x^2\right ) \, dx=f^{a} \left (\int f^{b x} \sinh \left (f \,x^{2}+e x +d \right )^{3}d x \right ) \] Input:

int(f^(b*x+a)*sinh(f*x^2+e*x+d)^3,x)
 

Output:

f**a*int(f**(b*x)*sinh(d + e*x + f*x**2)**3,x)