\(\int f^{a+b x+c x^2} \sinh ^3(d+e x) \, dx\) [359]

Optimal result

Integrand size = 21, antiderivative size = 315 \[ \int f^{a+b x+c x^2} \sinh ^3(d+e x) \, dx=-\frac {3 e^{-d-\frac {(e-b \log (f))^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {e-b \log (f)-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{-3 d-\frac {(3 e-b \log (f))^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {3 e-b \log (f)-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}-\frac {3 e^{d-\frac {(e+b \log (f))^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {e+b \log (f)+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{3 d-\frac {(3 e+b \log (f))^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {3 e+b \log (f)+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {5623, 2325, 2266, 2235} \[ \int f^{a+b x+c x^2} \sinh ^3(d+e x) \, dx=-\frac {3 \sqrt {\pi } f^a e^{-\frac {(e-b \log (f))^2}{4 c \log (f)}-d} \text {erfi}\left (\frac {-b \log (f)-2 c x \log (f)+e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {\sqrt {\pi } f^a e^{-\frac {(3 e-b \log (f))^2}{4 c \log (f)}-3 d} \text {erfi}\left (\frac {-b \log (f)-2 c x \log (f)+3 e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}-\frac {3 \sqrt {\pi } f^a e^{d-\frac {(b \log (f)+e)^2}{4 c \log (f)}} \text {erfi}\left (\frac {b \log (f)+2 c x \log (f)+e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {\sqrt {\pi } f^a e^{3 d-\frac {(b \log (f)+3 e)^2}{4 c \log (f)}} \text {erfi}\left (\frac {b \log (f)+2 c x \log (f)+3 e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}} \]

[In]

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

[Out]

(-3*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]])])/(16*Sqrt[c]*Sqrt[Log[f]]) + (E^(-3*d - (3*e - b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(3*e - b*Log
[f] - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]]) - (3*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]])])/(16*Sqrt[c]*Sqrt[Log[f]]) + (
E^(3*d - (3*e + b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(3*e + b*Log[f] + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Lo
g[f]])])/(16*Sqrt[c]*Sqrt[Log[f]])

Rule 2235

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2
]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2266

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[F^(a - b^2/(4*c)), Int[F^((b + 2*c*x)^2/(4*c))
, x], x] /; FreeQ[{F, a, b, c}, x]

Rule 2325

Int[(u_.)*(F_)^(v_)*(G_)^(w_), x_Symbol] :> With[{z = v*Log[F] + w*Log[G]}, Int[u*NormalizeIntegrand[E^z, x],
x] /; BinomialQ[z, x] || (PolynomialQ[z, x] && LeQ[Exponent[z, x], 2])] /; FreeQ[{F, G}, x]

Rule 5623

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

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {1}{8} e^{-3 d-3 e x} f^{a+b x+c x^2}+\frac {3}{8} e^{-d-e x} f^{a+b x+c x^2}-\frac {3}{8} e^{d+e x} f^{a+b x+c x^2}+\frac {1}{8} e^{3 d+3 e x} f^{a+b x+c x^2}\right ) \, dx \\ & = -\left (\frac {1}{8} \int e^{-3 d-3 e x} f^{a+b x+c x^2} \, dx\right )+\frac {1}{8} \int e^{3 d+3 e x} f^{a+b x+c x^2} \, dx+\frac {3}{8} \int e^{-d-e x} f^{a+b x+c x^2} \, dx-\frac {3}{8} \int e^{d+e x} f^{a+b x+c x^2} \, dx \\ & = -\left (\frac {1}{8} \int \exp \left (-3 d+a \log (f)+c x^2 \log (f)-x (3 e-b \log (f))\right ) \, dx\right )+\frac {1}{8} \int \exp \left (3 d+a \log (f)+c x^2 \log (f)+x (3 e+b \log (f))\right ) \, dx+\frac {3}{8} \int \exp \left (-d+a \log (f)+c x^2 \log (f)-x (e-b \log (f))\right ) \, dx-\frac {3}{8} \int \exp \left (d+a \log (f)+c x^2 \log (f)+x (e+b \log (f))\right ) \, dx \\ & = \frac {1}{8} \left (3 e^{-d-\frac {(e-b \log (f))^2}{4 c \log (f)}} f^a\right ) \int \exp \left (\frac {(-e+b \log (f)+2 c x \log (f))^2}{4 c \log (f)}\right ) \, dx-\frac {1}{8} \left (e^{-3 d-\frac {(3 e-b \log (f))^2}{4 c \log (f)}} f^a\right ) \int \exp \left (\frac {(-3 e+b \log (f)+2 c x \log (f))^2}{4 c \log (f)}\right ) \, dx-\frac {1}{8} \left (3 e^{d-\frac {(e+b \log (f))^2}{4 c \log (f)}} f^a\right ) \int e^{\frac {(e+b \log (f)+2 c x \log (f))^2}{4 c \log (f)}} \, dx+\frac {1}{8} \left (e^{3 d-\frac {(3 e+b \log (f))^2}{4 c \log (f)}} f^a\right ) \int \exp \left (\frac {(3 e+b \log (f)+2 c x \log (f))^2}{4 c \log (f)}\right ) \, dx \\ & = -\frac {3 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 )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{-3 d-\frac {(3 e-b \log (f))^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {3 e-b \log (f)-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}-\frac {3 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 )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{3 d-\frac {(3 e+b \log (f))^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {3 e+b \log (f)+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.64 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.83 \[ \int f^{a+b x+c x^2} \sinh ^3(d+e x) \, dx=\frac {e^{-\frac {3 e (3 e+2 b \log (f))}{4 c \log (f)}} f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \left ((\cosh (d)+\sinh (d)) \left (-3 e^{\frac {e (2 e+b \log (f))}{c \log (f)}} \text {erfi}\left (\frac {e+(b+2 c x) \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )+3 e^{\frac {2 e (e+b \log (f))}{c \log (f)}} \text {erfi}\left (\frac {-e+(b+2 c x) \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right ) (\cosh (2 d)-\sinh (2 d))+\text {erfi}\left (\frac {3 e+(b+2 c x) \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right ) (\cosh (2 d)+\sinh (2 d))\right )-e^{\frac {3 b e}{c}} \text {erfi}\left (\frac {-3 e+(b+2 c x) \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right ) (\cosh (3 d)-\sinh (3 d))\right )}{16 \sqrt {c} \sqrt {\log (f)}} \]

[In]

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

[Out]

(f^(a - b^2/(4*c))*Sqrt[Pi]*((Cosh[d] + Sinh[d])*(-3*E^((e*(2*e + b*Log[f]))/(c*Log[f]))*Erfi[(e + (b + 2*c*x)
*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])] + 3*E^((2*e*(e + b*Log[f]))/(c*Log[f]))*Erfi[(-e + (b + 2*c*x)*Log[f])/(2*S
qrt[c]*Sqrt[Log[f]])]*(Cosh[2*d] - Sinh[2*d]) + Erfi[(3*e + (b + 2*c*x)*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])]*(Cos
h[2*d] + Sinh[2*d])) - E^((3*b*e)/c)*Erfi[(-3*e + (b + 2*c*x)*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])]*(Cosh[3*d] - S
inh[3*d])))/(16*Sqrt[c]*E^((3*e*(3*e + 2*b*Log[f]))/(4*c*Log[f]))*Sqrt[Log[f]])

Maple [A] (verified)

Time = 1.00 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.03

method	result	size
risch	\(-\frac {\operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {3 e +b \ln \left (f \right )}{2 \sqrt {-c \ln \left (f \right )}}\right ) \sqrt {\pi }\, f^{a} f^{-\frac {b^{2}}{4 c}} {\mathrm e}^{-\frac {3 \left (2 \ln \left (f \right ) b e -4 d \ln \left (f \right ) c +3 e^{2}\right )}{4 \ln \left (f \right ) c}}}{16 \sqrt {-c \ln \left (f \right )}}+\frac {\operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {b \ln \left (f \right )-3 e}{2 \sqrt {-c \ln \left (f \right )}}\right ) \sqrt {\pi }\, f^{a} f^{-\frac {b^{2}}{4 c}} {\mathrm e}^{\frac {\frac {3 \ln \left (f \right ) b e}{2}-3 d \ln \left (f \right ) c -\frac {9 e^{2}}{4}}{c \ln \left (f \right )}}}{16 \sqrt {-c \ln \left (f \right )}}-\frac {3 \,\operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {b \ln \left (f \right )-e}{2 \sqrt {-c \ln \left (f \right )}}\right ) \sqrt {\pi }\, f^{a} f^{-\frac {b^{2}}{4 c}} {\mathrm e}^{\frac {2 \ln \left (f \right ) b e -4 d \ln \left (f \right ) c -e^{2}}{4 \ln \left (f \right ) c}}}{16 \sqrt {-c \ln \left (f \right )}}+\frac {3 \,\operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {e +b \ln \left (f \right )}{2 \sqrt {-c \ln \left (f \right )}}\right ) \sqrt {\pi }\, f^{a} f^{-\frac {b^{2}}{4 c}} {\mathrm e}^{-\frac {2 \ln \left (f \right ) b e -4 d \ln \left (f \right ) c +e^{2}}{4 \ln \left (f \right ) c}}}{16 \sqrt {-c \ln \left (f \right )}}\)	\(326\)

[In]

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

[Out]

-1/16*erf(-(-c*ln(f))^(1/2)*x+1/2*(3*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(-3/4*(2*ln(f)*b*e-4*d*ln(f)*c+3*e^2)/ln(f)/c)+1/16*erf(-(-c*ln(f))^(1/2)*x+1/2*(b*ln(f)-3*e)/(-c*ln(f))^(
1/2))/(-c*ln(f))^(1/2)*Pi^(1/2)*f^a*f^(-1/4*b^2/c)*exp(3/4*(2*ln(f)*b*e-4*d*ln(f)*c-3*e^2)/ln(f)/c)-3/16*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*l
n(f)*b*e-4*d*ln(f)*c-e^2)/ln(f)/c)+3/16*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)/ln(f)/c)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 527 vs. \(2 (247) = 494\).

Time = 0.34 (sec) , antiderivative size = 527, normalized size of antiderivative = 1.67 \[ \int f^{a+b x+c x^2} \sinh ^3(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} + 9 \, e^{2} - 6 \, {\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} + 9 \, e^{2} - 6 \, {\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 ) + 3 \, e\right )} \sqrt {-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) - 3 \, \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 ) + 3 \, \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} + 9 \, e^{2} + 6 \, {\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} + 9 \, e^{2} + 6 \, {\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 ) - 3 \, e\right )} \sqrt {-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right )}{16 \, c \log \left (f\right )} \]

[In]

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

[Out]

-1/16*(sqrt(-c*log(f))*(sqrt(pi)*cosh(-1/4*((b^2 - 4*a*c)*log(f)^2 + 9*e^2 - 6*(2*c*d - b*e)*log(f))/(c*log(f)
)) + sqrt(pi)*sinh(-1/4*((b^2 - 4*a*c)*log(f)^2 + 9*e^2 - 6*(2*c*d - b*e)*log(f))/(c*log(f))))*erf(1/2*((2*c*x
+ b)*log(f) + 3*e)*sqrt(-c*log(f))/(c*log(f))) - 3*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)*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))) + 3*sqrt(-c*log(f))*(sq
rt(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)*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))) - sqrt(-c*log(f))*(sqrt(pi)*cosh(-1/4*((b^2 - 4*a*c)*log(f)^2 + 9*e^2 + 6*(2*c*d - b*e)*log(f)
)/(c*log(f))) + sqrt(pi)*sinh(-1/4*((b^2 - 4*a*c)*log(f)^2 + 9*e^2 + 6*(2*c*d - b*e)*log(f))/(c*log(f))))*erf(
1/2*((2*c*x + b)*log(f) - 3*e)*sqrt(-c*log(f))/(c*log(f))))/(c*log(f))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.83 \[ \int f^{a+b x+c x^2} \sinh ^3(d+e x) \, dx=\frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x - \frac {b \log \left (f\right ) + 3 \, e}{2 \, \sqrt {-c \log \left (f\right )}}\right ) e^{\left (3 \, d - \frac {{\left (b \log \left (f\right ) + 3 \, e\right )}^{2}}{4 \, c \log \left (f\right )}\right )}}{16 \, \sqrt {-c \log \left (f\right )}} - \frac {3 \, \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x - \frac {b \log \left (f\right ) + e}{2 \, \sqrt {-c \log \left (f\right )}}\right ) e^{\left (d - \frac {{\left (b \log \left (f\right ) + e\right )}^{2}}{4 \, c \log \left (f\right )}\right )}}{16 \, \sqrt {-c \log \left (f\right )}} + \frac {3 \, \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x - \frac {b \log \left (f\right ) - e}{2 \, \sqrt {-c \log \left (f\right )}}\right ) e^{\left (-d - \frac {{\left (b \log \left (f\right ) - e\right )}^{2}}{4 \, c \log \left (f\right )}\right )}}{16 \, \sqrt {-c \log \left (f\right )}} - \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x - \frac {b \log \left (f\right ) - 3 \, e}{2 \, \sqrt {-c \log \left (f\right )}}\right ) e^{\left (-3 \, d - \frac {{\left (b \log \left (f\right ) - 3 \, e\right )}^{2}}{4 \, c \log \left (f\right )}\right )}}{16 \, \sqrt {-c \log \left (f\right )}} \]

[In]

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

[Out]

1/16*sqrt(pi)*f^a*erf(sqrt(-c*log(f))*x - 1/2*(b*log(f) + 3*e)/sqrt(-c*log(f)))*e^(3*d - 1/4*(b*log(f) + 3*e)^
2/(c*log(f)))/sqrt(-c*log(f)) - 3/16*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)) + 3/16*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)) - 1/16*sqrt(pi)*f^a*erf(sqr
t(-c*log(f))*x - 1/2*(b*log(f) - 3*e)/sqrt(-c*log(f)))*e^(-3*d - 1/4*(b*log(f) - 3*e)^2/(c*log(f)))/sqrt(-c*lo
g(f))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.08 \[ \int f^{a+b x+c x^2} \sinh ^3(d+e x) \, dx=\frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right )} {\left (2 \, x + \frac {b \log \left (f\right ) - 3 \, e}{c \log \left (f\right )}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2} - 4 \, a c \log \left (f\right )^{2} + 12 \, c d \log \left (f\right ) - 6 \, b e \log \left (f\right ) + 9 \, e^{2}}{4 \, c \log \left (f\right )}\right )}}{16 \, \sqrt {-c \log \left (f\right )}} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right )} {\left (2 \, x + \frac {b \log \left (f\right ) - e}{c \log \left (f\right )}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2} - 4 \, a c \log \left (f\right )^{2} + 4 \, c d \log \left (f\right ) - 2 \, b e \log \left (f\right ) + e^{2}}{4 \, c \log \left (f\right )}\right )}}{16 \, \sqrt {-c \log \left (f\right )}} + \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right )} {\left (2 \, x + \frac {b \log \left (f\right ) + e}{c \log \left (f\right )}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2} - 4 \, a c \log \left (f\right )^{2} - 4 \, c d \log \left (f\right ) + 2 \, b e \log \left (f\right ) + e^{2}}{4 \, c \log \left (f\right )}\right )}}{16 \, \sqrt {-c \log \left (f\right )}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right )} {\left (2 \, x + \frac {b \log \left (f\right ) + 3 \, e}{c \log \left (f\right )}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2} - 4 \, a c \log \left (f\right )^{2} - 12 \, c d \log \left (f\right ) + 6 \, b e \log \left (f\right ) + 9 \, e^{2}}{4 \, c \log \left (f\right )}\right )}}{16 \, \sqrt {-c \log \left (f\right )}} \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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