\(\int f^{a+c x^2} \sinh ^3(d+e x) \, dx\) [350]

Optimal result

Integrand size = 18, antiderivative size = 271 \[ \int f^{a+c x^2} \sinh ^3(d+e x) \, dx=-\frac {3 e^{-d-\frac {e^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {e-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{-3 d-\frac {9 e^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {3 e-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}-\frac {3 e^{d-\frac {e^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {e+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{3 d-\frac {9 e^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {3 e+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}} \]

[Out]

3/16*exp(-d-1/4*e^2/c/ln(f))*f^a*erfi(1/2*(-e+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-9/4*e^2/c/ln(f))*f^a*erfi(1/2*(-3*e+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^2/c/ln(f))*f^a*erfi(1/2*(e+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-9/4*e^2/c/ln(f))*f^a*erfi(1/2*(3*e+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 = 0.25 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5623, 2325, 2266, 2235} \[ \int f^{a+c x^2} \sinh ^3(d+e x) \, dx=-\frac {3 \sqrt {\pi } f^a e^{-\frac {e^2}{4 c \log (f)}-d} \text {erfi}\left (\frac {e-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {\sqrt {\pi } f^a e^{-\frac {9 e^2}{4 c \log (f)}-3 d} \text {erfi}\left (\frac {3 e-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}-\frac {3 \sqrt {\pi } f^a e^{d-\frac {e^2}{4 c \log (f)}} \text {erfi}\left (\frac {2 c x \log (f)+e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {\sqrt {\pi } f^a e^{3 d-\frac {9 e^2}{4 c \log (f)}} \text {erfi}\left (\frac {2 c x \log (f)+3 e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}} \]

[In]

Int[f^(a + c*x^2)*Sinh[d + e*x]^3,x]

[Out]

(-3*E^(-d - e^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(e - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c]*Sqrt
[Log[f]]) + (E^(-3*d - (9*e^2)/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(3*e - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])
/(16*Sqrt[c]*Sqrt[Log[f]]) - (3*E^(d - e^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(e + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[
Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]]) + (E^(3*d - (9*e^2)/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(3*e + 2*c*x*Log[f])/
(2*Sqrt[c]*Sqrt[Log[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+c x^2}+\frac {3}{8} e^{-d-e x} f^{a+c x^2}-\frac {3}{8} e^{d+e x} f^{a+c x^2}+\frac {1}{8} e^{3 d+3 e x} f^{a+c x^2}\right ) \, dx \\ & = -\left (\frac {1}{8} \int e^{-3 d-3 e x} f^{a+c x^2} \, dx\right )+\frac {1}{8} \int e^{3 d+3 e x} f^{a+c x^2} \, dx+\frac {3}{8} \int e^{-d-e x} f^{a+c x^2} \, dx-\frac {3}{8} \int e^{d+e x} f^{a+c x^2} \, dx \\ & = -\left (\frac {1}{8} \int e^{-3 d-3 e x+a \log (f)+c x^2 \log (f)} \, dx\right )+\frac {1}{8} \int e^{3 d+3 e x+a \log (f)+c x^2 \log (f)} \, dx+\frac {3}{8} \int e^{-d-e x+a \log (f)+c x^2 \log (f)} \, dx-\frac {3}{8} \int e^{d+e x+a \log (f)+c x^2 \log (f)} \, dx \\ & = -\left (\frac {1}{8} \left (e^{-3 d-\frac {9 e^2}{4 c \log (f)}} f^a\right ) \int e^{\frac {(-3 e+2 c x \log (f))^2}{4 c \log (f)}} \, dx\right )+\frac {1}{8} \left (e^{3 d-\frac {9 e^2}{4 c \log (f)}} f^a\right ) \int e^{\frac {(3 e+2 c x \log (f))^2}{4 c \log (f)}} \, dx+\frac {1}{8} \left (3 e^{-d-\frac {e^2}{4 c \log (f)}} f^a\right ) \int e^{\frac {(-e+2 c x \log (f))^2}{4 c \log (f)}} \, dx-\frac {1}{8} \left (3 e^{d-\frac {e^2}{4 c \log (f)}} f^a\right ) \int e^{\frac {(e+2 c x \log (f))^2}{4 c \log (f)}} \, dx \\ & = -\frac {3 e^{-d-\frac {e^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {e-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{-3 d-\frac {9 e^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {3 e-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}-\frac {3 e^{d-\frac {e^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {e+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{3 d-\frac {9 e^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {3 e+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.79 \[ \int f^{a+c x^2} \sinh ^3(d+e x) \, dx=\frac {e^{-\frac {9 e^2}{4 c \log (f)}} f^a \sqrt {\pi } \left ((\cosh (d)+\sinh (d)) \left (-3 e^{\frac {2 e^2}{c \log (f)}} \text {erfi}\left (\frac {e+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )+3 e^{\frac {2 e^2}{c \log (f)}} \text {erfi}\left (\frac {-e+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right ) (\cosh (2 d)-\sinh (2 d))+\text {erfi}\left (\frac {3 e+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right ) (\cosh (2 d)+\sinh (2 d))\right )+\text {erfi}\left (\frac {-3 e+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right ) (-\cosh (3 d)+\sinh (3 d))\right )}{16 \sqrt {c} \sqrt {\log (f)}} \]

[In]

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

[Out]

(f^a*Sqrt[Pi]*((Cosh[d] + Sinh[d])*(-3*E^((2*e^2)/(c*Log[f]))*Erfi[(e + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])
] + 3*E^((2*e^2)/(c*Log[f]))*Erfi[(-e + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])]*(Cosh[2*d] - Sinh[2*d]) + Erfi
[(3*e + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])]*(Cosh[2*d] + Sinh[2*d])) + Erfi[(-3*e + 2*c*x*Log[f])/(2*Sqrt[
c]*Sqrt[Log[f]])]*(-Cosh[3*d] + Sinh[3*d])))/(16*Sqrt[c]*E^((9*e^2)/(4*c*Log[f]))*Sqrt[Log[f]])

Maple [A] (verified)

Time = 0.98 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.86

method	result	size
risch	\(-\frac {\operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {3 e}{2 \sqrt {-c \ln \left (f \right )}}\right ) \sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {3 d \ln \left (f \right ) c -\frac {9 e^{2}}{4}}{c \ln \left (f \right )}}}{16 \sqrt {-c \ln \left (f \right )}}-\frac {\operatorname {erf}\left (\sqrt {-c \ln \left (f \right )}\, x +\frac {3 e}{2 \sqrt {-c \ln \left (f \right )}}\right ) \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {3 \left (4 d \ln \left (f \right ) c +3 e^{2}\right )}{4 \ln \left (f \right ) c}}}{16 \sqrt {-c \ln \left (f \right )}}+\frac {3 \,\operatorname {erf}\left (\sqrt {-c \ln \left (f \right )}\, x +\frac {e}{2 \sqrt {-c \ln \left (f \right )}}\right ) \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {4 d \ln \left (f \right ) c +e^{2}}{4 \ln \left (f \right ) c}}}{16 \sqrt {-c \ln \left (f \right )}}+\frac {3 \,\operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {e}{2 \sqrt {-c \ln \left (f \right )}}\right ) \sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {4 d \ln \left (f \right ) c -e^{2}}{4 \ln \left (f \right ) c}}}{16 \sqrt {-c \ln \left (f \right )}}\)	\(234\)

[In]

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

[Out]

-1/16*erf(-(-c*ln(f))^(1/2)*x+3/2*e/(-c*ln(f))^(1/2))/(-c*ln(f))^(1/2)*Pi^(1/2)*f^a*exp(3/4*(4*d*ln(f)*c-3*e^2
)/ln(f)/c)-1/16*erf((-c*ln(f))^(1/2)*x+3/2*e/(-c*ln(f))^(1/2))/(-c*ln(f))^(1/2)*Pi^(1/2)*f^a*exp(-3/4*(4*d*ln(
f)*c+3*e^2)/ln(f)/c)+3/16*erf((-c*ln(f))^(1/2)*x+1/2*e/(-c*ln(f))^(1/2))/(-c*ln(f))^(1/2)*Pi^(1/2)*f^a*exp(-1/
4*(4*d*ln(f)*c+e^2)/ln(f)/c)+3/16*erf(-(-c*ln(f))^(1/2)*x+1/2*e/(-c*ln(f))^(1/2))/(-c*ln(f))^(1/2)*Pi^(1/2)*f^
a*exp(1/4*(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. 427 vs. \(2 (205) = 410\).

Time = 0.31 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.58 \[ \int f^{a+c x^2} \sinh ^3(d+e x) \, dx=-\frac {\sqrt {-c \log \left (f\right )} {\left (\sqrt {\pi } \cosh \left (\frac {4 \, a c \log \left (f\right )^{2} + 12 \, c d \log \left (f\right ) - 9 \, e^{2}}{4 \, c \log \left (f\right )}\right ) + \sqrt {\pi } \sinh \left (\frac {4 \, a c \log \left (f\right )^{2} + 12 \, c d \log \left (f\right ) - 9 \, e^{2}}{4 \, c \log \left (f\right )}\right )\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x \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 {4 \, a c \log \left (f\right )^{2} + 4 \, c d \log \left (f\right ) - e^{2}}{4 \, c \log \left (f\right )}\right ) + \sqrt {\pi } \sinh \left (\frac {4 \, a c \log \left (f\right )^{2} + 4 \, c d \log \left (f\right ) - e^{2}}{4 \, c \log \left (f\right )}\right )\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x \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 {4 \, a c \log \left (f\right )^{2} - 4 \, c d \log \left (f\right ) - e^{2}}{4 \, c \log \left (f\right )}\right ) + \sqrt {\pi } \sinh \left (\frac {4 \, a c \log \left (f\right )^{2} - 4 \, c d \log \left (f\right ) - e^{2}}{4 \, c \log \left (f\right )}\right )\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x \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 {4 \, a c \log \left (f\right )^{2} - 12 \, c d \log \left (f\right ) - 9 \, e^{2}}{4 \, c \log \left (f\right )}\right ) + \sqrt {\pi } \sinh \left (\frac {4 \, a c \log \left (f\right )^{2} - 12 \, c d \log \left (f\right ) - 9 \, e^{2}}{4 \, c \log \left (f\right )}\right )\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x \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+a)*sinh(e*x+d)^3,x, algorithm="fricas")

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.78 \[ \int f^{a+c x^2} \sinh ^3(d+e x) \, dx=\frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x - \frac {3 \, e}{2 \, \sqrt {-c \log \left (f\right )}}\right ) e^{\left (3 \, d - \frac {9 \, e^{2}}{4 \, c \log \left (f\right )}\right )}}{16 \, \sqrt {-c \log \left (f\right )}} - \frac {3 \, \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x - \frac {e}{2 \, \sqrt {-c \log \left (f\right )}}\right ) e^{\left (d - \frac {e^{2}}{4 \, c \log \left (f\right )}\right )}}{16 \, \sqrt {-c \log \left (f\right )}} + \frac {3 \, \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x + \frac {e}{2 \, \sqrt {-c \log \left (f\right )}}\right ) e^{\left (-d - \frac {e^{2}}{4 \, c \log \left (f\right )}\right )}}{16 \, \sqrt {-c \log \left (f\right )}} - \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x + \frac {3 \, e}{2 \, \sqrt {-c \log \left (f\right )}}\right ) e^{\left (-3 \, d - \frac {9 \, e^{2}}{4 \, c \log \left (f\right )}\right )}}{16 \, \sqrt {-c \log \left (f\right )}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.97 \[ \int f^{a+c x^2} \sinh ^3(d+e x) \, dx=-\frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right )} {\left (2 \, x + \frac {3 \, e}{c \log \left (f\right )}\right )}\right ) e^{\left (\frac {4 \, a c \log \left (f\right )^{2} + 12 \, c d \log \left (f\right ) - 9 \, e^{2}}{4 \, c \log \left (f\right )}\right )}}{16 \, \sqrt {-c \log \left (f\right )}} + \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right )} {\left (2 \, x + \frac {e}{c \log \left (f\right )}\right )}\right ) e^{\left (\frac {4 \, a c \log \left (f\right )^{2} + 4 \, c d \log \left (f\right ) - e^{2}}{4 \, c \log \left (f\right )}\right )}}{16 \, \sqrt {-c \log \left (f\right )}} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right )} {\left (2 \, x - \frac {e}{c \log \left (f\right )}\right )}\right ) e^{\left (\frac {4 \, a c \log \left (f\right )^{2} - 4 \, c d \log \left (f\right ) - e^{2}}{4 \, c \log \left (f\right )}\right )}}{16 \, \sqrt {-c \log \left (f\right )}} + \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right )} {\left (2 \, x - \frac {3 \, e}{c \log \left (f\right )}\right )}\right ) e^{\left (\frac {4 \, a c \log \left (f\right )^{2} - 12 \, c d \log \left (f\right ) - 9 \, e^{2}}{4 \, c \log \left (f\right )}\right )}}{16 \, \sqrt {-c \log \left (f\right )}} \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(f^(a + c*x^2)*sinh(d + e*x)^3,x)

[Out]

int(f^(a + c*x^2)*sinh(d + e*x)^3, x)