3.319 \(\int f^{a+c x^2} \cosh ^3(d+e x+f x^2) \, dx\)

Optimal. Leaf size=300 \[ \frac{3 \sqrt{\pi } f^a e^{\frac{e^2}{4 f-4 c \log (f)}-d} \text{Erf}\left (\frac{2 x (f-c \log (f))+e}{2 \sqrt{f-c \log (f)}}\right )}{16 \sqrt{f-c \log (f)}}+\frac{\sqrt{\pi } f^a e^{\frac{9 e^2}{12 f-4 c \log (f)}-3 d} \text{Erf}\left (\frac{2 x (3 f-c \log (f))+3 e}{2 \sqrt{3 f-c \log (f)}}\right )}{16 \sqrt{3 f-c \log (f)}}+\frac{3 \sqrt{\pi } f^a e^{d-\frac{e^2}{4 (c \log (f)+f)}} \text{Erfi}\left (\frac{2 x (c \log (f)+f)+e}{2 \sqrt{c \log (f)+f}}\right )}{16 \sqrt{c \log (f)+f}}+\frac{\sqrt{\pi } f^a e^{3 d-\frac{9 e^2}{4 (c \log (f)+3 f)}} \text{Erfi}\left (\frac{2 x (c \log (f)+3 f)+3 e}{2 \sqrt{c \log (f)+3 f}}\right )}{16 \sqrt{c \log (f)+3 f}} \]

[Out]

(3*E^(-d + e^2/(4*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(e + 2*x*(f - c*Log[f]))/(2*Sqrt[f - c*Log[f]])])/(16*Sqrt
[f - c*Log[f]]) + (E^(-3*d + (9*e^2)/(12*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(3*e + 2*x*(3*f - c*Log[f]))/(2*Sqr
t[3*f - c*Log[f]])])/(16*Sqrt[3*f - c*Log[f]]) + (3*E^(d - e^2/(4*(f + c*Log[f])))*f^a*Sqrt[Pi]*Erfi[(e + 2*x*
(f + c*Log[f]))/(2*Sqrt[f + c*Log[f]])])/(16*Sqrt[f + c*Log[f]]) + (E^(3*d - (9*e^2)/(4*(3*f + c*Log[f])))*f^a
*Sqrt[Pi]*Erfi[(3*e + 2*x*(3*f + c*Log[f]))/(2*Sqrt[3*f + c*Log[f]])])/(16*Sqrt[3*f + c*Log[f]])

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Rubi [A]  time = 0.5812, antiderivative size = 300, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {5513, 2287, 2234, 2205, 2204} \[ \frac{3 \sqrt{\pi } f^a e^{\frac{e^2}{4 f-4 c \log (f)}-d} \text{Erf}\left (\frac{2 x (f-c \log (f))+e}{2 \sqrt{f-c \log (f)}}\right )}{16 \sqrt{f-c \log (f)}}+\frac{\sqrt{\pi } f^a e^{\frac{9 e^2}{12 f-4 c \log (f)}-3 d} \text{Erf}\left (\frac{2 x (3 f-c \log (f))+3 e}{2 \sqrt{3 f-c \log (f)}}\right )}{16 \sqrt{3 f-c \log (f)}}+\frac{3 \sqrt{\pi } f^a e^{d-\frac{e^2}{4 (c \log (f)+f)}} \text{Erfi}\left (\frac{2 x (c \log (f)+f)+e}{2 \sqrt{c \log (f)+f}}\right )}{16 \sqrt{c \log (f)+f}}+\frac{\sqrt{\pi } f^a e^{3 d-\frac{9 e^2}{4 (c \log (f)+3 f)}} \text{Erfi}\left (\frac{2 x (c \log (f)+3 f)+3 e}{2 \sqrt{c \log (f)+3 f}}\right )}{16 \sqrt{c \log (f)+3 f}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*E^(-d + e^2/(4*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(e + 2*x*(f - c*Log[f]))/(2*Sqrt[f - c*Log[f]])])/(16*Sqrt
[f - c*Log[f]]) + (E^(-3*d + (9*e^2)/(12*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(3*e + 2*x*(3*f - c*Log[f]))/(2*Sqr
t[3*f - c*Log[f]])])/(16*Sqrt[3*f - c*Log[f]]) + (3*E^(d - e^2/(4*(f + c*Log[f])))*f^a*Sqrt[Pi]*Erfi[(e + 2*x*
(f + c*Log[f]))/(2*Sqrt[f + c*Log[f]])])/(16*Sqrt[f + c*Log[f]]) + (E^(3*d - (9*e^2)/(4*(3*f + c*Log[f])))*f^a
*Sqrt[Pi]*Erfi[(3*e + 2*x*(3*f + c*Log[f]))/(2*Sqrt[3*f + c*Log[f]])])/(16*Sqrt[3*f + c*Log[f]])

Rule 5513

Int[Cosh[v_]^(n_.)*(F_)^(u_), x_Symbol] :> Int[ExpandTrigToExp[F^u, Cosh[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]

Rule 2287

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 2234

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 2205

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

Rule 2204

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]

Rubi steps

\begin{align*} \int f^{a+c x^2} \cosh ^3\left (d+e x+f x^2\right ) \, dx &=\int \left (\frac{1}{8} e^{-3 \left (d+e x+f x^2\right )} f^{a+c x^2}+\frac{3}{8} \exp \left (2 d+2 e x+2 f x^2-3 \left (d+e x+f x^2\right )\right ) f^{a+c x^2}+\frac{3}{8} \exp \left (4 d+4 e x+4 f x^2-3 \left (d+e x+f x^2\right )\right ) f^{a+c x^2}+\frac{1}{8} \exp \left (6 d+6 e x+6 f x^2-3 \left (d+e x+f x^2\right )\right ) f^{a+c x^2}\right ) \, dx\\ &=\frac{1}{8} \int e^{-3 \left (d+e x+f x^2\right )} f^{a+c x^2} \, dx+\frac{1}{8} \int \exp \left (6 d+6 e x+6 f x^2-3 \left (d+e x+f x^2\right )\right ) f^{a+c x^2} \, dx+\frac{3}{8} \int \exp \left (2 d+2 e x+2 f x^2-3 \left (d+e x+f x^2\right )\right ) f^{a+c x^2} \, dx+\frac{3}{8} \int \exp \left (4 d+4 e x+4 f x^2-3 \left (d+e x+f x^2\right )\right ) f^{a+c x^2} \, dx\\ &=\frac{1}{8} \int \exp \left (-3 d-3 e x+a \log (f)-x^2 (3 f-c \log (f))\right ) \, dx+\frac{1}{8} \int \exp \left (3 d+3 e x+a \log (f)+x^2 (3 f+c \log (f))\right ) \, dx+\frac{3}{8} \int e^{-d-e x+a \log (f)-x^2 (f-c \log (f))} \, dx+\frac{3}{8} \int e^{d+e x+a \log (f)+x^2 (f+c \log (f))} \, dx\\ &=\frac{1}{8} \left (3 e^{-d+\frac{e^2}{4 f-4 c \log (f)}} f^a\right ) \int \exp \left (\frac{(-e+2 x (-f+c \log (f)))^2}{4 (-f+c \log (f))}\right ) \, dx+\frac{1}{8} \left (e^{-3 d+\frac{9 e^2}{12 f-4 c \log (f)}} f^a\right ) \int \exp \left (\frac{(-3 e+2 x (-3 f+c \log (f)))^2}{4 (-3 f+c \log (f))}\right ) \, dx+\frac{1}{8} \left (3 e^{d-\frac{e^2}{4 (f+c \log (f))}} f^a\right ) \int \exp \left (\frac{(e+2 x (f+c \log (f)))^2}{4 (f+c \log (f))}\right ) \, dx+\frac{1}{8} \left (e^{3 d-\frac{9 e^2}{4 (3 f+c \log (f))}} f^a\right ) \int \exp \left (\frac{(3 e+2 x (3 f+c \log (f)))^2}{4 (3 f+c \log (f))}\right ) \, dx\\ &=\frac{3 e^{-d+\frac{e^2}{4 f-4 c \log (f)}} f^a \sqrt{\pi } \text{erf}\left (\frac{e+2 x (f-c \log (f))}{2 \sqrt{f-c \log (f)}}\right )}{16 \sqrt{f-c \log (f)}}+\frac{e^{-3 d+\frac{9 e^2}{12 f-4 c \log (f)}} f^a \sqrt{\pi } \text{erf}\left (\frac{3 e+2 x (3 f-c \log (f))}{2 \sqrt{3 f-c \log (f)}}\right )}{16 \sqrt{3 f-c \log (f)}}+\frac{3 e^{d-\frac{e^2}{4 (f+c \log (f))}} f^a \sqrt{\pi } \text{erfi}\left (\frac{e+2 x (f+c \log (f))}{2 \sqrt{f+c \log (f)}}\right )}{16 \sqrt{f+c \log (f)}}+\frac{e^{3 d-\frac{9 e^2}{4 (3 f+c \log (f))}} f^a \sqrt{\pi } \text{erfi}\left (\frac{3 e+2 x (3 f+c \log (f))}{2 \sqrt{3 f+c \log (f)}}\right )}{16 \sqrt{3 f+c \log (f)}}\\ \end{align*}

Mathematica [A]  time = 5.74456, size = 478, normalized size = 1.59 \[ \frac{\sqrt{\pi } f^a \exp \left (-\frac{1}{4} e^2 \left (\frac{9}{c \log (f)+3 f}+\frac{1}{c \log (f)+f}\right )\right ) \left ((f-c \log (f)) \left (\sqrt{3 f-c \log (f)} \left (c^2 \log ^2(f)+4 c f \log (f)+3 f^2\right ) (\cosh (3 d)-\sinh (3 d)) \exp \left (\frac{1}{4} e^2 \left (\frac{1}{c \log (f)+f}+\frac{9}{c \log (f)+3 f}+\frac{9}{3 f-c \log (f)}\right )\right ) \text{Erf}\left (\frac{-2 c x \log (f)+3 e+6 f x}{2 \sqrt{3 f-c \log (f)}}\right )+(3 f-c \log (f)) \left (3 \sqrt{c \log (f)+f} (c \log (f)+3 f) (\sinh (d)+\cosh (d)) e^{\frac{9 e^2}{4 (c \log (f)+3 f)}} \text{Erfi}\left (\frac{2 c x \log (f)+e+2 f x}{2 \sqrt{c \log (f)+f}}\right )+(c \log (f)+f) \sqrt{c \log (f)+3 f} (\sinh (3 d)+\cosh (3 d)) e^{\frac{e^2}{4 (c \log (f)+f)}} \text{Erfi}\left (\frac{2 c x \log (f)+3 e+6 f x}{2 \sqrt{c \log (f)+3 f}}\right )\right )\right )+3 \sqrt{f-c \log (f)} \left (-c^2 f \log ^2(f)-c^3 \log ^3(f)+9 c f^2 \log (f)+9 f^3\right ) (\cosh (d)-\sinh (d)) \exp \left (\frac{1}{4} e^2 \left (\frac{1}{c \log (f)+f}+\frac{9}{c \log (f)+3 f}+\frac{1}{f-c \log (f)}\right )\right ) \text{Erf}\left (\frac{-2 c x \log (f)+e+2 f x}{2 \sqrt{f-c \log (f)}}\right )\right )}{16 \left (-10 c^2 f^2 \log ^2(f)+c^4 \log ^4(f)+9 f^4\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(f^a*Sqrt[Pi]*(3*E^((e^2*((f - c*Log[f])^(-1) + (f + c*Log[f])^(-1) + 9/(3*f + c*Log[f])))/4)*Erf[(e + 2*f*x -
2*c*x*Log[f])/(2*Sqrt[f - c*Log[f]])]*Sqrt[f - c*Log[f]]*(9*f^3 + 9*c*f^2*Log[f] - c^2*f*Log[f]^2 - c^3*Log[f
]^3)*(Cosh[d] - Sinh[d]) + (f - c*Log[f])*(E^((e^2*(9/(3*f - c*Log[f]) + (f + c*Log[f])^(-1) + 9/(3*f + c*Log[
f])))/4)*Erf[(3*e + 6*f*x - 2*c*x*Log[f])/(2*Sqrt[3*f - c*Log[f]])]*Sqrt[3*f - c*Log[f]]*(3*f^2 + 4*c*f*Log[f]
+ c^2*Log[f]^2)*(Cosh[3*d] - Sinh[3*d]) + (3*f - c*Log[f])*(3*E^((9*e^2)/(4*(3*f + c*Log[f])))*Erfi[(e + 2*f*
x + 2*c*x*Log[f])/(2*Sqrt[f + c*Log[f]])]*Sqrt[f + c*Log[f]]*(3*f + c*Log[f])*(Cosh[d] + Sinh[d]) + E^(e^2/(4*
(f + c*Log[f])))*Erfi[(3*e + 6*f*x + 2*c*x*Log[f])/(2*Sqrt[3*f + c*Log[f]])]*(f + c*Log[f])*Sqrt[3*f + c*Log[f
]]*(Cosh[3*d] + Sinh[3*d])))))/(16*E^((e^2*((f + c*Log[f])^(-1) + 9/(3*f + c*Log[f])))/4)*(9*f^4 - 10*c^2*f^2*
Log[f]^2 + c^4*Log[f]^4))

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Maple [A]  time = 0.236, size = 302, normalized size = 1. \begin{align*}{\frac{{f}^{a}\sqrt{\pi }}{16}{{\rm e}^{-{\frac{12\,d\ln \left ( f \right ) c-36\,df+9\,{e}^{2}}{4\,c\ln \left ( f \right ) -12\,f}}}}{\it Erf} \left ( x\sqrt{3\,f-c\ln \left ( f \right ) }+{\frac{3\,e}{2}{\frac{1}{\sqrt{3\,f-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{3\,f-c\ln \left ( f \right ) }}}}-{\frac{{f}^{a}\sqrt{\pi }}{16}{{\rm e}^{{\frac{12\,d\ln \left ( f \right ) c+36\,df-9\,{e}^{2}}{4\,c\ln \left ( f \right ) +12\,f}}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) -3\,f}x+{\frac{3\,e}{2}{\frac{1}{\sqrt{-c\ln \left ( f \right ) -3\,f}}}} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) -3\,f}}}}+{\frac{3\,{f}^{a}\sqrt{\pi }}{16}{{\rm e}^{-{\frac{4\,d\ln \left ( f \right ) c-4\,df+{e}^{2}}{4\,c\ln \left ( f \right ) -4\,f}}}}{\it Erf} \left ( x\sqrt{f-c\ln \left ( f \right ) }+{\frac{e}{2}{\frac{1}{\sqrt{f-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{f-c\ln \left ( f \right ) }}}}-{\frac{3\,{f}^{a}\sqrt{\pi }}{16}{{\rm e}^{{\frac{4\,d\ln \left ( f \right ) c+4\,df-{e}^{2}}{4\,c\ln \left ( f \right ) +4\,f}}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) -f}x+{\frac{e}{2}{\frac{1}{\sqrt{-c\ln \left ( f \right ) -f}}}} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) -f}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*Pi^(1/2)*f^a*exp(-3/4*(4*d*ln(f)*c-12*d*f+3*e^2)/(-3*f+c*ln(f)))/(3*f-c*ln(f))^(1/2)*erf(x*(3*f-c*ln(f))^
(1/2)+3/2*e/(3*f-c*ln(f))^(1/2))-1/16*Pi^(1/2)*f^a*exp(3/4*(4*d*ln(f)*c+12*d*f-3*e^2)/(3*f+c*ln(f)))/(-c*ln(f)
-3*f)^(1/2)*erf(-(-c*ln(f)-3*f)^(1/2)*x+3/2*e/(-c*ln(f)-3*f)^(1/2))+3/16*Pi^(1/2)*f^a*exp(-1/4*(4*d*ln(f)*c-4*
d*f+e^2)/(-f+c*ln(f)))/(f-c*ln(f))^(1/2)*erf(x*(f-c*ln(f))^(1/2)+1/2*e/(f-c*ln(f))^(1/2))-3/16*Pi^(1/2)*f^a*ex
p(1/4*(4*d*ln(f)*c+4*d*f-e^2)/(f+c*ln(f)))/(-c*ln(f)-f)^(1/2)*erf(-(-c*ln(f)-f)^(1/2)*x+1/2*e/(-c*ln(f)-f)^(1/
2))

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Maxima [A]  time = 1.13557, size = 355, normalized size = 1.18 \begin{align*} \frac{\sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{-c \log \left (f\right ) - 3 \, f} x - \frac{3 \, e}{2 \, \sqrt{-c \log \left (f\right ) - 3 \, f}}\right ) e^{\left (3 \, d - \frac{9 \, e^{2}}{4 \,{\left (c \log \left (f\right ) + 3 \, f\right )}}\right )}}{16 \, \sqrt{-c \log \left (f\right ) - 3 \, f}} + \frac{3 \, \sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{-c \log \left (f\right ) - f} x - \frac{e}{2 \, \sqrt{-c \log \left (f\right ) - f}}\right ) e^{\left (d - \frac{e^{2}}{4 \,{\left (c \log \left (f\right ) + f\right )}}\right )}}{16 \, \sqrt{-c \log \left (f\right ) - f}} + \frac{3 \, \sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{-c \log \left (f\right ) + f} x + \frac{e}{2 \, \sqrt{-c \log \left (f\right ) + f}}\right ) e^{\left (-d - \frac{e^{2}}{4 \,{\left (c \log \left (f\right ) - f\right )}}\right )}}{16 \, \sqrt{-c \log \left (f\right ) + f}} + \frac{\sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{-c \log \left (f\right ) + 3 \, f} x + \frac{3 \, e}{2 \, \sqrt{-c \log \left (f\right ) + 3 \, f}}\right ) e^{\left (-3 \, d - \frac{9 \, e^{2}}{4 \,{\left (c \log \left (f\right ) - 3 \, f\right )}}\right )}}{16 \, \sqrt{-c \log \left (f\right ) + 3 \, f}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.15398, size = 2219, normalized size = 7.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*((sqrt(pi)*(c^3*log(f)^3 + 3*c^2*f*log(f)^2 - c*f^2*log(f) - 3*f^3)*cosh(1/4*(4*a*c*log(f)^2 - 9*e^2 + 3
6*d*f - 12*(c*d + a*f)*log(f))/(c*log(f) - 3*f)) + sqrt(pi)*(c^3*log(f)^3 + 3*c^2*f*log(f)^2 - c*f^2*log(f) -
3*f^3)*sinh(1/4*(4*a*c*log(f)^2 - 9*e^2 + 36*d*f - 12*(c*d + a*f)*log(f))/(c*log(f) - 3*f)))*sqrt(-c*log(f) +
3*f)*erf(1/2*(2*c*x*log(f) - 6*f*x - 3*e)*sqrt(-c*log(f) + 3*f)/(c*log(f) - 3*f)) + 3*(sqrt(pi)*(c^3*log(f)^3
+ c^2*f*log(f)^2 - 9*c*f^2*log(f) - 9*f^3)*cosh(1/4*(4*a*c*log(f)^2 - e^2 + 4*d*f - 4*(c*d + a*f)*log(f))/(c*l
og(f) - f)) + sqrt(pi)*(c^3*log(f)^3 + c^2*f*log(f)^2 - 9*c*f^2*log(f) - 9*f^3)*sinh(1/4*(4*a*c*log(f)^2 - e^2
+ 4*d*f - 4*(c*d + a*f)*log(f))/(c*log(f) - f)))*sqrt(-c*log(f) + f)*erf(1/2*(2*c*x*log(f) - 2*f*x - e)*sqrt(
-c*log(f) + f)/(c*log(f) - f)) + 3*(sqrt(pi)*(c^3*log(f)^3 - c^2*f*log(f)^2 - 9*c*f^2*log(f) + 9*f^3)*cosh(1/4
*(4*a*c*log(f)^2 - e^2 + 4*d*f + 4*(c*d + a*f)*log(f))/(c*log(f) + f)) + sqrt(pi)*(c^3*log(f)^3 - c^2*f*log(f)
^2 - 9*c*f^2*log(f) + 9*f^3)*sinh(1/4*(4*a*c*log(f)^2 - e^2 + 4*d*f + 4*(c*d + a*f)*log(f))/(c*log(f) + f)))*s
qrt(-c*log(f) - f)*erf(1/2*(2*c*x*log(f) + 2*f*x + e)*sqrt(-c*log(f) - f)/(c*log(f) + f)) + (sqrt(pi)*(c^3*log
(f)^3 - 3*c^2*f*log(f)^2 - c*f^2*log(f) + 3*f^3)*cosh(1/4*(4*a*c*log(f)^2 - 9*e^2 + 36*d*f + 12*(c*d + a*f)*lo
g(f))/(c*log(f) + 3*f)) + sqrt(pi)*(c^3*log(f)^3 - 3*c^2*f*log(f)^2 - c*f^2*log(f) + 3*f^3)*sinh(1/4*(4*a*c*lo
g(f)^2 - 9*e^2 + 36*d*f + 12*(c*d + a*f)*log(f))/(c*log(f) + 3*f)))*sqrt(-c*log(f) - 3*f)*erf(1/2*(2*c*x*log(f
) + 6*f*x + 3*e)*sqrt(-c*log(f) - 3*f)/(c*log(f) + 3*f)))/(c^4*log(f)^4 - 10*c^2*f^2*log(f)^2 + 9*f^4)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.38782, size = 475, normalized size = 1.58 \begin{align*} -\frac{\sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{-c \log \left (f\right ) - 3 \, f}{\left (2 \, x + \frac{3 \, e}{c \log \left (f\right ) + 3 \, f}\right )}\right ) e^{\left (\frac{4 \, a c \log \left (f\right )^{2} + 12 \, c d \log \left (f\right ) + 12 \, a f \log \left (f\right ) + 36 \, d f - 9 \, e^{2}}{4 \,{\left (c \log \left (f\right ) + 3 \, f\right )}}\right )}}{16 \, \sqrt{-c \log \left (f\right ) - 3 \, f}} - \frac{3 \, \sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{-c \log \left (f\right ) - f}{\left (2 \, x + \frac{e}{c \log \left (f\right ) + f}\right )}\right ) e^{\left (\frac{4 \, a c \log \left (f\right )^{2} + 4 \, c d \log \left (f\right ) + 4 \, a f \log \left (f\right ) + 4 \, d f - e^{2}}{4 \,{\left (c \log \left (f\right ) + f\right )}}\right )}}{16 \, \sqrt{-c \log \left (f\right ) - f}} - \frac{3 \, \sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{-c \log \left (f\right ) + f}{\left (2 \, x - \frac{e}{c \log \left (f\right ) - f}\right )}\right ) e^{\left (\frac{4 \, a c \log \left (f\right )^{2} - 4 \, c d \log \left (f\right ) - 4 \, a f \log \left (f\right ) + 4 \, d f - e^{2}}{4 \,{\left (c \log \left (f\right ) - f\right )}}\right )}}{16 \, \sqrt{-c \log \left (f\right ) + f}} - \frac{\sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{-c \log \left (f\right ) + 3 \, f}{\left (2 \, x - \frac{3 \, e}{c \log \left (f\right ) - 3 \, f}\right )}\right ) e^{\left (\frac{4 \, a c \log \left (f\right )^{2} - 12 \, c d \log \left (f\right ) - 12 \, a f \log \left (f\right ) + 36 \, d f - 9 \, e^{2}}{4 \,{\left (c \log \left (f\right ) - 3 \, f\right )}}\right )}}{16 \, \sqrt{-c \log \left (f\right ) + 3 \, f}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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