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]

3/16*exp(-d+e^2/(4*f-4*c*ln(f)))*f^a*erf(1/2*(e+2*x*(f-c*ln(f)))/(f-c*ln(f))^(1/2))*Pi^(1/2)/(f-c*ln(f))^(1/2)
+1/16*exp(-3*d+9*e^2/(12*f-4*c*ln(f)))*f^a*erf(1/2*(3*e+2*x*(3*f-c*ln(f)))/(3*f-c*ln(f))^(1/2))*Pi^(1/2)/(3*f-
c*ln(f))^(1/2)+3/16*exp(d-1/4*e^2/(f+c*ln(f)))*f^a*erfi(1/2*(e+2*x*(f+c*ln(f)))/(f+c*ln(f))^(1/2))*Pi^(1/2)/(f
+c*ln(f))^(1/2)+1/16*exp(3*d-9/4*e^2/(3*f+c*ln(f)))*f^a*erfi(1/2*(3*e+2*x*(3*f+c*ln(f)))/(3*f+c*ln(f))^(1/2))*
Pi^(1/2)/(3*f+c*ln(f))^(1/2)

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Rubi [A]  time = 0.58, antiderivative size = 300, normalized size of antiderivative = 1.00, 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 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]

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 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 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 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]

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*}

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Mathematica [A]  time = 6.00, size = 478, normalized size = 1.59 \[ \frac {\sqrt {\pi } f^a \exp \left (-\frac {1}{4} e^2 \left (\frac {9}{c \log (f)+3 f}+\frac {1}{c \log (f)+f}\right )\right ) \left ((f-c \log (f)) \left (\sqrt {3 f-c \log (f)} \left (c^2 \log ^2(f)+4 c f \log (f)+3 f^2\right ) (\cosh (3 d)-\sinh (3 d)) \exp \left (\frac {1}{4} e^2 \left (\frac {1}{c \log (f)+f}+\frac {9}{c \log (f)+3 f}+\frac {9}{3 f-c \log (f)}\right )\right ) \text {erf}\left (\frac {-2 c x \log (f)+3 e+6 f x}{2 \sqrt {3 f-c \log (f)}}\right )+(3 f-c \log (f)) \left (3 \sqrt {c \log (f)+f} (c \log (f)+3 f) (\sinh (d)+\cosh (d)) e^{\frac {9 e^2}{4 (c \log (f)+3 f)}} \text {erfi}\left (\frac {2 c x \log (f)+e+2 f x}{2 \sqrt {c \log (f)+f}}\right )+(c \log (f)+f) \sqrt {c \log (f)+3 f} (\sinh (3 d)+\cosh (3 d)) e^{\frac {e^2}{4 (c \log (f)+f)}} \text {erfi}\left (\frac {2 c x \log (f)+3 e+6 f x}{2 \sqrt {c \log (f)+3 f}}\right )\right )\right )+3 \sqrt {f-c \log (f)} \left (-c^3 \log ^3(f)-c^2 f \log ^2(f)+9 c f^2 \log (f)+9 f^3\right ) (\cosh (d)-\sinh (d)) \exp \left (\frac {1}{4} e^2 \left (\frac {1}{c \log (f)+f}+\frac {9}{c \log (f)+3 f}+\frac {1}{f-c \log (f)}\right )\right ) \text {erf}\left (\frac {-2 c x \log (f)+e+2 f x}{2 \sqrt {f-c \log (f)}}\right )\right )}{16 \left (c^4 \log ^4(f)-10 c^2 f^2 \log ^2(f)+9 f^4\right )} \]

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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fricas [B]  time = 0.53, size = 847, normalized size = 2.82 \[ \text {result too large to display} \]

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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giac [A]  time = 0.19, size = 352, normalized size = 1.17 \[ -\frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \relax (f) - 3 \, f} {\left (2 \, x + \frac {3 \, e}{c \log \relax (f) + 3 \, f}\right )}\right ) e^{\left (\frac {4 \, a c \log \relax (f)^{2} + 12 \, c d \log \relax (f) + 12 \, a f \log \relax (f) + 36 \, d f - 9 \, e^{2}}{4 \, {\left (c \log \relax (f) + 3 \, f\right )}}\right )}}{16 \, \sqrt {-c \log \relax (f) - 3 \, f}} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \relax (f) - f} {\left (2 \, x + \frac {e}{c \log \relax (f) + f}\right )}\right ) e^{\left (\frac {4 \, a c \log \relax (f)^{2} + 4 \, c d \log \relax (f) + 4 \, a f \log \relax (f) + 4 \, d f - e^{2}}{4 \, {\left (c \log \relax (f) + f\right )}}\right )}}{16 \, \sqrt {-c \log \relax (f) - f}} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \relax (f) + f} {\left (2 \, x - \frac {e}{c \log \relax (f) - f}\right )}\right ) e^{\left (\frac {4 \, a c \log \relax (f)^{2} - 4 \, c d \log \relax (f) - 4 \, a f \log \relax (f) + 4 \, d f - e^{2}}{4 \, {\left (c \log \relax (f) - f\right )}}\right )}}{16 \, \sqrt {-c \log \relax (f) + f}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \relax (f) + 3 \, f} {\left (2 \, x - \frac {3 \, e}{c \log \relax (f) - 3 \, f}\right )}\right ) e^{\left (\frac {4 \, a c \log \relax (f)^{2} - 12 \, c d \log \relax (f) - 12 \, a f \log \relax (f) + 36 \, d f - 9 \, e^{2}}{4 \, {\left (c \log \relax (f) - 3 \, f\right )}}\right )}}{16 \, \sqrt {-c \log \relax (f) + 3 \, f}} \]

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)

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maple [A]  time = 0.42, size = 302, normalized size = 1.01 \[ \frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {3 \left (4 d \ln \relax (f ) c -12 d f +3 e^{2}\right )}{4 \left (-3 f +c \ln \relax (f )\right )}} \erf \left (x \sqrt {3 f -c \ln \relax (f )}+\frac {3 e}{2 \sqrt {3 f -c \ln \relax (f )}}\right )}{16 \sqrt {3 f -c \ln \relax (f )}}-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {3 d \ln \relax (f ) c +9 d f -\frac {9 e^{2}}{4}}{3 f +c \ln \relax (f )}} \erf \left (-\sqrt {-c \ln \relax (f )-3 f}\, x +\frac {3 e}{2 \sqrt {-c \ln \relax (f )-3 f}}\right )}{16 \sqrt {-c \ln \relax (f )-3 f}}+\frac {3 \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {4 d \ln \relax (f ) c -4 d f +e^{2}}{4 \left (-f +c \ln \relax (f )\right )}} \erf \left (x \sqrt {f -c \ln \relax (f )}+\frac {e}{2 \sqrt {f -c \ln \relax (f )}}\right )}{16 \sqrt {f -c \ln \relax (f )}}-\frac {3 \sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {4 d \ln \relax (f ) c +4 d f -e^{2}}{4 c \ln \relax (f )+4 f}} \erf \left (-\sqrt {-c \ln \relax (f )-f}\, x +\frac {e}{2 \sqrt {-c \ln \relax (f )-f}}\right )}{16 \sqrt {-c \ln \relax (f )-f}} \]

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 = 0.34, size = 263, normalized size = 0.88 \[ \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f) - 3 \, f} x - \frac {3 \, e}{2 \, \sqrt {-c \log \relax (f) - 3 \, f}}\right ) e^{\left (3 \, d - \frac {9 \, e^{2}}{4 \, {\left (c \log \relax (f) + 3 \, f\right )}}\right )}}{16 \, \sqrt {-c \log \relax (f) - 3 \, f}} + \frac {3 \, \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f) - f} x - \frac {e}{2 \, \sqrt {-c \log \relax (f) - f}}\right ) e^{\left (d - \frac {e^{2}}{4 \, {\left (c \log \relax (f) + f\right )}}\right )}}{16 \, \sqrt {-c \log \relax (f) - f}} + \frac {3 \, \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f) + f} x + \frac {e}{2 \, \sqrt {-c \log \relax (f) + f}}\right ) e^{\left (-d - \frac {e^{2}}{4 \, {\left (c \log \relax (f) - f\right )}}\right )}}{16 \, \sqrt {-c \log \relax (f) + f}} + \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f) + 3 \, f} x + \frac {3 \, e}{2 \, \sqrt {-c \log \relax (f) + 3 \, f}}\right ) e^{\left (-3 \, d - \frac {9 \, e^{2}}{4 \, {\left (c \log \relax (f) - 3 \, f\right )}}\right )}}{16 \, \sqrt {-c \log \relax (f) + 3 \, f}} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int f^{c\,x^2+a}\,{\mathrm {cosh}\left (f\,x^2+e\,x+d\right )}^3 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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