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

Optimal. Leaf size=257 \[ \frac {3}{16} \sqrt {\pi } f^{a-\frac {1}{2}} e^{\frac {(e-b \log (f))^2}{4 f}-d} \text {erf}\left (\frac {-b \log (f)+e+2 f x}{2 \sqrt {f}}\right )+\frac {1}{16} \sqrt {\frac {\pi }{3}} f^{a-\frac {1}{2}} e^{\frac {(3 e-b \log (f))^2}{12 f}-3 d} \text {erf}\left (\frac {-b \log (f)+3 e+6 f x}{2 \sqrt {3} \sqrt {f}}\right )+\frac {3}{16} \sqrt {\pi } f^{a-\frac {1}{2}} e^{d-\frac {(b \log (f)+e)^2}{4 f}} \text {erfi}\left (\frac {b \log (f)+e+2 f x}{2 \sqrt {f}}\right )+\frac {1}{16} \sqrt {\frac {\pi }{3}} f^{a-\frac {1}{2}} e^{3 d-\frac {(b \log (f)+3 e)^2}{12 f}} \text {erfi}\left (\frac {b \log (f)+3 e+6 f x}{2 \sqrt {3} \sqrt {f}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.47, antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {5513, 2287, 2234, 2205, 2204} \[ \frac {3}{16} \sqrt {\pi } f^{a-\frac {1}{2}} e^{\frac {(e-b \log (f))^2}{4 f}-d} \text {Erf}\left (\frac {-b \log (f)+e+2 f x}{2 \sqrt {f}}\right )+\frac {1}{16} \sqrt {\frac {\pi }{3}} f^{a-\frac {1}{2}} e^{\frac {(3 e-b \log (f))^2}{12 f}-3 d} \text {Erf}\left (\frac {-b \log (f)+3 e+6 f x}{2 \sqrt {3} \sqrt {f}}\right )+\frac {3}{16} \sqrt {\pi } f^{a-\frac {1}{2}} e^{d-\frac {(b \log (f)+e)^2}{4 f}} \text {Erfi}\left (\frac {b \log (f)+e+2 f x}{2 \sqrt {f}}\right )+\frac {1}{16} \sqrt {\frac {\pi }{3}} f^{a-\frac {1}{2}} e^{3 d-\frac {(b \log (f)+3 e)^2}{12 f}} \text {Erfi}\left (\frac {b \log (f)+3 e+6 f x}{2 \sqrt {3} \sqrt {f}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(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

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+b x} \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+b x}+\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+b x}+\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+b x}+\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+b x}\right ) \, dx\\ &=\frac {1}{8} \int e^{-3 \left (d+e x+f x^2\right )} f^{a+b x} \, 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+b x} \, 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+b x} \, 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+b x} \, dx\\ &=\frac {1}{8} \int \exp \left (-3 d-3 f x^2+a \log (f)-x (3 e-b \log (f))\right ) \, dx+\frac {1}{8} \int \exp \left (3 d+3 f x^2+a \log (f)+x (3 e+b \log (f))\right ) \, dx+\frac {3}{8} \int e^{-d-f x^2+a \log (f)-x (e-b \log (f))} \, dx+\frac {3}{8} \int e^{d+f x^2+a \log (f)+x (e+b \log (f))} \, dx\\ &=\frac {1}{8} \left (3 e^{-d+\frac {(e-b \log (f))^2}{4 f}} f^a\right ) \int e^{-\frac {(-e-2 f x+b \log (f))^2}{4 f}} \, dx+\frac {1}{8} \left (e^{-3 d+\frac {(3 e-b \log (f))^2}{12 f}} f^a\right ) \int e^{-\frac {(-3 e-6 f x+b \log (f))^2}{12 f}} \, dx+\frac {1}{8} \left (3 e^{d-\frac {(e+b \log (f))^2}{4 f}} f^a\right ) \int e^{\frac {(e+2 f x+b \log (f))^2}{4 f}} \, dx+\frac {1}{8} \left (e^{3 d-\frac {(3 e+b \log (f))^2}{12 f}} f^a\right ) \int e^{\frac {(3 e+6 f x+b \log (f))^2}{12 f}} \, 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 )\\ \end {align*}

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Mathematica [A]  time = 0.80, size = 353, normalized size = 1.37 \[ \frac {1}{16} \sqrt {\frac {\pi }{3}} f^{a-\frac {b e+f}{2 f}} e^{-\frac {b^2 \log ^2(f)+3 e^2}{4 f}} \left (3 \sqrt {3} (\cosh (d)-\sinh (d)) e^{\frac {b^2 \log ^2(f)+2 e^2}{2 f}} \text {erf}\left (\frac {-b \log (f)+e+2 f x}{2 \sqrt {f}}\right )+(\cosh (3 d)-\sinh (3 d)) e^{\frac {2 b^2 \log ^2(f)+9 e^2}{6 f}} \text {erf}\left (\frac {-b \log (f)+3 e+6 f x}{2 \sqrt {3} \sqrt {f}}\right )+\sinh (3 d) e^{\frac {b^2 \log ^2(f)}{6 f}} \text {erfi}\left (\frac {b \log (f)+3 e+6 f x}{2 \sqrt {3} \sqrt {f}}\right )+\cosh (3 d) e^{\frac {b^2 \log ^2(f)}{6 f}} \text {erfi}\left (\frac {b \log (f)+3 e+6 f x}{2 \sqrt {3} \sqrt {f}}\right )+3 \sqrt {3} \sinh (d) e^{\frac {e^2}{2 f}} \text {erfi}\left (\frac {b \log (f)+e+2 f x}{2 \sqrt {f}}\right )+3 \sqrt {3} \cosh (d) e^{\frac {e^2}{2 f}} \text {erfi}\left (\frac {b \log (f)+e+2 f x}{2 \sqrt {f}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(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] - Sinh[d]) + 3*Sqrt[3]*E^(e^2/(2*f))*E
rfi[(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))*Erfi[(3*e + 6*f*x + b*Log[f])/(2*S
qrt[3]*Sqrt[f])]*Sinh[3*d]))/(16*E^((3*e^2 + b^2*Log[f]^2)/(4*f)))

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fricas [B]  time = 0.52, size = 539, normalized size = 2.10 \[ -\frac {\sqrt {3} \sqrt {\pi } \sqrt {-f} \cosh \left (\frac {b^{2} \log \relax (f)^{2} + 9 \, e^{2} - 36 \, d f + 6 \, {\left (b e - 2 \, a f\right )} \log \relax (f)}{12 \, f}\right ) \operatorname {erf}\left (\frac {\sqrt {3} {\left (6 \, f x + b \log \relax (f) + 3 \, e\right )} \sqrt {-f}}{6 \, f}\right ) + \sqrt {3} \sqrt {\pi } \sqrt {f} \cosh \left (\frac {b^{2} \log \relax (f)^{2} + 9 \, e^{2} - 36 \, d f - 6 \, {\left (b e - 2 \, a f\right )} \log \relax (f)}{12 \, f}\right ) \operatorname {erf}\left (-\frac {\sqrt {3} {\left (6 \, f x - b \log \relax (f) + 3 \, e\right )}}{6 \, \sqrt {f}}\right ) - \sqrt {3} \sqrt {\pi } \sqrt {-f} \operatorname {erf}\left (\frac {\sqrt {3} {\left (6 \, f x + b \log \relax (f) + 3 \, e\right )} \sqrt {-f}}{6 \, f}\right ) \sinh \left (\frac {b^{2} \log \relax (f)^{2} + 9 \, e^{2} - 36 \, d f + 6 \, {\left (b e - 2 \, a f\right )} \log \relax (f)}{12 \, f}\right ) + \sqrt {3} \sqrt {\pi } \sqrt {f} \operatorname {erf}\left (-\frac {\sqrt {3} {\left (6 \, f x - b \log \relax (f) + 3 \, e\right )}}{6 \, \sqrt {f}}\right ) \sinh \left (\frac {b^{2} \log \relax (f)^{2} + 9 \, e^{2} - 36 \, d f - 6 \, {\left (b e - 2 \, a f\right )} \log \relax (f)}{12 \, f}\right ) + 9 \, \sqrt {\pi } \sqrt {-f} \cosh \left (\frac {b^{2} \log \relax (f)^{2} + e^{2} - 4 \, d f + 2 \, {\left (b e - 2 \, a f\right )} \log \relax (f)}{4 \, f}\right ) \operatorname {erf}\left (\frac {{\left (2 \, f x + b \log \relax (f) + e\right )} \sqrt {-f}}{2 \, f}\right ) + 9 \, \sqrt {\pi } \sqrt {f} \cosh \left (\frac {b^{2} \log \relax (f)^{2} + e^{2} - 4 \, d f - 2 \, {\left (b e - 2 \, a f\right )} \log \relax (f)}{4 \, f}\right ) \operatorname {erf}\left (-\frac {2 \, f x - b \log \relax (f) + e}{2 \, \sqrt {f}}\right ) - 9 \, \sqrt {\pi } \sqrt {-f} \operatorname {erf}\left (\frac {{\left (2 \, f x + b \log \relax (f) + e\right )} \sqrt {-f}}{2 \, f}\right ) \sinh \left (\frac {b^{2} \log \relax (f)^{2} + e^{2} - 4 \, d f + 2 \, {\left (b e - 2 \, a f\right )} \log \relax (f)}{4 \, f}\right ) + 9 \, \sqrt {\pi } \sqrt {f} \operatorname {erf}\left (-\frac {2 \, f x - b \log \relax (f) + e}{2 \, \sqrt {f}}\right ) \sinh \left (\frac {b^{2} \log \relax (f)^{2} + e^{2} - 4 \, d f - 2 \, {\left (b e - 2 \, a f\right )} \log \relax (f)}{4 \, f}\right )}{48 \, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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)/sqrt(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*lo
g(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*sqrt(pi)*sqrt(f)*cosh(1/4*(b^2*l
og(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*log(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

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giac [A]  time = 0.16, size = 285, normalized size = 1.11 \[ -\frac {\sqrt {3} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{6} \, \sqrt {3} \sqrt {f} {\left (6 \, x - \frac {b \log \relax (f) - 3 \, e}{f}\right )}\right ) e^{\left (\frac {b^{2} \log \relax (f)^{2} + 12 \, a f \log \relax (f) - 6 \, b e \log \relax (f) - 36 \, d f + 9 \, e^{2}}{12 \, f}\right )}}{48 \, \sqrt {f}} - \frac {\sqrt {3} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{6} \, \sqrt {3} \sqrt {-f} {\left (6 \, x + \frac {b \log \relax (f) + 3 \, e}{f}\right )}\right ) e^{\left (-\frac {b^{2} \log \relax (f)^{2} - 12 \, a f \log \relax (f) + 6 \, b e \log \relax (f) - 36 \, d f + 9 \, e^{2}}{12 \, f}\right )}}{48 \, \sqrt {-f}} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {f} {\left (2 \, x - \frac {b \log \relax (f) - e}{f}\right )}\right ) e^{\left (\frac {b^{2} \log \relax (f)^{2} + 4 \, a f \log \relax (f) - 2 \, b e \log \relax (f) - 4 \, d f + e^{2}}{4 \, f}\right )}}{16 \, \sqrt {f}} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-f} {\left (2 \, x + \frac {b \log \relax (f) + e}{f}\right )}\right ) e^{\left (-\frac {b^{2} \log \relax (f)^{2} - 4 \, a f \log \relax (f) + 2 \, b e \log \relax (f) - 4 \, d f + e^{2}}{4 \, f}\right )}}{16 \, \sqrt {-f}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.43, size = 265, normalized size = 1.03 \[ -\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {\ln \relax (f )^{2} b^{2}-6 \ln \relax (f ) b e -36 d f +9 e^{2}}{12 f}} \sqrt {3}\, \erf \left (-\sqrt {3}\, \sqrt {f}\, x +\frac {\left (b \ln \relax (f )-3 e \right ) \sqrt {3}}{6 \sqrt {f}}\right )}{48 \sqrt {f}}-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {\ln \relax (f )^{2} b^{2}+6 \ln \relax (f ) b e -36 d f +9 e^{2}}{12 f}} \erf \left (-\sqrt {-3 f}\, x +\frac {3 e +b \ln \relax (f )}{2 \sqrt {-3 f}}\right )}{16 \sqrt {-3 f}}-\frac {3 \sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {\ln \relax (f )^{2} b^{2}-2 \ln \relax (f ) b e -4 d f +e^{2}}{4 f}} \erf \left (-\sqrt {f}\, x +\frac {b \ln \relax (f )-e}{2 \sqrt {f}}\right )}{16 \sqrt {f}}-\frac {3 \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {\ln \relax (f )^{2} b^{2}+2 \ln \relax (f ) b e -4 d f +e^{2}}{4 f}} \erf \left (-\sqrt {-f}\, x +\frac {e +b \ln \relax (f )}{2 \sqrt {-f}}\right )}{16 \sqrt {-f}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.44, size = 228, normalized size = 0.89 \[ \frac {\sqrt {3} \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {3} \sqrt {-f} x - \frac {\sqrt {3} {\left (b \log \relax (f) + 3 \, e\right )}}{6 \, \sqrt {-f}}\right ) e^{\left (3 \, d - \frac {{\left (b \log \relax (f) + 3 \, e\right )}^{2}}{12 \, f}\right )}}{48 \, \sqrt {-f}} + \frac {3}{16} \, \sqrt {\pi } f^{a - \frac {1}{2}} \operatorname {erf}\left (\sqrt {f} x - \frac {b \log \relax (f) - e}{2 \, \sqrt {f}}\right ) e^{\left (-d + \frac {{\left (b \log \relax (f) - e\right )}^{2}}{4 \, f}\right )} + \frac {\sqrt {3} \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {3} \sqrt {f} x - \frac {\sqrt {3} {\left (b \log \relax (f) - 3 \, e\right )}}{6 \, \sqrt {f}}\right ) e^{\left (-3 \, d + \frac {{\left (b \log \relax (f) - 3 \, e\right )}^{2}}{12 \, f}\right )}}{48 \, \sqrt {f}} + \frac {3 \, \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-f} x - \frac {b \log \relax (f) + e}{2 \, \sqrt {-f}}\right ) e^{\left (d - \frac {{\left (b \log \relax (f) + e\right )}^{2}}{4 \, f}\right )}}{16 \, \sqrt {-f}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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(pi)*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*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(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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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