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

Optimal. Leaf size=239 \[ \frac {3}{16} \sqrt {\pi } f^{a-\frac {1}{2}} e^{\frac {b^2 \log ^2(f)}{4 f}-d} \text {erf}\left (\frac {2 f x-b \log (f)}{2 \sqrt {f}}\right )+\frac {1}{16} \sqrt {\frac {\pi }{3}} f^{a-\frac {1}{2}} e^{\frac {b^2 \log ^2(f)}{12 f}-3 d} \text {erf}\left (\frac {6 f x-b \log (f)}{2 \sqrt {3} \sqrt {f}}\right )+\frac {3}{16} \sqrt {\pi } f^{a-\frac {1}{2}} e^{d-\frac {b^2 \log ^2(f)}{4 f}} \text {erfi}\left (\frac {b \log (f)+2 f x}{2 \sqrt {f}}\right )+\frac {1}{16} \sqrt {\frac {\pi }{3}} f^{a-\frac {1}{2}} e^{3 d-\frac {b^2 \log ^2(f)}{12 f}} \text {erfi}\left (\frac {b \log (f)+6 f x}{2 \sqrt {3} \sqrt {f}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.29, antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {5513, 2287, 2234, 2205, 2204} \[ \frac {3}{16} \sqrt {\pi } f^{a-\frac {1}{2}} e^{\frac {b^2 \log ^2(f)}{4 f}-d} \text {Erf}\left (\frac {2 f x-b \log (f)}{2 \sqrt {f}}\right )+\frac {1}{16} \sqrt {\frac {\pi }{3}} f^{a-\frac {1}{2}} e^{\frac {b^2 \log ^2(f)}{12 f}-3 d} \text {Erf}\left (\frac {6 f x-b \log (f)}{2 \sqrt {3} \sqrt {f}}\right )+\frac {3}{16} \sqrt {\pi } f^{a-\frac {1}{2}} e^{d-\frac {b^2 \log ^2(f)}{4 f}} \text {Erfi}\left (\frac {b \log (f)+2 f x}{2 \sqrt {f}}\right )+\frac {1}{16} \sqrt {\frac {\pi }{3}} f^{a-\frac {1}{2}} e^{3 d-\frac {b^2 \log ^2(f)}{12 f}} \text {Erfi}\left (\frac {b \log (f)+6 f x}{2 \sqrt {3} \sqrt {f}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x)*Cosh[d + f*x^2]^3,x]

[Out]

(3*E^(-d + (b^2*Log[f]^2)/(4*f))*f^(-1/2 + a)*Sqrt[Pi]*Erf[(2*f*x - b*Log[f])/(2*Sqrt[f])])/16 + (E^(-3*d + (b
^2*Log[f]^2)/(12*f))*f^(-1/2 + a)*Sqrt[Pi/3]*Erf[(6*f*x - b*Log[f])/(2*Sqrt[3]*Sqrt[f])])/16 + (3*E^(d - (b^2*
Log[f]^2)/(4*f))*f^(-1/2 + a)*Sqrt[Pi]*Erfi[(2*f*x + b*Log[f])/(2*Sqrt[f])])/16 + (E^(3*d - (b^2*Log[f]^2)/(12
*f))*f^(-1/2 + a)*Sqrt[Pi/3]*Erfi[(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+f x^2\right ) \, dx &=\int \left (\frac {1}{8} e^{-3 d-3 f x^2} f^{a+b x}+\frac {3}{8} e^{-d-f x^2} f^{a+b x}+\frac {3}{8} e^{d+f x^2} f^{a+b x}+\frac {1}{8} e^{3 d+3 f x^2} f^{a+b x}\right ) \, dx\\ &=\frac {1}{8} \int e^{-3 d-3 f x^2} f^{a+b x} \, dx+\frac {1}{8} \int e^{3 d+3 f x^2} f^{a+b x} \, dx+\frac {3}{8} \int e^{-d-f x^2} f^{a+b x} \, dx+\frac {3}{8} \int e^{d+f x^2} f^{a+b x} \, dx\\ &=\frac {1}{8} \int e^{-3 d-3 f x^2+a \log (f)+b x \log (f)} \, dx+\frac {1}{8} \int e^{3 d+3 f x^2+a \log (f)+b x \log (f)} \, dx+\frac {3}{8} \int e^{-d-f x^2+a \log (f)+b x \log (f)} \, dx+\frac {3}{8} \int e^{d+f x^2+a \log (f)+b x \log (f)} \, dx\\ &=\frac {1}{8} \left (3 e^{d-\frac {b^2 \log ^2(f)}{4 f}} f^a\right ) \int e^{\frac {(2 f x+b \log (f))^2}{4 f}} \, dx+\frac {1}{8} \left (e^{3 d-\frac {b^2 \log ^2(f)}{12 f}} f^a\right ) \int e^{\frac {(6 f x+b \log (f))^2}{12 f}} \, dx+\frac {1}{8} \left (e^{-3 d+\frac {b^2 \log ^2(f)}{12 f}} f^a\right ) \int e^{-\frac {(-6 f x+b \log (f))^2}{12 f}} \, dx+\frac {1}{8} \left (3 e^{-d+\frac {b^2 \log ^2(f)}{4 f}} f^a\right ) \int e^{-\frac {(-2 f x+b \log (f))^2}{4 f}} \, dx\\ &=\frac {3}{16} e^{-d+\frac {b^2 \log ^2(f)}{4 f}} f^{-\frac {1}{2}+a} \sqrt {\pi } \text {erf}\left (\frac {2 f x-b \log (f)}{2 \sqrt {f}}\right )+\frac {1}{16} e^{-3 d+\frac {b^2 \log ^2(f)}{12 f}} f^{-\frac {1}{2}+a} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {6 f x-b \log (f)}{2 \sqrt {3} \sqrt {f}}\right )+\frac {3}{16} e^{d-\frac {b^2 \log ^2(f)}{4 f}} f^{-\frac {1}{2}+a} \sqrt {\pi } \text {erfi}\left (\frac {2 f x+b \log (f)}{2 \sqrt {f}}\right )+\frac {1}{16} e^{3 d-\frac {b^2 \log ^2(f)}{12 f}} f^{-\frac {1}{2}+a} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {6 f x+b \log (f)}{2 \sqrt {3} \sqrt {f}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.43, size = 286, normalized size = 1.20 \[ \frac {1}{16} \sqrt {\frac {\pi }{3}} f^{a-\frac {1}{2}} e^{-\frac {b^2 \log ^2(f)}{4 f}} \left (3 \sqrt {3} e^{\frac {b^2 \log ^2(f)}{2 f}} (\cosh (d)-\sinh (d)) \text {erf}\left (\frac {2 f x-b \log (f)}{2 \sqrt {f}}\right )+e^{\frac {b^2 \log ^2(f)}{3 f}} (\cosh (3 d)-\sinh (3 d)) \text {erf}\left (\frac {6 f x-b \log (f)}{2 \sqrt {3} \sqrt {f}}\right )+\sinh (3 d) e^{\frac {b^2 \log ^2(f)}{6 f}} \text {erfi}\left (\frac {b \log (f)+6 f x}{2 \sqrt {3} \sqrt {f}}\right )+\cosh (3 d) e^{\frac {b^2 \log ^2(f)}{6 f}} \text {erfi}\left (\frac {b \log (f)+6 f x}{2 \sqrt {3} \sqrt {f}}\right )+3 \sqrt {3} \sinh (d) \text {erfi}\left (\frac {b \log (f)+2 f x}{2 \sqrt {f}}\right )+3 \sqrt {3} \cosh (d) \text {erfi}\left (\frac {b \log (f)+2 f x}{2 \sqrt {f}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(f^(-1/2 + a)*Sqrt[Pi/3]*(3*Sqrt[3]*Cosh[d]*Erfi[(2*f*x + b*Log[f])/(2*Sqrt[f])] + E^((b^2*Log[f]^2)/(6*f))*Co
sh[3*d]*Erfi[(6*f*x + b*Log[f])/(2*Sqrt[3]*Sqrt[f])] + 3*Sqrt[3]*E^((b^2*Log[f]^2)/(2*f))*Erf[(2*f*x - b*Log[f
])/(2*Sqrt[f])]*(Cosh[d] - Sinh[d]) + 3*Sqrt[3]*Erfi[(2*f*x + b*Log[f])/(2*Sqrt[f])]*Sinh[d] + E^((b^2*Log[f]^
2)/(3*f))*Erf[(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[
(6*f*x + b*Log[f])/(2*Sqrt[3]*Sqrt[f])]*Sinh[3*d]))/(16*E^((b^2*Log[f]^2)/(4*f)))

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fricas [B]  time = 0.45, size = 443, normalized size = 1.85 \[ -\frac {\sqrt {3} \sqrt {\pi } \sqrt {-f} \cosh \left (\frac {b^{2} \log \relax (f)^{2} - 12 \, a f \log \relax (f) - 36 \, d f}{12 \, f}\right ) \operatorname {erf}\left (\frac {\sqrt {3} {\left (6 \, f x + b \log \relax (f)\right )} \sqrt {-f}}{6 \, f}\right ) + \sqrt {3} \sqrt {\pi } \sqrt {f} \cosh \left (\frac {b^{2} \log \relax (f)^{2} + 12 \, a f \log \relax (f) - 36 \, d f}{12 \, f}\right ) \operatorname {erf}\left (-\frac {\sqrt {3} {\left (6 \, f x - b \log \relax (f)\right )}}{6 \, \sqrt {f}}\right ) + \sqrt {3} \sqrt {\pi } \sqrt {f} \operatorname {erf}\left (-\frac {\sqrt {3} {\left (6 \, f x - b \log \relax (f)\right )}}{6 \, \sqrt {f}}\right ) \sinh \left (\frac {b^{2} \log \relax (f)^{2} + 12 \, a f \log \relax (f) - 36 \, d f}{12 \, f}\right ) - \sqrt {3} \sqrt {\pi } \sqrt {-f} \operatorname {erf}\left (\frac {\sqrt {3} {\left (6 \, f x + b \log \relax (f)\right )} \sqrt {-f}}{6 \, f}\right ) \sinh \left (\frac {b^{2} \log \relax (f)^{2} - 12 \, a f \log \relax (f) - 36 \, d f}{12 \, f}\right ) + 9 \, \sqrt {\pi } \sqrt {-f} \cosh \left (\frac {b^{2} \log \relax (f)^{2} - 4 \, a f \log \relax (f) - 4 \, d f}{4 \, f}\right ) \operatorname {erf}\left (\frac {{\left (2 \, f x + b \log \relax (f)\right )} \sqrt {-f}}{2 \, f}\right ) + 9 \, \sqrt {\pi } \sqrt {f} \cosh \left (\frac {b^{2} \log \relax (f)^{2} + 4 \, a f \log \relax (f) - 4 \, d f}{4 \, f}\right ) \operatorname {erf}\left (-\frac {2 \, f x - b \log \relax (f)}{2 \, \sqrt {f}}\right ) + 9 \, \sqrt {\pi } \sqrt {f} \operatorname {erf}\left (-\frac {2 \, f x - b \log \relax (f)}{2 \, \sqrt {f}}\right ) \sinh \left (\frac {b^{2} \log \relax (f)^{2} + 4 \, a f \log \relax (f) - 4 \, d f}{4 \, f}\right ) - 9 \, \sqrt {\pi } \sqrt {-f} \operatorname {erf}\left (\frac {{\left (2 \, f x + b \log \relax (f)\right )} \sqrt {-f}}{2 \, f}\right ) \sinh \left (\frac {b^{2} \log \relax (f)^{2} - 4 \, a f \log \relax (f) - 4 \, d f}{4 \, f}\right )}{48 \, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/48*(sqrt(3)*sqrt(pi)*sqrt(-f)*cosh(1/12*(b^2*log(f)^2 - 12*a*f*log(f) - 36*d*f)/f)*erf(1/6*sqrt(3)*(6*f*x +
b*log(f))*sqrt(-f)/f) + sqrt(3)*sqrt(pi)*sqrt(f)*cosh(1/12*(b^2*log(f)^2 + 12*a*f*log(f) - 36*d*f)/f)*erf(-1/
6*sqrt(3)*(6*f*x - b*log(f))/sqrt(f)) + sqrt(3)*sqrt(pi)*sqrt(f)*erf(-1/6*sqrt(3)*(6*f*x - b*log(f))/sqrt(f))*
sinh(1/12*(b^2*log(f)^2 + 12*a*f*log(f) - 36*d*f)/f) - sqrt(3)*sqrt(pi)*sqrt(-f)*erf(1/6*sqrt(3)*(6*f*x + b*lo
g(f))*sqrt(-f)/f)*sinh(1/12*(b^2*log(f)^2 - 12*a*f*log(f) - 36*d*f)/f) + 9*sqrt(pi)*sqrt(-f)*cosh(1/4*(b^2*log
(f)^2 - 4*a*f*log(f) - 4*d*f)/f)*erf(1/2*(2*f*x + b*log(f))*sqrt(-f)/f) + 9*sqrt(pi)*sqrt(f)*cosh(1/4*(b^2*log
(f)^2 + 4*a*f*log(f) - 4*d*f)/f)*erf(-1/2*(2*f*x - b*log(f))/sqrt(f)) + 9*sqrt(pi)*sqrt(f)*erf(-1/2*(2*f*x - b
*log(f))/sqrt(f))*sinh(1/4*(b^2*log(f)^2 + 4*a*f*log(f) - 4*d*f)/f) - 9*sqrt(pi)*sqrt(-f)*erf(1/2*(2*f*x + b*l
og(f))*sqrt(-f)/f)*sinh(1/4*(b^2*log(f)^2 - 4*a*f*log(f) - 4*d*f)/f))/f

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giac [A]  time = 0.17, size = 223, normalized size = 0.93 \[ -\frac {\sqrt {3} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{6} \, \sqrt {3} \sqrt {f} {\left (6 \, x - \frac {b \log \relax (f)}{f}\right )}\right ) e^{\left (\frac {b^{2} \log \relax (f)^{2} + 12 \, a f \log \relax (f) - 36 \, d f}{12 \, f}\right )}}{48 \, \sqrt {f}} - \frac {\sqrt {3} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{6} \, \sqrt {3} \sqrt {-f} {\left (6 \, x + \frac {b \log \relax (f)}{f}\right )}\right ) e^{\left (-\frac {b^{2} \log \relax (f)^{2} - 12 \, a f \log \relax (f) - 36 \, d f}{12 \, f}\right )}}{48 \, \sqrt {-f}} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {f} {\left (2 \, x - \frac {b \log \relax (f)}{f}\right )}\right ) e^{\left (\frac {b^{2} \log \relax (f)^{2} + 4 \, a f \log \relax (f) - 4 \, d f}{4 \, f}\right )}}{16 \, \sqrt {f}} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-f} {\left (2 \, x + \frac {b \log \relax (f)}{f}\right )}\right ) e^{\left (-\frac {b^{2} \log \relax (f)^{2} - 4 \, a f \log \relax (f) - 4 \, d f}{4 \, f}\right )}}{16 \, \sqrt {-f}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/48*sqrt(3)*sqrt(pi)*erf(-1/6*sqrt(3)*sqrt(f)*(6*x - b*log(f)/f))*e^(1/12*(b^2*log(f)^2 + 12*a*f*log(f) - 36
*d*f)/f)/sqrt(f) - 1/48*sqrt(3)*sqrt(pi)*erf(-1/6*sqrt(3)*sqrt(-f)*(6*x + b*log(f)/f))*e^(-1/12*(b^2*log(f)^2
- 12*a*f*log(f) - 36*d*f)/f)/sqrt(-f) - 3/16*sqrt(pi)*erf(-1/2*sqrt(f)*(2*x - b*log(f)/f))*e^(1/4*(b^2*log(f)^
2 + 4*a*f*log(f) - 4*d*f)/f)/sqrt(f) - 3/16*sqrt(pi)*erf(-1/2*sqrt(-f)*(2*x + b*log(f)/f))*e^(-1/4*(b^2*log(f)
^2 - 4*a*f*log(f) - 4*d*f)/f)/sqrt(-f)

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maple [A]  time = 0.42, size = 207, normalized size = 0.87 \[ -\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {\ln \relax (f )^{2} b^{2}-36 d f}{12 f}} \sqrt {3}\, \erf \left (-\sqrt {3}\, \sqrt {f}\, x +\frac {\ln \relax (f ) b \sqrt {3}}{6 \sqrt {f}}\right )}{48 \sqrt {f}}-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {\ln \relax (f )^{2} b^{2}-36 d f}{12 f}} \erf \left (-\sqrt {-3 f}\, x +\frac {\ln \relax (f ) b}{2 \sqrt {-3 f}}\right )}{16 \sqrt {-3 f}}-\frac {3 \sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {\ln \relax (f )^{2} b^{2}-4 d f}{4 f}} \erf \left (-\sqrt {f}\, x +\frac {\ln \relax (f ) b}{2 \sqrt {f}}\right )}{16 \sqrt {f}}-\frac {3 \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {\ln \relax (f )^{2} b^{2}-4 d f}{4 f}} \erf \left (-\sqrt {-f}\, x +\frac {\ln \relax (f ) b}{2 \sqrt {-f}}\right )}{16 \sqrt {-f}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.43, size = 200, normalized size = 0.84 \[ \frac {3}{16} \, \sqrt {\pi } f^{a - \frac {1}{2}} \operatorname {erf}\left (\sqrt {f} x - \frac {b \log \relax (f)}{2 \, \sqrt {f}}\right ) e^{\left (\frac {b^{2} \log \relax (f)^{2}}{4 \, f} - d\right )} + \frac {\sqrt {3} \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {3} \sqrt {f} x - \frac {\sqrt {3} b \log \relax (f)}{6 \, \sqrt {f}}\right ) e^{\left (\frac {b^{2} \log \relax (f)^{2}}{12 \, f} - 3 \, d\right )}}{48 \, \sqrt {f}} + \frac {\sqrt {3} \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {3} \sqrt {-f} x - \frac {\sqrt {3} b \log \relax (f)}{6 \, \sqrt {-f}}\right ) e^{\left (-\frac {b^{2} \log \relax (f)^{2}}{12 \, f} + 3 \, d\right )}}{48 \, \sqrt {-f}} + \frac {3 \, \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-f} x - \frac {b \log \relax (f)}{2 \, \sqrt {-f}}\right ) e^{\left (-\frac {b^{2} \log \relax (f)^{2}}{4 \, f} + d\right )}}{16 \, \sqrt {-f}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/16*sqrt(pi)*f^(a - 1/2)*erf(sqrt(f)*x - 1/2*b*log(f)/sqrt(f))*e^(1/4*b^2*log(f)^2/f - d) + 1/48*sqrt(3)*sqrt
(pi)*f^a*erf(sqrt(3)*sqrt(f)*x - 1/6*sqrt(3)*b*log(f)/sqrt(f))*e^(1/12*b^2*log(f)^2/f - 3*d)/sqrt(f) + 1/48*sq
rt(3)*sqrt(pi)*f^a*erf(sqrt(3)*sqrt(-f)*x - 1/6*sqrt(3)*b*log(f)/sqrt(-f))*e^(-1/12*b^2*log(f)^2/f + 3*d)/sqrt
(-f) + 3/16*sqrt(pi)*f^a*erf(sqrt(-f)*x - 1/2*b*log(f)/sqrt(-f))*e^(-1/4*b^2*log(f)^2/f + d)/sqrt(-f)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a + b*x)*cosh(d + f*x^2)^3,x)

[Out]

int(f^(a + b*x)*cosh(d + f*x^2)^3, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{a + b x} \cosh ^{3}{\left (d + f x^{2} \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(f**(a + b*x)*cosh(d + f*x**2)**3, x)

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