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

Optimal. Leaf size=81 \[ \frac {\sqrt {\pi } e^{-d} f^a \text {erf}\left (x \sqrt {f-c \log (f)}\right )}{4 \sqrt {f-c \log (f)}}+\frac {\sqrt {\pi } e^d f^a \text {erfi}\left (x \sqrt {c \log (f)+f}\right )}{4 \sqrt {c \log (f)+f}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.16, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5513, 2287, 2205, 2204} \[ \frac {\sqrt {\pi } e^{-d} f^a \text {Erf}\left (x \sqrt {f-c \log (f)}\right )}{4 \sqrt {f-c \log (f)}}+\frac {\sqrt {\pi } e^d f^a \text {Erfi}\left (x \sqrt {c \log (f)+f}\right )}{4 \sqrt {c \log (f)+f}} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + c*x^2)*Cosh[d + f*x^2],x]

[Out]

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

________________________________________________________________________________________

Mathematica [A]  time = 0.33, size = 75, normalized size = 0.93 \[ \frac {1}{4} \sqrt {\pi } f^a \left (\frac {(\cosh (d)-\sinh (d)) \text {erf}\left (x \sqrt {f-c \log (f)}\right )}{\sqrt {f-c \log (f)}}+\frac {(\sinh (d)+\cosh (d)) \text {erfi}\left (x \sqrt {c \log (f)+f}\right )}{\sqrt {c \log (f)+f}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(f^a*Sqrt[Pi]*((Erf[x*Sqrt[f - c*Log[f]]]*(Cosh[d] - Sinh[d]))/Sqrt[f - c*Log[f]] + (Erfi[x*Sqrt[f + c*Log[f]]
]*(Cosh[d] + Sinh[d]))/Sqrt[f + c*Log[f]]))/4

________________________________________________________________________________________

fricas [B]  time = 0.43, size = 145, normalized size = 1.79 \[ -\frac {{\left (\sqrt {\pi } {\left (c \log \relax (f) + f\right )} \cosh \left (a \log \relax (f) - d\right ) + \sqrt {\pi } {\left (c \log \relax (f) + f\right )} \sinh \left (a \log \relax (f) - d\right )\right )} \sqrt {-c \log \relax (f) + f} \operatorname {erf}\left (\sqrt {-c \log \relax (f) + f} x\right ) + {\left (\sqrt {\pi } {\left (c \log \relax (f) - f\right )} \cosh \left (a \log \relax (f) + d\right ) + \sqrt {\pi } {\left (c \log \relax (f) - f\right )} \sinh \left (a \log \relax (f) + d\right )\right )} \sqrt {-c \log \relax (f) - f} \operatorname {erf}\left (\sqrt {-c \log \relax (f) - f} x\right )}{4 \, {\left (c^{2} \log \relax (f)^{2} - f^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*((sqrt(pi)*(c*log(f) + f)*cosh(a*log(f) - d) + sqrt(pi)*(c*log(f) + f)*sinh(a*log(f) - d))*sqrt(-c*log(f)
+ f)*erf(sqrt(-c*log(f) + f)*x) + (sqrt(pi)*(c*log(f) - f)*cosh(a*log(f) + d) + sqrt(pi)*(c*log(f) - f)*sinh(
a*log(f) + d))*sqrt(-c*log(f) - f)*erf(sqrt(-c*log(f) - f)*x))/(c^2*log(f)^2 - f^2)

________________________________________________________________________________________

giac [A]  time = 0.13, size = 75, normalized size = 0.93 \[ -\frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-c \log \relax (f) - f} x\right ) e^{\left (a \log \relax (f) + d\right )}}{4 \, \sqrt {-c \log \relax (f) - f}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-c \log \relax (f) + f} x\right ) e^{\left (a \log \relax (f) - d\right )}}{4 \, \sqrt {-c \log \relax (f) + f}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.14, size = 70, normalized size = 0.86 \[ \frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-d} \erf \left (x \sqrt {f -c \ln \relax (f )}\right )}{4 \sqrt {f -c \ln \relax (f )}}+\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{d} \erf \left (\sqrt {-c \ln \relax (f )-f}\, x \right )}{4 \sqrt {-c \ln \relax (f )-f}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*Pi^(1/2)*f^a*exp(-d)/(f-c*ln(f))^(1/2)*erf(x*(f-c*ln(f))^(1/2))+1/4*Pi^(1/2)*f^a*exp(d)/(-c*ln(f)-f)^(1/2)
*erf((-c*ln(f)-f)^(1/2)*x)

________________________________________________________________________________________

maxima [A]  time = 0.33, size = 69, normalized size = 0.85 \[ \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f) + f} x\right ) e^{\left (-d\right )}}{4 \, \sqrt {-c \log \relax (f) + f}} + \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f) - f} x\right ) e^{d}}{4 \, \sqrt {-c \log \relax (f) - f}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int f^{c\,x^2+a}\,\mathrm {cosh}\left (f\,x^2+d\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a + c*x^2)*cosh(d + f*x^2),x)

[Out]

int(f^(a + c*x^2)*cosh(d + f*x^2), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________