3.121 \(\int f^{a+c x^2} \cos ^3(d+f x^2) \, dx\)

Optimal. Leaf size=205 \[ \frac{3 \sqrt{\pi } e^{-i d} f^a \text{Erf}\left (x \sqrt{-c \log (f)+i f}\right )}{16 \sqrt{-c \log (f)+i f}}+\frac{\sqrt{\pi } e^{-3 i d} f^a \text{Erf}\left (x \sqrt{-c \log (f)+3 i f}\right )}{16 \sqrt{-c \log (f)+3 i f}}+\frac{3 \sqrt{\pi } e^{i d} f^a \text{Erfi}\left (x \sqrt{c \log (f)+i f}\right )}{16 \sqrt{c \log (f)+i f}}+\frac{\sqrt{\pi } e^{3 i d} f^a \text{Erfi}\left (x \sqrt{c \log (f)+3 i f}\right )}{16 \sqrt{c \log (f)+3 i f}} \]

[Out]

(3*f^a*Sqrt[Pi]*Erf[x*Sqrt[I*f - c*Log[f]]])/(16*E^(I*d)*Sqrt[I*f - c*Log[f]]) + (f^a*Sqrt[Pi]*Erf[x*Sqrt[(3*I
)*f - c*Log[f]]])/(16*E^((3*I)*d)*Sqrt[(3*I)*f - c*Log[f]]) + (3*E^(I*d)*f^a*Sqrt[Pi]*Erfi[x*Sqrt[I*f + c*Log[
f]]])/(16*Sqrt[I*f + c*Log[f]]) + (E^((3*I)*d)*f^a*Sqrt[Pi]*Erfi[x*Sqrt[(3*I)*f + c*Log[f]]])/(16*Sqrt[(3*I)*f
+ c*Log[f]])

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Rubi [A]  time = 0.287936, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4473, 2287, 2205, 2204} \[ \frac{3 \sqrt{\pi } e^{-i d} f^a \text{Erf}\left (x \sqrt{-c \log (f)+i f}\right )}{16 \sqrt{-c \log (f)+i f}}+\frac{\sqrt{\pi } e^{-3 i d} f^a \text{Erf}\left (x \sqrt{-c \log (f)+3 i f}\right )}{16 \sqrt{-c \log (f)+3 i f}}+\frac{3 \sqrt{\pi } e^{i d} f^a \text{Erfi}\left (x \sqrt{c \log (f)+i f}\right )}{16 \sqrt{c \log (f)+i f}}+\frac{\sqrt{\pi } e^{3 i d} f^a \text{Erfi}\left (x \sqrt{c \log (f)+3 i f}\right )}{16 \sqrt{c \log (f)+3 i f}} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + c*x^2)*Cos[d + f*x^2]^3,x]

[Out]

(3*f^a*Sqrt[Pi]*Erf[x*Sqrt[I*f - c*Log[f]]])/(16*E^(I*d)*Sqrt[I*f - c*Log[f]]) + (f^a*Sqrt[Pi]*Erf[x*Sqrt[(3*I
)*f - c*Log[f]]])/(16*E^((3*I)*d)*Sqrt[(3*I)*f - c*Log[f]]) + (3*E^(I*d)*f^a*Sqrt[Pi]*Erfi[x*Sqrt[I*f + c*Log[
f]]])/(16*Sqrt[I*f + c*Log[f]]) + (E^((3*I)*d)*f^a*Sqrt[Pi]*Erfi[x*Sqrt[(3*I)*f + c*Log[f]]])/(16*Sqrt[(3*I)*f
+ c*Log[f]])

Rule 4473

Int[Cos[v_]^(n_.)*(F_)^(u_), x_Symbol] :> Int[ExpandTrigToExp[F^u, Cos[v]^n, x], x] /; FreeQ[F, x] && (LinearQ
[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 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} \cos ^3\left (d+f x^2\right ) \, dx &=\int \left (\frac{3}{8} e^{-i d-i f x^2} f^{a+c x^2}+\frac{3}{8} e^{i d+i f x^2} f^{a+c x^2}+\frac{1}{8} e^{-3 i d-3 i f x^2} f^{a+c x^2}+\frac{1}{8} e^{3 i d+3 i f x^2} f^{a+c x^2}\right ) \, dx\\ &=\frac{1}{8} \int e^{-3 i d-3 i f x^2} f^{a+c x^2} \, dx+\frac{1}{8} \int e^{3 i d+3 i f x^2} f^{a+c x^2} \, dx+\frac{3}{8} \int e^{-i d-i f x^2} f^{a+c x^2} \, dx+\frac{3}{8} \int e^{i d+i f x^2} f^{a+c x^2} \, dx\\ &=\frac{1}{8} \int \exp \left (-3 i d+a \log (f)-x^2 (3 i f-c \log (f))\right ) \, dx+\frac{1}{8} \int \exp \left (3 i d+a \log (f)+x^2 (3 i f+c \log (f))\right ) \, dx+\frac{3}{8} \int e^{-i d+a \log (f)-x^2 (i f-c \log (f))} \, dx+\frac{3}{8} \int e^{i d+a \log (f)+x^2 (i f+c \log (f))} \, dx\\ &=\frac{3 e^{-i d} f^a \sqrt{\pi } \text{erf}\left (x \sqrt{i f-c \log (f)}\right )}{16 \sqrt{i f-c \log (f)}}+\frac{e^{-3 i d} f^a \sqrt{\pi } \text{erf}\left (x \sqrt{3 i f-c \log (f)}\right )}{16 \sqrt{3 i f-c \log (f)}}+\frac{3 e^{i d} f^a \sqrt{\pi } \text{erfi}\left (x \sqrt{i f+c \log (f)}\right )}{16 \sqrt{i f+c \log (f)}}+\frac{e^{3 i d} f^a \sqrt{\pi } \text{erfi}\left (x \sqrt{3 i f+c \log (f)}\right )}{16 \sqrt{3 i f+c \log (f)}}\\ \end{align*}

Mathematica [A]  time = 2.20998, size = 389, normalized size = 1.9 \[ \frac{\sqrt [4]{-1} \sqrt{\pi } f^a \left ((f-i c \log (f)) \left (\sqrt{3 f-i c \log (f)} \left (i c^2 \log ^2(f)+4 c f \log (f)-3 i f^2\right ) (\cos (3 d)+i \sin (3 d)) \text{Erfi}\left (\sqrt [4]{-1} x \sqrt{3 f-i c \log (f)}\right )-(3 f-i c \log (f)) \left (9 f \sin (d) \sqrt{f+i c \log (f)} \text{Erf}\left (\frac{(1+i) x \sqrt{f+i c \log (f)}}{\sqrt{2}}\right )+3 \sqrt{f+i c \log (f)} \text{Erfi}\left ((-1)^{3/4} x \sqrt{f+i c \log (f)}\right ) (c \sin (d) \log (f)+\cos (d) (3 f+i c \log (f)))+(f+i c \log (f)) \sqrt{3 f+i c \log (f)} (\cos (3 d)-i \sin (3 d)) \text{Erfi}\left ((-1)^{3/4} x \sqrt{3 f+i c \log (f)}\right )\right )\right )+3 \sqrt{f-i c \log (f)} \left (-i c^2 f \log ^2(f)+c^3 \log ^3(f)+9 c f^2 \log (f)-9 i f^3\right ) (\cos (d)+i \sin (d)) \text{Erfi}\left (\sqrt [4]{-1} x \sqrt{f-i c \log (f)}\right )\right )}{16 \left (10 c^2 f^2 \log ^2(f)+c^4 \log ^4(f)+9 f^4\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-1)^(1/4)*f^a*Sqrt[Pi]*(3*Erfi[(-1)^(1/4)*x*Sqrt[f - I*c*Log[f]]]*Sqrt[f - I*c*Log[f]]*((-9*I)*f^3 + 9*c*f^2
*Log[f] - I*c^2*f*Log[f]^2 + c^3*Log[f]^3)*(Cos[d] + I*Sin[d]) + (f - I*c*Log[f])*(-((3*f - I*c*Log[f])*(9*f*E
rf[((1 + I)*x*Sqrt[f + I*c*Log[f]])/Sqrt[2]]*Sqrt[f + I*c*Log[f]]*Sin[d] + 3*Erfi[(-1)^(3/4)*x*Sqrt[f + I*c*Lo
g[f]]]*Sqrt[f + I*c*Log[f]]*(Cos[d]*(3*f + I*c*Log[f]) + c*Log[f]*Sin[d]) + Erfi[(-1)^(3/4)*x*Sqrt[3*f + I*c*L
og[f]]]*(f + I*c*Log[f])*Sqrt[3*f + I*c*Log[f]]*(Cos[3*d] - I*Sin[3*d]))) + Erfi[(-1)^(1/4)*x*Sqrt[3*f - I*c*L
og[f]]]*Sqrt[3*f - I*c*Log[f]]*((-3*I)*f^2 + 4*c*f*Log[f] + I*c^2*Log[f]^2)*(Cos[3*d] + I*Sin[3*d]))))/(16*(9*
f^4 + 10*c^2*f^2*Log[f]^2 + c^4*Log[f]^4))

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Maple [A]  time = 0.195, size = 162, normalized size = 0.8 \begin{align*}{\frac{{f}^{a}\sqrt{\pi }{{\rm e}^{-3\,id}}}{16}{\it Erf} \left ( x\sqrt{3\,if-c\ln \left ( f \right ) } \right ){\frac{1}{\sqrt{3\,if-c\ln \left ( f \right ) }}}}+{\frac{3\,{f}^{a}\sqrt{\pi }{{\rm e}^{-id}}}{16}{\it Erf} \left ( x\sqrt{if-c\ln \left ( f \right ) } \right ){\frac{1}{\sqrt{if-c\ln \left ( f \right ) }}}}+{\frac{3\,{f}^{a}\sqrt{\pi }{{\rm e}^{id}}}{16}{\it Erf} \left ( \sqrt{-c\ln \left ( f \right ) -if}x \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) -if}}}}+{\frac{{f}^{a}\sqrt{\pi }{{\rm e}^{3\,id}}}{16}{\it Erf} \left ( \sqrt{-c\ln \left ( f \right ) -3\,if}x \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) -3\,if}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*Pi^(1/2)*f^a*exp(-3*I*d)/(3*I*f-c*ln(f))^(1/2)*erf(x*(3*I*f-c*ln(f))^(1/2))+3/16*Pi^(1/2)*f^a*exp(-I*d)/(
I*f-c*ln(f))^(1/2)*erf(x*(I*f-c*ln(f))^(1/2))+3/16*Pi^(1/2)*f^a*exp(I*d)/(-c*ln(f)-I*f)^(1/2)*erf((-c*ln(f)-I*
f)^(1/2)*x)+1/16*Pi^(1/2)*f^a*exp(3*I*d)/(-c*ln(f)-3*I*f)^(1/2)*erf((-c*ln(f)-3*I*f)^(1/2)*x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: IndexError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: IndexError

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Fricas [B]  time = 0.547518, size = 869, normalized size = 4.24 \begin{align*} -\frac{\sqrt{\pi }{\left (c^{3} \log \left (f\right )^{3} - 3 i \, c^{2} f \log \left (f\right )^{2} + c f^{2} \log \left (f\right ) - 3 i \, f^{3}\right )} \sqrt{-c \log \left (f\right ) - 3 i \, f} \operatorname{erf}\left (\sqrt{-c \log \left (f\right ) - 3 i \, f} x\right ) e^{\left (a \log \left (f\right ) + 3 i \, d\right )} + \sqrt{\pi }{\left (3 \, c^{3} \log \left (f\right )^{3} - 3 i \, c^{2} f \log \left (f\right )^{2} + 27 \, c f^{2} \log \left (f\right ) - 27 i \, f^{3}\right )} \sqrt{-c \log \left (f\right ) - i \, f} \operatorname{erf}\left (\sqrt{-c \log \left (f\right ) - i \, f} x\right ) e^{\left (a \log \left (f\right ) + i \, d\right )} + \sqrt{\pi }{\left (3 \, c^{3} \log \left (f\right )^{3} + 3 i \, c^{2} f \log \left (f\right )^{2} + 27 \, c f^{2} \log \left (f\right ) + 27 i \, f^{3}\right )} \sqrt{-c \log \left (f\right ) + i \, f} \operatorname{erf}\left (\sqrt{-c \log \left (f\right ) + i \, f} x\right ) e^{\left (a \log \left (f\right ) - i \, d\right )} + \sqrt{\pi }{\left (c^{3} \log \left (f\right )^{3} + 3 i \, c^{2} f \log \left (f\right )^{2} + c f^{2} \log \left (f\right ) + 3 i \, f^{3}\right )} \sqrt{-c \log \left (f\right ) + 3 i \, f} \operatorname{erf}\left (\sqrt{-c \log \left (f\right ) + 3 i \, f} x\right ) e^{\left (a \log \left (f\right ) - 3 i \, d\right )}}{16 \,{\left (c^{4} \log \left (f\right )^{4} + 10 \, c^{2} f^{2} \log \left (f\right )^{2} + 9 \, f^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(sqrt(pi)*(c^3*log(f)^3 - 3*I*c^2*f*log(f)^2 + c*f^2*log(f) - 3*I*f^3)*sqrt(-c*log(f) - 3*I*f)*erf(sqrt(
-c*log(f) - 3*I*f)*x)*e^(a*log(f) + 3*I*d) + sqrt(pi)*(3*c^3*log(f)^3 - 3*I*c^2*f*log(f)^2 + 27*c*f^2*log(f) -
27*I*f^3)*sqrt(-c*log(f) - I*f)*erf(sqrt(-c*log(f) - I*f)*x)*e^(a*log(f) + I*d) + sqrt(pi)*(3*c^3*log(f)^3 +
3*I*c^2*f*log(f)^2 + 27*c*f^2*log(f) + 27*I*f^3)*sqrt(-c*log(f) + I*f)*erf(sqrt(-c*log(f) + I*f)*x)*e^(a*log(f
) - I*d) + sqrt(pi)*(c^3*log(f)^3 + 3*I*c^2*f*log(f)^2 + c*f^2*log(f) + 3*I*f^3)*sqrt(-c*log(f) + 3*I*f)*erf(s
qrt(-c*log(f) + 3*I*f)*x)*e^(a*log(f) - 3*I*d))/(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)*cos(f*x**2+d)**3,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{c x^{2} + a} \cos \left (f x^{2} + d\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^(c*x^2 + a)*cos(f*x^2 + d)^3, x)