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

Optimal. Leaf size=213 \[ \frac{3 i \sqrt{\pi } e^{-i d} f^a \text{Erf}\left (x \sqrt{-c \log (f)+i f}\right )}{16 \sqrt{-c \log (f)+i f}}-\frac{i \sqrt{\pi } e^{-3 i d} f^a \text{Erf}\left (x \sqrt{-c \log (f)+3 i f}\right )}{16 \sqrt{-c \log (f)+3 i f}}-\frac{3 i \sqrt{\pi } e^{i d} f^a \text{Erfi}\left (x \sqrt{c \log (f)+i f}\right )}{16 \sqrt{c \log (f)+i f}}+\frac{i \sqrt{\pi } e^{3 i d} f^a \text{Erfi}\left (x \sqrt{c \log (f)+3 i f}\right )}{16 \sqrt{c \log (f)+3 i f}} \]

[Out]

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

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Rubi [A]  time = 0.332318, antiderivative size = 213, 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 = {4472, 2287, 2205, 2204} \[ \frac{3 i \sqrt{\pi } e^{-i d} f^a \text{Erf}\left (x \sqrt{-c \log (f)+i f}\right )}{16 \sqrt{-c \log (f)+i f}}-\frac{i \sqrt{\pi } e^{-3 i d} f^a \text{Erf}\left (x \sqrt{-c \log (f)+3 i f}\right )}{16 \sqrt{-c \log (f)+3 i f}}-\frac{3 i \sqrt{\pi } e^{i d} f^a \text{Erfi}\left (x \sqrt{c \log (f)+i f}\right )}{16 \sqrt{c \log (f)+i f}}+\frac{i \sqrt{\pi } e^{3 i d} f^a \text{Erfi}\left (x \sqrt{c \log (f)+3 i f}\right )}{16 \sqrt{c \log (f)+3 i f}} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + c*x^2)*Sin[d + f*x^2]^3,x]

[Out]

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

Rule 4472

Int[(F_)^(u_)*Sin[v_]^(n_.), x_Symbol] :> Int[ExpandTrigToExp[F^u, Sin[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} \sin ^3\left (d+f x^2\right ) \, dx &=\int \left (\frac{3}{8} i e^{-i d-i f x^2} f^{a+c x^2}-\frac{3}{8} i e^{i d+i f x^2} f^{a+c x^2}-\frac{1}{8} i e^{-3 i d-3 i f x^2} f^{a+c x^2}+\frac{1}{8} i e^{3 i d+3 i f x^2} f^{a+c x^2}\right ) \, dx\\ &=-\left (\frac{1}{8} i \int e^{-3 i d-3 i f x^2} f^{a+c x^2} \, dx\right )+\frac{1}{8} i \int e^{3 i d+3 i f x^2} f^{a+c x^2} \, dx+\frac{3}{8} i \int e^{-i d-i f x^2} f^{a+c x^2} \, dx-\frac{3}{8} i \int e^{i d+i f x^2} f^{a+c x^2} \, dx\\ &=-\left (\frac{1}{8} i \int \exp \left (-3 i d+a \log (f)-x^2 (3 i f-c \log (f))\right ) \, dx\right )+\frac{1}{8} i \int \exp \left (3 i d+a \log (f)+x^2 (3 i f+c \log (f))\right ) \, dx+\frac{3}{8} i \int e^{-i d+a \log (f)-x^2 (i f-c \log (f))} \, dx-\frac{3}{8} i \int e^{i d+a \log (f)+x^2 (i f+c \log (f))} \, dx\\ &=\frac{3 i 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{i 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 i 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{i 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 = 2.28787, size = 386, normalized size = 1.81 \[ \frac{\sqrt [4]{-1} \sqrt{\pi } f^a \left ((f-i c \log (f)) \left (\sqrt{3 f-i c \log (f)} \left (-c^2 \log ^2(f)+4 i c f \log (f)+3 f^2\right ) (\cos (3 d)+i \sin (3 d)) \text{Erfi}\left (\sqrt [4]{-1} x \sqrt{3 f-i c \log (f)}\right )+(3 f-i c \log (f)) \left (3 \sqrt{f+i c \log (f)} \text{Erf}\left (\frac{(1+i) x \sqrt{f+i c \log (f)}}{\sqrt{2}}\right ) (c \sin (d) \log (f)+3 f \cos (d))+3 \sqrt{f+i c \log (f)} \text{Erfi}\left ((-1)^{3/4} x \sqrt{f+i c \log (f)}\right ) (c \cos (d) \log (f)-3 f \sin (d))+(f+i c \log (f)) \sqrt{3 f+i c \log (f)} (\sin (3 d)+i \cos (3 d)) \text{Erfi}\left ((-1)^{3/4} x \sqrt{3 f+i c \log (f)}\right )\right )\right )-3 \sqrt{f-i c \log (f)} \left (c^2 f \log ^2(f)+i c^3 \log ^3(f)+9 i c f^2 \log (f)+9 f^3\right ) (\cos (d)+i \sin (d)) \text{Erfi}\left (\sqrt [4]{-1} x \sqrt{f-i c \log (f)}\right )\right )}{16 \left (10 c^2 f^2 \log ^2(f)+c^4 \log ^4(f)+9 f^4\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + c*x^2)*Sin[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*f^3 + (9*I)*c*f^2
*Log[f] + c^2*f*Log[f]^2 + I*c^3*Log[f]^3)*(Cos[d] + I*Sin[d]) + (f - I*c*Log[f])*(Erfi[(-1)^(1/4)*x*Sqrt[3*f
- I*c*Log[f]]]*Sqrt[3*f - I*c*Log[f]]*(3*f^2 + (4*I)*c*f*Log[f] - c^2*Log[f]^2)*(Cos[3*d] + I*Sin[3*d]) + (3*f
- I*c*Log[f])*(3*Erfi[(-1)^(3/4)*x*Sqrt[f + I*c*Log[f]]]*Sqrt[f + I*c*Log[f]]*(c*Cos[d]*Log[f] - 3*f*Sin[d])
+ 3*Erf[((1 + I)*x*Sqrt[f + I*c*Log[f]])/Sqrt[2]]*Sqrt[f + I*c*Log[f]]*(3*f*Cos[d] + c*Log[f]*Sin[d]) + Erfi[(
-1)^(3/4)*x*Sqrt[3*f + I*c*Log[f]]]*(f + I*c*Log[f])*Sqrt[3*f + I*c*Log[f]]*(I*Cos[3*d] + 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 = 0.271, size = 166, normalized size = 0.8 \begin{align*}{{\frac{i}{16}}{f}^{a}\sqrt{\pi }{{\rm e}^{3\,id}}{\it Erf} \left ( \sqrt{-c\ln \left ( f \right ) -3\,if}x \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) -3\,if}}}}-{{\frac{i}{16}}{f}^{a}\sqrt{\pi }{{\rm e}^{-3\,id}}{\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\,i}{16}}{f}^{a}\sqrt{\pi }{{\rm e}^{-id}}{\it Erf} \left ( x\sqrt{if-c\ln \left ( f \right ) } \right ){\frac{1}{\sqrt{if-c\ln \left ( f \right ) }}}}-{{\frac{3\,i}{16}}{f}^{a}\sqrt{\pi }{{\rm e}^{id}}{\it Erf} \left ( \sqrt{-c\ln \left ( f \right ) -if}x \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) -if}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*I*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)-1/16*I*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*I*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*I*Pi^(1/2)*f^a*exp(I*d)/(-c*ln(f)-I*f)^(1/2)*erf((-c*ln(f)-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)*sin(f*x^2+d)^3,x, algorithm="maxima")

[Out]

Exception raised: IndexError

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Fricas [B]  time = 0.549795, size = 871, normalized size = 4.09 \begin{align*} \frac{\sqrt{\pi }{\left (-i \, c^{3} \log \left (f\right )^{3} - 3 \, c^{2} f \log \left (f\right )^{2} - i \, c f^{2} \log \left (f\right ) - 3 \, 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 i \, c^{3} \log \left (f\right )^{3} + 3 \, c^{2} f \log \left (f\right )^{2} + 27 i \, c f^{2} \log \left (f\right ) + 27 \, 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 i \, c^{3} \log \left (f\right )^{3} + 3 \, c^{2} f \log \left (f\right )^{2} - 27 i \, c f^{2} \log \left (f\right ) + 27 \, 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 (i \, c^{3} \log \left (f\right )^{3} - 3 \, c^{2} f \log \left (f\right )^{2} + i \, c f^{2} \log \left (f\right ) - 3 \, 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)*sin(f*x^2+d)^3,x, algorithm="fricas")

[Out]

1/16*(sqrt(pi)*(-I*c^3*log(f)^3 - 3*c^2*f*log(f)^2 - I*c*f^2*log(f) - 3*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*I*c^3*log(f)^3 + 3*c^2*f*log(f)^2 + 27*I*c*f^2*log(f)
+ 27*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*I*c^3*log(f)^3
+ 3*c^2*f*log(f)^2 - 27*I*c*f^2*log(f) + 27*f^3)*sqrt(-c*log(f) + I*f)*erf(sqrt(-c*log(f) + I*f)*x)*e^(a*log(
f) - I*d) + sqrt(pi)*(I*c^3*log(f)^3 - 3*c^2*f*log(f)^2 + I*c*f^2*log(f) - 3*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))/(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)*sin(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} \sin \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)*sin(f*x^2+d)^3,x, algorithm="giac")

[Out]

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