\(\int f^{a+c x^2} \sin (d+f x^2) \, dx\) [88]

Optimal result

Integrand size = 18, antiderivative size = 107 \[ \int f^{a+c x^2} \sin \left (d+f x^2\right ) \, dx=\frac {i e^{-i d} f^a \sqrt {\pi } \text {erf}\left (x \sqrt {i f-c \log (f)}\right )}{4 \sqrt {i f-c \log (f)}}-\frac {i e^{i d} f^a \sqrt {\pi } \text {erfi}\left (x \sqrt {i f+c \log (f)}\right )}{4 \sqrt {i f+c \log (f)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4560, 2325, 2236, 2235} \[ \int f^{a+c x^2} \sin \left (d+f x^2\right ) \, dx=\frac {i \sqrt {\pi } e^{-i d} f^a \text {erf}\left (x \sqrt {-c \log (f)+i f}\right )}{4 \sqrt {-c \log (f)+i f}}-\frac {i \sqrt {\pi } e^{i d} f^a \text {erfi}\left (x \sqrt {c \log (f)+i f}\right )}{4 \sqrt {c \log (f)+i f}} \]

[In]

Int[f^(a + c*x^2)*Sin[d + f*x^2],x]

[Out]

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

Rule 2235

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 2236

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 2325

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 4560

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]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{2} i e^{-i d-i f x^2} f^{a+c x^2}-\frac {1}{2} i e^{i d+i f x^2} f^{a+c x^2}\right ) \, dx \\ & = \frac {1}{2} i \int e^{-i d-i f x^2} f^{a+c x^2} \, dx-\frac {1}{2} i \int e^{i d+i f x^2} f^{a+c x^2} \, dx \\ & = \frac {1}{2} i \int e^{-i d+a \log (f)-x^2 (i f-c \log (f))} \, dx-\frac {1}{2} i \int e^{i d+a \log (f)+x^2 (i f+c \log (f))} \, dx \\ & = \frac {i e^{-i d} f^a \sqrt {\pi } \text {erf}\left (x \sqrt {i f-c \log (f)}\right )}{4 \sqrt {i f-c \log (f)}}-\frac {i e^{i d} f^a \sqrt {\pi } \text {erfi}\left (x \sqrt {i f+c \log (f)}\right )}{4 \sqrt {i f+c \log (f)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.59 \[ \int f^{a+c x^2} \sin \left (d+f x^2\right ) \, dx=-\frac {\sqrt [4]{-1} f^a \sqrt {\pi } \left (\text {erfi}\left (\sqrt [4]{-1} x \sqrt {f-i c \log (f)}\right ) \sqrt {f-i c \log (f)} (f+i c \log (f)) (\cos (d)+i \sin (d))+\sqrt {f+i c \log (f)} \left (c \text {erf}\left (\frac {(1+i) x \sqrt {f+i c \log (f)}}{\sqrt {2}}\right ) \log (f) \sin (d)+\text {erfi}\left ((-1)^{3/4} x \sqrt {f+i c \log (f)}\right ) (\cos (d) (i f+c \log (f))+f \sin (d))\right )\right )}{4 \left (f^2+c^2 \log ^2(f)\right )} \]

[In]

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

[Out]

-1/4*((-1)^(1/4)*f^a*Sqrt[Pi]*(Erfi[(-1)^(1/4)*x*Sqrt[f - I*c*Log[f]]]*Sqrt[f - I*c*Log[f]]*(f + I*c*Log[f])*(
Cos[d] + I*Sin[d]) + Sqrt[f + I*c*Log[f]]*(c*Erf[((1 + I)*x*Sqrt[f + I*c*Log[f]])/Sqrt[2]]*Log[f]*Sin[d] + Erf
i[(-1)^(3/4)*x*Sqrt[f + I*c*Log[f]]]*(Cos[d]*(I*f + c*Log[f]) + f*Sin[d]))))/(f^2 + c^2*Log[f]^2)

Maple [A] (verified)

Time = 0.39 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.79

method	result	size
risch	\(-\frac {i \sqrt {\pi }\, f^{a} {\mathrm e}^{i d} \operatorname {erf}\left (\sqrt {-c \ln \left (f \right )-i f}\, x \right )}{4 \sqrt {-c \ln \left (f \right )-i f}}+\frac {i \sqrt {\pi }\, f^{a} {\mathrm e}^{-i d} \operatorname {erf}\left (x \sqrt {i f -c \ln \left (f \right )}\right )}{4 \sqrt {i f -c \ln \left (f \right )}}\)	\(84\)

[In]

int(f^(c*x^2+a)*sin(f*x^2+d),x,method=_RETURNVERBOSE)

[Out]

-1/4*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)+1/4*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))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00 \[ \int f^{a+c x^2} \sin \left (d+f x^2\right ) \, dx=\frac {\sqrt {\pi } {\left (i \, c \log \left (f\right ) + f\right )} \sqrt {-c \log \left (f\right ) - i \, f} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) - i \, f} x\right ) e^{\left (a \log \left (f\right ) + i \, d\right )} + \sqrt {\pi } {\left (-i \, c \log \left (f\right ) + f\right )} \sqrt {-c \log \left (f\right ) + i \, f} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) + i \, f} x\right ) e^{\left (a \log \left (f\right ) - i \, d\right )}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int f^{a+c x^2} \sin \left (d+f x^2\right ) \, dx=\int f^{a + c x^{2}} \sin {\left (d + f x^{2} \right )}\, dx \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 209 vs. \(2 (73) = 146\).

Time = 0.24 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.95 \[ \int f^{a+c x^2} \sin \left (d+f x^2\right ) \, dx=\frac {\sqrt {\pi } \sqrt {2 \, c^{2} \log \left (f\right )^{2} + 2 \, f^{2}} {\left (f^{a} {\left (\cos \left (d\right ) - i \, \sin \left (d\right )\right )} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) + i \, f} x\right ) + f^{a} {\left (\cos \left (d\right ) + i \, \sin \left (d\right )\right )} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) - i \, f} x\right )\right )} \sqrt {c \log \left (f\right ) + \sqrt {c^{2} \log \left (f\right )^{2} + f^{2}}} - \sqrt {\pi } \sqrt {2 \, c^{2} \log \left (f\right )^{2} + 2 \, f^{2}} {\left (f^{a} {\left (-i \, \cos \left (d\right ) - \sin \left (d\right )\right )} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) + i \, f} x\right ) + f^{a} {\left (i \, \cos \left (d\right ) - \sin \left (d\right )\right )} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) - i \, f} x\right )\right )} \sqrt {-c \log \left (f\right ) + \sqrt {c^{2} \log \left (f\right )^{2} + f^{2}}}}{8 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}} \]

[In]

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

[Out]

1/8*(sqrt(pi)*sqrt(2*c^2*log(f)^2 + 2*f^2)*(f^a*(cos(d) - I*sin(d))*erf(sqrt(-c*log(f) + I*f)*x) + f^a*(cos(d)
+ I*sin(d))*erf(sqrt(-c*log(f) - I*f)*x))*sqrt(c*log(f) + sqrt(c^2*log(f)^2 + f^2)) - sqrt(pi)*sqrt(2*c^2*log
(f)^2 + 2*f^2)*(f^a*(-I*cos(d) - sin(d))*erf(sqrt(-c*log(f) + I*f)*x) + f^a*(I*cos(d) - sin(d))*erf(sqrt(-c*lo
g(f) - I*f)*x))*sqrt(-c*log(f) + sqrt(c^2*log(f)^2 + f^2)))/(c^2*log(f)^2 + f^2)

Giac [F]

\[ \int f^{a+c x^2} \sin \left (d+f x^2\right ) \, dx=\int { f^{c x^{2} + a} \sin \left (f x^{2} + d\right ) \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int f^{a+c x^2} \sin \left (d+f x^2\right ) \, dx=\int f^{c\,x^2+a}\,\sin \left (f\,x^2+d\right ) \,d x \]

[In]

int(f^(a + c*x^2)*sin(d + f*x^2),x)

[Out]

int(f^(a + c*x^2)*sin(d + f*x^2), x)