3.182 \(\int (e+f x)^2 \sin (a+\frac{b}{(c+d x)^3}) \, dx\)

Optimal. Leaf size=330 \[ -\frac{i e^{i a} f (c+d x)^2 \left (-\frac{i b}{(c+d x)^3}\right )^{2/3} (d e-c f) \text{Gamma}\left (-\frac{2}{3},-\frac{i b}{(c+d x)^3}\right )}{3 d^3}+\frac{i e^{-i a} f (c+d x)^2 \left (\frac{i b}{(c+d x)^3}\right )^{2/3} (d e-c f) \text{Gamma}\left (-\frac{2}{3},\frac{i b}{(c+d x)^3}\right )}{3 d^3}-\frac{i e^{i a} (c+d x) \sqrt [3]{-\frac{i b}{(c+d x)^3}} (d e-c f)^2 \text{Gamma}\left (-\frac{1}{3},-\frac{i b}{(c+d x)^3}\right )}{6 d^3}+\frac{i e^{-i a} (c+d x) \sqrt [3]{\frac{i b}{(c+d x)^3}} (d e-c f)^2 \text{Gamma}\left (-\frac{1}{3},\frac{i b}{(c+d x)^3}\right )}{6 d^3}-\frac{b f^2 \cos (a) \text{CosIntegral}\left (\frac{b}{(c+d x)^3}\right )}{3 d^3}+\frac{b f^2 \sin (a) \text{Si}\left (\frac{b}{(c+d x)^3}\right )}{3 d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{(c+d x)^3}\right )}{3 d^3} \]

[Out]

-(b*f^2*Cos[a]*CosIntegral[b/(c + d*x)^3])/(3*d^3) - ((I/3)*E^(I*a)*f*(d*e - c*f)*(((-I)*b)/(c + d*x)^3)^(2/3)
*(c + d*x)^2*Gamma[-2/3, ((-I)*b)/(c + d*x)^3])/d^3 + ((I/3)*f*(d*e - c*f)*((I*b)/(c + d*x)^3)^(2/3)*(c + d*x)
^2*Gamma[-2/3, (I*b)/(c + d*x)^3])/(d^3*E^(I*a)) - ((I/6)*E^(I*a)*(d*e - c*f)^2*(((-I)*b)/(c + d*x)^3)^(1/3)*(
c + d*x)*Gamma[-1/3, ((-I)*b)/(c + d*x)^3])/d^3 + ((I/6)*(d*e - c*f)^2*((I*b)/(c + d*x)^3)^(1/3)*(c + d*x)*Gam
ma[-1/3, (I*b)/(c + d*x)^3])/(d^3*E^(I*a)) + (f^2*(c + d*x)^3*Sin[a + b/(c + d*x)^3])/(3*d^3) + (b*f^2*Sin[a]*
SinIntegral[b/(c + d*x)^3])/(3*d^3)

________________________________________________________________________________________

Rubi [A]  time = 0.300248, antiderivative size = 330, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3433, 3365, 2208, 3423, 2218, 3379, 3297, 3303, 3299, 3302} \[ -\frac{i e^{i a} f (c+d x)^2 \left (-\frac{i b}{(c+d x)^3}\right )^{2/3} (d e-c f) \text{Gamma}\left (-\frac{2}{3},-\frac{i b}{(c+d x)^3}\right )}{3 d^3}+\frac{i e^{-i a} f (c+d x)^2 \left (\frac{i b}{(c+d x)^3}\right )^{2/3} (d e-c f) \text{Gamma}\left (-\frac{2}{3},\frac{i b}{(c+d x)^3}\right )}{3 d^3}-\frac{i e^{i a} (c+d x) \sqrt [3]{-\frac{i b}{(c+d x)^3}} (d e-c f)^2 \text{Gamma}\left (-\frac{1}{3},-\frac{i b}{(c+d x)^3}\right )}{6 d^3}+\frac{i e^{-i a} (c+d x) \sqrt [3]{\frac{i b}{(c+d x)^3}} (d e-c f)^2 \text{Gamma}\left (-\frac{1}{3},\frac{i b}{(c+d x)^3}\right )}{6 d^3}-\frac{b f^2 \cos (a) \text{CosIntegral}\left (\frac{b}{(c+d x)^3}\right )}{3 d^3}+\frac{b f^2 \sin (a) \text{Si}\left (\frac{b}{(c+d x)^3}\right )}{3 d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{(c+d x)^3}\right )}{3 d^3} \]

Antiderivative was successfully verified.

[In]

Int[(e + f*x)^2*Sin[a + b/(c + d*x)^3],x]

[Out]

-(b*f^2*Cos[a]*CosIntegral[b/(c + d*x)^3])/(3*d^3) - ((I/3)*E^(I*a)*f*(d*e - c*f)*(((-I)*b)/(c + d*x)^3)^(2/3)
*(c + d*x)^2*Gamma[-2/3, ((-I)*b)/(c + d*x)^3])/d^3 + ((I/3)*f*(d*e - c*f)*((I*b)/(c + d*x)^3)^(2/3)*(c + d*x)
^2*Gamma[-2/3, (I*b)/(c + d*x)^3])/(d^3*E^(I*a)) - ((I/6)*E^(I*a)*(d*e - c*f)^2*(((-I)*b)/(c + d*x)^3)^(1/3)*(
c + d*x)*Gamma[-1/3, ((-I)*b)/(c + d*x)^3])/d^3 + ((I/6)*(d*e - c*f)^2*((I*b)/(c + d*x)^3)^(1/3)*(c + d*x)*Gam
ma[-1/3, (I*b)/(c + d*x)^3])/(d^3*E^(I*a)) + (f^2*(c + d*x)^3*Sin[a + b/(c + d*x)^3])/(3*d^3) + (b*f^2*Sin[a]*
SinIntegral[b/(c + d*x)^3])/(3*d^3)

Rule 3433

Int[((g_.) + (h_.)*(x_))^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)])^(p_.), x_Symbol] :
> Module[{k = If[FractionQ[n], Denominator[n], 1]}, Dist[k/f^(m + 1), Subst[Int[ExpandIntegrand[(a + b*Sin[c +
d*x^(k*n)])^p, x^(k - 1)*(f*g - e*h + h*x^k)^m, x], x], x, (e + f*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, f,
g, h}, x] && IGtQ[p, 0] && IGtQ[m, 0]

Rule 3365

Int[Sin[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)], x_Symbol] :> Dist[I/2, Int[E^(-(c*I) - d*I*(e + f*x)^n), x],
x] - Dist[I/2, Int[E^(c*I + d*I*(e + f*x)^n), x], x] /; FreeQ[{c, d, e, f, n}, x]

Rule 2208

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> -Simp[(F^a*(c + d*x)*Gamma[1/n, -(b*(c + d*x)
^n*Log[F])])/(d*n*(-(b*(c + d*x)^n*Log[F]))^(1/n)), x] /; FreeQ[{F, a, b, c, d, n}, x] &&  !IntegerQ[2/n]

Rule 3423

Int[((e_.)*(x_))^(m_.)*Sin[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> Dist[I/2, Int[(e*x)^m*E^(-(c*I) - d*I*x^n),
x], x] - Dist[I/2, Int[(e*x)^m*E^(c*I + d*I*x^n), x], x] /; FreeQ[{c, d, e, m, n}, x]

Rule 2218

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> -Simp[(F^a*(e + f*
x)^(m + 1)*Gamma[(m + 1)/n, -(b*(c + d*x)^n*Log[F])])/(f*n*(-(b*(c + d*x)^n*Log[F]))^((m + 1)/n)), x] /; FreeQ
[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

Rule 3379

Int[(x_)^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Sin[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simpl
ify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0]))

Rule 3297

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[((c + d*x)^(m + 1)*Sin[e + f*x])/(d*(
m + 1)), x] - Dist[f/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[
m, -1]

Rule 3303

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[(c*f)/d + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 3299

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 3302

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rubi steps

\begin{align*} \int (e+f x)^2 \sin \left (a+\frac{b}{(c+d x)^3}\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \left (d^2 e^2 \left (1+\frac{c f (-2 d e+c f)}{d^2 e^2}\right ) \sin \left (a+\frac{b}{x^3}\right )+2 d e f \left (1-\frac{c f}{d e}\right ) x \sin \left (a+\frac{b}{x^3}\right )+f^2 x^2 \sin \left (a+\frac{b}{x^3}\right )\right ) \, dx,x,c+d x\right )}{d^3}\\ &=\frac{f^2 \operatorname{Subst}\left (\int x^2 \sin \left (a+\frac{b}{x^3}\right ) \, dx,x,c+d x\right )}{d^3}+\frac{(2 f (d e-c f)) \operatorname{Subst}\left (\int x \sin \left (a+\frac{b}{x^3}\right ) \, dx,x,c+d x\right )}{d^3}+\frac{(d e-c f)^2 \operatorname{Subst}\left (\int \sin \left (a+\frac{b}{x^3}\right ) \, dx,x,c+d x\right )}{d^3}\\ &=-\frac{f^2 \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^2} \, dx,x,\frac{1}{(c+d x)^3}\right )}{3 d^3}+\frac{(i f (d e-c f)) \operatorname{Subst}\left (\int e^{-i a-\frac{i b}{x^3}} x \, dx,x,c+d x\right )}{d^3}-\frac{(i f (d e-c f)) \operatorname{Subst}\left (\int e^{i a+\frac{i b}{x^3}} x \, dx,x,c+d x\right )}{d^3}+\frac{\left (i (d e-c f)^2\right ) \operatorname{Subst}\left (\int e^{-i a-\frac{i b}{x^3}} \, dx,x,c+d x\right )}{2 d^3}-\frac{\left (i (d e-c f)^2\right ) \operatorname{Subst}\left (\int e^{i a+\frac{i b}{x^3}} \, dx,x,c+d x\right )}{2 d^3}\\ &=-\frac{i e^{i a} f (d e-c f) \left (-\frac{i b}{(c+d x)^3}\right )^{2/3} (c+d x)^2 \Gamma \left (-\frac{2}{3},-\frac{i b}{(c+d x)^3}\right )}{3 d^3}+\frac{i e^{-i a} f (d e-c f) \left (\frac{i b}{(c+d x)^3}\right )^{2/3} (c+d x)^2 \Gamma \left (-\frac{2}{3},\frac{i b}{(c+d x)^3}\right )}{3 d^3}-\frac{i e^{i a} (d e-c f)^2 \sqrt [3]{-\frac{i b}{(c+d x)^3}} (c+d x) \Gamma \left (-\frac{1}{3},-\frac{i b}{(c+d x)^3}\right )}{6 d^3}+\frac{i e^{-i a} (d e-c f)^2 \sqrt [3]{\frac{i b}{(c+d x)^3}} (c+d x) \Gamma \left (-\frac{1}{3},\frac{i b}{(c+d x)^3}\right )}{6 d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{(c+d x)^3}\right )}{3 d^3}-\frac{\left (b f^2\right ) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x} \, dx,x,\frac{1}{(c+d x)^3}\right )}{3 d^3}\\ &=-\frac{i e^{i a} f (d e-c f) \left (-\frac{i b}{(c+d x)^3}\right )^{2/3} (c+d x)^2 \Gamma \left (-\frac{2}{3},-\frac{i b}{(c+d x)^3}\right )}{3 d^3}+\frac{i e^{-i a} f (d e-c f) \left (\frac{i b}{(c+d x)^3}\right )^{2/3} (c+d x)^2 \Gamma \left (-\frac{2}{3},\frac{i b}{(c+d x)^3}\right )}{3 d^3}-\frac{i e^{i a} (d e-c f)^2 \sqrt [3]{-\frac{i b}{(c+d x)^3}} (c+d x) \Gamma \left (-\frac{1}{3},-\frac{i b}{(c+d x)^3}\right )}{6 d^3}+\frac{i e^{-i a} (d e-c f)^2 \sqrt [3]{\frac{i b}{(c+d x)^3}} (c+d x) \Gamma \left (-\frac{1}{3},\frac{i b}{(c+d x)^3}\right )}{6 d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{(c+d x)^3}\right )}{3 d^3}-\frac{\left (b f^2 \cos (a)\right ) \operatorname{Subst}\left (\int \frac{\cos (b x)}{x} \, dx,x,\frac{1}{(c+d x)^3}\right )}{3 d^3}+\frac{\left (b f^2 \sin (a)\right ) \operatorname{Subst}\left (\int \frac{\sin (b x)}{x} \, dx,x,\frac{1}{(c+d x)^3}\right )}{3 d^3}\\ &=-\frac{b f^2 \cos (a) \text{Ci}\left (\frac{b}{(c+d x)^3}\right )}{3 d^3}-\frac{i e^{i a} f (d e-c f) \left (-\frac{i b}{(c+d x)^3}\right )^{2/3} (c+d x)^2 \Gamma \left (-\frac{2}{3},-\frac{i b}{(c+d x)^3}\right )}{3 d^3}+\frac{i e^{-i a} f (d e-c f) \left (\frac{i b}{(c+d x)^3}\right )^{2/3} (c+d x)^2 \Gamma \left (-\frac{2}{3},\frac{i b}{(c+d x)^3}\right )}{3 d^3}-\frac{i e^{i a} (d e-c f)^2 \sqrt [3]{-\frac{i b}{(c+d x)^3}} (c+d x) \Gamma \left (-\frac{1}{3},-\frac{i b}{(c+d x)^3}\right )}{6 d^3}+\frac{i e^{-i a} (d e-c f)^2 \sqrt [3]{\frac{i b}{(c+d x)^3}} (c+d x) \Gamma \left (-\frac{1}{3},\frac{i b}{(c+d x)^3}\right )}{6 d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{(c+d x)^3}\right )}{3 d^3}+\frac{b f^2 \sin (a) \text{Si}\left (\frac{b}{(c+d x)^3}\right )}{3 d^3}\\ \end{align*}

Mathematica [A]  time = 2.52395, size = 405, normalized size = 1.23 \[ \frac{\frac{3 b f (d e-c f) \left ((\cos (a)-i \sin (a)) \sqrt [3]{-\frac{i b}{(c+d x)^3}} \text{Gamma}\left (\frac{1}{3},\frac{i b}{(c+d x)^3}\right )+(\cos (a)+i \sin (a)) \sqrt [3]{\frac{i b}{(c+d x)^3}} \text{Gamma}\left (\frac{1}{3},-\frac{i b}{(c+d x)^3}\right )\right )}{2 (c+d x) \sqrt [3]{\frac{b^2}{(c+d x)^6}}}+\frac{3 b (d e-c f)^2 \left ((\cos (a)-i \sin (a)) \left (-\frac{i b}{(c+d x)^3}\right )^{2/3} \text{Gamma}\left (\frac{2}{3},\frac{i b}{(c+d x)^3}\right )+(\cos (a)+i \sin (a)) \left (\frac{i b}{(c+d x)^3}\right )^{2/3} \text{Gamma}\left (\frac{2}{3},-\frac{i b}{(c+d x)^3}\right )\right )}{2 (c+d x)^2 \left (\frac{b^2}{(c+d x)^6}\right )^{2/3}}+\sin (a) (c+d x) \cos \left (\frac{b}{(c+d x)^3}\right ) \left (c^2 f^2-c d f (3 e+f x)+d^2 \left (3 e^2+3 e f x+f^2 x^2\right )\right )+\cos (a) (c+d x) \sin \left (\frac{b}{(c+d x)^3}\right ) \left (c^2 f^2-c d f (3 e+f x)+d^2 \left (3 e^2+3 e f x+f^2 x^2\right )\right )-b f^2 \left (\cos (a) \text{CosIntegral}\left (\frac{b}{(c+d x)^3}\right )-\sin (a) \text{Si}\left (\frac{b}{(c+d x)^3}\right )\right )}{3 d^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(e + f*x)^2*Sin[a + b/(c + d*x)^3],x]

[Out]

((3*b*f*(d*e - c*f)*((((-I)*b)/(c + d*x)^3)^(1/3)*Gamma[1/3, (I*b)/(c + d*x)^3]*(Cos[a] - I*Sin[a]) + ((I*b)/(
c + d*x)^3)^(1/3)*Gamma[1/3, ((-I)*b)/(c + d*x)^3]*(Cos[a] + I*Sin[a])))/(2*(b^2/(c + d*x)^6)^(1/3)*(c + d*x))
+ (3*b*(d*e - c*f)^2*((((-I)*b)/(c + d*x)^3)^(2/3)*Gamma[2/3, (I*b)/(c + d*x)^3]*(Cos[a] - I*Sin[a]) + ((I*b)
/(c + d*x)^3)^(2/3)*Gamma[2/3, ((-I)*b)/(c + d*x)^3]*(Cos[a] + I*Sin[a])))/(2*(b^2/(c + d*x)^6)^(2/3)*(c + d*x
)^2) + (c + d*x)*(c^2*f^2 - c*d*f*(3*e + f*x) + d^2*(3*e^2 + 3*e*f*x + f^2*x^2))*Cos[b/(c + d*x)^3]*Sin[a] + (
c + d*x)*(c^2*f^2 - c*d*f*(3*e + f*x) + d^2*(3*e^2 + 3*e*f*x + f^2*x^2))*Cos[a]*Sin[b/(c + d*x)^3] - b*f^2*(Co
s[a]*CosIntegral[b/(c + d*x)^3] - Sin[a]*SinIntegral[b/(c + d*x)^3]))/(3*d^3)

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Maple [F]  time = 0.289, size = 0, normalized size = 0. \begin{align*} \int \left ( fx+e \right ) ^{2}\sin \left ( a+{\frac{b}{ \left ( dx+c \right ) ^{3}}} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{3} \,{\left (f^{2} x^{3} + 3 \, e f x^{2} + 3 \, e^{2} x\right )} \sin \left (\frac{a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) + \int \frac{{\left (b d f^{2} x^{3} + 3 \, b d e f x^{2} + 3 \, b d e^{2} x\right )} \cos \left (\frac{a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}{2 \,{\left (d^{4} x^{4} + 4 \, c d^{3} x^{3} + 6 \, c^{2} d^{2} x^{2} + 4 \, c^{3} d x + c^{4}\right )}}\,{d x} + \int \frac{{\left (b d f^{2} x^{3} + 3 \, b d e f x^{2} + 3 \, b d e^{2} x\right )} \cos \left (\frac{a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}{2 \,{\left ({\left (d^{4} x^{4} + 4 \, c d^{3} x^{3} + 6 \, c^{2} d^{2} x^{2} + 4 \, c^{3} d x + c^{4}\right )} \cos \left (\frac{a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )^{2} +{\left (d^{4} x^{4} + 4 \, c d^{3} x^{3} + 6 \, c^{2} d^{2} x^{2} + 4 \, c^{3} d x + c^{4}\right )} \sin \left (\frac{a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )^{2}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^2*sin(a+b/(d*x+c)^3),x, algorithm="maxima")

[Out]

1/3*(f^2*x^3 + 3*e*f*x^2 + 3*e^2*x)*sin((a*d^3*x^3 + 3*a*c*d^2*x^2 + 3*a*c^2*d*x + a*c^3 + b)/(d^3*x^3 + 3*c*d
^2*x^2 + 3*c^2*d*x + c^3)) + integrate(1/2*(b*d*f^2*x^3 + 3*b*d*e*f*x^2 + 3*b*d*e^2*x)*cos((a*d^3*x^3 + 3*a*c*
d^2*x^2 + 3*a*c^2*d*x + a*c^3 + b)/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3))/(d^4*x^4 + 4*c*d^3*x^3 + 6*c^2*d
^2*x^2 + 4*c^3*d*x + c^4), x) + integrate(1/2*(b*d*f^2*x^3 + 3*b*d*e*f*x^2 + 3*b*d*e^2*x)*cos((a*d^3*x^3 + 3*a
*c*d^2*x^2 + 3*a*c^2*d*x + a*c^3 + b)/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3))/((d^4*x^4 + 4*c*d^3*x^3 + 6*c
^2*d^2*x^2 + 4*c^3*d*x + c^4)*cos((a*d^3*x^3 + 3*a*c*d^2*x^2 + 3*a*c^2*d*x + a*c^3 + b)/(d^3*x^3 + 3*c*d^2*x^2
+ 3*c^2*d*x + c^3))^2 + (d^4*x^4 + 4*c*d^3*x^3 + 6*c^2*d^2*x^2 + 4*c^3*d*x + c^4)*sin((a*d^3*x^3 + 3*a*c*d^2*
x^2 + 3*a*c^2*d*x + a*c^3 + b)/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3))^2), x)

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Fricas [A]  time = 2.08244, size = 1116, normalized size = 3.38 \begin{align*} -\frac{b f^{2}{\rm Ei}\left (\frac{i \, b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) e^{\left (i \, a\right )} + b f^{2}{\rm Ei}\left (-\frac{i \, b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) e^{\left (-i \, a\right )} -{\left (-3 i \, d^{3} e f + 3 i \, c d^{2} f^{2}\right )} \left (\frac{i \, b}{d^{3}}\right )^{\frac{2}{3}} e^{\left (-i \, a\right )} \Gamma \left (\frac{1}{3}, \frac{i \, b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) -{\left (3 i \, d^{3} e f - 3 i \, c d^{2} f^{2}\right )} \left (-\frac{i \, b}{d^{3}}\right )^{\frac{2}{3}} e^{\left (i \, a\right )} \Gamma \left (\frac{1}{3}, -\frac{i \, b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) -{\left (-3 i \, d^{3} e^{2} + 6 i \, c d^{2} e f - 3 i \, c^{2} d f^{2}\right )} \left (\frac{i \, b}{d^{3}}\right )^{\frac{1}{3}} e^{\left (-i \, a\right )} \Gamma \left (\frac{2}{3}, \frac{i \, b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) -{\left (3 i \, d^{3} e^{2} - 6 i \, c d^{2} e f + 3 i \, c^{2} d f^{2}\right )} \left (-\frac{i \, b}{d^{3}}\right )^{\frac{1}{3}} e^{\left (i \, a\right )} \Gamma \left (\frac{2}{3}, -\frac{i \, b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) - 2 \,{\left (d^{3} f^{2} x^{3} + 3 \, d^{3} e f x^{2} + 3 \, d^{3} e^{2} x + 3 \, c d^{2} e^{2} - 3 \, c^{2} d e f + c^{3} f^{2}\right )} \sin \left (\frac{a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}{6 \, d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^2*sin(a+b/(d*x+c)^3),x, algorithm="fricas")

[Out]

-1/6*(b*f^2*Ei(I*b/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3))*e^(I*a) + b*f^2*Ei(-I*b/(d^3*x^3 + 3*c*d^2*x^2 +
3*c^2*d*x + c^3))*e^(-I*a) - (-3*I*d^3*e*f + 3*I*c*d^2*f^2)*(I*b/d^3)^(2/3)*e^(-I*a)*gamma(1/3, I*b/(d^3*x^3
+ 3*c*d^2*x^2 + 3*c^2*d*x + c^3)) - (3*I*d^3*e*f - 3*I*c*d^2*f^2)*(-I*b/d^3)^(2/3)*e^(I*a)*gamma(1/3, -I*b/(d^
3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)) - (-3*I*d^3*e^2 + 6*I*c*d^2*e*f - 3*I*c^2*d*f^2)*(I*b/d^3)^(1/3)*e^(-I
*a)*gamma(2/3, I*b/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)) - (3*I*d^3*e^2 - 6*I*c*d^2*e*f + 3*I*c^2*d*f^2)*
(-I*b/d^3)^(1/3)*e^(I*a)*gamma(2/3, -I*b/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)) - 2*(d^3*f^2*x^3 + 3*d^3*e
*f*x^2 + 3*d^3*e^2*x + 3*c*d^2*e^2 - 3*c^2*d*e*f + c^3*f^2)*sin((a*d^3*x^3 + 3*a*c*d^2*x^2 + 3*a*c^2*d*x + a*c
^3 + b)/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)))/d^3

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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*x+e)**2*sin(a+b/(d*x+c)**3),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^2*sin(a+b/(d*x+c)^3),x, algorithm="giac")

[Out]

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