3.3.22 \(\int (e+f x)^2 \sin (a+\frac {b}{(c+d x)^{2/3}}) \, dx\) [222]

Optimal. Leaf size=630 \[ \frac {2 b (d e-c f)^2 \sqrt [3]{c+d x} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}-\frac {8 b^3 f^2 (c+d x) \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{315 d^3}+\frac {b f (d e-c f) (c+d x)^{4/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac {2 b f^2 (c+d x)^{7/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{21 d^3}+\frac {b^3 f (d e-c f) \cos (a) \text {Ci}\left (\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}-\frac {16 b^{9/2} f^2 \sqrt {2 \pi } \cos (a) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{315 d^3}+\frac {2 b^{3/2} (d e-c f)^2 \sqrt {2 \pi } \cos (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{d^3}+\frac {2 b^{3/2} (d e-c f)^2 \sqrt {2 \pi } C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right ) \sin (a)}{d^3}+\frac {16 b^{9/2} f^2 \sqrt {2 \pi } S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right ) \sin (a)}{315 d^3}+\frac {16 b^4 f^2 \sqrt [3]{c+d x} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{315 d^3}-\frac {b^2 f (d e-c f) (c+d x)^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac {(d e-c f)^2 (c+d x) \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}-\frac {4 b^2 f^2 (c+d x)^{5/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{105 d^3}+\frac {f (d e-c f) (c+d x)^2 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}+\frac {f^2 (c+d x)^3 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{3 d^3}-\frac {b^3 f (d e-c f) \sin (a) \text {Si}\left (\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3} \]

[Out]

1/2*b^3*f*(-c*f+d*e)*Ci(b/(d*x+c)^(2/3))*cos(a)/d^3+2*b*(-c*f+d*e)^2*(d*x+c)^(1/3)*cos(a+b/(d*x+c)^(2/3))/d^3-
8/315*b^3*f^2*(d*x+c)*cos(a+b/(d*x+c)^(2/3))/d^3+1/2*b*f*(-c*f+d*e)*(d*x+c)^(4/3)*cos(a+b/(d*x+c)^(2/3))/d^3+2
/21*b*f^2*(d*x+c)^(7/3)*cos(a+b/(d*x+c)^(2/3))/d^3-1/2*b^3*f*(-c*f+d*e)*Si(b/(d*x+c)^(2/3))*sin(a)/d^3+16/315*
b^4*f^2*(d*x+c)^(1/3)*sin(a+b/(d*x+c)^(2/3))/d^3-1/2*b^2*f*(-c*f+d*e)*(d*x+c)^(2/3)*sin(a+b/(d*x+c)^(2/3))/d^3
+(-c*f+d*e)^2*(d*x+c)*sin(a+b/(d*x+c)^(2/3))/d^3-4/105*b^2*f^2*(d*x+c)^(5/3)*sin(a+b/(d*x+c)^(2/3))/d^3+f*(-c*
f+d*e)*(d*x+c)^2*sin(a+b/(d*x+c)^(2/3))/d^3+1/3*f^2*(d*x+c)^3*sin(a+b/(d*x+c)^(2/3))/d^3-16/315*b^(9/2)*f^2*co
s(a)*FresnelC(b^(1/2)*2^(1/2)/Pi^(1/2)/(d*x+c)^(1/3))*2^(1/2)*Pi^(1/2)/d^3+2*b^(3/2)*(-c*f+d*e)^2*cos(a)*Fresn
elS(b^(1/2)*2^(1/2)/Pi^(1/2)/(d*x+c)^(1/3))*2^(1/2)*Pi^(1/2)/d^3+2*b^(3/2)*(-c*f+d*e)^2*FresnelC(b^(1/2)*2^(1/
2)/Pi^(1/2)/(d*x+c)^(1/3))*sin(a)*2^(1/2)*Pi^(1/2)/d^3+16/315*b^(9/2)*f^2*FresnelS(b^(1/2)*2^(1/2)/Pi^(1/2)/(d
*x+c)^(1/3))*sin(a)*2^(1/2)*Pi^(1/2)/d^3

________________________________________________________________________________________

Rubi [A]
time = 0.52, antiderivative size = 630, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 13, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {3514, 3490, 3468, 3469, 3434, 3433, 3432, 3460, 3378, 3384, 3380, 3383, 3435} \begin {gather*} \frac {2 \sqrt {2 \pi } b^{3/2} \sin (a) (d e-c f)^2 \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {b}}{\sqrt [3]{c+d x}}\right )}{d^3}+\frac {2 \sqrt {2 \pi } b^{3/2} \cos (a) (d e-c f)^2 S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{d^3}-\frac {16 \sqrt {2 \pi } b^{9/2} f^2 \cos (a) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {b}}{\sqrt [3]{c+d x}}\right )}{315 d^3}+\frac {16 \sqrt {2 \pi } b^{9/2} f^2 \sin (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{315 d^3}+\frac {16 b^4 f^2 \sqrt [3]{c+d x} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{315 d^3}+\frac {b^3 f \cos (a) (d e-c f) \text {CosIntegral}\left (\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}-\frac {b^3 f \sin (a) (d e-c f) \text {Si}\left (\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}-\frac {8 b^3 f^2 (c+d x) \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{315 d^3}-\frac {b^2 f (c+d x)^{2/3} (d e-c f) \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}-\frac {4 b^2 f^2 (c+d x)^{5/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{105 d^3}+\frac {f (c+d x)^2 (d e-c f) \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}+\frac {(c+d x) (d e-c f)^2 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}+\frac {b f (c+d x)^{4/3} (d e-c f) \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac {2 b \sqrt [3]{c+d x} (d e-c f)^2 \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}+\frac {f^2 (c+d x)^3 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{3 d^3}+\frac {2 b f^2 (c+d x)^{7/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{21 d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b*(d*e - c*f)^2*(c + d*x)^(1/3)*Cos[a + b/(c + d*x)^(2/3)])/d^3 - (8*b^3*f^2*(c + d*x)*Cos[a + b/(c + d*x)^
(2/3)])/(315*d^3) + (b*f*(d*e - c*f)*(c + d*x)^(4/3)*Cos[a + b/(c + d*x)^(2/3)])/(2*d^3) + (2*b*f^2*(c + d*x)^
(7/3)*Cos[a + b/(c + d*x)^(2/3)])/(21*d^3) + (b^3*f*(d*e - c*f)*Cos[a]*CosIntegral[b/(c + d*x)^(2/3)])/(2*d^3)
- (16*b^(9/2)*f^2*Sqrt[2*Pi]*Cos[a]*FresnelC[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)^(1/3)])/(315*d^3) + (2*b^(3/2)*(d
*e - c*f)^2*Sqrt[2*Pi]*Cos[a]*FresnelS[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)^(1/3)])/d^3 + (2*b^(3/2)*(d*e - c*f)^2*S
qrt[2*Pi]*FresnelC[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)^(1/3)]*Sin[a])/d^3 + (16*b^(9/2)*f^2*Sqrt[2*Pi]*FresnelS[(Sq
rt[b]*Sqrt[2/Pi])/(c + d*x)^(1/3)]*Sin[a])/(315*d^3) + (16*b^4*f^2*(c + d*x)^(1/3)*Sin[a + b/(c + d*x)^(2/3)])
/(315*d^3) - (b^2*f*(d*e - c*f)*(c + d*x)^(2/3)*Sin[a + b/(c + d*x)^(2/3)])/(2*d^3) + ((d*e - c*f)^2*(c + d*x)
*Sin[a + b/(c + d*x)^(2/3)])/d^3 - (4*b^2*f^2*(c + d*x)^(5/3)*Sin[a + b/(c + d*x)^(2/3)])/(105*d^3) + (f*(d*e
- c*f)*(c + d*x)^2*Sin[a + b/(c + d*x)^(2/3)])/d^3 + (f^2*(c + d*x)^3*Sin[a + b/(c + d*x)^(2/3)])/(3*d^3) - (b
^3*f*(d*e - c*f)*Sin[a]*SinIntegral[b/(c + d*x)^(2/3)])/(2*d^3)

Rule 3378

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 3380

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 3383

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]

Rule 3384

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 3432

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2
]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]

Rule 3433

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[d, 2]))*FresnelC[Sqrt[2/Pi]*Rt[d, 2
]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]

Rule 3434

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

Rule 3435

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

Rule 3460

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 3468

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

Rule 3469

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

Rule 3490

Int[(x_)^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> -Subst[Int[(a + b*Sin[c + d/x^
n])^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, c, d}, x] && IGtQ[p, 0] && ILtQ[n, 0] && IntegerQ[m] && EqQ[n, -2
]

Rule 3514

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]

Rubi steps

\begin {align*} \int (e+f x)^2 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right ) \, dx &=\frac {3 \text {Subst}\left (\int \left ((d e-c f)^2 x^2 \sin \left (a+\frac {b}{x^2}\right )-2 f (-d e+c f) x^5 \sin \left (a+\frac {b}{x^2}\right )+f^2 x^8 \sin \left (a+\frac {b}{x^2}\right )\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d^3}\\ &=\frac {\left (3 f^2\right ) \text {Subst}\left (\int x^8 \sin \left (a+\frac {b}{x^2}\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d^3}+\frac {(6 f (d e-c f)) \text {Subst}\left (\int x^5 \sin \left (a+\frac {b}{x^2}\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d^3}+\frac {\left (3 (d e-c f)^2\right ) \text {Subst}\left (\int x^2 \sin \left (a+\frac {b}{x^2}\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d^3}\\ &=-\frac {\left (3 f^2\right ) \text {Subst}\left (\int \frac {\sin \left (a+b x^2\right )}{x^{10}} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{d^3}-\frac {(3 f (d e-c f)) \text {Subst}\left (\int \frac {\sin (a+b x)}{x^4} \, dx,x,\frac {1}{(c+d x)^{2/3}}\right )}{d^3}-\frac {\left (3 (d e-c f)^2\right ) \text {Subst}\left (\int \frac {\sin \left (a+b x^2\right )}{x^4} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{d^3}\\ &=\frac {(d e-c f)^2 (c+d x) \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}+\frac {f (d e-c f) (c+d x)^2 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}+\frac {f^2 (c+d x)^3 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{3 d^3}-\frac {\left (2 b f^2\right ) \text {Subst}\left (\int \frac {\cos \left (a+b x^2\right )}{x^8} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{3 d^3}-\frac {(b f (d e-c f)) \text {Subst}\left (\int \frac {\cos (a+b x)}{x^3} \, dx,x,\frac {1}{(c+d x)^{2/3}}\right )}{d^3}-\frac {\left (2 b (d e-c f)^2\right ) \text {Subst}\left (\int \frac {\cos \left (a+b x^2\right )}{x^2} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{d^3}\\ &=\frac {2 b (d e-c f)^2 \sqrt [3]{c+d x} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}+\frac {b f (d e-c f) (c+d x)^{4/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac {2 b f^2 (c+d x)^{7/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{21 d^3}+\frac {(d e-c f)^2 (c+d x) \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}+\frac {f (d e-c f) (c+d x)^2 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}+\frac {f^2 (c+d x)^3 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{3 d^3}+\frac {\left (4 b^2 f^2\right ) \text {Subst}\left (\int \frac {\sin \left (a+b x^2\right )}{x^6} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{21 d^3}+\frac {\left (b^2 f (d e-c f)\right ) \text {Subst}\left (\int \frac {\sin (a+b x)}{x^2} \, dx,x,\frac {1}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac {\left (4 b^2 (d e-c f)^2\right ) \text {Subst}\left (\int \sin \left (a+b x^2\right ) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{d^3}\\ &=\frac {2 b (d e-c f)^2 \sqrt [3]{c+d x} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}+\frac {b f (d e-c f) (c+d x)^{4/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac {2 b f^2 (c+d x)^{7/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{21 d^3}-\frac {b^2 f (d e-c f) (c+d x)^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac {(d e-c f)^2 (c+d x) \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}-\frac {4 b^2 f^2 (c+d x)^{5/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{105 d^3}+\frac {f (d e-c f) (c+d x)^2 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}+\frac {f^2 (c+d x)^3 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{3 d^3}+\frac {\left (8 b^3 f^2\right ) \text {Subst}\left (\int \frac {\cos \left (a+b x^2\right )}{x^4} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{105 d^3}+\frac {\left (b^3 f (d e-c f)\right ) \text {Subst}\left (\int \frac {\cos (a+b x)}{x} \, dx,x,\frac {1}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac {\left (4 b^2 (d e-c f)^2 \cos (a)\right ) \text {Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{d^3}+\frac {\left (4 b^2 (d e-c f)^2 \sin (a)\right ) \text {Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{d^3}\\ &=\frac {2 b (d e-c f)^2 \sqrt [3]{c+d x} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}-\frac {8 b^3 f^2 (c+d x) \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{315 d^3}+\frac {b f (d e-c f) (c+d x)^{4/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac {2 b f^2 (c+d x)^{7/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{21 d^3}+\frac {2 b^{3/2} (d e-c f)^2 \sqrt {2 \pi } \cos (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{d^3}+\frac {2 b^{3/2} (d e-c f)^2 \sqrt {2 \pi } C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right ) \sin (a)}{d^3}-\frac {b^2 f (d e-c f) (c+d x)^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac {(d e-c f)^2 (c+d x) \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}-\frac {4 b^2 f^2 (c+d x)^{5/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{105 d^3}+\frac {f (d e-c f) (c+d x)^2 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}+\frac {f^2 (c+d x)^3 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{3 d^3}-\frac {\left (16 b^4 f^2\right ) \text {Subst}\left (\int \frac {\sin \left (a+b x^2\right )}{x^2} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{315 d^3}+\frac {\left (b^3 f (d e-c f) \cos (a)\right ) \text {Subst}\left (\int \frac {\cos (b x)}{x} \, dx,x,\frac {1}{(c+d x)^{2/3}}\right )}{2 d^3}-\frac {\left (b^3 f (d e-c f) \sin (a)\right ) \text {Subst}\left (\int \frac {\sin (b x)}{x} \, dx,x,\frac {1}{(c+d x)^{2/3}}\right )}{2 d^3}\\ &=\frac {2 b (d e-c f)^2 \sqrt [3]{c+d x} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}-\frac {8 b^3 f^2 (c+d x) \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{315 d^3}+\frac {b f (d e-c f) (c+d x)^{4/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac {2 b f^2 (c+d x)^{7/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{21 d^3}+\frac {b^3 f (d e-c f) \cos (a) \text {Ci}\left (\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac {2 b^{3/2} (d e-c f)^2 \sqrt {2 \pi } \cos (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{d^3}+\frac {2 b^{3/2} (d e-c f)^2 \sqrt {2 \pi } C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right ) \sin (a)}{d^3}+\frac {16 b^4 f^2 \sqrt [3]{c+d x} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{315 d^3}-\frac {b^2 f (d e-c f) (c+d x)^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac {(d e-c f)^2 (c+d x) \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}-\frac {4 b^2 f^2 (c+d x)^{5/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{105 d^3}+\frac {f (d e-c f) (c+d x)^2 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}+\frac {f^2 (c+d x)^3 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{3 d^3}-\frac {b^3 f (d e-c f) \sin (a) \text {Si}\left (\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}-\frac {\left (32 b^5 f^2\right ) \text {Subst}\left (\int \cos \left (a+b x^2\right ) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{315 d^3}\\ &=\frac {2 b (d e-c f)^2 \sqrt [3]{c+d x} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}-\frac {8 b^3 f^2 (c+d x) \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{315 d^3}+\frac {b f (d e-c f) (c+d x)^{4/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac {2 b f^2 (c+d x)^{7/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{21 d^3}+\frac {b^3 f (d e-c f) \cos (a) \text {Ci}\left (\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac {2 b^{3/2} (d e-c f)^2 \sqrt {2 \pi } \cos (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{d^3}+\frac {2 b^{3/2} (d e-c f)^2 \sqrt {2 \pi } C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right ) \sin (a)}{d^3}+\frac {16 b^4 f^2 \sqrt [3]{c+d x} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{315 d^3}-\frac {b^2 f (d e-c f) (c+d x)^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac {(d e-c f)^2 (c+d x) \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}-\frac {4 b^2 f^2 (c+d x)^{5/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{105 d^3}+\frac {f (d e-c f) (c+d x)^2 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}+\frac {f^2 (c+d x)^3 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{3 d^3}-\frac {b^3 f (d e-c f) \sin (a) \text {Si}\left (\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}-\frac {\left (32 b^5 f^2 \cos (a)\right ) \text {Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{315 d^3}+\frac {\left (32 b^5 f^2 \sin (a)\right ) \text {Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{315 d^3}\\ &=\frac {2 b (d e-c f)^2 \sqrt [3]{c+d x} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}-\frac {8 b^3 f^2 (c+d x) \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{315 d^3}+\frac {b f (d e-c f) (c+d x)^{4/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac {2 b f^2 (c+d x)^{7/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{21 d^3}+\frac {b^3 f (d e-c f) \cos (a) \text {Ci}\left (\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}-\frac {16 b^{9/2} f^2 \sqrt {2 \pi } \cos (a) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{315 d^3}+\frac {2 b^{3/2} (d e-c f)^2 \sqrt {2 \pi } \cos (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{d^3}+\frac {2 b^{3/2} (d e-c f)^2 \sqrt {2 \pi } C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right ) \sin (a)}{d^3}+\frac {16 b^{9/2} f^2 \sqrt {2 \pi } S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right ) \sin (a)}{315 d^3}+\frac {16 b^4 f^2 \sqrt [3]{c+d x} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{315 d^3}-\frac {b^2 f (d e-c f) (c+d x)^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac {(d e-c f)^2 (c+d x) \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}-\frac {4 b^2 f^2 (c+d x)^{5/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{105 d^3}+\frac {f (d e-c f) (c+d x)^2 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}+\frac {f^2 (c+d x)^3 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{3 d^3}-\frac {b^3 f (d e-c f) \sin (a) \text {Si}\left (\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 1.78, size = 613, normalized size = 0.97 \begin {gather*} \frac {i e^{-i a} \left (e^{-\frac {i b}{(c+d x)^{2/3}}} \sqrt [3]{c+d x} \left (32 b^4 f^2+16 i b^3 f^2 (c+d x)^{2/3}+3 b^2 f \sqrt [3]{c+d x} (-105 d e+97 c f-8 d f x)-15 i b \left (84 d^2 e^2+21 d e f (-7 c+d x)+f^2 \left (67 c^2-13 c d x+4 d^2 x^2\right )\right )+210 (c+d x)^{2/3} \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 )\right )-e^{i \left (2 a+\frac {b}{(c+d x)^{2/3}}\right )} \sqrt [3]{c+d x} \left (32 b^4 f^2-16 i b^3 f^2 (c+d x)^{2/3}+3 b^2 f \sqrt [3]{c+d x} (-105 d e+97 c f-8 d f x)+15 i b \left (84 d^2 e^2+21 d e f (-7 c+d x)+f^2 \left (67 c^2-13 c d x+4 d^2 x^2\right )\right )+210 (c+d x)^{2/3} \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 )\right )+4 \sqrt [4]{-1} b^{3/2} e^{2 i a} \left (315 i d^2 e^2-630 i c d e f+\left (8 b^3+315 i c^2\right ) f^2\right ) \sqrt {\pi } \text {erfi}\left (\frac {\sqrt [4]{-1} \sqrt {b}}{\sqrt [3]{c+d x}}\right )-4 \sqrt [4]{-1} b^{3/2} \left (315 d^2 e^2-630 c d e f+\left (8 i b^3+315 c^2\right ) f^2\right ) \sqrt {\pi } \text {erfi}\left (\frac {(-1)^{3/4} \sqrt {b}}{\sqrt [3]{c+d x}}\right )+315 i b^3 f (-d e+c f) \text {Ei}\left (-\frac {i b}{(c+d x)^{2/3}}\right )+315 i b^3 e^{2 i a} f (-d e+c f) \text {Ei}\left (\frac {i b}{(c+d x)^{2/3}}\right )\right )}{1260 d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((I/1260)*(((c + d*x)^(1/3)*(32*b^4*f^2 + (16*I)*b^3*f^2*(c + d*x)^(2/3) + 3*b^2*f*(c + d*x)^(1/3)*(-105*d*e +
97*c*f - 8*d*f*x) - (15*I)*b*(84*d^2*e^2 + 21*d*e*f*(-7*c + d*x) + f^2*(67*c^2 - 13*c*d*x + 4*d^2*x^2)) + 210
*(c + d*x)^(2/3)*(c^2*f^2 - c*d*f*(3*e + f*x) + d^2*(3*e^2 + 3*e*f*x + f^2*x^2))))/E^((I*b)/(c + d*x)^(2/3)) -
E^(I*(2*a + b/(c + d*x)^(2/3)))*(c + d*x)^(1/3)*(32*b^4*f^2 - (16*I)*b^3*f^2*(c + d*x)^(2/3) + 3*b^2*f*(c + d
*x)^(1/3)*(-105*d*e + 97*c*f - 8*d*f*x) + (15*I)*b*(84*d^2*e^2 + 21*d*e*f*(-7*c + d*x) + f^2*(67*c^2 - 13*c*d*
x + 4*d^2*x^2)) + 210*(c + d*x)^(2/3)*(c^2*f^2 - c*d*f*(3*e + f*x) + d^2*(3*e^2 + 3*e*f*x + f^2*x^2))) + 4*(-1
)^(1/4)*b^(3/2)*E^((2*I)*a)*((315*I)*d^2*e^2 - (630*I)*c*d*e*f + (8*b^3 + (315*I)*c^2)*f^2)*Sqrt[Pi]*Erfi[((-1
)^(1/4)*Sqrt[b])/(c + d*x)^(1/3)] - 4*(-1)^(1/4)*b^(3/2)*(315*d^2*e^2 - 630*c*d*e*f + ((8*I)*b^3 + 315*c^2)*f^
2)*Sqrt[Pi]*Erfi[((-1)^(3/4)*Sqrt[b])/(c + d*x)^(1/3)] + (315*I)*b^3*f*(-(d*e) + c*f)*ExpIntegralEi[((-I)*b)/(
c + d*x)^(2/3)] + (315*I)*b^3*E^((2*I)*a)*f*(-(d*e) + c*f)*ExpIntegralEi[(I*b)/(c + d*x)^(2/3)]))/(d^3*E^(I*a)
)

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Maple [A]
time = 0.06, size = 452, normalized size = 0.72

method	result	size
derivativedivides	\(\frac {\left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) \left (d x +c \right ) \sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )-2 \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) b \left (-\left (d x +c \right )^{\frac {1}{3}} \cos \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )-\sqrt {b}\, \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (a \right ) \mathrm {S}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )^{\frac {1}{3}}}\right )+\sin \left (a \right ) \FresnelC \left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )^{\frac {1}{3}}}\right )\right )\right )+\frac {\left (-2 c \,f^{2}+2 d e f \right ) \left (d x +c \right )^{2} \sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )}{2}-\left (-2 c \,f^{2}+2 d e f \right ) b \left (-\frac {\left (d x +c \right )^{\frac {4}{3}} \cos \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )}{4}-\frac {b \left (-\frac {\left (d x +c \right )^{\frac {2}{3}} \sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )}{2}+b \left (\frac {\cos \left (a \right ) \cosineIntegral \left (\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )}{2}-\frac {\sin \left (a \right ) \sinIntegral \left (\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )}{2}\right )\right )}{2}\right )+\frac {f^{2} \left (d x +c \right )^{3} \sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )}{3}-\frac {2 f^{2} b \left (-\frac {\left (d x +c \right )^{\frac {7}{3}} \cos \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )}{7}-\frac {2 b \left (-\frac {\left (d x +c \right )^{\frac {5}{3}} \sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )}{5}+\frac {2 b \left (-\frac {\left (d x +c \right ) \cos \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )}{3}-\frac {2 b \left (-\left (d x +c \right )^{\frac {1}{3}} \sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )+\sqrt {b}\, \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (a \right ) \FresnelC \left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )^{\frac {1}{3}}}\right )-\sin \left (a \right ) \mathrm {S}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )^{\frac {1}{3}}}\right )\right )\right )}{3}\right )}{5}\right )}{7}\right )}{3}}{d^{3}}\)	\(452\)
default	\(\frac {\left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) \left (d x +c \right ) \sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )-2 \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) b \left (-\left (d x +c \right )^{\frac {1}{3}} \cos \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )-\sqrt {b}\, \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (a \right ) \mathrm {S}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )^{\frac {1}{3}}}\right )+\sin \left (a \right ) \FresnelC \left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )^{\frac {1}{3}}}\right )\right )\right )+\frac {\left (-2 c \,f^{2}+2 d e f \right ) \left (d x +c \right )^{2} \sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )}{2}-\left (-2 c \,f^{2}+2 d e f \right ) b \left (-\frac {\left (d x +c \right )^{\frac {4}{3}} \cos \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )}{4}-\frac {b \left (-\frac {\left (d x +c \right )^{\frac {2}{3}} \sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )}{2}+b \left (\frac {\cos \left (a \right ) \cosineIntegral \left (\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )}{2}-\frac {\sin \left (a \right ) \sinIntegral \left (\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )}{2}\right )\right )}{2}\right )+\frac {f^{2} \left (d x +c \right )^{3} \sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )}{3}-\frac {2 f^{2} b \left (-\frac {\left (d x +c \right )^{\frac {7}{3}} \cos \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )}{7}-\frac {2 b \left (-\frac {\left (d x +c \right )^{\frac {5}{3}} \sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )}{5}+\frac {2 b \left (-\frac {\left (d x +c \right ) \cos \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )}{3}-\frac {2 b \left (-\left (d x +c \right )^{\frac {1}{3}} \sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )+\sqrt {b}\, \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (a \right ) \FresnelC \left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )^{\frac {1}{3}}}\right )-\sin \left (a \right ) \mathrm {S}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )^{\frac {1}{3}}}\right )\right )\right )}{3}\right )}{5}\right )}{7}\right )}{3}}{d^{3}}\)	\(452\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/d^3*(1/3*(c^2*f^2-2*c*d*e*f+d^2*e^2)*(d*x+c)*sin(a+b/(d*x+c)^(2/3))-2/3*(c^2*f^2-2*c*d*e*f+d^2*e^2)*b*(-(d*x
+c)^(1/3)*cos(a+b/(d*x+c)^(2/3))-b^(1/2)*2^(1/2)*Pi^(1/2)*(cos(a)*FresnelS(b^(1/2)*2^(1/2)/Pi^(1/2)/(d*x+c)^(1
/3))+sin(a)*FresnelC(b^(1/2)*2^(1/2)/Pi^(1/2)/(d*x+c)^(1/3))))+1/6*(-2*c*f^2+2*d*e*f)*(d*x+c)^2*sin(a+b/(d*x+c
)^(2/3))-1/3*(-2*c*f^2+2*d*e*f)*b*(-1/4*(d*x+c)^(4/3)*cos(a+b/(d*x+c)^(2/3))-1/2*b*(-1/2*(d*x+c)^(2/3)*sin(a+b
/(d*x+c)^(2/3))+b*(1/2*cos(a)*Ci(b/(d*x+c)^(2/3))-1/2*sin(a)*Si(b/(d*x+c)^(2/3)))))+1/9*f^2*(d*x+c)^3*sin(a+b/
(d*x+c)^(2/3))-2/9*f^2*b*(-1/7*(d*x+c)^(7/3)*cos(a+b/(d*x+c)^(2/3))-2/7*b*(-1/5*(d*x+c)^(5/3)*sin(a+b/(d*x+c)^
(2/3))+2/5*b*(-1/3*(d*x+c)*cos(a+b/(d*x+c)^(2/3))-2/3*b*(-(d*x+c)^(1/3)*sin(a+b/(d*x+c)^(2/3))+b^(1/2)*2^(1/2)
*Pi^(1/2)*(cos(a)*FresnelC(b^(1/2)*2^(1/2)/Pi^(1/2)/(d*x+c)^(1/3))-sin(a)*FresnelS(b^(1/2)*2^(1/2)/Pi^(1/2)/(d
*x+c)^(1/3))))))))

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Maxima [C] Result contains complex when optimal does not.
time = 0.80, size = 1261, normalized size = 2.00 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1260*(630*sqrt(2)*(2*sqrt(2)*(d*x + c)^(2/3)*sqrt((d*x + c)^(-4/3))*b^2*cos(((d*x + c)^(2/3)*a + b)/(d*x + c
)^(2/3)) + sqrt(2)*(d*x + c)^(4/3)*sqrt((d*x + c)^(-4/3))*b*sin(((d*x + c)^(2/3)*a + b)/(d*x + c)^(2/3)) + (((
I + 1)*sqrt(pi)*(erf(sqrt(I*b/(d*x + c)^(2/3))) - 1) - (I - 1)*sqrt(pi)*(erf(sqrt(-I*b/(d*x + c)^(2/3))) - 1))
*cos(a) + (-(I - 1)*sqrt(pi)*(erf(sqrt(I*b/(d*x + c)^(2/3))) - 1) + (I + 1)*sqrt(pi)*(erf(sqrt(-I*b/(d*x + c)^
(2/3))) - 1))*sin(a))*b^2*(b^2/(d*x + c)^(4/3))^(1/4))*sqrt((d*x + c)^(4/3))*c^2*f^2/((d*x + c)^(1/3)*b*d^2) -
1260*sqrt(2)*(2*sqrt(2)*(d*x + c)^(2/3)*sqrt((d*x + c)^(-4/3))*b^2*cos(((d*x + c)^(2/3)*a + b)/(d*x + c)^(2/3
)) + sqrt(2)*(d*x + c)^(4/3)*sqrt((d*x + c)^(-4/3))*b*sin(((d*x + c)^(2/3)*a + b)/(d*x + c)^(2/3)) + (((I + 1)
*sqrt(pi)*(erf(sqrt(I*b/(d*x + c)^(2/3))) - 1) - (I - 1)*sqrt(pi)*(erf(sqrt(-I*b/(d*x + c)^(2/3))) - 1))*cos(a
) + (-(I - 1)*sqrt(pi)*(erf(sqrt(I*b/(d*x + c)^(2/3))) - 1) + (I + 1)*sqrt(pi)*(erf(sqrt(-I*b/(d*x + c)^(2/3))
) - 1))*sin(a))*b^2*(b^2/(d*x + c)^(4/3))^(1/4))*sqrt((d*x + c)^(4/3))*c*f*e/((d*x + c)^(1/3)*b*d) - 315*(((Ei
(I*b/(d*x + c)^(2/3)) + Ei(-I*b/(d*x + c)^(2/3)))*cos(a) + (I*Ei(I*b/(d*x + c)^(2/3)) - I*Ei(-I*b/(d*x + c)^(2
/3)))*sin(a))*b^3 + 2*(d*x + c)^(4/3)*b*cos(((d*x + c)^(2/3)*a + b)/(d*x + c)^(2/3)) - 2*((d*x + c)^(2/3)*b^2
- 2*(d*x + c)^2)*sin(((d*x + c)^(2/3)*a + b)/(d*x + c)^(2/3)))*c*f^2/d^2 + 315*(((Ei(I*b/(d*x + c)^(2/3)) + Ei
(-I*b/(d*x + c)^(2/3)))*cos(a) + (I*Ei(I*b/(d*x + c)^(2/3)) - I*Ei(-I*b/(d*x + c)^(2/3)))*sin(a))*b^3 + 2*(d*x
+ c)^(4/3)*b*cos(((d*x + c)^(2/3)*a + b)/(d*x + c)^(2/3)) - 2*((d*x + c)^(2/3)*b^2 - 2*(d*x + c)^2)*sin(((d*x
+ c)^(2/3)*a + b)/(d*x + c)^(2/3)))*f*e/d + 630*sqrt(2)*(2*sqrt(2)*(d*x + c)^(2/3)*sqrt((d*x + c)^(-4/3))*b^2
*cos(((d*x + c)^(2/3)*a + b)/(d*x + c)^(2/3)) + sqrt(2)*(d*x + c)^(4/3)*sqrt((d*x + c)^(-4/3))*b*sin(((d*x + c
)^(2/3)*a + b)/(d*x + c)^(2/3)) + (((I + 1)*sqrt(pi)*(erf(sqrt(I*b/(d*x + c)^(2/3))) - 1) - (I - 1)*sqrt(pi)*(
erf(sqrt(-I*b/(d*x + c)^(2/3))) - 1))*cos(a) + (-(I - 1)*sqrt(pi)*(erf(sqrt(I*b/(d*x + c)^(2/3))) - 1) + (I +
1)*sqrt(pi)*(erf(sqrt(-I*b/(d*x + c)^(2/3))) - 1))*sin(a))*b^2*(b^2/(d*x + c)^(4/3))^(1/4))*sqrt((d*x + c)^(4/
3))*e^2/((d*x + c)^(1/3)*b) - 2*sqrt(2)*(8*((-(I - 1)*sqrt(pi)*(erf(sqrt(I*b/(d*x + c)^(2/3))) - 1) + (I + 1)*
sqrt(pi)*(erf(sqrt(-I*b/(d*x + c)^(2/3))) - 1))*cos(a) + (-(I + 1)*sqrt(pi)*(erf(sqrt(I*b/(d*x + c)^(2/3))) -
1) + (I - 1)*sqrt(pi)*(erf(sqrt(-I*b/(d*x + c)^(2/3))) - 1))*sin(a))*b^5*(b^2/(d*x + c)^(4/3))^(1/4) + 2*(4*sq
rt(2)*(d*x + c)^(4/3)*sqrt((d*x + c)^(-4/3))*b^4 - 15*sqrt(2)*(d*x + c)^(8/3)*sqrt((d*x + c)^(-4/3))*b^2)*cos(
((d*x + c)^(2/3)*a + b)/(d*x + c)^(2/3)) - (16*sqrt(2)*(d*x + c)^(2/3)*sqrt((d*x + c)^(-4/3))*b^5 - 12*sqrt(2)
*(d*x + c)^2*sqrt((d*x + c)^(-4/3))*b^3 + 105*sqrt(2)*(d*x + c)^(10/3)*sqrt((d*x + c)^(-4/3))*b)*sin(((d*x + c
)^(2/3)*a + b)/(d*x + c)^(2/3)))*sqrt((d*x + c)^(4/3))*f^2/((d*x + c)^(1/3)*b*d^2))/d

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Fricas [A]
time = 0.42, size = 501, normalized size = 0.80 \begin {gather*} -\frac {315 \, {\left (b^{3} c f^{2} - b^{3} d f e\right )} \cos \left (a\right ) \operatorname {Ci}\left (\frac {b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + 315 \, {\left (b^{3} c f^{2} - b^{3} d f e\right )} \cos \left (a\right ) \operatorname {Ci}\left (-\frac {b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + 8 \, \sqrt {2} {\left (8 \, \pi b^{4} f^{2} \cos \left (a\right ) - 315 \, {\left (\pi b c^{2} f^{2} - 2 \, \pi b c d f e + \pi b d^{2} e^{2}\right )} \sin \left (a\right )\right )} \sqrt {\frac {b}{\pi }} \operatorname {C}\left (\frac {\sqrt {2} \sqrt {\frac {b}{\pi }}}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) - 8 \, \sqrt {2} {\left (8 \, \pi b^{4} f^{2} \sin \left (a\right ) + 315 \, {\left (\pi b c^{2} f^{2} - 2 \, \pi b c d f e + \pi b d^{2} e^{2}\right )} \cos \left (a\right )\right )} \sqrt {\frac {b}{\pi }} \operatorname {S}\left (\frac {\sqrt {2} \sqrt {\frac {b}{\pi }}}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) - 630 \, {\left (b^{3} c f^{2} - b^{3} d f e\right )} \sin \left (a\right ) \operatorname {Si}\left (\frac {b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + 2 \, {\left (16 \, b^{3} d f^{2} x + 16 \, b^{3} c f^{2} - 15 \, {\left (4 \, b d^{2} f^{2} x^{2} - 13 \, b c d f^{2} x + 67 \, b c^{2} f^{2} + 84 \, b d^{2} e^{2} + 21 \, {\left (b d^{2} f x - 7 \, b c d f\right )} e\right )} {\left (d x + c\right )}^{\frac {1}{3}}\right )} \cos \left (\frac {a d x + a c + {\left (d x + c\right )}^{\frac {1}{3}} b}{d x + c}\right ) - 2 \, {\left (210 \, d^{3} f^{2} x^{3} + 32 \, {\left (d x + c\right )}^{\frac {1}{3}} b^{4} f^{2} + 210 \, c^{3} f^{2} + 630 \, {\left (d^{3} x + c d^{2}\right )} e^{2} + 630 \, {\left (d^{3} f x^{2} - c^{2} d f\right )} e - 3 \, {\left (8 \, b^{2} d f^{2} x - 97 \, b^{2} c f^{2} + 105 \, b^{2} d f e\right )} {\left (d x + c\right )}^{\frac {2}{3}}\right )} \sin \left (\frac {a d x + a c + {\left (d x + c\right )}^{\frac {1}{3}} b}{d x + c}\right )}{1260 \, d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1260*(315*(b^3*c*f^2 - b^3*d*f*e)*cos(a)*cos_integral(b/(d*x + c)^(2/3)) + 315*(b^3*c*f^2 - b^3*d*f*e)*cos(
a)*cos_integral(-b/(d*x + c)^(2/3)) + 8*sqrt(2)*(8*pi*b^4*f^2*cos(a) - 315*(pi*b*c^2*f^2 - 2*pi*b*c*d*f*e + pi
*b*d^2*e^2)*sin(a))*sqrt(b/pi)*fresnel_cos(sqrt(2)*sqrt(b/pi)/(d*x + c)^(1/3)) - 8*sqrt(2)*(8*pi*b^4*f^2*sin(a
) + 315*(pi*b*c^2*f^2 - 2*pi*b*c*d*f*e + pi*b*d^2*e^2)*cos(a))*sqrt(b/pi)*fresnel_sin(sqrt(2)*sqrt(b/pi)/(d*x
+ c)^(1/3)) - 630*(b^3*c*f^2 - b^3*d*f*e)*sin(a)*sin_integral(b/(d*x + c)^(2/3)) + 2*(16*b^3*d*f^2*x + 16*b^3*
c*f^2 - 15*(4*b*d^2*f^2*x^2 - 13*b*c*d*f^2*x + 67*b*c^2*f^2 + 84*b*d^2*e^2 + 21*(b*d^2*f*x - 7*b*c*d*f)*e)*(d*
x + c)^(1/3))*cos((a*d*x + a*c + (d*x + c)^(1/3)*b)/(d*x + c)) - 2*(210*d^3*f^2*x^3 + 32*(d*x + c)^(1/3)*b^4*f
^2 + 210*c^3*f^2 + 630*(d^3*x + c*d^2)*e^2 + 630*(d^3*f*x^2 - c^2*d*f)*e - 3*(8*b^2*d*f^2*x - 97*b^2*c*f^2 + 1
05*b^2*d*f*e)*(d*x + c)^(2/3))*sin((a*d*x + a*c + (d*x + c)^(1/3)*b)/(d*x + c)))/d^3

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (e + f x\right )^{2} \sin {\left (a + \frac {b}{\left (c + d x\right )^{\frac {2}{3}}} \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \sin \left (a+\frac {b}{{\left (c+d\,x\right )}^{2/3}}\right )\,{\left (e+f\,x\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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