Integrand size = 22, antiderivative size = 630 \[ \int (e+f x)^2 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right ) \, dx=\frac {2 b (d e-c f)^2 \sqrt [3]{c+d x} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}-\frac {8 b^3 f^2 (c+d x) \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{315 d^3}+\frac {b f (d e-c f) (c+d x)^{4/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac {2 b f^2 (c+d x)^{7/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{21 d^3}+\frac {b^3 f (d e-c f) \cos (a) \operatorname {CosIntegral}\left (\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}-\frac {16 b^{9/2} f^2 \sqrt {2 \pi } \cos (a) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{315 d^3}+\frac {2 b^{3/2} (d e-c f)^2 \sqrt {2 \pi } \cos (a) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{d^3}+\frac {2 b^{3/2} (d e-c f)^2 \sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right ) \sin (a)}{d^3}+\frac {16 b^{9/2} f^2 \sqrt {2 \pi } \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right ) \sin (a)}{315 d^3}+\frac {16 b^4 f^2 \sqrt [3]{c+d x} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{315 d^3}-\frac {b^2 f (d e-c f) (c+d x)^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac {(d e-c f)^2 (c+d x) \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}-\frac {4 b^2 f^2 (c+d x)^{5/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{105 d^3}+\frac {f (d e-c f) (c+d x)^2 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}+\frac {f^2 (c+d x)^3 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{3 d^3}-\frac {b^3 f (d e-c f) \sin (a) \text {Si}\left (\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3} \]
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+2/2 1*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)*s in(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*cos(a)*F resnelC(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)*FresnelS(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
Result contains complex when optimal does not.
Time = 1.95 (sec) , antiderivative size = 613, normalized size of antiderivative = 0.97 \[ \int (e+f x)^2 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right ) \, dx=\frac {i e^{-i a} \left (e^{-\frac {i b}{(c+d x)^{2/3}}} \sqrt [3]{c+d x} \left (32 b^4 f^2+16 i b^3 f^2 (c+d x)^{2/3}+3 b^2 f \sqrt [3]{c+d x} (-105 d e+97 c f-8 d f x)-15 i b \left (84 d^2 e^2+21 d e f (-7 c+d x)+f^2 \left (67 c^2-13 c d x+4 d^2 x^2\right )\right )+210 (c+d x)^{2/3} \left (c^2 f^2-c d f (3 e+f x)+d^2 \left (3 e^2+3 e f x+f^2 x^2\right )\right )\right )-e^{i \left (2 a+\frac {b}{(c+d x)^{2/3}}\right )} \sqrt [3]{c+d x} \left (32 b^4 f^2-16 i b^3 f^2 (c+d x)^{2/3}+3 b^2 f \sqrt [3]{c+d x} (-105 d e+97 c f-8 d f x)+15 i b \left (84 d^2 e^2+21 d e f (-7 c+d x)+f^2 \left (67 c^2-13 c d x+4 d^2 x^2\right )\right )+210 (c+d x)^{2/3} \left (c^2 f^2-c d f (3 e+f x)+d^2 \left (3 e^2+3 e f x+f^2 x^2\right )\right )\right )+4 \sqrt [4]{-1} b^{3/2} e^{2 i a} \left (315 i d^2 e^2-630 i c d e f+\left (8 b^3+315 i c^2\right ) f^2\right ) \sqrt {\pi } \text {erfi}\left (\frac {\sqrt [4]{-1} \sqrt {b}}{\sqrt [3]{c+d x}}\right )-4 \sqrt [4]{-1} b^{3/2} \left (315 d^2 e^2-630 c d e f+\left (8 i b^3+315 c^2\right ) f^2\right ) \sqrt {\pi } \text {erfi}\left (\frac {(-1)^{3/4} \sqrt {b}}{\sqrt [3]{c+d x}}\right )+315 i b^3 f (-d e+c f) \operatorname {ExpIntegralEi}\left (-\frac {i b}{(c+d x)^{2/3}}\right )+315 i b^3 e^{2 i a} f (-d e+c f) \operatorname {ExpIntegralEi}\left (\frac {i b}{(c+d x)^{2/3}}\right )\right )}{1260 d^3} \]
((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))
Time = 0.83 (sec) , antiderivative size = 599, normalized size of antiderivative = 0.95, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3914, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (e+f x)^2 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right ) \, dx\) |
| \(\Big \downarrow \) 3914 |
| \(\displaystyle \frac {3 \int \left (f^2 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right ) (c+d x)^{8/3}+2 f (d e-c f) \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right ) (c+d x)^{5/3}+(d e-c f)^2 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right ) (c+d x)^{2/3}\right )d\sqrt [3]{c+d x}}{d^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 \left (\frac {2}{3} \sqrt {2 \pi } b^{3/2} \sin (a) (d e-c f)^2 \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )+\frac {2}{3} \sqrt {2 \pi } b^{3/2} \cos (a) (d e-c f)^2 \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )-\frac {16}{945} \sqrt {2 \pi } b^{9/2} f^2 \cos (a) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )+\frac {16}{945} \sqrt {2 \pi } b^{9/2} f^2 \sin (a) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )+\frac {16}{945} b^4 f^2 \sqrt [3]{c+d x} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )+\frac {1}{6} b^3 f \cos (a) (d e-c f) \operatorname {CosIntegral}\left (\frac {b}{(c+d x)^{2/3}}\right )-\frac {1}{6} b^3 f \sin (a) (d e-c f) \text {Si}\left (\frac {b}{(c+d x)^{2/3}}\right )-\frac {8}{945} b^3 f^2 (c+d x) \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )-\frac {1}{6} b^2 f (c+d x)^{2/3} (d e-c f) \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )-\frac {4}{315} b^2 f^2 (c+d x)^{5/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )+\frac {1}{3} f (c+d x)^2 (d e-c f) \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )+\frac {1}{3} (c+d x) (d e-c f)^2 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )+\frac {1}{6} b f (c+d x)^{4/3} (d e-c f) \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )+\frac {2}{3} b \sqrt [3]{c+d x} (d e-c f)^2 \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )+\frac {1}{9} f^2 (c+d x)^3 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )+\frac {2}{63} b f^2 (c+d x)^{7/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )\right )}{d^3}\) |
(3*((2*b*(d*e - c*f)^2*(c + d*x)^(1/3)*Cos[a + b/(c + d*x)^(2/3)])/3 - (8* b^3*f^2*(c + d*x)*Cos[a + b/(c + d*x)^(2/3)])/945 + (b*f*(d*e - c*f)*(c + d*x)^(4/3)*Cos[a + b/(c + d*x)^(2/3)])/6 + (2*b*f^2*(c + d*x)^(7/3)*Cos[a + b/(c + d*x)^(2/3)])/63 + (b^3*f*(d*e - c*f)*Cos[a]*CosIntegral[b/(c + d* x)^(2/3)])/6 - (16*b^(9/2)*f^2*Sqrt[2*Pi]*Cos[a]*FresnelC[(Sqrt[b]*Sqrt[2/ Pi])/(c + d*x)^(1/3)])/945 + (2*b^(3/2)*(d*e - c*f)^2*Sqrt[2*Pi]*Cos[a]*Fr esnelS[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)^(1/3)])/3 + (2*b^(3/2)*(d*e - c*f)^2 *Sqrt[2*Pi]*FresnelC[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)^(1/3)]*Sin[a])/3 + (16 *b^(9/2)*f^2*Sqrt[2*Pi]*FresnelS[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)^(1/3)]*Sin [a])/945 + (16*b^4*f^2*(c + d*x)^(1/3)*Sin[a + b/(c + d*x)^(2/3)])/945 - ( b^2*f*(d*e - c*f)*(c + d*x)^(2/3)*Sin[a + b/(c + d*x)^(2/3)])/6 + ((d*e - c*f)^2*(c + d*x)*Sin[a + b/(c + d*x)^(2/3)])/3 - (4*b^2*f^2*(c + d*x)^(5/3 )*Sin[a + b/(c + d*x)^(2/3)])/315 + (f*(d*e - c*f)*(c + d*x)^2*Sin[a + b/( c + d*x)^(2/3)])/3 + (f^2*(c + d*x)^3*Sin[a + b/(c + d*x)^(2/3)])/9 - (b^3 *f*(d*e - c*f)*Sin[a]*SinIntegral[b/(c + d*x)^(2/3)])/6))/d^3
3.3.22.3.1 Defintions of rubi rules used
Int[((g_.) + (h_.)*(x_))^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f _.)*(x_))^(n_)])^(p_.), x_Symbol] :> Module[{k = If[FractionQ[n], Denominat or[n], 1]}, Simp[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]
Time = 1.92 (sec) , antiderivative size = 424, normalized size of antiderivative = 0.67
| method | result | size |
| derivativedivides | \(\frac {\left (c f -d e \right )^{2} \left (d x +c \right ) \sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )-2 \left (c f -d e \right )^{2} 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 ) \operatorname {S}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )^{\frac {1}{3}}}\right )+\sin \left (a \right ) \operatorname {C}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )^{\frac {1}{3}}}\right )\right )\right )-f \left (c f -d e \right ) \left (d x +c \right )^{2} \sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )+2 f \left (c f -d e \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 ) \operatorname {Ci}\left (\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )}{2}-\frac {\sin \left (a \right ) \operatorname {Si}\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 ) \operatorname {C}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )^{\frac {1}{3}}}\right )-\sin \left (a \right ) \operatorname {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}}\) | \(424\) |
| default | \(\frac {\left (c f -d e \right )^{2} \left (d x +c \right ) \sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )-2 \left (c f -d e \right )^{2} 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 ) \operatorname {S}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )^{\frac {1}{3}}}\right )+\sin \left (a \right ) \operatorname {C}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )^{\frac {1}{3}}}\right )\right )\right )-f \left (c f -d e \right ) \left (d x +c \right )^{2} \sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )+2 f \left (c f -d e \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 ) \operatorname {Ci}\left (\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )}{2}-\frac {\sin \left (a \right ) \operatorname {Si}\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 ) \operatorname {C}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )^{\frac {1}{3}}}\right )-\sin \left (a \right ) \operatorname {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}}\) | \(424\) |
| parts | \(\text {Expression too large to display}\) | \(1510\) |
3/d^3*(1/3*(c*f-d*e)^2*(d*x+c)*sin(a+b/(d*x+c)^(2/3))-2/3*(c*f-d*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)*Fres nelS(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))))-1/3*f*(c*f-d*e)*(d*x+c)^2*sin(a+b/(d*x+c)^(2/ 3))+2/3*f*(c*f-d*e)*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))))))))
Time = 0.31 (sec) , antiderivative size = 461, normalized size of antiderivative = 0.73 \[ \int (e+f x)^2 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right ) \, dx=\frac {315 \, {\left (b^{3} d e f - b^{3} c f^{2}\right )} \cos \left (a\right ) \operatorname {Ci}\left (\frac {b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right ) - 4 \, \sqrt {2} {\left (8 \, \pi b^{4} f^{2} \cos \left (a\right ) - 315 \, \pi {\left (b d^{2} e^{2} - 2 \, b c d e f + b c^{2} f^{2}\right )} \sin \left (a\right )\right )} \sqrt {\frac {b}{\pi }} \operatorname {C}\left (\frac {\sqrt {2} \sqrt {\frac {b}{\pi }}}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + 4 \, \sqrt {2} {\left (8 \, \pi b^{4} f^{2} \sin \left (a\right ) + 315 \, \pi {\left (b d^{2} e^{2} - 2 \, b c d e f + b c^{2} f^{2}\right )} \cos \left (a\right )\right )} \sqrt {\frac {b}{\pi }} \operatorname {S}\left (\frac {\sqrt {2} \sqrt {\frac {b}{\pi }}}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) - 315 \, {\left (b^{3} d e f - b^{3} c f^{2}\right )} \sin \left (a\right ) \operatorname {Si}\left (\frac {b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right ) - {\left (16 \, b^{3} d f^{2} x + 16 \, b^{3} c f^{2} - 15 \, {\left (4 \, b d^{2} f^{2} x^{2} + 84 \, b d^{2} e^{2} - 147 \, b c d e f + 67 \, b c^{2} f^{2} + {\left (21 \, b d^{2} e f - 13 \, b c d f^{2}\right )} x\right )} {\left (d x + c\right )}^{\frac {1}{3}}\right )} \cos \left (\frac {a d x + a c + {\left (d x + c\right )}^{\frac {1}{3}} b}{d x + c}\right ) + {\left (210 \, d^{3} f^{2} x^{3} + 630 \, d^{3} e f x^{2} + 32 \, {\left (d x + c\right )}^{\frac {1}{3}} b^{4} f^{2} + 630 \, d^{3} e^{2} x + 630 \, c d^{2} e^{2} - 630 \, c^{2} d e f + 210 \, c^{3} f^{2} - 3 \, {\left (8 \, b^{2} d f^{2} x + 105 \, b^{2} d e f - 97 \, b^{2} c f^{2}\right )} {\left (d x + c\right )}^{\frac {2}{3}}\right )} \sin \left (\frac {a d x + a c + {\left (d x + c\right )}^{\frac {1}{3}} b}{d x + c}\right )}{630 \, d^{3}} \]
1/630*(315*(b^3*d*e*f - b^3*c*f^2)*cos(a)*cos_integral(b/(d*x + c)^(2/3)) - 4*sqrt(2)*(8*pi*b^4*f^2*cos(a) - 315*pi*(b*d^2*e^2 - 2*b*c*d*e*f + b*c^2 *f^2)*sin(a))*sqrt(b/pi)*fresnel_cos(sqrt(2)*sqrt(b/pi)/(d*x + c)^(1/3)) + 4*sqrt(2)*(8*pi*b^4*f^2*sin(a) + 315*pi*(b*d^2*e^2 - 2*b*c*d*e*f + b*c^2* f^2)*cos(a))*sqrt(b/pi)*fresnel_sin(sqrt(2)*sqrt(b/pi)/(d*x + c)^(1/3)) - 315*(b^3*d*e*f - b^3*c*f^2)*sin(a)*sin_integral(b/(d*x + c)^(2/3)) - (16*b ^3*d*f^2*x + 16*b^3*c*f^2 - 15*(4*b*d^2*f^2*x^2 + 84*b*d^2*e^2 - 147*b*c*d *e*f + 67*b*c^2*f^2 + (21*b*d^2*e*f - 13*b*c*d*f^2)*x)*(d*x + c)^(1/3))*co s((a*d*x + a*c + (d*x + c)^(1/3)*b)/(d*x + c)) + (210*d^3*f^2*x^3 + 630*d^ 3*e*f*x^2 + 32*(d*x + c)^(1/3)*b^4*f^2 + 630*d^3*e^2*x + 630*c*d^2*e^2 - 6 30*c^2*d*e*f + 210*c^3*f^2 - 3*(8*b^2*d*f^2*x + 105*b^2*d*e*f - 97*b^2*c*f ^2)*(d*x + c)^(2/3))*sin((a*d*x + a*c + (d*x + c)^(1/3)*b)/(d*x + c)))/d^3
\[ \int (e+f x)^2 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right ) \, dx=\int \left (e + f x\right )^{2} \sin {\left (a + \frac {b}{\left (c + d x\right )^{\frac {2}{3}}} \right )}\, dx \]
Result contains complex when optimal does not.
Time = 0.78 (sec) , antiderivative size = 1260, normalized size of antiderivative = 2.00 \[ \int (e+f x)^2 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right ) \, dx=\text {Too large to display} \]
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)*sqr t((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) - 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)*sq rt(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*e*f/((d*x + c)^(1/3)*b*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 +...
\[ \int (e+f x)^2 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right ) \, dx=\int { {\left (f x + e\right )}^{2} \sin \left (a + \frac {b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right ) \,d x } \]
Timed out. \[ \int (e+f x)^2 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right ) \, dx=\int \sin \left (a+\frac {b}{{\left (c+d\,x\right )}^{2/3}}\right )\,{\left (e+f\,x\right )}^2 \,d x \]