Integrand size = 27, antiderivative size = 299 \[ \int (c e+d e x)^{4/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right ) \, dx=-\frac {8 b^3 e \sqrt [3]{e (c+d x)} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{35 d}+\frac {6 b e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{35 d}-\frac {8 b^{7/2} e \sqrt {2 \pi } \sqrt [3]{e (c+d x)} \cos (a) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{35 d \sqrt [3]{c+d x}}-\frac {8 b^{7/2} e \sqrt {2 \pi } \sqrt [3]{e (c+d x)} \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right ) \sin (a)}{35 d \sqrt [3]{c+d x}}-\frac {4 b^2 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{35 d}+\frac {3 e (c+d x)^2 \sqrt [3]{e (c+d x)} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{7 d} \]
-8/35*b^3*e*(e*(d*x+c))^(1/3)*cos(a+b/(d*x+c)^(2/3))/d+6/35*b*e*(d*x+c)^(4 /3)*(e*(d*x+c))^(1/3)*cos(a+b/(d*x+c)^(2/3))/d-4/35*b^2*e*(d*x+c)^(2/3)*(e *(d*x+c))^(1/3)*sin(a+b/(d*x+c)^(2/3))/d+3/7*e*(d*x+c)^2*(e*(d*x+c))^(1/3) *sin(a+b/(d*x+c)^(2/3))/d-8/35*b^(7/2)*e*(e*(d*x+c))^(1/3)*cos(a)*FresnelS (b^(1/2)*2^(1/2)/Pi^(1/2)/(d*x+c)^(1/3))*2^(1/2)*Pi^(1/2)/d/(d*x+c)^(1/3)- 8/35*b^(7/2)*e*(e*(d*x+c))^(1/3)*FresnelC(b^(1/2)*2^(1/2)/Pi^(1/2)/(d*x+c) ^(1/3))*sin(a)*2^(1/2)*Pi^(1/2)/d/(d*x+c)^(1/3)
Time = 0.70 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.79 \[ \int (c e+d e x)^{4/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right ) \, dx=\frac {(e (c+d x))^{4/3} \left (\frac {\cos \left (\frac {b}{(c+d x)^{2/3}}\right ) \left (-8 b^3 \cos (a)+6 b (c+d x)^{4/3} \cos (a)-4 b^2 (c+d x)^{2/3} \sin (a)+15 (c+d x)^2 \sin (a)\right )}{c+d x}-\frac {8 b^{7/2} \sqrt {2 \pi } \left (\cos (a) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )+\operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right ) \sin (a)\right )}{(c+d x)^{4/3}}+\frac {\left (-4 b^2 (c+d x)^{2/3} \cos (a)+15 (c+d x)^2 \cos (a)+8 b^3 \sin (a)-6 b (c+d x)^{4/3} \sin (a)\right ) \sin \left (\frac {b}{(c+d x)^{2/3}}\right )}{c+d x}\right )}{35 d} \]
((e*(c + d*x))^(4/3)*((Cos[b/(c + d*x)^(2/3)]*(-8*b^3*Cos[a] + 6*b*(c + d* x)^(4/3)*Cos[a] - 4*b^2*(c + d*x)^(2/3)*Sin[a] + 15*(c + d*x)^2*Sin[a]))/( c + d*x) - (8*b^(7/2)*Sqrt[2*Pi]*(Cos[a]*FresnelS[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)^(1/3)] + FresnelC[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)^(1/3)]*Sin[a]))/(c + d*x)^(4/3) + ((-4*b^2*(c + d*x)^(2/3)*Cos[a] + 15*(c + d*x)^2*Cos[a] + 8*b^3*Sin[a] - 6*b*(c + d*x)^(4/3)*Sin[a])*Sin[b/(c + d*x)^(2/3)])/(c + d* x)))/(35*d)
Time = 0.75 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.79, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {3916, 3898, 3896, 3890, 3868, 3869, 3868, 3869, 3834, 3832, 3833}
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.
\(\displaystyle \int (c e+d e x)^{4/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right ) \, dx\) |
\(\Big \downarrow \) 3916 |
\(\displaystyle \frac {\int (e (c+d x))^{4/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )d(c+d x)}{d}\) |
\(\Big \downarrow \) 3898 |
\(\displaystyle \frac {e \sqrt [3]{e (c+d x)} \int (c+d x)^{4/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )d(c+d x)}{d \sqrt [3]{c+d x}}\) |
\(\Big \downarrow \) 3896 |
\(\displaystyle \frac {3 e \sqrt [3]{e (c+d x)} \int (c+d x)^2 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )d\sqrt [3]{c+d x}}{d \sqrt [3]{c+d x}}\) |
\(\Big \downarrow \) 3890 |
\(\displaystyle -\frac {3 e \sqrt [3]{e (c+d x)} \int \frac {\sin \left (a+b (c+d x)^{2/3}\right )}{(c+d x)^{8/3}}d\frac {1}{\sqrt [3]{c+d x}}}{d \sqrt [3]{c+d x}}\) |
\(\Big \downarrow \) 3868 |
\(\displaystyle -\frac {3 e \sqrt [3]{e (c+d x)} \left (\frac {2}{7} b \int \frac {\cos \left (a+b (c+d x)^{2/3}\right )}{(c+d x)^2}d\frac {1}{\sqrt [3]{c+d x}}-\frac {\sin \left (a+b (c+d x)^{2/3}\right )}{7 (c+d x)^{7/3}}\right )}{d \sqrt [3]{c+d x}}\) |
\(\Big \downarrow \) 3869 |
\(\displaystyle -\frac {3 e \sqrt [3]{e (c+d x)} \left (\frac {2}{7} b \left (-\frac {2}{5} b \int \frac {\sin \left (a+b (c+d x)^{2/3}\right )}{(c+d x)^{4/3}}d\frac {1}{\sqrt [3]{c+d x}}-\frac {\cos \left (a+b (c+d x)^{2/3}\right )}{5 (c+d x)^{5/3}}\right )-\frac {\sin \left (a+b (c+d x)^{2/3}\right )}{7 (c+d x)^{7/3}}\right )}{d \sqrt [3]{c+d x}}\) |
\(\Big \downarrow \) 3868 |
\(\displaystyle -\frac {3 e \sqrt [3]{e (c+d x)} \left (\frac {2}{7} b \left (-\frac {2}{5} b \left (\frac {2}{3} b \int \frac {\cos \left (a+b (c+d x)^{2/3}\right )}{(c+d x)^{2/3}}d\frac {1}{\sqrt [3]{c+d x}}-\frac {\sin \left (a+b (c+d x)^{2/3}\right )}{3 (c+d x)}\right )-\frac {\cos \left (a+b (c+d x)^{2/3}\right )}{5 (c+d x)^{5/3}}\right )-\frac {\sin \left (a+b (c+d x)^{2/3}\right )}{7 (c+d x)^{7/3}}\right )}{d \sqrt [3]{c+d x}}\) |
\(\Big \downarrow \) 3869 |
\(\displaystyle -\frac {3 e \sqrt [3]{e (c+d x)} \left (\frac {2}{7} b \left (-\frac {2}{5} b \left (\frac {2}{3} b \left (-2 b \int \sin \left (a+b (c+d x)^{2/3}\right )d\frac {1}{\sqrt [3]{c+d x}}-\frac {\cos \left (a+b (c+d x)^{2/3}\right )}{\sqrt [3]{c+d x}}\right )-\frac {\sin \left (a+b (c+d x)^{2/3}\right )}{3 (c+d x)}\right )-\frac {\cos \left (a+b (c+d x)^{2/3}\right )}{5 (c+d x)^{5/3}}\right )-\frac {\sin \left (a+b (c+d x)^{2/3}\right )}{7 (c+d x)^{7/3}}\right )}{d \sqrt [3]{c+d x}}\) |
\(\Big \downarrow \) 3834 |
\(\displaystyle -\frac {3 e \sqrt [3]{e (c+d x)} \left (\frac {2}{7} b \left (-\frac {2}{5} b \left (\frac {2}{3} b \left (-2 b \left (\sin (a) \int \cos \left (b (c+d x)^{2/3}\right )d\frac {1}{\sqrt [3]{c+d x}}+\cos (a) \int \sin \left (b (c+d x)^{2/3}\right )d\frac {1}{\sqrt [3]{c+d x}}\right )-\frac {\cos \left (a+b (c+d x)^{2/3}\right )}{\sqrt [3]{c+d x}}\right )-\frac {\sin \left (a+b (c+d x)^{2/3}\right )}{3 (c+d x)}\right )-\frac {\cos \left (a+b (c+d x)^{2/3}\right )}{5 (c+d x)^{5/3}}\right )-\frac {\sin \left (a+b (c+d x)^{2/3}\right )}{7 (c+d x)^{7/3}}\right )}{d \sqrt [3]{c+d x}}\) |
\(\Big \downarrow \) 3832 |
\(\displaystyle -\frac {3 e \sqrt [3]{e (c+d x)} \left (\frac {2}{7} b \left (-\frac {2}{5} b \left (\frac {2}{3} b \left (-2 b \left (\sin (a) \int \cos \left (b (c+d x)^{2/3}\right )d\frac {1}{\sqrt [3]{c+d x}}+\frac {\sqrt {\frac {\pi }{2}} \cos (a) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{\sqrt {b}}\right )-\frac {\cos \left (a+b (c+d x)^{2/3}\right )}{\sqrt [3]{c+d x}}\right )-\frac {\sin \left (a+b (c+d x)^{2/3}\right )}{3 (c+d x)}\right )-\frac {\cos \left (a+b (c+d x)^{2/3}\right )}{5 (c+d x)^{5/3}}\right )-\frac {\sin \left (a+b (c+d x)^{2/3}\right )}{7 (c+d x)^{7/3}}\right )}{d \sqrt [3]{c+d x}}\) |
\(\Big \downarrow \) 3833 |
\(\displaystyle -\frac {3 e \sqrt [3]{e (c+d x)} \left (\frac {2}{7} b \left (-\frac {2}{5} b \left (\frac {2}{3} b \left (-2 b \left (\frac {\sqrt {\frac {\pi }{2}} \sin (a) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{\sqrt {b}}+\frac {\sqrt {\frac {\pi }{2}} \cos (a) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{\sqrt {b}}\right )-\frac {\cos \left (a+b (c+d x)^{2/3}\right )}{\sqrt [3]{c+d x}}\right )-\frac {\sin \left (a+b (c+d x)^{2/3}\right )}{3 (c+d x)}\right )-\frac {\cos \left (a+b (c+d x)^{2/3}\right )}{5 (c+d x)^{5/3}}\right )-\frac {\sin \left (a+b (c+d x)^{2/3}\right )}{7 (c+d x)^{7/3}}\right )}{d \sqrt [3]{c+d x}}\) |
(-3*e*(e*(c + d*x))^(1/3)*(-1/7*Sin[a + b*(c + d*x)^(2/3)]/(c + d*x)^(7/3) + (2*b*(-1/5*Cos[a + b*(c + d*x)^(2/3)]/(c + d*x)^(5/3) - (2*b*((2*b*(-(C os[a + b*(c + d*x)^(2/3)]/(c + d*x)^(1/3)) - 2*b*((Sqrt[Pi/2]*Cos[a]*Fresn elS[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)^(1/3)])/Sqrt[b] + (Sqrt[Pi/2]*FresnelC[ (Sqrt[b]*Sqrt[2/Pi])/(c + d*x)^(1/3)]*Sin[a])/Sqrt[b])))/3 - Sin[a + b*(c + d*x)^(2/3)]/(3*(c + d*x))))/5))/7))/(d*(c + d*x)^(1/3))
3.3.49.3.1 Defintions of rubi rules used
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]
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]
Int[Sin[(c_) + (d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[Sin[c] In t[Cos[d*(e + f*x)^2], x], x] + Simp[Cos[c] Int[Sin[d*(e + f*x)^2], x], x] /; FreeQ[{c, d, e, f}, x]
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] - Simp[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]
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] + Simp[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]
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]
Int[(x_)^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol ] :> Module[{k = Denominator[n]}, Simp[k Subst[Int[x^(k*(m + 1) - 1)*(a + b*Sin[c + d*x^(k*n)])^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[p] && FractionQ[n]
Int[((e_)*(x_))^(m_)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_ Symbol] :> Simp[e^IntPart[m]*((e*x)^FracPart[m]/x^FracPart[m]) Int[x^m*(a + b*Sin[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IntegerQ[ p] && FractionQ[n]
Int[((g_.) + (h_.)*(x_))^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f _.)*(x_))^(n_)])^(p_.), x_Symbol] :> Simp[1/f Subst[Int[(h*(x/f))^m*(a + b*Sin[c + d*x^n])^p, x], x, e + f*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IGtQ[p, 0] && EqQ[f*g - e*h, 0]
\[\int \left (d e x +c e \right )^{\frac {4}{3}} \sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )d x\]
\[ \int (c e+d e x)^{4/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right ) \, dx=\int { {\left (d e x + c e\right )}^{\frac {4}{3}} \sin \left (a + \frac {b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right ) \,d x } \]
Timed out. \[ \int (c e+d e x)^{4/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right ) \, dx=\text {Timed out} \]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.02 (sec) , antiderivative size = 1120, normalized size of antiderivative = 3.75 \[ \int (c e+d e x)^{4/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right ) \, dx=\text {Too large to display} \]
-3/8*((((-I*gamma(-7/2, I*b*conjugate((d*x + c)^(-2/3))) + I*gamma(-7/2, - I*b/(d*x + c)^(2/3)))*cos(7/4*pi + 7/3*arctan2(0, d*x + c)) + (I*gamma(-7/ 2, -I*b*conjugate((d*x + c)^(-2/3))) - I*gamma(-7/2, I*b/(d*x + c)^(2/3))) *cos(-7/4*pi + 7/3*arctan2(0, d*x + c)) + (gamma(-7/2, I*b*conjugate((d*x + c)^(-2/3))) + gamma(-7/2, -I*b/(d*x + c)^(2/3)))*sin(7/4*pi + 7/3*arctan 2(0, d*x + c)) - (gamma(-7/2, -I*b*conjugate((d*x + c)^(-2/3))) + gamma(-7 /2, I*b/(d*x + c)^(2/3)))*sin(-7/4*pi + 7/3*arctan2(0, d*x + c)))*cos(a) - ((gamma(-7/2, I*b*conjugate((d*x + c)^(-2/3))) + gamma(-7/2, -I*b/(d*x + c)^(2/3)))*cos(7/4*pi + 7/3*arctan2(0, d*x + c)) + (gamma(-7/2, -I*b*conju gate((d*x + c)^(-2/3))) + gamma(-7/2, I*b/(d*x + c)^(2/3)))*cos(-7/4*pi + 7/3*arctan2(0, d*x + c)) - (-I*gamma(-7/2, I*b*conjugate((d*x + c)^(-2/3)) ) + I*gamma(-7/2, -I*b/(d*x + c)^(2/3)))*sin(7/4*pi + 7/3*arctan2(0, d*x + c)) - (-I*gamma(-7/2, -I*b*conjugate((d*x + c)^(-2/3))) + I*gamma(-7/2, I *b/(d*x + c)^(2/3)))*sin(-7/4*pi + 7/3*arctan2(0, d*x + c)))*sin(a))*d^2*e ^(4/3)*x^2 + 2*(((-I*gamma(-7/2, I*b*conjugate((d*x + c)^(-2/3))) + I*gamm a(-7/2, -I*b/(d*x + c)^(2/3)))*cos(7/4*pi + 7/3*arctan2(0, d*x + c)) + (I* gamma(-7/2, -I*b*conjugate((d*x + c)^(-2/3))) - I*gamma(-7/2, I*b/(d*x + c )^(2/3)))*cos(-7/4*pi + 7/3*arctan2(0, d*x + c)) + (gamma(-7/2, I*b*conjug ate((d*x + c)^(-2/3))) + gamma(-7/2, -I*b/(d*x + c)^(2/3)))*sin(7/4*pi + 7 /3*arctan2(0, d*x + c)) - (gamma(-7/2, -I*b*conjugate((d*x + c)^(-2/3))...
Exception generated. \[ \int (c e+d e x)^{4/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right ) \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{1,[0,6,1,0,0,0]%%%}+%%%{-2,[0,3,1,1,1,0]%%%}+%%%{1,[0,0,1, 2,2,0]%%%
Timed out. \[ \int (c e+d e x)^{4/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right ) \, dx=\int \sin \left (a+\frac {b}{{\left (c+d\,x\right )}^{2/3}}\right )\,{\left (c\,e+d\,e\,x\right )}^{4/3} \,d x \]