Integrand size = 16, antiderivative size = 117 \[ \int \sqrt [3]{c \sin ^3\left (a+b x^2\right )} \, dx=\frac {\sqrt {\frac {\pi }{2}} \cos (a) \csc \left (a+b x^2\right ) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{\sqrt {b}}+\frac {\sqrt {\frac {\pi }{2}} \csc \left (a+b x^2\right ) \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right ) \sin (a) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{\sqrt {b}} \] Output:
1/2*2^(1/2)*Pi^(1/2)*cos(a)*csc(b*x^2+a)*FresnelS(b^(1/2)*2^(1/2)/Pi^(1/2) *x)*(c*sin(b*x^2+a)^3)^(1/3)/b^(1/2)+1/2*2^(1/2)*Pi^(1/2)*csc(b*x^2+a)*Fre snelC(b^(1/2)*2^(1/2)/Pi^(1/2)*x)*sin(a)*(c*sin(b*x^2+a)^3)^(1/3)/b^(1/2)
Time = 0.17 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.68 \[ \int \sqrt [3]{c \sin ^3\left (a+b x^2\right )} \, dx=\frac {\sqrt {\frac {\pi }{2}} \csc \left (a+b x^2\right ) \left (\cos (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )+\operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right ) \sin (a)\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{\sqrt {b}} \] Input:
Integrate[(c*Sin[a + b*x^2]^3)^(1/3),x]
Output:
(Sqrt[Pi/2]*Csc[a + b*x^2]*(Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*x] + Fresne lC[Sqrt[b]*Sqrt[2/Pi]*x]*Sin[a])*(c*Sin[a + b*x^2]^3)^(1/3))/Sqrt[b]
Time = 0.34 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.80, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {7271, 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.
\(\displaystyle \int \sqrt [3]{c \sin ^3\left (a+b x^2\right )} \, dx\) |
\(\Big \downarrow \) 7271 |
\(\displaystyle \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )} \int \sin \left (b x^2+a\right )dx\) |
\(\Big \downarrow \) 3834 |
\(\displaystyle \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )} \left (\sin (a) \int \cos \left (b x^2\right )dx+\cos (a) \int \sin \left (b x^2\right )dx\right )\) |
\(\Big \downarrow \) 3832 |
\(\displaystyle \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )} \left (\sin (a) \int \cos \left (b x^2\right )dx+\frac {\sqrt {\frac {\pi }{2}} \cos (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )}{\sqrt {b}}\right )\) |
\(\Big \downarrow \) 3833 |
\(\displaystyle \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )} \left (\frac {\sqrt {\frac {\pi }{2}} \sin (a) \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )}{\sqrt {b}}+\frac {\sqrt {\frac {\pi }{2}} \cos (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )}{\sqrt {b}}\right )\) |
Input:
Int[(c*Sin[a + b*x^2]^3)^(1/3),x]
Output:
Csc[a + b*x^2]*((Sqrt[Pi/2]*Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*x])/Sqrt[b] + (Sqrt[Pi/2]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*x]*Sin[a])/Sqrt[b])*(c*Sin[a + b*x^2]^3)^(1/3)
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[(u_.)*((a_.)*(v_)^(m_.))^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a*v^m)^ FracPart[p]/v^(m*FracPart[p])) Int[u*v^(m*p), x], x] /; FreeQ[{a, m, p}, x] && !IntegerQ[p] && !FreeQ[v, x] && !(EqQ[a, 1] && EqQ[m, 1]) && !(Eq Q[v, x] && EqQ[m, 1])
Result contains complex when optimal does not.
Time = 0.95 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.34
method | result | size |
risch | \(\frac {\operatorname {erf}\left (\sqrt {-i b}\, x \right ) \sqrt {\pi }\, \left (i c \,{\mathrm e}^{-3 i \left (b \,x^{2}+a \right )} \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{3}\right )^{\frac {1}{3}} {\mathrm e}^{i \left (b \,x^{2}+2 a \right )}}{4 \sqrt {-i b}\, \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )}-\frac {\left (i c \,{\mathrm e}^{-3 i \left (b \,x^{2}+a \right )} \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{3}\right )^{\frac {1}{3}} {\mathrm e}^{i b \,x^{2}} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {i b}\, x \right )}{4 \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right ) \sqrt {i b}}\) | \(157\) |
Input:
int((c*sin(b*x^2+a)^3)^(1/3),x,method=_RETURNVERBOSE)
Output:
1/4*erf((-I*b)^(1/2)*x)/(-I*b)^(1/2)*Pi^(1/2)/(exp(2*I*(b*x^2+a))-1)*(I*c* exp(-3*I*(b*x^2+a))*(exp(2*I*(b*x^2+a))-1)^3)^(1/3)*exp(I*(b*x^2+2*a))-1/4 *(I*c*exp(-3*I*(b*x^2+a))*(exp(2*I*(b*x^2+a))-1)^3)^(1/3)/(exp(2*I*(b*x^2+ a))-1)*exp(I*b*x^2)*Pi^(1/2)/(I*b)^(1/2)*erf((I*b)^(1/2)*x)
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.03 \[ \int \sqrt [3]{c \sin ^3\left (a+b x^2\right )} \, dx=\frac {{\left (\sqrt {2} {\left (-i \, \pi e^{\left (i \, a\right )} + i \, \pi e^{\left (-i \, a\right )}\right )} \sqrt {\frac {b}{\pi }} \operatorname {C}\left (\sqrt {2} x \sqrt {\frac {b}{\pi }}\right ) + \sqrt {2} {\left (\pi e^{\left (i \, a\right )} + \pi e^{\left (-i \, a\right )}\right )} \sqrt {\frac {b}{\pi }} \operatorname {S}\left (\sqrt {2} x \sqrt {\frac {b}{\pi }}\right )\right )} \left (-{\left (c \cos \left (b x^{2} + a\right )^{2} - c\right )} \sin \left (b x^{2} + a\right )\right )^{\frac {1}{3}}}{4 \, b \sin \left (b x^{2} + a\right )} \] Input:
integrate((c*sin(b*x^2+a)^3)^(1/3),x, algorithm="fricas")
Output:
1/4*(sqrt(2)*(-I*pi*e^(I*a) + I*pi*e^(-I*a))*sqrt(b/pi)*fresnel_cos(sqrt(2 )*x*sqrt(b/pi)) + sqrt(2)*(pi*e^(I*a) + pi*e^(-I*a))*sqrt(b/pi)*fresnel_si n(sqrt(2)*x*sqrt(b/pi)))*(-(c*cos(b*x^2 + a)^2 - c)*sin(b*x^2 + a))^(1/3)/ (b*sin(b*x^2 + a))
\[ \int \sqrt [3]{c \sin ^3\left (a+b x^2\right )} \, dx=\int \sqrt [3]{c \sin ^{3}{\left (a + b x^{2} \right )}}\, dx \] Input:
integrate((c*sin(b*x**2+a)**3)**(1/3),x)
Output:
Integral((c*sin(a + b*x**2)**3)**(1/3), x)
Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.44 \[ \int \sqrt [3]{c \sin ^3\left (a+b x^2\right )} \, dx=\frac {\sqrt {2} \sqrt {\pi } {\left ({\left (-\left (i + 1\right ) \, \cos \left (a\right ) + \left (i - 1\right ) \, \sin \left (a\right )\right )} \operatorname {erf}\left (\sqrt {i \, b} x\right ) + {\left (\left (i - 1\right ) \, \cos \left (a\right ) - \left (i + 1\right ) \, \sin \left (a\right )\right )} \operatorname {erf}\left (\sqrt {-i \, b} x\right )\right )} c^{\frac {1}{3}}}{16 \, \sqrt {b}} \] Input:
integrate((c*sin(b*x^2+a)^3)^(1/3),x, algorithm="maxima")
Output:
1/16*sqrt(2)*sqrt(pi)*((-(I + 1)*cos(a) + (I - 1)*sin(a))*erf(sqrt(I*b)*x) + ((I - 1)*cos(a) - (I + 1)*sin(a))*erf(sqrt(-I*b)*x))*c^(1/3)/sqrt(b)
\[ \int \sqrt [3]{c \sin ^3\left (a+b x^2\right )} \, dx=\int { \left (c \sin \left (b x^{2} + a\right )^{3}\right )^{\frac {1}{3}} \,d x } \] Input:
integrate((c*sin(b*x^2+a)^3)^(1/3),x, algorithm="giac")
Output:
integrate((c*sin(b*x^2 + a)^3)^(1/3), x)
Timed out. \[ \int \sqrt [3]{c \sin ^3\left (a+b x^2\right )} \, dx=\int {\left (c\,{\sin \left (b\,x^2+a\right )}^3\right )}^{1/3} \,d x \] Input:
int((c*sin(a + b*x^2)^3)^(1/3),x)
Output:
int((c*sin(a + b*x^2)^3)^(1/3), x)
\[ \int \sqrt [3]{c \sin ^3\left (a+b x^2\right )} \, dx=c^{\frac {1}{3}} \left (\int \sin \left (b \,x^{2}+a \right )d x \right ) \] Input:
int((c*sin(b*x^2+a)^3)^(1/3),x)
Output:
c**(1/3)*int(sin(a + b*x**2),x)