Integrand size = 14, antiderivative size = 66 \[ \int \frac {1}{\sqrt [3]{a+a \sin (c+d x)}} \, dx=-\frac {\sqrt [6]{2} \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{6},\frac {3}{2},\frac {1}{2} (1-\sin (c+d x))\right )}{d \sqrt [6]{1+\sin (c+d x)} \sqrt [3]{a+a \sin (c+d x)}} \] Output:
-cos(d*x+c)*hypergeom([1/2, 5/6],[3/2],1/2-1/2*sin(d*x+c))*2^(1/6)/d/(1+si n(d*x+c))^(1/6)/(a+a*sin(d*x+c))^(1/3)
Time = 0.07 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.06 \[ \int \frac {1}{\sqrt [3]{a+a \sin (c+d x)}} \, dx=\frac {3 \sqrt {2} \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},\sin ^2\left (\frac {1}{4} (2 c+\pi +2 d x)\right )\right )}{d \sqrt {1-\sin (c+d x)} \sqrt [3]{a (1+\sin (c+d x))}} \] Input:
Integrate[(a + a*Sin[c + d*x])^(-1/3),x]
Output:
(3*Sqrt[2]*Cos[c + d*x]*Hypergeometric2F1[1/6, 1/2, 7/6, Sin[(2*c + Pi + 2 *d*x)/4]^2])/(d*Sqrt[1 - Sin[c + d*x]]*(a*(1 + Sin[c + d*x]))^(1/3))
Time = 0.26 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 3131, 3042, 3130}
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.
\(\displaystyle \int \frac {1}{\sqrt [3]{a \sin (c+d x)+a}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sqrt [3]{a \sin (c+d x)+a}}dx\) |
\(\Big \downarrow \) 3131 |
\(\displaystyle \frac {\sqrt [3]{\sin (c+d x)+1} \int \frac {1}{\sqrt [3]{\sin (c+d x)+1}}dx}{\sqrt [3]{a \sin (c+d x)+a}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt [3]{\sin (c+d x)+1} \int \frac {1}{\sqrt [3]{\sin (c+d x)+1}}dx}{\sqrt [3]{a \sin (c+d x)+a}}\) |
\(\Big \downarrow \) 3130 |
\(\displaystyle -\frac {\sqrt [6]{2} \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{6},\frac {3}{2},\frac {1}{2} (1-\sin (c+d x))\right )}{d \sqrt [6]{\sin (c+d x)+1} \sqrt [3]{a \sin (c+d x)+a}}\) |
Input:
Int[(a + a*Sin[c + d*x])^(-1/3),x]
Output:
-((2^(1/6)*Cos[c + d*x]*Hypergeometric2F1[1/2, 5/6, 3/2, (1 - Sin[c + d*x] )/2])/(d*(1 + Sin[c + d*x])^(1/6)*(a + a*Sin[c + d*x])^(1/3)))
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-2^(n + 1/2))*a^(n - 1/2)*b*(Cos[c + d*x]/(d*Sqrt[a + b*Sin[c + d*x]]))*Hypergeome tric2F1[1/2, 1/2 - n, 3/2, (1/2)*(1 - b*(Sin[c + d*x]/a))], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 - b^2, 0] && !IntegerQ[2*n] && GtQ[a, 0]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[a^IntPar t[n]*((a + b*Sin[c + d*x])^FracPart[n]/(1 + (b/a)*Sin[c + d*x])^FracPart[n] ) Int[(1 + (b/a)*Sin[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 - b^2, 0] && !IntegerQ[2*n] && !GtQ[a, 0]
\[\int \frac {1}{\left (a +a \sin \left (d x +c \right )\right )^{\frac {1}{3}}}d x\]
Input:
int(1/(a+a*sin(d*x+c))^(1/3),x)
Output:
int(1/(a+a*sin(d*x+c))^(1/3),x)
\[ \int \frac {1}{\sqrt [3]{a+a \sin (c+d x)}} \, dx=\int { \frac {1}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {1}{3}}} \,d x } \] Input:
integrate(1/(a+a*sin(d*x+c))^(1/3),x, algorithm="fricas")
Output:
integral((a*sin(d*x + c) + a)^(-1/3), x)
\[ \int \frac {1}{\sqrt [3]{a+a \sin (c+d x)}} \, dx=\int \frac {1}{\sqrt [3]{a \sin {\left (c + d x \right )} + a}}\, dx \] Input:
integrate(1/(a+a*sin(d*x+c))**(1/3),x)
Output:
Integral((a*sin(c + d*x) + a)**(-1/3), x)
\[ \int \frac {1}{\sqrt [3]{a+a \sin (c+d x)}} \, dx=\int { \frac {1}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {1}{3}}} \,d x } \] Input:
integrate(1/(a+a*sin(d*x+c))^(1/3),x, algorithm="maxima")
Output:
integrate((a*sin(d*x + c) + a)^(-1/3), x)
\[ \int \frac {1}{\sqrt [3]{a+a \sin (c+d x)}} \, dx=\int { \frac {1}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {1}{3}}} \,d x } \] Input:
integrate(1/(a+a*sin(d*x+c))^(1/3),x, algorithm="giac")
Output:
integrate((a*sin(d*x + c) + a)^(-1/3), x)
Timed out. \[ \int \frac {1}{\sqrt [3]{a+a \sin (c+d x)}} \, dx=\int \frac {1}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{1/3}} \,d x \] Input:
int(1/(a + a*sin(c + d*x))^(1/3),x)
Output:
int(1/(a + a*sin(c + d*x))^(1/3), x)
\[ \int \frac {1}{\sqrt [3]{a+a \sin (c+d x)}} \, dx=\frac {\int \frac {1}{\left (\sin \left (d x +c \right )+1\right )^{\frac {1}{3}}}d x}{a^{\frac {1}{3}}} \] Input:
int(1/(a+a*sin(d*x+c))^(1/3),x)
Output:
int(1/(sin(c + d*x) + 1)**(1/3),x)/a**(1/3)