Integrand size = 14, antiderivative size = 67 \[ \int (a+a \cos (c+d x))^{4/3} \, dx=\frac {2\ 2^{5/6} a \sqrt [3]{a+a \cos (c+d x)} \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},\frac {1}{2},\frac {3}{2},\frac {1}{2} (1-\cos (c+d x))\right ) \sin (c+d x)}{d (1+\cos (c+d x))^{5/6}} \] Output:
2*2^(5/6)*a*(a+a*cos(d*x+c))^(1/3)*hypergeom([-5/6, 1/2],[3/2],1/2-1/2*cos (d*x+c))*sin(d*x+c)/d/(1+cos(d*x+c))^(5/6)
Time = 0.15 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.03 \[ \int (a+a \cos (c+d x))^{4/3} \, dx=-\frac {6 (a (1+\cos (c+d x)))^{4/3} \cot \left (\frac {1}{2} (c+d x)\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {11}{6},\frac {17}{6},\cos ^2\left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {\sin ^2\left (\frac {1}{2} (c+d x)\right )}}{11 d} \] Input:
Integrate[(a + a*Cos[c + d*x])^(4/3),x]
Output:
(-6*(a*(1 + Cos[c + d*x]))^(4/3)*Cot[(c + d*x)/2]*Hypergeometric2F1[1/2, 1 1/6, 17/6, Cos[(c + d*x)/2]^2]*Sqrt[Sin[(c + d*x)/2]^2])/(11*d)
Time = 0.28 (sec) , antiderivative size = 67, 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.
\(\displaystyle \int (a \cos (c+d x)+a)^{4/3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^{4/3}dx\) |
\(\Big \downarrow \) 3131 |
\(\displaystyle \frac {a \sqrt [3]{a \cos (c+d x)+a} \int (\cos (c+d x)+1)^{4/3}dx}{\sqrt [3]{\cos (c+d x)+1}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {a \sqrt [3]{a \cos (c+d x)+a} \int \left (\sin \left (c+d x+\frac {\pi }{2}\right )+1\right )^{4/3}dx}{\sqrt [3]{\cos (c+d x)+1}}\) |
\(\Big \downarrow \) 3130 |
\(\displaystyle \frac {2\ 2^{5/6} a \sin (c+d x) \sqrt [3]{a \cos (c+d x)+a} \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},\frac {1}{2},\frac {3}{2},\frac {1}{2} (1-\cos (c+d x))\right )}{d (\cos (c+d x)+1)^{5/6}}\) |
Input:
Int[(a + a*Cos[c + d*x])^(4/3),x]
Output:
(2*2^(5/6)*a*(a + a*Cos[c + d*x])^(1/3)*Hypergeometric2F1[-5/6, 1/2, 3/2, (1 - Cos[c + d*x])/2]*Sin[c + d*x])/(d*(1 + Cos[c + d*x])^(5/6))
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]
\[\int \left (a +a \cos \left (d x +c \right )\right )^{\frac {4}{3}}d x\]
Input:
int((a+a*cos(d*x+c))^(4/3),x)
Output:
int((a+a*cos(d*x+c))^(4/3),x)
\[ \int (a+a \cos (c+d x))^{4/3} \, dx=\int { {\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {4}{3}} \,d x } \] Input:
integrate((a+a*cos(d*x+c))^(4/3),x, algorithm="fricas")
Output:
integral((a*cos(d*x + c) + a)^(4/3), x)
\[ \int (a+a \cos (c+d x))^{4/3} \, dx=\int \left (a \cos {\left (c + d x \right )} + a\right )^{\frac {4}{3}}\, dx \] Input:
integrate((a+a*cos(d*x+c))**(4/3),x)
Output:
Integral((a*cos(c + d*x) + a)**(4/3), x)
\[ \int (a+a \cos (c+d x))^{4/3} \, dx=\int { {\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {4}{3}} \,d x } \] Input:
integrate((a+a*cos(d*x+c))^(4/3),x, algorithm="maxima")
Output:
integrate((a*cos(d*x + c) + a)^(4/3), x)
\[ \int (a+a \cos (c+d x))^{4/3} \, dx=\int { {\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {4}{3}} \,d x } \] Input:
integrate((a+a*cos(d*x+c))^(4/3),x, algorithm="giac")
Output:
integrate((a*cos(d*x + c) + a)^(4/3), x)
Timed out. \[ \int (a+a \cos (c+d x))^{4/3} \, dx=\int {\left (a+a\,\cos \left (c+d\,x\right )\right )}^{4/3} \,d x \] Input:
int((a + a*cos(c + d*x))^(4/3),x)
Output:
int((a + a*cos(c + d*x))^(4/3), x)
\[ \int (a+a \cos (c+d x))^{4/3} \, dx=a^{\frac {4}{3}} \left (\int \left (\cos \left (d x +c \right )+1\right )^{\frac {1}{3}}d x +\int \left (\cos \left (d x +c \right )+1\right )^{\frac {1}{3}} \cos \left (d x +c \right )d x \right ) \] Input:
int((a+a*cos(d*x+c))^(4/3),x)
Output:
a**(1/3)*a*(int((cos(c + d*x) + 1)**(1/3),x) + int((cos(c + d*x) + 1)**(1/ 3)*cos(c + d*x),x))