Integrand size = 21, antiderivative size = 58 \[ \int \frac {\cos ^2(c+d x)}{\sqrt [3]{b \cos (c+d x)}} \, dx=-\frac {3 (b \cos (c+d x))^{8/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4}{3},\frac {7}{3},\cos ^2(c+d x)\right ) \sin (c+d x)}{8 b^3 d \sqrt {\sin ^2(c+d x)}} \] Output:
-3/8*(b*cos(d*x+c))^(8/3)*hypergeom([1/2, 4/3],[7/3],cos(d*x+c)^2)*sin(d*x +c)/b^3/d/(sin(d*x+c)^2)^(1/2)
Time = 0.09 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.09 \[ \int \frac {\cos ^2(c+d x)}{\sqrt [3]{b \cos (c+d x)}} \, dx=-\frac {3 \cos ^2(c+d x) \cot (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4}{3},\frac {7}{3},\cos ^2(c+d x)\right ) \sqrt {\sin ^2(c+d x)}}{8 d \sqrt [3]{b \cos (c+d x)}} \] Input:
Integrate[Cos[c + d*x]^2/(b*Cos[c + d*x])^(1/3),x]
Output:
(-3*Cos[c + d*x]^2*Cot[c + d*x]*Hypergeometric2F1[1/2, 4/3, 7/3, Cos[c + d *x]^2]*Sqrt[Sin[c + d*x]^2])/(8*d*(b*Cos[c + d*x])^(1/3))
Time = 0.22 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2030, 3042, 3122}
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.
\(\displaystyle \int \frac {\cos ^2(c+d x)}{\sqrt [3]{b \cos (c+d x)}} \, dx\) |
\(\Big \downarrow \) 2030 |
\(\displaystyle \frac {\int (b \cos (c+d x))^{5/3}dx}{b^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \left (b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/3}dx}{b^2}\) |
\(\Big \downarrow \) 3122 |
\(\displaystyle -\frac {3 \sin (c+d x) (b \cos (c+d x))^{8/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4}{3},\frac {7}{3},\cos ^2(c+d x)\right )}{8 b^3 d \sqrt {\sin ^2(c+d x)}}\) |
Input:
Int[Cos[c + d*x]^2/(b*Cos[c + d*x])^(1/3),x]
Output:
(-3*(b*Cos[c + d*x])^(8/3)*Hypergeometric2F1[1/2, 4/3, 7/3, Cos[c + d*x]^2 ]*Sin[c + d*x])/(8*b^3*d*Sqrt[Sin[c + d*x]^2])
Int[(Fx_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Simp[1/b^m Int[(b*v) ^(m + n)*Fx, x], x] /; FreeQ[{b, n}, x] && IntegerQ[m]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1)*Sqrt[Cos[c + d*x]^2]))*Hypergeometric2 F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2], x] /; FreeQ[{b, c, d, n}, x] && !IntegerQ[2*n]
\[\int \frac {\cos \left (d x +c \right )^{2}}{\left (\cos \left (d x +c \right ) b \right )^{\frac {1}{3}}}d x\]
Input:
int(cos(d*x+c)^2/(cos(d*x+c)*b)^(1/3),x)
Output:
int(cos(d*x+c)^2/(cos(d*x+c)*b)^(1/3),x)
\[ \int \frac {\cos ^2(c+d x)}{\sqrt [3]{b \cos (c+d x)}} \, dx=\int { \frac {\cos \left (d x + c\right )^{2}}{\left (b \cos \left (d x + c\right )\right )^{\frac {1}{3}}} \,d x } \] Input:
integrate(cos(d*x+c)^2/(b*cos(d*x+c))^(1/3),x, algorithm="fricas")
Output:
integral((b*cos(d*x + c))^(2/3)*cos(d*x + c)/b, x)
\[ \int \frac {\cos ^2(c+d x)}{\sqrt [3]{b \cos (c+d x)}} \, dx=\int \frac {\cos ^{2}{\left (c + d x \right )}}{\sqrt [3]{b \cos {\left (c + d x \right )}}}\, dx \] Input:
integrate(cos(d*x+c)**2/(b*cos(d*x+c))**(1/3),x)
Output:
Integral(cos(c + d*x)**2/(b*cos(c + d*x))**(1/3), x)
\[ \int \frac {\cos ^2(c+d x)}{\sqrt [3]{b \cos (c+d x)}} \, dx=\int { \frac {\cos \left (d x + c\right )^{2}}{\left (b \cos \left (d x + c\right )\right )^{\frac {1}{3}}} \,d x } \] Input:
integrate(cos(d*x+c)^2/(b*cos(d*x+c))^(1/3),x, algorithm="maxima")
Output:
integrate(cos(d*x + c)^2/(b*cos(d*x + c))^(1/3), x)
\[ \int \frac {\cos ^2(c+d x)}{\sqrt [3]{b \cos (c+d x)}} \, dx=\int { \frac {\cos \left (d x + c\right )^{2}}{\left (b \cos \left (d x + c\right )\right )^{\frac {1}{3}}} \,d x } \] Input:
integrate(cos(d*x+c)^2/(b*cos(d*x+c))^(1/3),x, algorithm="giac")
Output:
integrate(cos(d*x + c)^2/(b*cos(d*x + c))^(1/3), x)
Timed out. \[ \int \frac {\cos ^2(c+d x)}{\sqrt [3]{b \cos (c+d x)}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^2}{{\left (b\,\cos \left (c+d\,x\right )\right )}^{1/3}} \,d x \] Input:
int(cos(c + d*x)^2/(b*cos(c + d*x))^(1/3),x)
Output:
int(cos(c + d*x)^2/(b*cos(c + d*x))^(1/3), x)
\[ \int \frac {\cos ^2(c+d x)}{\sqrt [3]{b \cos (c+d x)}} \, dx=\frac {\int \cos \left (d x +c \right )^{\frac {5}{3}}d x}{b^{\frac {1}{3}}} \] Input:
int(cos(d*x+c)^2/(b*cos(d*x+c))^(1/3),x)
Output:
int(cos(c + d*x)**2/cos(c + d*x)**(1/3),x)/b**(1/3)