Integrand size = 19, antiderivative size = 58 \[ \int \frac {\cos (c+d x)}{(b \cos (c+d x))^{4/3}} \, dx=-\frac {3 (b \cos (c+d x))^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},\cos ^2(c+d x)\right ) \sin (c+d x)}{2 b^2 d \sqrt {\sin ^2(c+d x)}} \] Output:
-3/2*(b*cos(d*x+c))^(2/3)*hypergeom([1/3, 1/2],[4/3],cos(d*x+c)^2)*sin(d*x +c)/b^2/d/(sin(d*x+c)^2)^(1/2)
Time = 0.01 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00 \[ \int \frac {\cos (c+d x)}{(b \cos (c+d x))^{4/3}} \, dx=-\frac {3 \cot (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},\cos ^2(c+d x)\right ) \sqrt {\sin ^2(c+d x)}}{2 b d \sqrt [3]{b \cos (c+d x)}} \] Input:
Integrate[Cos[c + d*x]/(b*Cos[c + d*x])^(4/3),x]
Output:
(-3*Cot[c + d*x]*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sqrt[Sin [c + d*x]^2])/(2*b*d*(b*Cos[c + d*x])^(1/3))
Time = 0.21 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, 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.
\(\displaystyle \int \frac {\cos (c+d x)}{(b \cos (c+d x))^{4/3}} \, dx\) |
\(\Big \downarrow \) 2030 |
\(\displaystyle \frac {\int \frac {1}{\sqrt [3]{b \cos (c+d x)}}dx}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {1}{\sqrt [3]{b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}\) |
\(\Big \downarrow \) 3122 |
\(\displaystyle -\frac {3 \sin (c+d x) (b \cos (c+d x))^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},\cos ^2(c+d x)\right )}{2 b^2 d \sqrt {\sin ^2(c+d x)}}\) |
Input:
Int[Cos[c + d*x]/(b*Cos[c + d*x])^(4/3),x]
Output:
(-3*(b*Cos[c + d*x])^(2/3)*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2 ]*Sin[c + d*x])/(2*b^2*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]
\[\int \frac {\cos \left (d x +c \right )}{\left (\cos \left (d x +c \right ) b \right )^{\frac {4}{3}}}d x\]
Input:
int(cos(d*x+c)/(cos(d*x+c)*b)^(4/3),x)
Output:
int(cos(d*x+c)/(cos(d*x+c)*b)^(4/3),x)
\[ \int \frac {\cos (c+d x)}{(b \cos (c+d x))^{4/3}} \, dx=\int { \frac {\cos \left (d x + c\right )}{\left (b \cos \left (d x + c\right )\right )^{\frac {4}{3}}} \,d x } \] Input:
integrate(cos(d*x+c)/(b*cos(d*x+c))^(4/3),x, algorithm="fricas")
Output:
integral((b*cos(d*x + c))^(2/3)/(b^2*cos(d*x + c)), x)
\[ \int \frac {\cos (c+d x)}{(b \cos (c+d x))^{4/3}} \, dx=\int \frac {\cos {\left (c + d x \right )}}{\left (b \cos {\left (c + d x \right )}\right )^{\frac {4}{3}}}\, dx \] Input:
integrate(cos(d*x+c)/(b*cos(d*x+c))**(4/3),x)
Output:
Integral(cos(c + d*x)/(b*cos(c + d*x))**(4/3), x)
\[ \int \frac {\cos (c+d x)}{(b \cos (c+d x))^{4/3}} \, dx=\int { \frac {\cos \left (d x + c\right )}{\left (b \cos \left (d x + c\right )\right )^{\frac {4}{3}}} \,d x } \] Input:
integrate(cos(d*x+c)/(b*cos(d*x+c))^(4/3),x, algorithm="maxima")
Output:
integrate(cos(d*x + c)/(b*cos(d*x + c))^(4/3), x)
\[ \int \frac {\cos (c+d x)}{(b \cos (c+d x))^{4/3}} \, dx=\int { \frac {\cos \left (d x + c\right )}{\left (b \cos \left (d x + c\right )\right )^{\frac {4}{3}}} \,d x } \] Input:
integrate(cos(d*x+c)/(b*cos(d*x+c))^(4/3),x, algorithm="giac")
Output:
integrate(cos(d*x + c)/(b*cos(d*x + c))^(4/3), x)
Timed out. \[ \int \frac {\cos (c+d x)}{(b \cos (c+d x))^{4/3}} \, dx=\int \frac {\cos \left (c+d\,x\right )}{{\left (b\,\cos \left (c+d\,x\right )\right )}^{4/3}} \,d x \] Input:
int(cos(c + d*x)/(b*cos(c + d*x))^(4/3),x)
Output:
int(cos(c + d*x)/(b*cos(c + d*x))^(4/3), x)
\[ \int \frac {\cos (c+d x)}{(b \cos (c+d x))^{4/3}} \, dx=\frac {\int \frac {1}{\cos \left (d x +c \right )^{\frac {1}{3}}}d x}{b^{\frac {4}{3}}} \] Input:
int(cos(d*x+c)/(b*cos(d*x+c))^(4/3),x)
Output:
int(1/cos(c + d*x)**(1/3),x)/(b**(1/3)*b)