\(\int \frac {(b \cos (c+d x))^n}{\cos ^{\frac {3}{2}}(c+d x)} \, dx\) [256]

Optimal result

Integrand size = 21, antiderivative size = 80 \[ \int \frac {(b \cos (c+d x))^n}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\frac {2 (b \cos (c+d x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (-1+2 n),\frac {1}{4} (3+2 n),\cos ^2(c+d x)\right ) \sin (c+d x)}{d (1-2 n) \sqrt {\cos (c+d x)} \sqrt {\sin ^2(c+d x)}} \] Output:

2*(b*cos(d*x+c))^n*hypergeom([1/2, -1/4+1/2*n],[3/4+1/2*n],cos(d*x+c)^2)*s 
in(d*x+c)/d/(1-2*n)/cos(d*x+c)^(1/2)/(sin(d*x+c)^2)^(1/2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00 \[ \int \frac {(b \cos (c+d x))^n}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {(b \cos (c+d x))^n \csc (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} \left (-\frac {1}{2}+n\right ),\frac {1}{2} \left (\frac {3}{2}+n\right ),\cos ^2(c+d x)\right ) \sqrt {\sin ^2(c+d x)}}{d \left (-\frac {1}{2}+n\right ) \sqrt {\cos (c+d x)}} \] Input:

Integrate[(b*Cos[c + d*x])^n/Cos[c + d*x]^(3/2),x]
 

Output:

-(((b*Cos[c + d*x])^n*Csc[c + d*x]*Hypergeometric2F1[1/2, (-1/2 + n)/2, (3 
/2 + n)/2, Cos[c + d*x]^2]*Sqrt[Sin[c + d*x]^2])/(d*(-1/2 + n)*Sqrt[Cos[c 
+ d*x]]))
 

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 80, 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 = {2034, 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.

\(\displaystyle \int \frac {(b \cos (c+d x))^n}{\cos ^{\frac {3}{2}}(c+d x)} \, dx\)

\(\Big \downarrow \) 2034

\(\displaystyle \cos ^{-n}(c+d x) (b \cos (c+d x))^n \int \cos ^{n-\frac {3}{2}}(c+d x)dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \cos ^{-n}(c+d x) (b \cos (c+d x))^n \int \sin \left (c+d x+\frac {\pi }{2}\right )^{n-\frac {3}{2}}dx\)

\(\Big \downarrow \) 3122

\(\displaystyle \frac {2 \sin (c+d x) (b \cos (c+d x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (2 n-1),\frac {1}{4} (2 n+3),\cos ^2(c+d x)\right )}{d (1-2 n) \sqrt {\sin ^2(c+d x)} \sqrt {\cos (c+d x)}}\)

Input:

Int[(b*Cos[c + d*x])^n/Cos[c + d*x]^(3/2),x]
 

Output:

(2*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (-1 + 2*n)/4, (3 + 2*n)/4, Co 
s[c + d*x]^2]*Sin[c + d*x])/(d*(1 - 2*n)*Sqrt[Cos[c + d*x]]*Sqrt[Sin[c + d 
*x]^2])
 

Defintions of rubi rules used

rule 2034

Int[(Fx_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Simp[b^IntPart 
[n]*((b*v)^FracPart[n]/(a^IntPart[n]*(a*v)^FracPart[n]))   Int[(a*v)^(m + n 
)*Fx, x], x] /; FreeQ[{a, b, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] && 
  !IntegerQ[m + n]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3122

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]
 

Maple [F]

\[\int \frac {\left (\cos \left (d x +c \right ) b \right )^{n}}{\cos \left (d x +c \right )^{\frac {3}{2}}}d x\]

Input:

int((cos(d*x+c)*b)^n/cos(d*x+c)^(3/2),x)
 

Output:

int((cos(d*x+c)*b)^n/cos(d*x+c)^(3/2),x)
 

Fricas [F]

\[ \int \frac {(b \cos (c+d x))^n}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {\left (b \cos \left (d x + c\right )\right )^{n}}{\cos \left (d x + c\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate((b*cos(d*x+c))^n/cos(d*x+c)^(3/2),x, algorithm="fricas")
 

Output:

integral((b*cos(d*x + c))^n/cos(d*x + c)^(3/2), x)
 

Sympy [F]

\[ \int \frac {(b \cos (c+d x))^n}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {\left (b \cos {\left (c + d x \right )}\right )^{n}}{\cos ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \] Input:

integrate((b*cos(d*x+c))**n/cos(d*x+c)**(3/2),x)
 

Output:

Integral((b*cos(c + d*x))**n/cos(c + d*x)**(3/2), x)
 

Maxima [F]

\[ \int \frac {(b \cos (c+d x))^n}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {\left (b \cos \left (d x + c\right )\right )^{n}}{\cos \left (d x + c\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate((b*cos(d*x+c))^n/cos(d*x+c)^(3/2),x, algorithm="maxima")
 

Output:

integrate((b*cos(d*x + c))^n/cos(d*x + c)^(3/2), x)
 

Giac [F]

\[ \int \frac {(b \cos (c+d x))^n}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {\left (b \cos \left (d x + c\right )\right )^{n}}{\cos \left (d x + c\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate((b*cos(d*x+c))^n/cos(d*x+c)^(3/2),x, algorithm="giac")
 

Output:

integrate((b*cos(d*x + c))^n/cos(d*x + c)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(b \cos (c+d x))^n}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {{\left (b\,\cos \left (c+d\,x\right )\right )}^n}{{\cos \left (c+d\,x\right )}^{3/2}} \,d x \] Input:

int((b*cos(c + d*x))^n/cos(c + d*x)^(3/2),x)
 

Output:

int((b*cos(c + d*x))^n/cos(c + d*x)^(3/2), x)
 

Reduce [F]

\[ \int \frac {(b \cos (c+d x))^n}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=b^{n} \left (\int \frac {\cos \left (d x +c \right )^{n +\frac {1}{2}}}{\cos \left (d x +c \right )^{2}}d x \right ) \] Input:

int((b*cos(d*x+c))^n/cos(d*x+c)^(3/2),x)
 

Output:

b**n*int(cos(c + d*x)**((2*n + 1)/2)/cos(c + d*x)**2,x)