Integrand size = 21, antiderivative size = 80 \[ \int \cos ^{\frac {3}{2}}(c+d x) (b \cos (c+d x))^n \, dx=-\frac {2 \cos ^{\frac {5}{2}}(c+d x) (b \cos (c+d x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (5+2 n),\frac {1}{4} (9+2 n),\cos ^2(c+d x)\right ) \sin (c+d x)}{d (5+2 n) \sqrt {\sin ^2(c+d x)}} \] Output:
-2*cos(d*x+c)^(5/2)*(b*cos(d*x+c))^n*hypergeom([1/2, 5/4+1/2*n],[9/4+1/2*n ],cos(d*x+c)^2)*sin(d*x+c)/d/(5+2*n)/(sin(d*x+c)^2)^(1/2)
Time = 0.13 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00 \[ \int \cos ^{\frac {3}{2}}(c+d x) (b \cos (c+d x))^n \, dx=-\frac {\cos ^{\frac {5}{2}}(c+d x) (b \cos (c+d x))^n \csc (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} \left (\frac {5}{2}+n\right ),\frac {1}{2} \left (\frac {9}{2}+n\right ),\cos ^2(c+d x)\right ) \sqrt {\sin ^2(c+d x)}}{d \left (\frac {5}{2}+n\right )} \] Input:
Integrate[Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^n,x]
Output:
-((Cos[c + d*x]^(5/2)*(b*Cos[c + d*x])^n*Csc[c + d*x]*Hypergeometric2F1[1/ 2, (5/2 + n)/2, (9/2 + n)/2, Cos[c + d*x]^2]*Sqrt[Sin[c + d*x]^2])/(d*(5/2 + n)))
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.
\(\displaystyle \int \cos ^{\frac {3}{2}}(c+d x) (b \cos (c+d x))^n \, dx\) |
\(\Big \downarrow \) 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 |
\(\displaystyle -\frac {2 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (b \cos (c+d x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (2 n+5),\frac {1}{4} (2 n+9),\cos ^2(c+d x)\right )}{d (2 n+5) \sqrt {\sin ^2(c+d x)}}\) |
Input:
Int[Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^n,x]
Output:
(-2*Cos[c + d*x]^(5/2)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (5 + 2*n) /4, (9 + 2*n)/4, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(5 + 2*n)*Sqrt[Sin[c + d *x]^2])
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]
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]
\[\int \cos \left (d x +c \right )^{\frac {3}{2}} \left (\cos \left (d x +c \right ) b \right )^{n}d x\]
Input:
int(cos(d*x+c)^(3/2)*(cos(d*x+c)*b)^n,x)
Output:
int(cos(d*x+c)^(3/2)*(cos(d*x+c)*b)^n,x)
\[ \int \cos ^{\frac {3}{2}}(c+d x) (b \cos (c+d x))^n \, dx=\int { \left (b \cos \left (d x + c\right )\right )^{n} \cos \left (d x + c\right )^{\frac {3}{2}} \,d x } \] Input:
integrate(cos(d*x+c)^(3/2)*(b*cos(d*x+c))^n,x, algorithm="fricas")
Output:
integral((b*cos(d*x + c))^n*cos(d*x + c)^(3/2), x)
\[ \int \cos ^{\frac {3}{2}}(c+d x) (b \cos (c+d x))^n \, dx=\int \left (b \cos {\left (c + d x \right )}\right )^{n} \cos ^{\frac {3}{2}}{\left (c + d x \right )}\, dx \] Input:
integrate(cos(d*x+c)**(3/2)*(b*cos(d*x+c))**n,x)
Output:
Integral((b*cos(c + d*x))**n*cos(c + d*x)**(3/2), x)
\[ \int \cos ^{\frac {3}{2}}(c+d x) (b \cos (c+d x))^n \, dx=\int { \left (b \cos \left (d x + c\right )\right )^{n} \cos \left (d x + c\right )^{\frac {3}{2}} \,d x } \] Input:
integrate(cos(d*x+c)^(3/2)*(b*cos(d*x+c))^n,x, algorithm="maxima")
Output:
integrate((b*cos(d*x + c))^n*cos(d*x + c)^(3/2), x)
\[ \int \cos ^{\frac {3}{2}}(c+d x) (b \cos (c+d x))^n \, dx=\int { \left (b \cos \left (d x + c\right )\right )^{n} \cos \left (d x + c\right )^{\frac {3}{2}} \,d x } \] Input:
integrate(cos(d*x+c)^(3/2)*(b*cos(d*x+c))^n,x, algorithm="giac")
Output:
integrate((b*cos(d*x + c))^n*cos(d*x + c)^(3/2), x)
Timed out. \[ \int \cos ^{\frac {3}{2}}(c+d x) (b \cos (c+d x))^n \, dx=\int {\cos \left (c+d\,x\right )}^{3/2}\,{\left (b\,\cos \left (c+d\,x\right )\right )}^n \,d x \] Input:
int(cos(c + d*x)^(3/2)*(b*cos(c + d*x))^n,x)
Output:
int(cos(c + d*x)^(3/2)*(b*cos(c + d*x))^n, x)
\[ \int \cos ^{\frac {3}{2}}(c+d x) (b \cos (c+d x))^n \, dx=b^{n} \left (\int \cos \left (d x +c \right )^{n +\frac {1}{2}} \cos \left (d x +c \right )d x \right ) \] Input:
int(cos(d*x+c)^(3/2)*(b*cos(d*x+c))^n,x)
Output:
b**n*int(cos(c + d*x)**((2*n + 1)/2)*cos(c + d*x),x)