\(\int (d \cos (a+b x))^n (c \sin (a+b x))^{3/2} \, dx\) [367]

Optimal result

Integrand size = 23, antiderivative size = 76 \[ \int (d \cos (a+b x))^n (c \sin (a+b x))^{3/2} \, dx=-\frac {c (d \cos (a+b x))^{1+n} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1+n}{2},\frac {3+n}{2},\cos ^2(a+b x)\right ) \sqrt {c \sin (a+b x)}}{b d (1+n) \sqrt [4]{\sin ^2(a+b x)}} \] Output:

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

Mathematica [A] (verified)

Time = 0.58 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00 \[ \int (d \cos (a+b x))^n (c \sin (a+b x))^{3/2} \, dx=-\frac {(d \cos (a+b x))^n \cot (a+b x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1+n}{2},\frac {3+n}{2},\cos ^2(a+b x)\right ) (c \sin (a+b x))^{3/2}}{b (1+n) \sqrt [4]{\sin ^2(a+b x)}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3042, 3056}

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.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int (c \sin (a+b x))^{3/2} (d \cos (a+b x))^ndx\)

\(\Big \downarrow \) 3056

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 3042

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

rule 3056

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n 
_), x_Symbol] :> Simp[(-b^(2*IntPart[(n - 1)/2] + 1))*(b*Sin[e + f*x])^(2*F 
racPart[(n - 1)/2])*((a*Cos[e + f*x])^(m + 1)/(a*f*(m + 1)*(Sin[e + f*x]^2) 
^FracPart[(n - 1)/2]))*Hypergeometric2F1[(1 + m)/2, (1 - n)/2, (3 + m)/2, C 
os[e + f*x]^2], x] /; FreeQ[{a, b, e, f, m, n}, x] && SimplerQ[n, m]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(sqrt(c*sin(b*x + a))*(d*cos(b*x + a))^n*c*sin(b*x + a), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

d**n*sqrt(c)*int(sqrt(sin(a + b*x))*cos(a + b*x)**n*sin(a + b*x),x)*c