\(\int \frac {(d \cos (a+b x))^n}{(c \sin (a+b x))^{3/2}} \, dx\) [370]

Optimal result

Integrand size = 23, antiderivative size = 78 \[ \int \frac {(d \cos (a+b x))^n}{(c \sin (a+b x))^{3/2}} \, dx=-\frac {(d \cos (a+b x))^{1+n} \operatorname {Hypergeometric2F1}\left (\frac {5}{4},\frac {1+n}{2},\frac {3+n}{2},\cos ^2(a+b x)\right ) \sqrt [4]{\sin ^2(a+b x)}}{b c d (1+n) \sqrt {c \sin (a+b x)}} \]

[Out]

-(d*cos(b*x+a))^(1+n)*hypergeom([5/4, 1/2+1/2*n],[3/2+1/2*n],cos(b*x+a)^2)*(sin(b*x+a)^2)^(1/4)/b/c/d/(1+n)/(c
*sin(b*x+a))^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {2656} \[ \int \frac {(d \cos (a+b x))^n}{(c \sin (a+b x))^{3/2}} \, dx=-\frac {\sqrt [4]{\sin ^2(a+b x)} (d \cos (a+b x))^{n+1} \operatorname {Hypergeometric2F1}\left (\frac {5}{4},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(a+b x)\right )}{b c d (n+1) \sqrt {c \sin (a+b x)}} \]

[In]

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

[Out]

-(((d*Cos[a + b*x])^(1 + n)*Hypergeometric2F1[5/4, (1 + n)/2, (3 + n)/2, Cos[a + b*x]^2]*(Sin[a + b*x]^2)^(1/4
))/(b*c*d*(1 + n)*Sqrt[c*Sin[a + b*x]]))

Rule 2656

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^(2*IntPar
t[(n - 1)/2] + 1))*(b*Sin[e + f*x])^(2*FracPart[(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, Cos[e + f*x]^2], x] /; FreeQ[{a
, b, e, f, m, n}, x] && SimplerQ[n, m]

Rubi steps \begin{align*} \text {integral}& = -\frac {(d \cos (a+b x))^{1+n} \operatorname {Hypergeometric2F1}\left (\frac {5}{4},\frac {1+n}{2},\frac {3+n}{2},\cos ^2(a+b x)\right ) \sqrt [4]{\sin ^2(a+b x)}}{b c d (1+n) \sqrt {c \sin (a+b x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.01 \[ \int \frac {(d \cos (a+b x))^n}{(c \sin (a+b x))^{3/2}} \, dx=-\frac {(d \cos (a+b x))^n \cot (a+b x) \operatorname {Hypergeometric2F1}\left (\frac {5}{4},\frac {1+n}{2},\frac {3+n}{2},\cos ^2(a+b x)\right ) \sqrt {c \sin (a+b x)} \sqrt [4]{\sin ^2(a+b x)}}{b c^2 (1+n)} \]

[In]

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

[Out]

-(((d*Cos[a + b*x])^n*Cot[a + b*x]*Hypergeometric2F1[5/4, (1 + n)/2, (3 + n)/2, Cos[a + b*x]^2]*Sqrt[c*Sin[a +
b*x]]*(Sin[a + b*x]^2)^(1/4))/(b*c^2*(1 + n)))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(-sqrt(c*sin(b*x + a))*(d*cos(b*x + a))^n/(c^2*cos(b*x + a)^2 - c^2), x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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