\(\int (d \cos (e+f x))^n (b \sin (e+f x))^m \, dx\) [339]

Optimal result

Integrand size = 21, antiderivative size = 88 \[ \int (d \cos (e+f x))^n (b \sin (e+f x))^m \, dx=-\frac {b (d \cos (e+f x))^{1+n} \operatorname {Hypergeometric2F1}\left (\frac {1-m}{2},\frac {1+n}{2},\frac {3+n}{2},\cos ^2(e+f x)\right ) (b \sin (e+f x))^{-1+m} \sin ^2(e+f x)^{\frac {1-m}{2}}}{d f (1+n)} \]

[Out]

-b*(d*cos(f*x+e))^(1+n)*hypergeom([1/2+1/2*n, 1/2-1/2*m],[3/2+1/2*n],cos(f*x+e)^2)*(b*sin(f*x+e))^(-1+m)*(sin(
f*x+e)^2)^(1/2-1/2*m)/d/f/(1+n)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2656} \[ \int (d \cos (e+f x))^n (b \sin (e+f x))^m \, dx=-\frac {b \sin ^2(e+f x)^{\frac {1-m}{2}} (b \sin (e+f x))^{m-1} (d \cos (e+f x))^{n+1} \operatorname {Hypergeometric2F1}\left (\frac {1-m}{2},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(e+f x)\right )}{d f (n+1)} \]

[In]

Int[(d*Cos[e + f*x])^n*(b*Sin[e + f*x])^m,x]

[Out]

-((b*(d*Cos[e + f*x])^(1 + n)*Hypergeometric2F1[(1 - m)/2, (1 + n)/2, (3 + n)/2, Cos[e + f*x]^2]*(b*Sin[e + f*
x])^(-1 + m)*(Sin[e + f*x]^2)^((1 - m)/2))/(d*f*(1 + n)))

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 {b (d \cos (e+f x))^{1+n} \operatorname {Hypergeometric2F1}\left (\frac {1-m}{2},\frac {1+n}{2},\frac {3+n}{2},\cos ^2(e+f x)\right ) (b \sin (e+f x))^{-1+m} \sin ^2(e+f x)^{\frac {1-m}{2}}}{d f (1+n)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.97 \[ \int (d \cos (e+f x))^n (b \sin (e+f x))^m \, dx=\frac {(d \cos (e+f x))^n \cos ^2(e+f x)^{\frac {1-n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},\frac {1-n}{2},\frac {3+m}{2},\sin ^2(e+f x)\right ) (b \sin (e+f x))^m \tan (e+f x)}{f (1+m)} \]

[In]

Integrate[(d*Cos[e + f*x])^n*(b*Sin[e + f*x])^m,x]

[Out]

((d*Cos[e + f*x])^n*(Cos[e + f*x]^2)^((1 - n)/2)*Hypergeometric2F1[(1 + m)/2, (1 - n)/2, (3 + m)/2, Sin[e + f*
x]^2]*(b*Sin[e + f*x])^m*Tan[e + f*x])/(f*(1 + m))

Maple [F]

\[\int \left (d \cos \left (f x +e \right )\right )^{n} \left (b \sin \left (f x +e \right )\right )^{m}d x\]

[In]

int((d*cos(f*x+e))^n*(b*sin(f*x+e))^m,x)

[Out]

int((d*cos(f*x+e))^n*(b*sin(f*x+e))^m,x)

Fricas [F]

\[ \int (d \cos (e+f x))^n (b \sin (e+f x))^m \, dx=\int { \left (d \cos \left (f x + e\right )\right )^{n} \left (b \sin \left (f x + e\right )\right )^{m} \,d x } \]

[In]

integrate((d*cos(f*x+e))^n*(b*sin(f*x+e))^m,x, algorithm="fricas")

[Out]

integral((d*cos(f*x + e))^n*(b*sin(f*x + e))^m, x)

Sympy [F]

\[ \int (d \cos (e+f x))^n (b \sin (e+f x))^m \, dx=\int \left (b \sin {\left (e + f x \right )}\right )^{m} \left (d \cos {\left (e + f x \right )}\right )^{n}\, dx \]

[In]

integrate((d*cos(f*x+e))**n*(b*sin(f*x+e))**m,x)

[Out]

Integral((b*sin(e + f*x))**m*(d*cos(e + f*x))**n, x)

Maxima [F]

\[ \int (d \cos (e+f x))^n (b \sin (e+f x))^m \, dx=\int { \left (d \cos \left (f x + e\right )\right )^{n} \left (b \sin \left (f x + e\right )\right )^{m} \,d x } \]

[In]

integrate((d*cos(f*x+e))^n*(b*sin(f*x+e))^m,x, algorithm="maxima")

[Out]

integrate((d*cos(f*x + e))^n*(b*sin(f*x + e))^m, x)

Giac [F]

\[ \int (d \cos (e+f x))^n (b \sin (e+f x))^m \, dx=\int { \left (d \cos \left (f x + e\right )\right )^{n} \left (b \sin \left (f x + e\right )\right )^{m} \,d x } \]

[In]

integrate((d*cos(f*x+e))^n*(b*sin(f*x+e))^m,x, algorithm="giac")

[Out]

integrate((d*cos(f*x + e))^n*(b*sin(f*x + e))^m, x)

Mupad [F(-1)]

Timed out. \[ \int (d \cos (e+f x))^n (b \sin (e+f x))^m \, dx=\int {\left (d\,\cos \left (e+f\,x\right )\right )}^n\,{\left (b\,\sin \left (e+f\,x\right )\right )}^m \,d x \]

[In]

int((d*cos(e + f*x))^n*(b*sin(e + f*x))^m,x)

[Out]

int((d*cos(e + f*x))^n*(b*sin(e + f*x))^m, x)