\(\int (d \cos (a+b x))^n \sin ^4(a+b x) \, dx\) [361]

Optimal result

Integrand size = 19, antiderivative size = 69 \[ \int (d \cos (a+b x))^n \sin ^4(a+b x) \, dx=-\frac {(d \cos (a+b x))^{1+n} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1+n}{2},\frac {3+n}{2},\cos ^2(a+b x)\right ) \sin (a+b x)}{b d (1+n) \sqrt {\sin ^2(a+b x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2656} \[ \int (d \cos (a+b x))^n \sin ^4(a+b x) \, dx=-\frac {\sin (a+b x) (d \cos (a+b x))^{n+1} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(a+b x)\right )}{b d (n+1) \sqrt {\sin ^2(a+b x)}} \]

[In]

Int[(d*Cos[a + b*x])^n*Sin[a + b*x]^4,x]

[Out]

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

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 {3}{2},\frac {1+n}{2},\frac {3+n}{2},\cos ^2(a+b x)\right ) \sin (a+b x)}{b d (1+n) \sqrt {\sin ^2(a+b x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.99 \[ \int (d \cos (a+b x))^n \sin ^4(a+b x) \, dx=-\frac {(d \cos (a+b x))^n \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1+n}{2},\frac {3+n}{2},\cos ^2(a+b x)\right ) \sin (2 (a+b x))}{2 b (1+n) \sqrt {\sin ^2(a+b x)}} \]

[In]

Integrate[(d*Cos[a + b*x])^n*Sin[a + b*x]^4,x]

[Out]

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

Maple [F]

\[\int \left (d \cos \left (b x +a \right )\right )^{n} \left (\sin ^{4}\left (b x +a \right )\right )d x\]

[In]

int((d*cos(b*x+a))^n*sin(b*x+a)^4,x)

[Out]

int((d*cos(b*x+a))^n*sin(b*x+a)^4,x)

Fricas [F]

\[ \int (d \cos (a+b x))^n \sin ^4(a+b x) \, dx=\int { \left (d \cos \left (b x + a\right )\right )^{n} \sin \left (b x + a\right )^{4} \,d x } \]

[In]

integrate((d*cos(b*x+a))^n*sin(b*x+a)^4,x, algorithm="fricas")

[Out]

integral((cos(b*x + a)^4 - 2*cos(b*x + a)^2 + 1)*(d*cos(b*x + a))^n, x)

Sympy [F(-1)]

Timed out. \[ \int (d \cos (a+b x))^n \sin ^4(a+b x) \, dx=\text {Timed out} \]

[In]

integrate((d*cos(b*x+a))**n*sin(b*x+a)**4,x)

[Out]

Timed out

Maxima [F]

\[ \int (d \cos (a+b x))^n \sin ^4(a+b x) \, dx=\int { \left (d \cos \left (b x + a\right )\right )^{n} \sin \left (b x + a\right )^{4} \,d x } \]

[In]

integrate((d*cos(b*x+a))^n*sin(b*x+a)^4,x, algorithm="maxima")

[Out]

integrate((d*cos(b*x + a))^n*sin(b*x + a)^4, x)

Giac [F]

\[ \int (d \cos (a+b x))^n \sin ^4(a+b x) \, dx=\int { \left (d \cos \left (b x + a\right )\right )^{n} \sin \left (b x + a\right )^{4} \,d x } \]

[In]

integrate((d*cos(b*x+a))^n*sin(b*x+a)^4,x, algorithm="giac")

[Out]

integrate((d*cos(b*x + a))^n*sin(b*x + a)^4, x)

Mupad [F(-1)]

Timed out. \[ \int (d \cos (a+b x))^n \sin ^4(a+b x) \, dx=\int {\sin \left (a+b\,x\right )}^4\,{\left (d\,\cos \left (a+b\,x\right )\right )}^n \,d x \]

[In]

int(sin(a + b*x)^4*(d*cos(a + b*x))^n,x)

[Out]

int(sin(a + b*x)^4*(d*cos(a + b*x))^n, x)