3.360 \(\int (d \cos (a+b x))^n \csc ^5(a+b x) \, dx\)

Optimal. Leaf size=49 \[ -\frac {(d \cos (a+b x))^{n+1} \, _2F_1\left (3,\frac {n+1}{2};\frac {n+3}{2};\cos ^2(a+b x)\right )}{b d (n+1)} \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2565, 364} \[ -\frac {(d \cos (a+b x))^{n+1} \, _2F_1\left (3,\frac {n+1}{2};\frac {n+3}{2};\cos ^2(a+b x)\right )}{b d (n+1)} \]

Antiderivative was successfully verified.

[In]

Int[(d*Cos[a + b*x])^n*Csc[a + b*x]^5,x]

[Out]

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

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 2565

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(a*f)^(-1), Subst[
Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2]
&&  !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])

Rubi steps

\begin {align*} \int (d \cos (a+b x))^n \csc ^5(a+b x) \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {x^n}{\left (1-\frac {x^2}{d^2}\right )^3} \, dx,x,d \cos (a+b x)\right )}{b d}\\ &=-\frac {(d \cos (a+b x))^{1+n} \, _2F_1\left (3,\frac {1+n}{2};\frac {3+n}{2};\cos ^2(a+b x)\right )}{b d (1+n)}\\ \end {align*}

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Mathematica [B]  time = 4.02, size = 244, normalized size = 4.98 \[ -\frac {2^{-n-5} \cos (a+b x) (d \cos (a+b x))^n \left (3\ 2^{n+1} \, _2F_1(1,n+1;n+2;\cos (a+b x))+3\ 2^{n+1} \, _2F_1(2,n+1;n+2;\cos (a+b x))+2^{n+2} \, _2F_1(3,n+1;n+2;\cos (a+b x))+2 \sec ^2\left (\frac {1}{2} (a+b x)\right )^{n+1} \, _2F_1\left (n-1,n+1;n+2;\frac {1}{2} \cos (a+b x) \sec ^2\left (\frac {1}{2} (a+b x)\right )\right )+3 \sec ^2\left (\frac {1}{2} (a+b x)\right )^{n+1} \, _2F_1\left (n,n+1;n+2;\frac {1}{2} \cos (a+b x) \sec ^2\left (\frac {1}{2} (a+b x)\right )\right )+3 \sec ^2\left (\frac {1}{2} (a+b x)\right )^{n+1} \, _2F_1\left (n+1,n+1;n+2;\frac {1}{2} \cos (a+b x) \sec ^2\left (\frac {1}{2} (a+b x)\right )\right )\right )}{b (n+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*Cos[a + b*x])^n*Csc[a + b*x]^5,x]

[Out]

-((2^(-5 - n)*Cos[a + b*x]*(d*Cos[a + b*x])^n*(3*2^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, Cos[a + b*x]] +
3*2^(1 + n)*Hypergeometric2F1[2, 1 + n, 2 + n, Cos[a + b*x]] + 2^(2 + n)*Hypergeometric2F1[3, 1 + n, 2 + n, Co
s[a + b*x]] + 2*Hypergeometric2F1[-1 + n, 1 + n, 2 + n, (Cos[a + b*x]*Sec[(a + b*x)/2]^2)/2]*(Sec[(a + b*x)/2]
^2)^(1 + n) + 3*Hypergeometric2F1[n, 1 + n, 2 + n, (Cos[a + b*x]*Sec[(a + b*x)/2]^2)/2]*(Sec[(a + b*x)/2]^2)^(
1 + n) + 3*Hypergeometric2F1[1 + n, 1 + n, 2 + n, (Cos[a + b*x]*Sec[(a + b*x)/2]^2)/2]*(Sec[(a + b*x)/2]^2)^(1
+ n)))/(b*(1 + n)))

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fricas [F]  time = 0.89, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (d \cos \left (b x + a\right )\right )^{n} \csc \left (b x + a\right )^{5}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cos(b*x+a))^n*csc(b*x+a)^5,x, algorithm="fricas")

[Out]

integral((d*cos(b*x + a))^n*csc(b*x + a)^5, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d \cos \left (b x + a\right )\right )^{n} \csc \left (b x + a\right )^{5}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cos(b*x+a))^n*csc(b*x+a)^5,x, algorithm="giac")

[Out]

integrate((d*cos(b*x + a))^n*csc(b*x + a)^5, x)

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maple [F]  time = 0.30, size = 0, normalized size = 0.00 \[ \int \left (d \cos \left (b x +a \right )\right )^{n} \left (\csc ^{5}\left (b x +a \right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*cos(b*x+a))^n*csc(b*x+a)^5,x)

[Out]

int((d*cos(b*x+a))^n*csc(b*x+a)^5,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d \cos \left (b x + a\right )\right )^{n} \csc \left (b x + a\right )^{5}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cos(b*x+a))^n*csc(b*x+a)^5,x, algorithm="maxima")

[Out]

integrate((d*cos(b*x + a))^n*csc(b*x + a)^5, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (d\,\cos \left (a+b\,x\right )\right )}^n}{{\sin \left (a+b\,x\right )}^5} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*cos(a + b*x))^n/sin(a + b*x)^5,x)

[Out]

int((d*cos(a + b*x))^n/sin(a + b*x)^5, x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cos(b*x+a))**n*csc(b*x+a)**5,x)

[Out]

Timed out

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