3.234 \(\int (d \cos (a+b x))^{7/2} \csc ^2(a+b x) \, dx\)

Optimal. Leaf size=96 \[ -\frac {5 d^4 \sqrt {\cos (a+b x)} F\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{3 b \sqrt {d \cos (a+b x)}}-\frac {5 d^3 \sin (a+b x) \sqrt {d \cos (a+b x)}}{3 b}-\frac {d \csc (a+b x) (d \cos (a+b x))^{5/2}}{b} \]

[Out]

-d*(d*cos(b*x+a))^(5/2)*csc(b*x+a)/b-5/3*d^4*(cos(1/2*a+1/2*b*x)^2)^(1/2)/cos(1/2*a+1/2*b*x)*EllipticF(sin(1/2
*a+1/2*b*x),2^(1/2))*cos(b*x+a)^(1/2)/b/(d*cos(b*x+a))^(1/2)-5/3*d^3*sin(b*x+a)*(d*cos(b*x+a))^(1/2)/b

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Rubi [A]  time = 0.08, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2567, 2635, 2642, 2641} \[ -\frac {5 d^3 \sin (a+b x) \sqrt {d \cos (a+b x)}}{3 b}-\frac {5 d^4 \sqrt {\cos (a+b x)} F\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{3 b \sqrt {d \cos (a+b x)}}-\frac {d \csc (a+b x) (d \cos (a+b x))^{5/2}}{b} \]

Antiderivative was successfully verified.

[In]

Int[(d*Cos[a + b*x])^(7/2)*Csc[a + b*x]^2,x]

[Out]

-((d*(d*Cos[a + b*x])^(5/2)*Csc[a + b*x])/b) - (5*d^4*Sqrt[Cos[a + b*x]]*EllipticF[(a + b*x)/2, 2])/(3*b*Sqrt[
d*Cos[a + b*x]]) - (5*d^3*Sqrt[d*Cos[a + b*x]]*Sin[a + b*x])/(3*b)

Rule 2567

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(a*(a*Cos[e +
f*x])^(m - 1)*(b*Sin[e + f*x])^(n + 1))/(b*f*(n + 1)), x] + Dist[(a^2*(m - 1))/(b^2*(n + 1)), Int[(a*Cos[e +
f*x])^(m - 2)*(b*Sin[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f}, x] && GtQ[m, 1] && LtQ[n, -1] && (Intege
rsQ[2*m, 2*n] || EqQ[m + n, 0])

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 2641

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{c, d}, x]

Rule 2642

Int[1/Sqrt[(b_)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[Sin[c + d*x]]/Sqrt[b*Sin[c + d*x]], Int[1/Sqr
t[Sin[c + d*x]], x], x] /; FreeQ[{b, c, d}, x]

Rubi steps

\begin {align*} \int (d \cos (a+b x))^{7/2} \csc ^2(a+b x) \, dx &=-\frac {d (d \cos (a+b x))^{5/2} \csc (a+b x)}{b}-\frac {1}{2} \left (5 d^2\right ) \int (d \cos (a+b x))^{3/2} \, dx\\ &=-\frac {d (d \cos (a+b x))^{5/2} \csc (a+b x)}{b}-\frac {5 d^3 \sqrt {d \cos (a+b x)} \sin (a+b x)}{3 b}-\frac {1}{6} \left (5 d^4\right ) \int \frac {1}{\sqrt {d \cos (a+b x)}} \, dx\\ &=-\frac {d (d \cos (a+b x))^{5/2} \csc (a+b x)}{b}-\frac {5 d^3 \sqrt {d \cos (a+b x)} \sin (a+b x)}{3 b}-\frac {\left (5 d^4 \sqrt {\cos (a+b x)}\right ) \int \frac {1}{\sqrt {\cos (a+b x)}} \, dx}{6 \sqrt {d \cos (a+b x)}}\\ &=-\frac {d (d \cos (a+b x))^{5/2} \csc (a+b x)}{b}-\frac {5 d^4 \sqrt {\cos (a+b x)} F\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{3 b \sqrt {d \cos (a+b x)}}-\frac {5 d^3 \sqrt {d \cos (a+b x)} \sin (a+b x)}{3 b}\\ \end {align*}

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Mathematica [A]  time = 0.22, size = 73, normalized size = 0.76 \[ \frac {d^3 \sqrt {d \cos (a+b x)} \left (\sqrt {\cos (a+b x)} (\cos (2 (a+b x))-4) \csc (a+b x)-5 F\left (\left .\frac {1}{2} (a+b x)\right |2\right )\right )}{3 b \sqrt {\cos (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*Cos[a + b*x])^(7/2)*Csc[a + b*x]^2,x]

[Out]

(d^3*Sqrt[d*Cos[a + b*x]]*(Sqrt[Cos[a + b*x]]*(-4 + Cos[2*(a + b*x)])*Csc[a + b*x] - 5*EllipticF[(a + b*x)/2,
2]))/(3*b*Sqrt[Cos[a + b*x]])

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fricas [F]  time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {d \cos \left (b x + a\right )} d^{3} \cos \left (b x + a\right )^{3} \csc \left (b x + a\right )^{2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cos(b*x+a))^(7/2)*csc(b*x+a)^2,x, algorithm="fricas")

[Out]

integral(sqrt(d*cos(b*x + a))*d^3*cos(b*x + a)^3*csc(b*x + a)^2, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d \cos \left (b x + a\right )\right )^{\frac {7}{2}} \csc \left (b x + a\right )^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cos(b*x+a))^(7/2)*csc(b*x+a)^2,x, algorithm="giac")

[Out]

integrate((d*cos(b*x + a))^(7/2)*csc(b*x + a)^2, x)

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maple [A]  time = 0.24, size = 216, normalized size = 2.25 \[ -\frac {\sqrt {d \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, d^{5} \sin \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (-32 \left (\sin ^{8}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+10 \left (2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right )^{\frac {3}{2}} \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \EllipticF \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )+64 \left (\sin ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-28 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-4 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+3\right )}{6 \left (-2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) d +\left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) d \right )^{\frac {3}{2}} \cos \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {d \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right )}\, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*cos(b*x+a))^(7/2)*csc(b*x+a)^2,x)

[Out]

-1/6*(d*(2*cos(1/2*b*x+1/2*a)^2-1)*sin(1/2*b*x+1/2*a)^2)^(1/2)*d^5/(-2*sin(1/2*b*x+1/2*a)^4*d+sin(1/2*b*x+1/2*
a)^2*d)^(3/2)/cos(1/2*b*x+1/2*a)*sin(1/2*b*x+1/2*a)*(-32*sin(1/2*b*x+1/2*a)^8+10*(2*sin(1/2*b*x+1/2*a)^2-1)^(3
/2)*(sin(1/2*b*x+1/2*a)^2)^(1/2)*EllipticF(cos(1/2*b*x+1/2*a),2^(1/2))*cos(1/2*b*x+1/2*a)+64*sin(1/2*b*x+1/2*a
)^6-28*sin(1/2*b*x+1/2*a)^4-4*sin(1/2*b*x+1/2*a)^2+3)/(d*(2*cos(1/2*b*x+1/2*a)^2-1))^(1/2)/b

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d \cos \left (b x + a\right )\right )^{\frac {7}{2}} \csc \left (b x + a\right )^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cos(b*x+a))^(7/2)*csc(b*x+a)^2,x, algorithm="maxima")

[Out]

integrate((d*cos(b*x + a))^(7/2)*csc(b*x + a)^2, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (d\,\cos \left (a+b\,x\right )\right )}^{7/2}}{{\sin \left (a+b\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*cos(a + b*x))^(7/2)/sin(a + b*x)^2,x)

[Out]

int((d*cos(a + b*x))^(7/2)/sin(a + b*x)^2, 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))**(7/2)*csc(b*x+a)**2,x)

[Out]

Timed out

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