3.314 \(\int (\csc (x)-\sin (x))^{7/2} \, dx\)

Optimal. Leaf size=73 \[ \frac {2}{7} \cos ^3(x) \cot ^2(x) \sqrt {\cos (x) \cot (x)}+\frac {8}{7} \cos (x) \cot ^2(x) \sqrt {\cos (x) \cot (x)}-\frac {64}{35} \cot (x) \csc (x) \sqrt {\cos (x) \cot (x)}+\frac {256}{35} \sec (x) \sqrt {\cos (x) \cot (x)} \]

[Out]

8/7*cos(x)*cot(x)^2*(cos(x)*cot(x))^(1/2)+2/7*cos(x)^3*cot(x)^2*(cos(x)*cot(x))^(1/2)-64/35*cot(x)*csc(x)*(cos
(x)*cot(x))^(1/2)+256/35*sec(x)*(cos(x)*cot(x))^(1/2)

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Rubi [A]  time = 0.15, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {4397, 4400, 2598, 2594, 2589} \[ \frac {2}{7} \cos ^3(x) \cot ^2(x) \sqrt {\cos (x) \cot (x)}+\frac {8}{7} \cos (x) \cot ^2(x) \sqrt {\cos (x) \cot (x)}-\frac {64}{35} \cot (x) \csc (x) \sqrt {\cos (x) \cot (x)}+\frac {256}{35} \sec (x) \sqrt {\cos (x) \cot (x)} \]

Antiderivative was successfully verified.

[In]

Int[(Csc[x] - Sin[x])^(7/2),x]

[Out]

(8*Cos[x]*Cot[x]^2*Sqrt[Cos[x]*Cot[x]])/7 + (2*Cos[x]^3*Cot[x]^2*Sqrt[Cos[x]*Cot[x]])/7 - (64*Cot[x]*Sqrt[Cos[
x]*Cot[x]]*Csc[x])/35 + (256*Sqrt[Cos[x]*Cot[x]]*Sec[x])/35

Rule 2589

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*(a*Sin[e
+ f*x])^m*(b*Tan[e + f*x])^(n - 1))/(f*m), x] /; FreeQ[{a, b, e, f, m, n}, x] && EqQ[m + n - 1, 0]

Rule 2594

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(a*Sin[e
+ f*x])^m*(b*Tan[e + f*x])^(n - 1))/(f*(n - 1)), x] - Dist[(b^2*(m + n - 1))/(n - 1), Int[(a*Sin[e + f*x])^m*(
b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && IntegersQ[2*m, 2*n] &&  !(GtQ[m,
1] &&  !IntegerQ[(m - 1)/2])

Rule 2598

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

Rule 4397

Int[u_, x_Symbol] :> Int[TrigSimplify[u], x] /; TrigSimplifyQ[u]

Rule 4400

Int[(u_.)*((v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> With[{uu = ActivateTrig[u], vv = ActivateTrig[v], ww = Ac
tivateTrig[w]}, Dist[(vv^m*ww^n)^FracPart[p]/(vv^(m*FracPart[p])*ww^(n*FracPart[p])), Int[uu*vv^(m*p)*ww^(n*p)
, x], x]] /; FreeQ[{m, n, p}, x] &&  !IntegerQ[p] && ( !InertTrigFreeQ[v] ||  !InertTrigFreeQ[w])

Rubi steps

\begin {align*} \int (\csc (x)-\sin (x))^{7/2} \, dx &=\int (\cos (x) \cot (x))^{7/2} \, dx\\ &=\frac {\sqrt {\cos (x) \cot (x)} \int \cos ^{\frac {7}{2}}(x) \cot ^{\frac {7}{2}}(x) \, dx}{\sqrt {\cos (x)} \sqrt {\cot (x)}}\\ &=\frac {2}{7} \cos ^3(x) \cot ^2(x) \sqrt {\cos (x) \cot (x)}+\frac {\left (12 \sqrt {\cos (x) \cot (x)}\right ) \int \cos ^{\frac {3}{2}}(x) \cot ^{\frac {7}{2}}(x) \, dx}{7 \sqrt {\cos (x)} \sqrt {\cot (x)}}\\ &=\frac {8}{7} \cos (x) \cot ^2(x) \sqrt {\cos (x) \cot (x)}+\frac {2}{7} \cos ^3(x) \cot ^2(x) \sqrt {\cos (x) \cot (x)}+\frac {\left (32 \sqrt {\cos (x) \cot (x)}\right ) \int \frac {\cot ^{\frac {7}{2}}(x)}{\sqrt {\cos (x)}} \, dx}{7 \sqrt {\cos (x)} \sqrt {\cot (x)}}\\ &=\frac {8}{7} \cos (x) \cot ^2(x) \sqrt {\cos (x) \cot (x)}+\frac {2}{7} \cos ^3(x) \cot ^2(x) \sqrt {\cos (x) \cot (x)}-\frac {64}{35} \cot (x) \sqrt {\cos (x) \cot (x)} \csc (x)-\frac {\left (128 \sqrt {\cos (x) \cot (x)}\right ) \int \frac {\cot ^{\frac {3}{2}}(x)}{\sqrt {\cos (x)}} \, dx}{35 \sqrt {\cos (x)} \sqrt {\cot (x)}}\\ &=\frac {8}{7} \cos (x) \cot ^2(x) \sqrt {\cos (x) \cot (x)}+\frac {2}{7} \cos ^3(x) \cot ^2(x) \sqrt {\cos (x) \cot (x)}-\frac {64}{35} \cot (x) \sqrt {\cos (x) \cot (x)} \csc (x)+\frac {256}{35} \sqrt {\cos (x) \cot (x)} \sec (x)\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 37, normalized size = 0.51 \[ -\frac {1}{70} \sec (x) \sqrt {\cos (x) \cot (x)} \left (115 \cos ^2(x)+5 \cos (3 x) \cos (x)+28 \cot ^2(x)-512\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(Csc[x] - Sin[x])^(7/2),x]

[Out]

-1/70*(Sqrt[Cos[x]*Cot[x]]*(-512 + 115*Cos[x]^2 + 5*Cos[x]*Cos[3*x] + 28*Cot[x]^2)*Sec[x])

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fricas [A]  time = 1.00, size = 44, normalized size = 0.60 \[ -\frac {2 \, {\left (5 \, \cos \relax (x)^{6} + 20 \, \cos \relax (x)^{4} - 160 \, \cos \relax (x)^{2} + 128\right )} \sqrt {\frac {\cos \relax (x)^{2}}{\sin \relax (x)}}}{35 \, {\left (\cos \relax (x)^{3} - \cos \relax (x)\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((csc(x)-sin(x))^(7/2),x, algorithm="fricas")

[Out]

-2/35*(5*cos(x)^6 + 20*cos(x)^4 - 160*cos(x)^2 + 128)*sqrt(cos(x)^2/sin(x))/(cos(x)^3 - cos(x))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (\csc \relax (x) - \sin \relax (x)\right )}^{\frac {7}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((csc(x)-sin(x))^(7/2),x, algorithm="giac")

[Out]

integrate((csc(x) - sin(x))^(7/2), x)

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maple [A]  time = 0.31, size = 40, normalized size = 0.55 \[ \frac {2 \left (5 \left (\cos ^{6}\relax (x )\right )+20 \left (\cos ^{4}\relax (x )\right )-160 \left (\cos ^{2}\relax (x )\right )+128\right ) \sin \relax (x ) \left (\frac {\cos ^{2}\relax (x )}{\sin \relax (x )}\right )^{\frac {7}{2}}}{35 \cos \relax (x )^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((csc(x)-sin(x))^(7/2),x)

[Out]

2/35*(5*cos(x)^6+20*cos(x)^4-160*cos(x)^2+128)*sin(x)*(cos(x)^2/sin(x))^(7/2)/cos(x)^7

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maxima [B]  time = 0.57, size = 578, normalized size = 7.92 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((csc(x)-sin(x))^(7/2),x, algorithm="maxima")

[Out]

-1/280*(cos(x)^2 + sin(x)^2 + 2*cos(x) + 1)^(1/4)*(cos(x)^2 + sin(x)^2 - 2*cos(x) + 1)^(1/4)*(((5*cos(21/2*x)
+ 105*cos(17/2*x) - 2275*cos(13/2*x) + 5817*cos(9/2*x) - 5*cos(7/2*x) - 5817*cos(5/2*x) - 105*cos(3/2*x) + 227
5*cos(1/2*x) - 5*sin(21/2*x) - 105*sin(17/2*x) + 2275*sin(13/2*x) - 5817*sin(9/2*x) - 5*sin(7/2*x) + 5817*sin(
5/2*x) - 105*sin(3/2*x) - 2275*sin(1/2*x))*cos(7/2*arctan2(sin(x), cos(x) - 1)) + (5*cos(21/2*x) + 105*cos(17/
2*x) - 2275*cos(13/2*x) + 5817*cos(9/2*x) - 5*cos(7/2*x) - 5817*cos(5/2*x) - 105*cos(3/2*x) + 2275*cos(1/2*x)
+ 5*sin(21/2*x) + 105*sin(17/2*x) - 2275*sin(13/2*x) + 5817*sin(9/2*x) + 5*sin(7/2*x) - 5817*sin(5/2*x) + 105*
sin(3/2*x) + 2275*sin(1/2*x))*sin(7/2*arctan2(sin(x), cos(x) - 1)))*cos(7/2*arctan2(sin(x), cos(x) + 1)) + ((5
*cos(21/2*x) + 105*cos(17/2*x) - 2275*cos(13/2*x) + 5817*cos(9/2*x) - 5*cos(7/2*x) - 5817*cos(5/2*x) - 105*cos
(3/2*x) + 2275*cos(1/2*x) + 5*sin(21/2*x) + 105*sin(17/2*x) - 2275*sin(13/2*x) + 5817*sin(9/2*x) + 5*sin(7/2*x
) - 5817*sin(5/2*x) + 105*sin(3/2*x) + 2275*sin(1/2*x))*cos(7/2*arctan2(sin(x), cos(x) - 1)) - (5*cos(21/2*x)
+ 105*cos(17/2*x) - 2275*cos(13/2*x) + 5817*cos(9/2*x) - 5*cos(7/2*x) - 5817*cos(5/2*x) - 105*cos(3/2*x) + 227
5*cos(1/2*x) - 5*sin(21/2*x) - 105*sin(17/2*x) + 2275*sin(13/2*x) - 5817*sin(9/2*x) - 5*sin(7/2*x) + 5817*sin(
5/2*x) - 105*sin(3/2*x) - 2275*sin(1/2*x))*sin(7/2*arctan2(sin(x), cos(x) - 1)))*sin(7/2*arctan2(sin(x), cos(x
) + 1)))/(cos(x)^8 + sin(x)^8 + 4*(cos(x)^2 + 1)*sin(x)^6 - 4*cos(x)^6 + 2*(3*cos(x)^4 + 2*cos(x)^2 + 3)*sin(x
)^4 + 6*cos(x)^4 + 4*(cos(x)^6 - cos(x)^4 - cos(x)^2 + 1)*sin(x)^2 - 4*cos(x)^2 + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1/sin(x) - sin(x))^(7/2),x)

[Out]

int((1/sin(x) - sin(x))^(7/2), 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((csc(x)-sin(x))**(7/2),x)

[Out]

Timed out

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