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*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

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Rubi [A]  time = 0.148391, antiderivative size = 73, normalized size of antiderivative = 1., 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 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])

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 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 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]

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*}

Mathematica [A]  time = 0.0795734, 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]

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

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Maple [A]  time = 0.157, size = 40, normalized size = 0.6 \begin{align*}{\frac{ \left ( 10\, \left ( \cos \left ( x \right ) \right ) ^{6}+40\, \left ( \cos \left ( x \right ) \right ) ^{4}-320\, \left ( \cos \left ( x \right ) \right ) ^{2}+256 \right ) \sin \left ( x \right ) }{35\, \left ( \cos \left ( x \right ) \right ) ^{7}} \left ({\frac{ \left ( \cos \left ( x \right ) \right ) ^{2}}{\sin \left ( x \right ) }} \right ) ^{{\frac{7}{2}}}} \end{align*}

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 = 1.97216, size = 780, normalized size = 10.68 \begin{align*} \text{result too large to display} \end{align*}

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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Fricas [A]  time = 2.33403, size = 131, normalized size = 1.79 \begin{align*} -\frac{2 \,{\left (5 \, \cos \left (x\right )^{6} + 20 \, \cos \left (x\right )^{4} - 160 \, \cos \left (x\right )^{2} + 128\right )} \sqrt{\frac{\cos \left (x\right )^{2}}{\sin \left (x\right )}}}{35 \,{\left (\cos \left (x\right )^{3} - \cos \left (x\right )\right )}} \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (\csc \left (x\right ) - \sin \left (x\right )\right )}^{\frac{7}{2}}\,{d x} \end{align*}

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)