\(\int (a \csc ^4(x))^{7/2} \, dx\) [61]

Optimal result

Integrand size = 10, antiderivative size = 164 \[ \int \left (a \csc ^4(x)\right )^{7/2} \, dx=-2 a^3 \cos ^2(x) \cot (x) \sqrt {a \csc ^4(x)}-3 a^3 \cos ^2(x) \cot ^3(x) \sqrt {a \csc ^4(x)}-\frac {20}{7} a^3 \cos ^2(x) \cot ^5(x) \sqrt {a \csc ^4(x)}-\frac {5}{3} a^3 \cos ^2(x) \cot ^7(x) \sqrt {a \csc ^4(x)}-\frac {6}{11} a^3 \cos ^2(x) \cot ^9(x) \sqrt {a \csc ^4(x)}-\frac {1}{13} a^3 \cos ^2(x) \cot ^{11}(x) \sqrt {a \csc ^4(x)}-a^3 \cos (x) \sqrt {a \csc ^4(x)} \sin (x) \]

[Out]

-2*a^3*cos(x)^2*cot(x)*(a*csc(x)^4)^(1/2)-3*a^3*cos(x)^2*cot(x)^3*(a*csc(x)^4)^(1/2)-20/7*a^3*cos(x)^2*cot(x)^
5*(a*csc(x)^4)^(1/2)-5/3*a^3*cos(x)^2*cot(x)^7*(a*csc(x)^4)^(1/2)-6/11*a^3*cos(x)^2*cot(x)^9*(a*csc(x)^4)^(1/2
)-1/13*a^3*cos(x)^2*cot(x)^11*(a*csc(x)^4)^(1/2)-a^3*cos(x)*sin(x)*(a*csc(x)^4)^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4208, 3852} \[ \int \left (a \csc ^4(x)\right )^{7/2} \, dx=-\frac {1}{13} a^3 \cos ^2(x) \cot ^{11}(x) \sqrt {a \csc ^4(x)}-\frac {6}{11} a^3 \cos ^2(x) \cot ^9(x) \sqrt {a \csc ^4(x)}-\frac {5}{3} a^3 \cos ^2(x) \cot ^7(x) \sqrt {a \csc ^4(x)}-\frac {20}{7} a^3 \cos ^2(x) \cot ^5(x) \sqrt {a \csc ^4(x)}-3 a^3 \cos ^2(x) \cot ^3(x) \sqrt {a \csc ^4(x)}-2 a^3 \cos ^2(x) \cot (x) \sqrt {a \csc ^4(x)}-a^3 \sin (x) \cos (x) \sqrt {a \csc ^4(x)} \]

[In]

Int[(a*Csc[x]^4)^(7/2),x]

[Out]

-2*a^3*Cos[x]^2*Cot[x]*Sqrt[a*Csc[x]^4] - 3*a^3*Cos[x]^2*Cot[x]^3*Sqrt[a*Csc[x]^4] - (20*a^3*Cos[x]^2*Cot[x]^5
*Sqrt[a*Csc[x]^4])/7 - (5*a^3*Cos[x]^2*Cot[x]^7*Sqrt[a*Csc[x]^4])/3 - (6*a^3*Cos[x]^2*Cot[x]^9*Sqrt[a*Csc[x]^4
])/11 - (a^3*Cos[x]^2*Cot[x]^11*Sqrt[a*Csc[x]^4])/13 - a^3*Cos[x]*Sqrt[a*Csc[x]^4]*Sin[x]

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 4208

Int[((b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Dist[b^IntPart[p]*((b*(c*Sec[e + f*x])^n)^
FracPart[p]/(c*Sec[e + f*x])^(n*FracPart[p])), Int[(c*Sec[e + f*x])^(n*p), x], x] /; FreeQ[{b, c, e, f, n, p},
x] &&  !IntegerQ[p]

Rubi steps \begin{align*} \text {integral}& = \left (a^3 \sqrt {a \csc ^4(x)} \sin ^2(x)\right ) \int \csc ^{14}(x) \, dx \\ & = -\left (\left (a^3 \sqrt {a \csc ^4(x)} \sin ^2(x)\right ) \text {Subst}\left (\int \left (1+6 x^2+15 x^4+20 x^6+15 x^8+6 x^{10}+x^{12}\right ) \, dx,x,\cot (x)\right )\right ) \\ & = -2 a^3 \cos ^2(x) \cot (x) \sqrt {a \csc ^4(x)}-3 a^3 \cos ^2(x) \cot ^3(x) \sqrt {a \csc ^4(x)}-\frac {20}{7} a^3 \cos ^2(x) \cot ^5(x) \sqrt {a \csc ^4(x)}-\frac {5}{3} a^3 \cos ^2(x) \cot ^7(x) \sqrt {a \csc ^4(x)}-\frac {6}{11} a^3 \cos ^2(x) \cot ^9(x) \sqrt {a \csc ^4(x)}-\frac {1}{13} a^3 \cos ^2(x) \cot ^{11}(x) \sqrt {a \csc ^4(x)}-a^3 \cos (x) \sqrt {a \csc ^4(x)} \sin (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.36 \[ \int \left (a \csc ^4(x)\right )^{7/2} \, dx=-\frac {a^3 \cos (x) \sqrt {a \csc ^4(x)} \left (1024+512 \csc ^2(x)+384 \csc ^4(x)+320 \csc ^6(x)+280 \csc ^8(x)+252 \csc ^{10}(x)+231 \csc ^{12}(x)\right ) \sin (x)}{3003} \]

[In]

Integrate[(a*Csc[x]^4)^(7/2),x]

[Out]

-1/3003*(a^3*Cos[x]*Sqrt[a*Csc[x]^4]*(1024 + 512*Csc[x]^2 + 384*Csc[x]^4 + 320*Csc[x]^6 + 280*Csc[x]^8 + 252*C
sc[x]^10 + 231*Csc[x]^12)*Sin[x])

Maple [A] (verified)

Time = 1.29 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.37

method	result	size
default	\(-\frac {\cot \left (x \right ) \csc \left (x \right )^{10} \sqrt {a \csc \left (x \right )^{4}}\, a^{3} \left (1024 \cos \left (x \right )^{12}-6656 \cos \left (x \right )^{10}+18304 \cos \left (x \right )^{8}-27456 \cos \left (x \right )^{6}+24024 \cos \left (x \right )^{4}-12012 \cos \left (x \right )^{2}+3003\right ) \sqrt {16}}{12012}\)	\(61\)
risch	\(\frac {2048 i a^{3} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}\, \left (1716 \,{\mathrm e}^{10 i x}-1287 \,{\mathrm e}^{8 i x}+715 \,{\mathrm e}^{6 i x}-286 \,{\mathrm e}^{4 i x}-13+79 \cos \left (2 x \right )+77 i \sin \left (2 x \right )\right )}{3003 \left ({\mathrm e}^{2 i x}-1\right )^{11}}\)	\(77\)

[In]

int((a*csc(x)^4)^(7/2),x,method=_RETURNVERBOSE)

[Out]

-1/12012*cot(x)*csc(x)^10*(a*csc(x)^4)^(1/2)*a^3*(1024*cos(x)^12-6656*cos(x)^10+18304*cos(x)^8-27456*cos(x)^6+
24024*cos(x)^4-12012*cos(x)^2+3003)*16^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.72 \[ \int \left (a \csc ^4(x)\right )^{7/2} \, dx=\frac {{\left (1024 \, a^{3} \cos \left (x\right )^{13} - 6656 \, a^{3} \cos \left (x\right )^{11} + 18304 \, a^{3} \cos \left (x\right )^{9} - 27456 \, a^{3} \cos \left (x\right )^{7} + 24024 \, a^{3} \cos \left (x\right )^{5} - 12012 \, a^{3} \cos \left (x\right )^{3} + 3003 \, a^{3} \cos \left (x\right )\right )} \sqrt {\frac {a}{\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1}}}{3003 \, {\left (\cos \left (x\right )^{10} - 5 \, \cos \left (x\right )^{8} + 10 \, \cos \left (x\right )^{6} - 10 \, \cos \left (x\right )^{4} + 5 \, \cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )} \]

[In]

integrate((a*csc(x)^4)^(7/2),x, algorithm="fricas")

[Out]

1/3003*(1024*a^3*cos(x)^13 - 6656*a^3*cos(x)^11 + 18304*a^3*cos(x)^9 - 27456*a^3*cos(x)^7 + 24024*a^3*cos(x)^5
- 12012*a^3*cos(x)^3 + 3003*a^3*cos(x))*sqrt(a/(cos(x)^4 - 2*cos(x)^2 + 1))/((cos(x)^10 - 5*cos(x)^8 + 10*cos
(x)^6 - 10*cos(x)^4 + 5*cos(x)^2 - 1)*sin(x))

Sympy [F(-1)]

Timed out. \[ \int \left (a \csc ^4(x)\right )^{7/2} \, dx=\text {Timed out} \]

[In]

integrate((a*csc(x)**4)**(7/2),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.40 \[ \int \left (a \csc ^4(x)\right )^{7/2} \, dx=-\frac {3003 \, a^{\frac {7}{2}} \tan \left (x\right )^{12} + 6006 \, a^{\frac {7}{2}} \tan \left (x\right )^{10} + 9009 \, a^{\frac {7}{2}} \tan \left (x\right )^{8} + 8580 \, a^{\frac {7}{2}} \tan \left (x\right )^{6} + 5005 \, a^{\frac {7}{2}} \tan \left (x\right )^{4} + 1638 \, a^{\frac {7}{2}} \tan \left (x\right )^{2} + 231 \, a^{\frac {7}{2}}}{3003 \, \tan \left (x\right )^{13}} \]

[In]

integrate((a*csc(x)^4)^(7/2),x, algorithm="maxima")

[Out]

-1/3003*(3003*a^(7/2)*tan(x)^12 + 6006*a^(7/2)*tan(x)^10 + 9009*a^(7/2)*tan(x)^8 + 8580*a^(7/2)*tan(x)^6 + 500
5*a^(7/2)*tan(x)^4 + 1638*a^(7/2)*tan(x)^2 + 231*a^(7/2))/tan(x)^13

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.42 \[ \int \left (a \csc ^4(x)\right )^{7/2} \, dx=-\frac {{\left (3003 \, a^{3} \tan \left (x\right )^{12} + 6006 \, a^{3} \tan \left (x\right )^{10} + 9009 \, a^{3} \tan \left (x\right )^{8} + 8580 \, a^{3} \tan \left (x\right )^{6} + 5005 \, a^{3} \tan \left (x\right )^{4} + 1638 \, a^{3} \tan \left (x\right )^{2} + 231 \, a^{3}\right )} \sqrt {a}}{3003 \, \tan \left (x\right )^{13}} \]

[In]

integrate((a*csc(x)^4)^(7/2),x, algorithm="giac")

[Out]

-1/3003*(3003*a^3*tan(x)^12 + 6006*a^3*tan(x)^10 + 9009*a^3*tan(x)^8 + 8580*a^3*tan(x)^6 + 5005*a^3*tan(x)^4 +
1638*a^3*tan(x)^2 + 231*a^3)*sqrt(a)/tan(x)^13

Mupad [B] (verification not implemented)

Time = 23.06 (sec) , antiderivative size = 603, normalized size of antiderivative = 3.68 \[ \int \left (a \csc ^4(x)\right )^{7/2} \, dx=\text {Too large to display} \]

[In]

int((a/sin(x)^4)^(7/2),x)

[Out]

(a^3*(a/((exp(-x*1i)*1i)/2 - (exp(x*1i)*1i)/2)^4)^(1/2)*(6*exp(x*4i) - 4*exp(x*2i) - 4*exp(x*6i) + exp(x*8i) +
1)*2048i)/(7*(exp(x*2i) - 1)^7*(exp(x*2i) - 2*exp(x*4i) + exp(x*6i))) + (a^3*(a/((exp(-x*1i)*1i)/2 - (exp(x*1
i)*1i)/2)^4)^(1/2)*(6*exp(x*4i) - 4*exp(x*2i) - 4*exp(x*6i) + exp(x*8i) + 1)*1536i)/((exp(x*2i) - 1)^8*(exp(x*
2i) - 2*exp(x*4i) + exp(x*6i))) + (a^3*(a/((exp(-x*1i)*1i)/2 - (exp(x*1i)*1i)/2)^4)^(1/2)*(6*exp(x*4i) - 4*exp
(x*2i) - 4*exp(x*6i) + exp(x*8i) + 1)*10240i)/(3*(exp(x*2i) - 1)^9*(exp(x*2i) - 2*exp(x*4i) + exp(x*6i))) + (a
^3*(a/((exp(-x*1i)*1i)/2 - (exp(x*1i)*1i)/2)^4)^(1/2)*(6*exp(x*4i) - 4*exp(x*2i) - 4*exp(x*6i) + exp(x*8i) + 1
)*4096i)/((exp(x*2i) - 1)^10*(exp(x*2i) - 2*exp(x*4i) + exp(x*6i))) + (a^3*(a/((exp(-x*1i)*1i)/2 - (exp(x*1i)*
1i)/2)^4)^(1/2)*(6*exp(x*4i) - 4*exp(x*2i) - 4*exp(x*6i) + exp(x*8i) + 1)*30720i)/(11*(exp(x*2i) - 1)^11*(exp(
x*2i) - 2*exp(x*4i) + exp(x*6i))) + (a^3*(a/((exp(-x*1i)*1i)/2 - (exp(x*1i)*1i)/2)^4)^(1/2)*(6*exp(x*4i) - 4*e
xp(x*2i) - 4*exp(x*6i) + exp(x*8i) + 1)*1024i)/((exp(x*2i) - 1)^12*(exp(x*2i) - 2*exp(x*4i) + exp(x*6i))) + (a
^3*(a/((exp(-x*1i)*1i)/2 - (exp(x*1i)*1i)/2)^4)^(1/2)*(6*exp(x*4i) - 4*exp(x*2i) - 4*exp(x*6i) + exp(x*8i) + 1
)*2048i)/(13*(exp(x*2i) - 1)^13*(exp(x*2i) - 2*exp(x*4i) + exp(x*6i)))