\(\int (a \csc ^2(x))^{7/2} \, dx\) [47]

Optimal result

Integrand size = 10, antiderivative size = 84 \[ \int \left (a \csc ^2(x)\right )^{7/2} \, dx=-\frac {5}{16} a^{7/2} \text {arctanh}\left (\frac {\sqrt {a} \cot (x)}{\sqrt {a \csc ^2(x)}}\right )-\frac {5}{16} a^3 \cot (x) \sqrt {a \csc ^2(x)}-\frac {5}{24} a^2 \cot (x) \left (a \csc ^2(x)\right )^{3/2}-\frac {1}{6} a \cot (x) \left (a \csc ^2(x)\right )^{5/2} \]

[Out]

-5/16*a^(7/2)*arctanh(cot(x)*a^(1/2)/(a*csc(x)^2)^(1/2))-5/24*a^2*cot(x)*(a*csc(x)^2)^(3/2)-1/6*a*cot(x)*(a*cs
c(x)^2)^(5/2)-5/16*a^3*cot(x)*(a*csc(x)^2)^(1/2)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4207, 201, 223, 212} \[ \int \left (a \csc ^2(x)\right )^{7/2} \, dx=-\frac {5}{16} a^{7/2} \text {arctanh}\left (\frac {\sqrt {a} \cot (x)}{\sqrt {a \csc ^2(x)}}\right )-\frac {5}{16} a^3 \cot (x) \sqrt {a \csc ^2(x)}-\frac {5}{24} a^2 \cot (x) \left (a \csc ^2(x)\right )^{3/2}-\frac {1}{6} a \cot (x) \left (a \csc ^2(x)\right )^{5/2} \]

[In]

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

[Out]

(-5*a^(7/2)*ArcTanh[(Sqrt[a]*Cot[x])/Sqrt[a*Csc[x]^2]])/16 - (5*a^3*Cot[x]*Sqrt[a*Csc[x]^2])/16 - (5*a^2*Cot[x
]*(a*Csc[x]^2)^(3/2))/24 - (a*Cot[x]*(a*Csc[x]^2)^(5/2))/6

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 4207

Int[((b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[b*(ff/
f), Subst[Int[(b + b*ff^2*x^2)^(p - 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{b, e, f, p}, x] &&  !IntegerQ[p
]

Rubi steps \begin{align*} \text {integral}& = -\left (a \text {Subst}\left (\int \left (a+a x^2\right )^{5/2} \, dx,x,\cot (x)\right )\right ) \\ & = -\frac {1}{6} a \cot (x) \left (a \csc ^2(x)\right )^{5/2}-\frac {1}{6} \left (5 a^2\right ) \text {Subst}\left (\int \left (a+a x^2\right )^{3/2} \, dx,x,\cot (x)\right ) \\ & = -\frac {5}{24} a^2 \cot (x) \left (a \csc ^2(x)\right )^{3/2}-\frac {1}{6} a \cot (x) \left (a \csc ^2(x)\right )^{5/2}-\frac {1}{8} \left (5 a^3\right ) \text {Subst}\left (\int \sqrt {a+a x^2} \, dx,x,\cot (x)\right ) \\ & = -\frac {5}{16} a^3 \cot (x) \sqrt {a \csc ^2(x)}-\frac {5}{24} a^2 \cot (x) \left (a \csc ^2(x)\right )^{3/2}-\frac {1}{6} a \cot (x) \left (a \csc ^2(x)\right )^{5/2}-\frac {1}{16} \left (5 a^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+a x^2}} \, dx,x,\cot (x)\right ) \\ & = -\frac {5}{16} a^3 \cot (x) \sqrt {a \csc ^2(x)}-\frac {5}{24} a^2 \cot (x) \left (a \csc ^2(x)\right )^{3/2}-\frac {1}{6} a \cot (x) \left (a \csc ^2(x)\right )^{5/2}-\frac {1}{16} \left (5 a^4\right ) \text {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\cot (x)}{\sqrt {a \csc ^2(x)}}\right ) \\ & = -\frac {5}{16} a^{7/2} \text {arctanh}\left (\frac {\sqrt {a} \cot (x)}{\sqrt {a \csc ^2(x)}}\right )-\frac {5}{16} a^3 \cot (x) \sqrt {a \csc ^2(x)}-\frac {5}{24} a^2 \cot (x) \left (a \csc ^2(x)\right )^{3/2}-\frac {1}{6} a \cot (x) \left (a \csc ^2(x)\right )^{5/2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.73 \[ \int \left (a \csc ^2(x)\right )^{7/2} \, dx=\frac {a^3 \csc ^5(x) \sqrt {a \csc ^2(x)} \left (-396 \cos (x)+170 \cos (3 x)-30 \cos (5 x)+480 \left (-\log \left (\cos \left (\frac {x}{2}\right )\right )+\log \left (\sin \left (\frac {x}{2}\right )\right )\right ) \sin ^6(x)\right )}{1536} \]

[In]

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

[Out]

(a^3*Csc[x]^5*Sqrt[a*Csc[x]^2]*(-396*Cos[x] + 170*Cos[3*x] - 30*Cos[5*x] + 480*(-Log[Cos[x/2]] + Log[Sin[x/2]]
)*Sin[x]^6))/1536

Maple [A] (verified)

Time = 0.69 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.62

method	result	size
default	\(\frac {\csc \left (x \right )^{5} \left (15 \ln \left (\csc \left (x \right )-\cot \left (x \right )\right ) \sin \left (x \right )^{6}-15 \cos \left (x \right )^{5}+40 \cos \left (x \right )^{3}-33 \cos \left (x \right )\right ) a^{3} \sqrt {a \csc \left (x \right )^{2}}\, \sqrt {4}}{96}\)	\(52\)
risch	\(-\frac {i a^{3} \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left (15 \,{\mathrm e}^{10 i x}-85 \,{\mathrm e}^{8 i x}+198 \,{\mathrm e}^{6 i x}+198 \,{\mathrm e}^{4 i x}-85 \,{\mathrm e}^{2 i x}+15\right )}{24 \left ({\mathrm e}^{2 i x}-1\right )^{5}}-\frac {5 a^{3} \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}+1\right ) \sin \left (x \right )}{8}+\frac {5 a^{3} \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}-1\right ) \sin \left (x \right )}{8}\)	\(141\)

[In]

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

[Out]

1/96*csc(x)^5*(15*ln(csc(x)-cot(x))*sin(x)^6-15*cos(x)^5+40*cos(x)^3-33*cos(x))*a^3*(a*csc(x)^2)^(1/2)*4^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.26 \[ \int \left (a \csc ^2(x)\right )^{7/2} \, dx=-\frac {{\left (30 \, a^{3} \cos \left (x\right )^{5} - 80 \, a^{3} \cos \left (x\right )^{3} + 66 \, a^{3} \cos \left (x\right ) + 15 \, {\left (a^{3} \cos \left (x\right )^{6} - 3 \, a^{3} \cos \left (x\right )^{4} + 3 \, a^{3} \cos \left (x\right )^{2} - a^{3}\right )} \log \left (-\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right )\right )} \sqrt {-\frac {a}{\cos \left (x\right )^{2} - 1}}}{96 \, {\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \sin \left (x\right )} \]

[In]

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

[Out]

-1/96*(30*a^3*cos(x)^5 - 80*a^3*cos(x)^3 + 66*a^3*cos(x) + 15*(a^3*cos(x)^6 - 3*a^3*cos(x)^4 + 3*a^3*cos(x)^2
- a^3)*log(-(cos(x) - 1)/(cos(x) + 1)))*sqrt(-a/(cos(x)^2 - 1))/((cos(x)^4 - 2*cos(x)^2 + 1)*sin(x))

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2183 vs. \(2 (64) = 128\).

Time = 2.22 (sec) , antiderivative size = 2183, normalized size of antiderivative = 25.99 \[ \int \left (a \csc ^2(x)\right )^{7/2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/48*(1020*a^3*cos(3*x)*sin(2*x) - 180*a^3*cos(x)*sin(2*x) + 180*a^3*cos(2*x)*sin(x) - 30*a^3*sin(x) + 15*(a^3
*cos(12*x)^2 + 36*a^3*cos(10*x)^2 + 225*a^3*cos(8*x)^2 + 400*a^3*cos(6*x)^2 + 225*a^3*cos(4*x)^2 + 36*a^3*cos(
2*x)^2 + a^3*sin(12*x)^2 + 36*a^3*sin(10*x)^2 + 225*a^3*sin(8*x)^2 + 400*a^3*sin(6*x)^2 + 225*a^3*sin(4*x)^2 -
180*a^3*sin(4*x)*sin(2*x) + 36*a^3*sin(2*x)^2 - 12*a^3*cos(2*x) + a^3 - 2*(6*a^3*cos(10*x) - 15*a^3*cos(8*x)
+ 20*a^3*cos(6*x) - 15*a^3*cos(4*x) + 6*a^3*cos(2*x) - a^3)*cos(12*x) - 12*(15*a^3*cos(8*x) - 20*a^3*cos(6*x)
+ 15*a^3*cos(4*x) - 6*a^3*cos(2*x) + a^3)*cos(10*x) - 30*(20*a^3*cos(6*x) - 15*a^3*cos(4*x) + 6*a^3*cos(2*x) -
a^3)*cos(8*x) - 40*(15*a^3*cos(4*x) - 6*a^3*cos(2*x) + a^3)*cos(6*x) - 30*(6*a^3*cos(2*x) - a^3)*cos(4*x) - 2
*(6*a^3*sin(10*x) - 15*a^3*sin(8*x) + 20*a^3*sin(6*x) - 15*a^3*sin(4*x) + 6*a^3*sin(2*x))*sin(12*x) - 12*(15*a
^3*sin(8*x) - 20*a^3*sin(6*x) + 15*a^3*sin(4*x) - 6*a^3*sin(2*x))*sin(10*x) - 30*(20*a^3*sin(6*x) - 15*a^3*sin
(4*x) + 6*a^3*sin(2*x))*sin(8*x) - 120*(5*a^3*sin(4*x) - 2*a^3*sin(2*x))*sin(6*x))*arctan2(sin(x), cos(x) + 1)
- 15*(a^3*cos(12*x)^2 + 36*a^3*cos(10*x)^2 + 225*a^3*cos(8*x)^2 + 400*a^3*cos(6*x)^2 + 225*a^3*cos(4*x)^2 + 3
6*a^3*cos(2*x)^2 + a^3*sin(12*x)^2 + 36*a^3*sin(10*x)^2 + 225*a^3*sin(8*x)^2 + 400*a^3*sin(6*x)^2 + 225*a^3*si
n(4*x)^2 - 180*a^3*sin(4*x)*sin(2*x) + 36*a^3*sin(2*x)^2 - 12*a^3*cos(2*x) + a^3 - 2*(6*a^3*cos(10*x) - 15*a^3
*cos(8*x) + 20*a^3*cos(6*x) - 15*a^3*cos(4*x) + 6*a^3*cos(2*x) - a^3)*cos(12*x) - 12*(15*a^3*cos(8*x) - 20*a^3
*cos(6*x) + 15*a^3*cos(4*x) - 6*a^3*cos(2*x) + a^3)*cos(10*x) - 30*(20*a^3*cos(6*x) - 15*a^3*cos(4*x) + 6*a^3*
cos(2*x) - a^3)*cos(8*x) - 40*(15*a^3*cos(4*x) - 6*a^3*cos(2*x) + a^3)*cos(6*x) - 30*(6*a^3*cos(2*x) - a^3)*co
s(4*x) - 2*(6*a^3*sin(10*x) - 15*a^3*sin(8*x) + 20*a^3*sin(6*x) - 15*a^3*sin(4*x) + 6*a^3*sin(2*x))*sin(12*x)
- 12*(15*a^3*sin(8*x) - 20*a^3*sin(6*x) + 15*a^3*sin(4*x) - 6*a^3*sin(2*x))*sin(10*x) - 30*(20*a^3*sin(6*x) -
15*a^3*sin(4*x) + 6*a^3*sin(2*x))*sin(8*x) - 120*(5*a^3*sin(4*x) - 2*a^3*sin(2*x))*sin(6*x))*arctan2(sin(x), c
os(x) - 1) - 2*(15*a^3*sin(11*x) - 85*a^3*sin(9*x) + 198*a^3*sin(7*x) + 198*a^3*sin(5*x) - 85*a^3*sin(3*x) + 1
5*a^3*sin(x))*cos(12*x) - 30*(6*a^3*sin(10*x) - 15*a^3*sin(8*x) + 20*a^3*sin(6*x) - 15*a^3*sin(4*x) + 6*a^3*si
n(2*x))*cos(11*x) - 12*(85*a^3*sin(9*x) - 198*a^3*sin(7*x) - 198*a^3*sin(5*x) + 85*a^3*sin(3*x) - 15*a^3*sin(x
))*cos(10*x) - 170*(15*a^3*sin(8*x) - 20*a^3*sin(6*x) + 15*a^3*sin(4*x) - 6*a^3*sin(2*x))*cos(9*x) - 30*(198*a
^3*sin(7*x) + 198*a^3*sin(5*x) - 85*a^3*sin(3*x) + 15*a^3*sin(x))*cos(8*x) - 396*(20*a^3*sin(6*x) - 15*a^3*sin
(4*x) + 6*a^3*sin(2*x))*cos(7*x) + 40*(198*a^3*sin(5*x) - 85*a^3*sin(3*x) + 15*a^3*sin(x))*cos(6*x) + 1188*(5*
a^3*sin(4*x) - 2*a^3*sin(2*x))*cos(5*x) + 150*(17*a^3*sin(3*x) - 3*a^3*sin(x))*cos(4*x) + 2*(15*a^3*cos(11*x)
- 85*a^3*cos(9*x) + 198*a^3*cos(7*x) + 198*a^3*cos(5*x) - 85*a^3*cos(3*x) + 15*a^3*cos(x))*sin(12*x) + 30*(6*a
^3*cos(10*x) - 15*a^3*cos(8*x) + 20*a^3*cos(6*x) - 15*a^3*cos(4*x) + 6*a^3*cos(2*x) - a^3)*sin(11*x) + 12*(85*
a^3*cos(9*x) - 198*a^3*cos(7*x) - 198*a^3*cos(5*x) + 85*a^3*cos(3*x) - 15*a^3*cos(x))*sin(10*x) + 170*(15*a^3*
cos(8*x) - 20*a^3*cos(6*x) + 15*a^3*cos(4*x) - 6*a^3*cos(2*x) + a^3)*sin(9*x) + 30*(198*a^3*cos(7*x) + 198*a^3
*cos(5*x) - 85*a^3*cos(3*x) + 15*a^3*cos(x))*sin(8*x) + 396*(20*a^3*cos(6*x) - 15*a^3*cos(4*x) + 6*a^3*cos(2*x
) - a^3)*sin(7*x) - 40*(198*a^3*cos(5*x) - 85*a^3*cos(3*x) + 15*a^3*cos(x))*sin(6*x) - 396*(15*a^3*cos(4*x) -
6*a^3*cos(2*x) + a^3)*sin(5*x) - 150*(17*a^3*cos(3*x) - 3*a^3*cos(x))*sin(4*x) - 170*(6*a^3*cos(2*x) - a^3)*si
n(3*x))*sqrt(-a)/(2*(6*cos(10*x) - 15*cos(8*x) + 20*cos(6*x) - 15*cos(4*x) + 6*cos(2*x) - 1)*cos(12*x) - cos(1
2*x)^2 + 12*(15*cos(8*x) - 20*cos(6*x) + 15*cos(4*x) - 6*cos(2*x) + 1)*cos(10*x) - 36*cos(10*x)^2 + 30*(20*cos
(6*x) - 15*cos(4*x) + 6*cos(2*x) - 1)*cos(8*x) - 225*cos(8*x)^2 + 40*(15*cos(4*x) - 6*cos(2*x) + 1)*cos(6*x) -
400*cos(6*x)^2 + 30*(6*cos(2*x) - 1)*cos(4*x) - 225*cos(4*x)^2 - 36*cos(2*x)^2 + 2*(6*sin(10*x) - 15*sin(8*x)
+ 20*sin(6*x) - 15*sin(4*x) + 6*sin(2*x))*sin(12*x) - sin(12*x)^2 + 12*(15*sin(8*x) - 20*sin(6*x) + 15*sin(4*
x) - 6*sin(2*x))*sin(10*x) - 36*sin(10*x)^2 + 30*(20*sin(6*x) - 15*sin(4*x) + 6*sin(2*x))*sin(8*x) - 225*sin(8
*x)^2 + 120*(5*sin(4*x) - 2*sin(2*x))*sin(6*x) - 400*sin(6*x)^2 - 225*sin(4*x)^2 + 180*sin(4*x)*sin(2*x) - 36*
sin(2*x)^2 + 12*cos(2*x) - 1)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 164 vs. \(2 (64) = 128\).

Time = 0.29 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.95 \[ \int \left (a \csc ^2(x)\right )^{7/2} \, dx=\frac {1}{384} \, {\left (60 \, a^{3} \log \left (-\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right ) \mathrm {sgn}\left (\sin \left (x\right )\right ) - \frac {45 \, a^{3} {\left (\cos \left (x\right ) - 1\right )} \mathrm {sgn}\left (\sin \left (x\right )\right )}{\cos \left (x\right ) + 1} + \frac {9 \, a^{3} {\left (\cos \left (x\right ) - 1\right )}^{2} \mathrm {sgn}\left (\sin \left (x\right )\right )}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac {a^{3} {\left (\cos \left (x\right ) - 1\right )}^{3} \mathrm {sgn}\left (\sin \left (x\right )\right )}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {{\left (a^{3} \mathrm {sgn}\left (\sin \left (x\right )\right ) - \frac {9 \, a^{3} {\left (\cos \left (x\right ) - 1\right )} \mathrm {sgn}\left (\sin \left (x\right )\right )}{\cos \left (x\right ) + 1} + \frac {45 \, a^{3} {\left (\cos \left (x\right ) - 1\right )}^{2} \mathrm {sgn}\left (\sin \left (x\right )\right )}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac {110 \, a^{3} {\left (\cos \left (x\right ) - 1\right )}^{3} \mathrm {sgn}\left (\sin \left (x\right )\right )}{{\left (\cos \left (x\right ) + 1\right )}^{3}}\right )} {\left (\cos \left (x\right ) + 1\right )}^{3}}{{\left (\cos \left (x\right ) - 1\right )}^{3}}\right )} \sqrt {a} \]

[In]

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

[Out]

1/384*(60*a^3*log(-(cos(x) - 1)/(cos(x) + 1))*sgn(sin(x)) - 45*a^3*(cos(x) - 1)*sgn(sin(x))/(cos(x) + 1) + 9*a
^3*(cos(x) - 1)^2*sgn(sin(x))/(cos(x) + 1)^2 - a^3*(cos(x) - 1)^3*sgn(sin(x))/(cos(x) + 1)^3 + (a^3*sgn(sin(x)
) - 9*a^3*(cos(x) - 1)*sgn(sin(x))/(cos(x) + 1) + 45*a^3*(cos(x) - 1)^2*sgn(sin(x))/(cos(x) + 1)^2 - 110*a^3*(
cos(x) - 1)^3*sgn(sin(x))/(cos(x) + 1)^3)*(cos(x) + 1)^3/(cos(x) - 1)^3)*sqrt(a)

Mupad [F(-1)]

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

[In]

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

[Out]

int((a/sin(x)^2)^(7/2), x)