3.47 \(\int (a \csc ^2(x))^{7/2} \, dx\)
Optimal. Leaf size=84 \[ -\frac{5}{16} a^3 \cot (x) \sqrt{a \csc ^2(x)}-\frac{5}{24} a^2 \cot (x) \left (a \csc ^2(x)\right )^{3/2}-\frac{5}{16} a^{7/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cot (x)}{\sqrt{a \csc ^2(x)}}\right )-\frac{1}{6} a \cot (x) \left (a \csc ^2(x)\right )^{5/2} \]
[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
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Rubi [A] time = 0.0395658, antiderivative size = 84, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used =
{4122, 195, 217, 206} \[ -\frac{5}{16} a^3 \cot (x) \sqrt{a \csc ^2(x)}-\frac{5}{24} a^2 \cot (x) \left (a \csc ^2(x)\right )^{3/2}-\frac{5}{16} a^{7/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cot (x)}{\sqrt{a \csc ^2(x)}}\right )-\frac{1}{6} a \cot (x) \left (a \csc ^2(x)\right )^{5/2} \]
Antiderivative was successfully verified.
[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 4122
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
]
Rule 195
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 217
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 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rubi steps
\begin{align*} \int \left (a \csc ^2(x)\right )^{7/2} \, dx &=-\left (a \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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} \tanh ^{-1}\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] time = 0.324462, size = 61, normalized size = 0.73 \[ \frac{a^3 \csc ^5(x) \sqrt{a \csc ^2(x)} \left (-396 \cos (x)+170 \cos (3 x)-30 \cos (5 x)+480 \sin ^6(x) \left (\log \left (\sin \left (\frac{x}{2}\right )\right )-\log \left (\cos \left (\frac{x}{2}\right )\right )\right )\right )}{1536} \]
Antiderivative was successfully verified.
[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
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Maple [A] time = 0.116, size = 102, normalized size = 1.2 \begin{align*} -{\frac{\sqrt{4}\sin \left ( x \right ) }{96} \left ( 15\, \left ( \cos \left ( x \right ) \right ) ^{6}\ln \left ( -{\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }} \right ) +15\, \left ( \cos \left ( x \right ) \right ) ^{5}-45\, \left ( \cos \left ( x \right ) \right ) ^{4}\ln \left ( -{\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }} \right ) -40\, \left ( \cos \left ( x \right ) \right ) ^{3}+45\, \left ( \cos \left ( x \right ) \right ) ^{2}\ln \left ( -{\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }} \right ) +33\,\cos \left ( x \right ) -15\,\ln \left ( -{\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }} \right ) \right ) \left ( -{\frac{a}{ \left ( \cos \left ( x \right ) \right ) ^{2}-1}} \right ) ^{{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a*csc(x)^2)^(7/2),x)
[Out]
-1/96*4^(1/2)*(15*cos(x)^6*ln(-(-1+cos(x))/sin(x))+15*cos(x)^5-45*cos(x)^4*ln(-(-1+cos(x))/sin(x))-40*cos(x)^3
+45*cos(x)^2*ln(-(-1+cos(x))/sin(x))+33*cos(x)-15*ln(-(-1+cos(x))/sin(x)))*sin(x)*(-a/(cos(x)^2-1))^(7/2)
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Maxima [B] time = 14.6467, size = 2947, normalized size = 35.08 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
________________________________________________________________________________________
Fricas [A] time = 0.508105, size = 286, normalized size = 3.4 \begin{align*} -\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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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))
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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((a*csc(x)**2)**(7/2),x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 1.43107, size = 221, normalized size = 2.63 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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)