3.253 \(\int \frac {1}{a-b \sin ^6(x)} \, dx\)

Optimal. Leaf size=175 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt [3]{a}-\sqrt [3]{b}} \tan (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}-\sqrt [3]{b}}}+\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}} \tan (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}}}+\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}} \tan (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}}} \]

[Out]

1/3*arctan((a^(1/3)-b^(1/3))^(1/2)*tan(x)/a^(1/6))/a^(5/6)/(a^(1/3)-b^(1/3))^(1/2)+1/3*arctan((a^(1/3)+(-1)^(1
/3)*b^(1/3))^(1/2)*tan(x)/a^(1/6))/a^(5/6)/(a^(1/3)+(-1)^(1/3)*b^(1/3))^(1/2)+1/3*arctan((a^(1/3)-(-1)^(2/3)*b
^(1/3))^(1/2)*tan(x)/a^(1/6))/a^(5/6)/(a^(1/3)-(-1)^(2/3)*b^(1/3))^(1/2)

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Rubi [A]  time = 0.26, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3211, 3181, 203} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt [3]{a}-\sqrt [3]{b}} \tan (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}-\sqrt [3]{b}}}+\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}} \tan (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}}}+\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}} \tan (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}}} \]

Antiderivative was successfully verified.

[In]

Int[(a - b*Sin[x]^6)^(-1),x]

[Out]

ArcTan[(Sqrt[a^(1/3) - b^(1/3)]*Tan[x])/a^(1/6)]/(3*a^(5/6)*Sqrt[a^(1/3) - b^(1/3)]) + ArcTan[(Sqrt[a^(1/3) +
(-1)^(1/3)*b^(1/3)]*Tan[x])/a^(1/6)]/(3*a^(5/6)*Sqrt[a^(1/3) + (-1)^(1/3)*b^(1/3)]) + ArcTan[(Sqrt[a^(1/3) - (
-1)^(2/3)*b^(1/3)]*Tan[x])/a^(1/6)]/(3*a^(5/6)*Sqrt[a^(1/3) - (-1)^(2/3)*b^(1/3)])

Rule 203

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

Rule 3181

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

Rule 3211

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(-1), x_Symbol] :> Module[{k}, Dist[2/(a*n), Sum[Int[1/(1 - Si
n[e + f*x]^2/((-1)^((4*k)/n)*Rt[-(a/b), n/2])), x], {k, 1, n/2}], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[n/
2]

Rubi steps

\begin {align*} \int \frac {1}{a-b \sin ^6(x)} \, dx &=\frac {\int \frac {1}{1-\frac {\sqrt [3]{b} \sin ^2(x)}{\sqrt [3]{a}}} \, dx}{3 a}+\frac {\int \frac {1}{1+\frac {\sqrt [3]{-1} \sqrt [3]{b} \sin ^2(x)}{\sqrt [3]{a}}} \, dx}{3 a}+\frac {\int \frac {1}{1-\frac {(-1)^{2/3} \sqrt [3]{b} \sin ^2(x)}{\sqrt [3]{a}}} \, dx}{3 a}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{1+\left (1-\frac {\sqrt [3]{b}}{\sqrt [3]{a}}\right ) x^2} \, dx,x,\tan (x)\right )}{3 a}+\frac {\operatorname {Subst}\left (\int \frac {1}{1+\left (1+\frac {\sqrt [3]{-1} \sqrt [3]{b}}{\sqrt [3]{a}}\right ) x^2} \, dx,x,\tan (x)\right )}{3 a}+\frac {\operatorname {Subst}\left (\int \frac {1}{1+\left (1-\frac {(-1)^{2/3} \sqrt [3]{b}}{\sqrt [3]{a}}\right ) x^2} \, dx,x,\tan (x)\right )}{3 a}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt [3]{a}-\sqrt [3]{b}} \tan (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}-\sqrt [3]{b}}}+\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}} \tan (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}}}+\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}} \tan (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}}}\\ \end {align*}

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Mathematica [C]  time = 0.18, size = 148, normalized size = 0.85 \[ \frac {8}{3} \text {RootSum}\left [\text {$\#$1}^6 b-6 \text {$\#$1}^5 b+15 \text {$\#$1}^4 b+64 \text {$\#$1}^3 a-20 \text {$\#$1}^3 b+15 \text {$\#$1}^2 b-6 \text {$\#$1} b+b\& ,\frac {2 \text {$\#$1}^2 \tan ^{-1}\left (\frac {\sin (2 x)}{\cos (2 x)-\text {$\#$1}}\right )-i \text {$\#$1}^2 \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (2 x)+1\right )}{\text {$\#$1}^5 b-5 \text {$\#$1}^4 b+10 \text {$\#$1}^3 b+32 \text {$\#$1}^2 a-10 \text {$\#$1}^2 b+5 \text {$\#$1} b-b}\& \right ] \]

Antiderivative was successfully verified.

[In]

Integrate[(a - b*Sin[x]^6)^(-1),x]

[Out]

(8*RootSum[b - 6*b*#1 + 15*b*#1^2 + 64*a*#1^3 - 20*b*#1^3 + 15*b*#1^4 - 6*b*#1^5 + b*#1^6 & , (2*ArcTan[Sin[2*
x]/(Cos[2*x] - #1)]*#1^2 - I*Log[1 - 2*Cos[2*x]*#1 + #1^2]*#1^2)/(-b + 5*b*#1 + 32*a*#1^2 - 10*b*#1^2 + 10*b*#
1^3 - 5*b*#1^4 + b*#1^5) & ])/3

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fricas [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(1/(a-b*sin(x)^6),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {1}{b \sin \relax (x)^{6} - a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-b*sin(x)^6),x, algorithm="giac")

[Out]

integrate(-1/(b*sin(x)^6 - a), x)

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maple [C]  time = 1.42, size = 71, normalized size = 0.41 \[ \frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (a -b \right ) \textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{4}+2 \textit {\_R}^{2}+1\right ) \ln \left (\tan \relax (x )-\textit {\_R} \right )}{\textit {\_R}^{5} a -\textit {\_R}^{5} b +2 \textit {\_R}^{3} a +\textit {\_R} a}\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a-b*sin(x)^6),x)

[Out]

1/6*sum((_R^4+2*_R^2+1)/(_R^5*a-_R^5*b+2*_R^3*a+_R*a)*ln(tan(x)-_R),_R=RootOf((a-b)*_Z^6+3*a*_Z^4+3*a*_Z^2+a))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {1}{b \sin \relax (x)^{6} - a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-b*sin(x)^6),x, algorithm="maxima")

[Out]

-integrate(1/(b*sin(x)^6 - a), x)

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mupad [B]  time = 16.07, size = 513, normalized size = 2.93 \[ \sum _{k=1}^6\ln \left (-\frac {b^3\,\left (a-b\right )\,\left (\mathrm {cot}\relax (x)-\mathrm {root}\left (46656\,a^5\,b\,d^6-46656\,a^6\,d^6-3888\,a^4\,d^4-108\,a^2\,d^2-1,d,k\right )\,a\,8+\mathrm {root}\left (46656\,a^5\,b\,d^6-46656\,a^6\,d^6-3888\,a^4\,d^4-108\,a^2\,d^2-1,d,k\right )\,b\,2-{\mathrm {root}\left (46656\,a^5\,b\,d^6-46656\,a^6\,d^6-3888\,a^4\,d^4-108\,a^2\,d^2-1,d,k\right )}^3\,a^3\,504-{\mathrm {root}\left (46656\,a^5\,b\,d^6-46656\,a^6\,d^6-3888\,a^4\,d^4-108\,a^2\,d^2-1,d,k\right )}^5\,a^5\,7776-{\mathrm {root}\left (46656\,a^5\,b\,d^6-46656\,a^6\,d^6-3888\,a^4\,d^4-108\,a^2\,d^2-1,d,k\right )}^3\,a^2\,b\,144+{\mathrm {root}\left (46656\,a^5\,b\,d^6-46656\,a^6\,d^6-3888\,a^4\,d^4-108\,a^2\,d^2-1,d,k\right )}^5\,a^4\,b\,7776+{\mathrm {root}\left (46656\,a^5\,b\,d^6-46656\,a^6\,d^6-3888\,a^4\,d^4-108\,a^2\,d^2-1,d,k\right )}^2\,a^2\,\mathrm {cot}\relax (x)\,60+{\mathrm {root}\left (46656\,a^5\,b\,d^6-46656\,a^6\,d^6-3888\,a^4\,d^4-108\,a^2\,d^2-1,d,k\right )}^4\,a^4\,\mathrm {cot}\relax (x)\,864-{\mathrm {root}\left (46656\,a^5\,b\,d^6-46656\,a^6\,d^6-3888\,a^4\,d^4-108\,a^2\,d^2-1,d,k\right )}^4\,a^3\,b\,\mathrm {cot}\relax (x)\,864+{\mathrm {root}\left (46656\,a^5\,b\,d^6-46656\,a^6\,d^6-3888\,a^4\,d^4-108\,a^2\,d^2-1,d,k\right )}^2\,a\,b\,\mathrm {cot}\relax (x)\,12\right )\,3}{\mathrm {cot}\relax (x)}\right )\,\mathrm {root}\left (46656\,a^5\,b\,d^6-46656\,a^6\,d^6-3888\,a^4\,d^4-108\,a^2\,d^2-1,d,k\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a - b*sin(x)^6),x)

[Out]

symsum(log(-(3*b^3*(a - b)*(cot(x) - 8*root(46656*a^5*b*d^6 - 46656*a^6*d^6 - 3888*a^4*d^4 - 108*a^2*d^2 - 1,
d, k)*a + 2*root(46656*a^5*b*d^6 - 46656*a^6*d^6 - 3888*a^4*d^4 - 108*a^2*d^2 - 1, d, k)*b - 504*root(46656*a^
5*b*d^6 - 46656*a^6*d^6 - 3888*a^4*d^4 - 108*a^2*d^2 - 1, d, k)^3*a^3 - 7776*root(46656*a^5*b*d^6 - 46656*a^6*
d^6 - 3888*a^4*d^4 - 108*a^2*d^2 - 1, d, k)^5*a^5 - 144*root(46656*a^5*b*d^6 - 46656*a^6*d^6 - 3888*a^4*d^4 -
108*a^2*d^2 - 1, d, k)^3*a^2*b + 7776*root(46656*a^5*b*d^6 - 46656*a^6*d^6 - 3888*a^4*d^4 - 108*a^2*d^2 - 1, d
, k)^5*a^4*b + 60*root(46656*a^5*b*d^6 - 46656*a^6*d^6 - 3888*a^4*d^4 - 108*a^2*d^2 - 1, d, k)^2*a^2*cot(x) +
864*root(46656*a^5*b*d^6 - 46656*a^6*d^6 - 3888*a^4*d^4 - 108*a^2*d^2 - 1, d, k)^4*a^4*cot(x) - 864*root(46656
*a^5*b*d^6 - 46656*a^6*d^6 - 3888*a^4*d^4 - 108*a^2*d^2 - 1, d, k)^4*a^3*b*cot(x) + 12*root(46656*a^5*b*d^6 -
46656*a^6*d^6 - 3888*a^4*d^4 - 108*a^2*d^2 - 1, d, k)^2*a*b*cot(x)))/cot(x))*root(46656*a^5*b*d^6 - 46656*a^6*
d^6 - 3888*a^4*d^4 - 108*a^2*d^2 - 1, d, k), k, 1, 6)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{a - b \sin ^{6}{\relax (x )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-b*sin(x)**6),x)

[Out]

Integral(1/(a - b*sin(x)**6), x)

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