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

Optimal. Leaf size=171 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt [3]{a}+\sqrt [3]{b}} \tan (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}+\sqrt [3]{b}}}+\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b}} \tan (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b}}}+\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b}} \tan (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b}}} \]

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.26, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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*}

________________________________________________________________________________________

Mathematica [C]  time = 0.22, size = 148, normalized size = 0.87 \[ -\frac {8}{3} \text {RootSum}\left [\text {$\#$1}^6 b-6 \text {$\#$1}^5 b+15 \text {$\#$1}^4 b-64 \text {$\#$1}^3 a-20 \text {$\#$1}^3 b+15 \text {$\#$1}^2 b-6 \text {$\#$1} b+b\& ,\frac {2 \text {$\#$1}^2 \tan ^{-1}\left (\frac {\sin (2 x)}{\cos (2 x)-\text {$\#$1}}\right )-i \text {$\#$1}^2 \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (2 x)+1\right )}{\text {$\#$1}^5 b-5 \text {$\#$1}^4 b+10 \text {$\#$1}^3 b-32 \text {$\#$1}^2 a-10 \text {$\#$1}^2 b+5 \text {$\#$1} b-b}\& \right ] \]

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

________________________________________________________________________________________

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

________________________________________________________________________________________

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)

________________________________________________________________________________________

maple [C]  time = 1.54, size = 68, normalized size = 0.40 \[ \frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (a +b \right ) \textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{4}+2 \textit {\_R}^{2}+1\right ) \ln \left (\tan \relax (x )-\textit {\_R} \right )}{\textit {\_R}^{5} a +\textit {\_R}^{5} b +2 \textit {\_R}^{3} a +\textit {\_R} a}\right )}{6} \]

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))

________________________________________________________________________________________

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)

________________________________________________________________________________________

mupad [B]  time = 15.71, size = 513, normalized size = 3.00 \[ \sum _{k=1}^6\ln \left (-\frac {b^3\,\left (a+b\right )\,\left (-\mathrm {cot}\relax (x)+\mathrm {root}\left (46656\,a^5\,b\,d^6+46656\,a^6\,d^6+3888\,a^4\,d^4+108\,a^2\,d^2+1,d,k\right )\,a\,8+\mathrm {root}\left (46656\,a^5\,b\,d^6+46656\,a^6\,d^6+3888\,a^4\,d^4+108\,a^2\,d^2+1,d,k\right )\,b\,2+{\mathrm {root}\left (46656\,a^5\,b\,d^6+46656\,a^6\,d^6+3888\,a^4\,d^4+108\,a^2\,d^2+1,d,k\right )}^3\,a^3\,504+{\mathrm {root}\left (46656\,a^5\,b\,d^6+46656\,a^6\,d^6+3888\,a^4\,d^4+108\,a^2\,d^2+1,d,k\right )}^5\,a^5\,7776-{\mathrm {root}\left (46656\,a^5\,b\,d^6+46656\,a^6\,d^6+3888\,a^4\,d^4+108\,a^2\,d^2+1,d,k\right )}^3\,a^2\,b\,144+{\mathrm {root}\left (46656\,a^5\,b\,d^6+46656\,a^6\,d^6+3888\,a^4\,d^4+108\,a^2\,d^2+1,d,k\right )}^5\,a^4\,b\,7776-{\mathrm {root}\left (46656\,a^5\,b\,d^6+46656\,a^6\,d^6+3888\,a^4\,d^4+108\,a^2\,d^2+1,d,k\right )}^2\,a^2\,\mathrm {cot}\relax (x)\,60-{\mathrm {root}\left (46656\,a^5\,b\,d^6+46656\,a^6\,d^6+3888\,a^4\,d^4+108\,a^2\,d^2+1,d,k\right )}^4\,a^4\,\mathrm {cot}\relax (x)\,864-{\mathrm {root}\left (46656\,a^5\,b\,d^6+46656\,a^6\,d^6+3888\,a^4\,d^4+108\,a^2\,d^2+1,d,k\right )}^4\,a^3\,b\,\mathrm {cot}\relax (x)\,864+{\mathrm {root}\left (46656\,a^5\,b\,d^6+46656\,a^6\,d^6+3888\,a^4\,d^4+108\,a^2\,d^2+1,d,k\right )}^2\,a\,b\,\mathrm {cot}\relax (x)\,12\right )\,3}{\mathrm {cot}\relax (x)}\right )\,\mathrm {root}\left (46656\,a^5\,b\,d^6+46656\,a^6\,d^6+3888\,a^4\,d^4+108\,a^2\,d^2+1,d,k\right ) \]

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)*(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 -
cot(x) + 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)

________________________________________________________________________________________

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)

________________________________________________________________________________________