3.3.51 \(\int \frac {1}{a+b \sin ^8(x)} \, dx\) [251]

Optimal. Leaf size=245 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt [4]{-a}-\sqrt [4]{b}} \tan (x)}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt {\sqrt [4]{-a}-\sqrt [4]{b}}}-\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt [4]{-a}-i \sqrt [4]{b}} \tan (x)}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt {\sqrt [4]{-a}-i \sqrt [4]{b}}}-\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt [4]{-a}+i \sqrt [4]{b}} \tan (x)}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt {\sqrt [4]{-a}+i \sqrt [4]{b}}}-\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt [4]{-a}+\sqrt [4]{b}} \tan (x)}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt {\sqrt [4]{-a}+\sqrt [4]{b}}} \]

[Out]

-1/4*arctan(((-a)^(1/4)-b^(1/4))^(1/2)*tan(x)/(-a)^(1/8))/(-a)^(7/8)/((-a)^(1/4)-b^(1/4))^(1/2)-1/4*arctan(((-
a)^(1/4)-I*b^(1/4))^(1/2)*tan(x)/(-a)^(1/8))/(-a)^(7/8)/((-a)^(1/4)-I*b^(1/4))^(1/2)-1/4*arctan(((-a)^(1/4)+I*
b^(1/4))^(1/2)*tan(x)/(-a)^(1/8))/(-a)^(7/8)/((-a)^(1/4)+I*b^(1/4))^(1/2)-1/4*arctan(((-a)^(1/4)+b^(1/4))^(1/2
)*tan(x)/(-a)^(1/8))/(-a)^(7/8)/((-a)^(1/4)+b^(1/4))^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.34, antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3290, 3260, 209} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt {\sqrt [4]{-a}-i \sqrt [4]{b}} \tan (x)}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt {\sqrt [4]{-a}-i \sqrt [4]{b}}}-\frac {\text {ArcTan}\left (\frac {\sqrt {\sqrt [4]{-a}+i \sqrt [4]{b}} \tan (x)}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt {\sqrt [4]{-a}+i \sqrt [4]{b}}}-\frac {\text {ArcTan}\left (\frac {\sqrt {\sqrt [4]{-a}+\sqrt [4]{b}} \tan (x)}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt {\sqrt [4]{-a}+\sqrt [4]{b}}}-\frac {\text {ArcTan}\left (\frac {\sqrt {a \sqrt [4]{b}+(-a)^{5/4}} \tan (x)}{(-a)^{5/8}}\right )}{4 (-a)^{3/8} \sqrt {a \sqrt [4]{b}+(-a)^{5/4}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Sin[x]^8)^(-1),x]

[Out]

-1/4*ArcTan[(Sqrt[(-a)^(1/4) - I*b^(1/4)]*Tan[x])/(-a)^(1/8)]/((-a)^(7/8)*Sqrt[(-a)^(1/4) - I*b^(1/4)]) - ArcT
an[(Sqrt[(-a)^(1/4) + I*b^(1/4)]*Tan[x])/(-a)^(1/8)]/(4*(-a)^(7/8)*Sqrt[(-a)^(1/4) + I*b^(1/4)]) - ArcTan[(Sqr
t[(-a)^(1/4) + b^(1/4)]*Tan[x])/(-a)^(1/8)]/(4*(-a)^(7/8)*Sqrt[(-a)^(1/4) + b^(1/4)]) - ArcTan[(Sqrt[(-a)^(5/4
) + a*b^(1/4)]*Tan[x])/(-a)^(5/8)]/(4*(-a)^(3/8)*Sqrt[(-a)^(5/4) + a*b^(1/4)])

Rule 209

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

Rule 3260

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 3290

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 ^8(x)} \, dx &=\frac {\int \frac {1}{1-\frac {\sqrt [4]{b} \sin ^2(x)}{\sqrt [4]{-a}}} \, dx}{4 a}+\frac {\int \frac {1}{1-\frac {i \sqrt [4]{b} \sin ^2(x)}{\sqrt [4]{-a}}} \, dx}{4 a}+\frac {\int \frac {1}{1+\frac {i \sqrt [4]{b} \sin ^2(x)}{\sqrt [4]{-a}}} \, dx}{4 a}+\frac {\int \frac {1}{1+\frac {\sqrt [4]{b} \sin ^2(x)}{\sqrt [4]{-a}}} \, dx}{4 a}\\ &=\frac {\text {Subst}\left (\int \frac {1}{1+\left (1-\frac {\sqrt [4]{b}}{\sqrt [4]{-a}}\right ) x^2} \, dx,x,\tan (x)\right )}{4 a}+\frac {\text {Subst}\left (\int \frac {1}{1+\left (1-\frac {i \sqrt [4]{b}}{\sqrt [4]{-a}}\right ) x^2} \, dx,x,\tan (x)\right )}{4 a}+\frac {\text {Subst}\left (\int \frac {1}{1+\left (1+\frac {i \sqrt [4]{b}}{\sqrt [4]{-a}}\right ) x^2} \, dx,x,\tan (x)\right )}{4 a}+\frac {\text {Subst}\left (\int \frac {1}{1+\left (1+\frac {\sqrt [4]{b}}{\sqrt [4]{-a}}\right ) x^2} \, dx,x,\tan (x)\right )}{4 a}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt [4]{-a}-i \sqrt [4]{b}} \tan (x)}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt {\sqrt [4]{-a}-i \sqrt [4]{b}}}-\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt [4]{-a}+i \sqrt [4]{b}} \tan (x)}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt {\sqrt [4]{-a}+i \sqrt [4]{b}}}-\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt [4]{-a}+\sqrt [4]{b}} \tan (x)}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt {\sqrt [4]{-a}+\sqrt [4]{b}}}-\frac {\tan ^{-1}\left (\frac {\sqrt {(-a)^{5/4}+a \sqrt [4]{b}} \tan (x)}{(-a)^{5/8}}\right )}{4 (-a)^{3/8} \sqrt {(-a)^{5/4}+a \sqrt [4]{b}}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.22, size = 174, normalized size = 0.71 \begin {gather*} 8 \text {RootSum}\left [b-8 b \text {$\#$1}+28 b \text {$\#$1}^2-56 b \text {$\#$1}^3+256 a \text {$\#$1}^4+70 b \text {$\#$1}^4-56 b \text {$\#$1}^5+28 b \text {$\#$1}^6-8 b \text {$\#$1}^7+b \text {$\#$1}^8\&,\frac {2 \tan ^{-1}\left (\frac {\sin (2 x)}{\cos (2 x)-\text {$\#$1}}\right ) \text {$\#$1}^3-i \log \left (1-2 \cos (2 x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3}{-b+7 b \text {$\#$1}-21 b \text {$\#$1}^2+128 a \text {$\#$1}^3+35 b \text {$\#$1}^3-35 b \text {$\#$1}^4+21 b \text {$\#$1}^5-7 b \text {$\#$1}^6+b \text {$\#$1}^7}\&\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*Sin[x]^8)^(-1),x]

[Out]

8*RootSum[b - 8*b*#1 + 28*b*#1^2 - 56*b*#1^3 + 256*a*#1^4 + 70*b*#1^4 - 56*b*#1^5 + 28*b*#1^6 - 8*b*#1^7 + b*#
1^8 & , (2*ArcTan[Sin[2*x]/(Cos[2*x] - #1)]*#1^3 - I*Log[1 - 2*Cos[2*x]*#1 + #1^2]*#1^3)/(-b + 7*b*#1 - 21*b*#
1^2 + 128*a*#1^3 + 35*b*#1^3 - 35*b*#1^4 + 21*b*#1^5 - 7*b*#1^6 + b*#1^7) & ]

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.48, size = 85, normalized size = 0.35

method	result	size
default	\(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (a +b \right ) \textit {\_Z}^{8}+4 a \,\textit {\_Z}^{6}+6 a \,\textit {\_Z}^{4}+4 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{6}+3 \textit {\_R}^{4}+3 \textit {\_R}^{2}+1\right ) \ln \left (\tan \left (x \right )-\textit {\_R} \right )}{\textit {\_R}^{7} a +\textit {\_R}^{7} b +3 \textit {\_R}^{5} a +3 \textit {\_R}^{3} a +\textit {\_R} a}\right )}{8}\)	\(85\)
risch	\(\munderset {\textit {\_R} =\RootOf \left (1+\left (16777216 a^{8}[phone] a^{7} b \right ) \textit {\_Z}^{8}+1048576 a^{6} \textit {\_Z}^{6}+24576 a^{4} \textit {\_Z}^{4}+256 a^{2} \textit {\_Z}^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i x}+\left (-\frac {4194304 i a^{8}}{b}-4194304 i a^{7}\right ) \textit {\_R}^{7}+\left (\frac {524288 a^{7}}{b}+524288 a^{6}\right ) \textit {\_R}^{6}+\left (-\frac {196608 i a^{6}}{b}+65536 i a^{5}\right ) \textit {\_R}^{5}+\left (\frac {24576 a^{5}}{b}-8192 a^{4}\right ) \textit {\_R}^{4}+\left (-\frac {3072 i a^{4}}{b}-1024 i a^{3}\right ) \textit {\_R}^{3}+\left (\frac {384 a^{3}}{b}+128 a^{2}\right ) \textit {\_R}^{2}+\left (-\frac {16 i a^{2}}{b}+16 i a \right ) \textit {\_R} +\frac {2 a}{b}-1\right )\)	\(193\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*sin(x)^8),x,method=_RETURNVERBOSE)

[Out]

1/8*sum((_R^6+3*_R^4+3*_R^2+1)/(_R^7*a+_R^7*b+3*_R^5*a+3*_R^3*a+_R*a)*ln(tan(x)-_R),_R=RootOf((a+b)*_Z^8+4*a*_
Z^6+6*a*_Z^4+4*a*_Z^2+a))

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sin(x)^8),x, algorithm="maxima")

[Out]

integrate(1/(b*sin(x)^8 + a), x)

________________________________________________________________________________________

Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 665483 vs. \(2 (165) = 330\).
time = 6.29, size = 665483, normalized size = 2716.26 \begin {gather*} \text {too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sin(x)^8),x, algorithm="fricas")

[Out]

-1/384*sqrt(1/2)*sqrt((-I*sqrt(3) + 1)*(((a^3*sqrt(-(2*a*b*sqrt(-b/a) + a*b - b^2)/((a^6 + 2*a^5*b + a^4*b^2)*
sqrt(-b/a))) + a^2*b*sqrt(-(2*a*b*sqrt(-b/a) + a*b - b^2)/((a^6 + 2*a^5*b + a^4*b^2)*sqrt(-b/a))) - 3*a)*sqrt(
-b/a) - b)^2*a/((a^3 + a^2*b)^2*b) - 3*(2*a^2*b*sqrt(-(2*a*b*sqrt(-b/a) + a*b - b^2)/((a^6 + 2*a^5*b + a^4*b^2
)*sqrt(-b/a))) + (2*a^3*sqrt(-(2*a*b*sqrt(-b/a) + a*b - b^2)/((a^6 + 2*a^5*b + a^4*b^2)*sqrt(-b/a))) - 3*a)*sq
rt(-b/a) - b)/((a^5 + a^4*b)*sqrt(-b/a)))/(-1/1572864*(2*a^2*b*sqrt(-(2*a*b*sqrt(-b/a) + a*b - b^2)/((a^6 + 2*
a^5*b + a^4*b^2)*sqrt(-b/a))) + (2*a^3*sqrt(-(2*a*b*sqrt(-b/a) + a*b - b^2)/((a^6 + 2*a^5*b + a^4*b^2)*sqrt(-b
/a))) - 3*a)*sqrt(-b/a) - b)*((a^3*sqrt(-(2*a*b*sqrt(-b/a) + a*b - b^2)/((a^6 + 2*a^5*b + a^4*b^2)*sqrt(-b/a))
) + a^2*b*sqrt(-(2*a*b*sqrt(-b/a) + a*b - b^2)/((a^6 + 2*a^5*b + a^4*b^2)*sqrt(-b/a))) - 3*a)*sqrt(-b/a) - b)*
a/((a^5 + a^4*b)*(a^3 + a^2*b)*b) + 1/524288*(2*a^2*b*sqrt(-(2*a*b*sqrt(-b/a) + a*b - b^2)/((a^6 + 2*a^5*b + a
^4*b^2)*sq ...

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{a + b \sin ^{8}{\left (x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sin(x)**8),x)

[Out]

Integral(1/(a + b*sin(x)**8), x)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sin(x)^8),x, algorithm="giac")

[Out]

integrate(1/(b*sin(x)^8 + a), x)

________________________________________________________________________________________

Mupad [B]
time = 16.95, size = 816, normalized size = 3.33 \begin {gather*} \sum _{k=1}^8\ln \left (-b^5\,\left (a+b\right )\,\left ({\mathrm {root}\left (16777216\,a^7\,b\,d^8+16777216\,a^8\,d^8+1048576\,a^6\,d^6+24576\,a^4\,d^4+256\,a^2\,d^2+1,d,k\right )}^2\,a^2\,800+{\mathrm {root}\left (16777216\,a^7\,b\,d^8+16777216\,a^8\,d^8+1048576\,a^6\,d^6+24576\,a^4\,d^4+256\,a^2\,d^2+1,d,k\right )}^4\,a^4\,43008+{\mathrm {root}\left (16777216\,a^7\,b\,d^8+16777216\,a^8\,d^8+1048576\,a^6\,d^6+24576\,a^4\,d^4+256\,a^2\,d^2+1,d,k\right )}^6\,a^6\,786432+\mathrm {root}\left (16777216\,a^7\,b\,d^8+16777216\,a^8\,d^8+1048576\,a^6\,d^6+24576\,a^4\,d^4+256\,a^2\,d^2+1,d,k\right )\,b\,\mathrm {tan}\left (x\right )\,4-{\mathrm {root}\left (16777216\,a^7\,b\,d^8+16777216\,a^8\,d^8+1048576\,a^6\,d^6+24576\,a^4\,d^4+256\,a^2\,d^2+1,d,k\right )}^4\,a^3\,b\,6144+{\mathrm {root}\left (16777216\,a^7\,b\,d^8+16777216\,a^8\,d^8+1048576\,a^6\,d^6+24576\,a^4\,d^4+256\,a^2\,d^2+1,d,k\right )}^6\,a^5\,b\,786432-{\mathrm {root}\left (16777216\,a^7\,b\,d^8+16777216\,a^8\,d^8+1048576\,a^6\,d^6+24576\,a^4\,d^4+256\,a^2\,d^2+1,d,k\right )}^3\,a^3\,\mathrm {tan}\left (x\right )\,9984-{\mathrm {root}\left (16777216\,a^7\,b\,d^8+16777216\,a^8\,d^8+1048576\,a^6\,d^6+24576\,a^4\,d^4+256\,a^2\,d^2+1,d,k\right )}^5\,a^5\,\mathrm {tan}\left (x\right )\,557056-{\mathrm {root}\left (16777216\,a^7\,b\,d^8+16777216\,a^8\,d^8+1048576\,a^6\,d^6+24576\,a^4\,d^4+256\,a^2\,d^2+1,d,k\right )}^7\,a^7\,\mathrm {tan}\left (x\right )\,10485760+{\mathrm {root}\left (16777216\,a^7\,b\,d^8+16777216\,a^8\,d^8+1048576\,a^6\,d^6+24576\,a^4\,d^4+256\,a^2\,d^2+1,d,k\right )}^2\,a\,b\,32-\mathrm {root}\left (16777216\,a^7\,b\,d^8+16777216\,a^8\,d^8+1048576\,a^6\,d^6+24576\,a^4\,d^4+256\,a^2\,d^2+1,d,k\right )\,a\,\mathrm {tan}\left (x\right )\,60-{\mathrm {root}\left (16777216\,a^7\,b\,d^8+16777216\,a^8\,d^8+1048576\,a^6\,d^6+24576\,a^4\,d^4+256\,a^2\,d^2+1,d,k\right )}^3\,a^2\,b\,\mathrm {tan}\left (x\right )\,768+{\mathrm {root}\left (16777216\,a^7\,b\,d^8+16777216\,a^8\,d^8+1048576\,a^6\,d^6+24576\,a^4\,d^4+256\,a^2\,d^2+1,d,k\right )}^5\,a^4\,b\,\mathrm {tan}\left (x\right )\,98304-{\mathrm {root}\left (16777216\,a^7\,b\,d^8+16777216\,a^8\,d^8+1048576\,a^6\,d^6+24576\,a^4\,d^4+256\,a^2\,d^2+1,d,k\right )}^7\,a^6\,b\,\mathrm {tan}\left (x\right )\,10485760+5\right )\,2\right )\,\mathrm {root}\left (16777216\,a^7\,b\,d^8+16777216\,a^8\,d^8+1048576\,a^6\,d^6+24576\,a^4\,d^4+256\,a^2\,d^2+1,d,k\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*sin(x)^8),x)

[Out]

symsum(log(-2*b^5*(a + b)*(800*root(16777216*a^7*b*d^8 + 16777216*a^8*d^8 + 1048576*a^6*d^6 + 24576*a^4*d^4 +
256*a^2*d^2 + 1, d, k)^2*a^2 + 43008*root(16777216*a^7*b*d^8 + 16777216*a^8*d^8 + 1048576*a^6*d^6 + 24576*a^4*
d^4 + 256*a^2*d^2 + 1, d, k)^4*a^4 + 786432*root(16777216*a^7*b*d^8 + 16777216*a^8*d^8 + 1048576*a^6*d^6 + 245
76*a^4*d^4 + 256*a^2*d^2 + 1, d, k)^6*a^6 + 4*root(16777216*a^7*b*d^8 + 16777216*a^8*d^8 + 1048576*a^6*d^6 + 2
4576*a^4*d^4 + 256*a^2*d^2 + 1, d, k)*b*tan(x) - 6144*root(16777216*a^7*b*d^8 + 16777216*a^8*d^8 + 1048576*a^6
*d^6 + 24576*a^4*d^4 + 256*a^2*d^2 + 1, d, k)^4*a^3*b + 786432*root(16777216*a^7*b*d^8 + 16777216*a^8*d^8 + 10
48576*a^6*d^6 + 24576*a^4*d^4 + 256*a^2*d^2 + 1, d, k)^6*a^5*b - 9984*root(16777216*a^7*b*d^8 + 16777216*a^8*d
^8 + 1048576*a^6*d^6 + 24576*a^4*d^4 + 256*a^2*d^2 + 1, d, k)^3*a^3*tan(x) - 557056*root(16777216*a^7*b*d^8 +
16777216*a^8*d^8 + 1048576*a^6*d^6 + 24576*a^4*d^4 + 256*a^2*d^2 + 1, d, k)^5*a^5*tan(x) - 10485760*root(16777
216*a^7*b*d^8 + 16777216*a^8*d^8 + 1048576*a^6*d^6 + 24576*a^4*d^4 + 256*a^2*d^2 + 1, d, k)^7*a^7*tan(x) + 32*
root(16777216*a^7*b*d^8 + 16777216*a^8*d^8 + 1048576*a^6*d^6 + 24576*a^4*d^4 + 256*a^2*d^2 + 1, d, k)^2*a*b -
60*root(16777216*a^7*b*d^8 + 16777216*a^8*d^8 + 1048576*a^6*d^6 + 24576*a^4*d^4 + 256*a^2*d^2 + 1, d, k)*a*tan
(x) - 768*root(16777216*a^7*b*d^8 + 16777216*a^8*d^8 + 1048576*a^6*d^6 + 24576*a^4*d^4 + 256*a^2*d^2 + 1, d, k
)^3*a^2*b*tan(x) + 98304*root(16777216*a^7*b*d^8 + 16777216*a^8*d^8 + 1048576*a^6*d^6 + 24576*a^4*d^4 + 256*a^
2*d^2 + 1, d, k)^5*a^4*b*tan(x) - 10485760*root(16777216*a^7*b*d^8 + 16777216*a^8*d^8 + 1048576*a^6*d^6 + 2457
6*a^4*d^4 + 256*a^2*d^2 + 1, d, k)^7*a^6*b*tan(x) + 5))*root(16777216*a^7*b*d^8 + 16777216*a^8*d^8 + 1048576*a
^6*d^6 + 24576*a^4*d^4 + 256*a^2*d^2 + 1, d, k), k, 1, 8)

________________________________________________________________________________________