3.252 \(\int \frac {1}{a-b \sin ^5(x)} \, dx\)

Optimal. Leaf size=379 \[ -\frac {2 \tan ^{-1}\left (\frac {\sqrt [5]{b}-\sqrt [5]{a} \tan \left (\frac {x}{2}\right )}{\sqrt {a^{2/5}-b^{2/5}}}\right )}{5 a^{4/5} \sqrt {a^{2/5}-b^{2/5}}}-\frac {2 \tan ^{-1}\left (\frac {(-1)^{2/5} \sqrt [5]{b}-\sqrt [5]{a} \tan \left (\frac {x}{2}\right )}{\sqrt {a^{2/5}-(-1)^{4/5} b^{2/5}}}\right )}{5 a^{4/5} \sqrt {a^{2/5}-(-1)^{4/5} b^{2/5}}}-\frac {2 \tan ^{-1}\left (\frac {(-1)^{4/5} \sqrt [5]{b}-\sqrt [5]{a} \tan \left (\frac {x}{2}\right )}{\sqrt {a^{2/5}+(-1)^{3/5} b^{2/5}}}\right )}{5 a^{4/5} \sqrt {a^{2/5}+(-1)^{3/5} b^{2/5}}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt [5]{a} \tan \left (\frac {x}{2}\right )+\sqrt [5]{-1} \sqrt [5]{b}}{\sqrt {a^{2/5}-(-1)^{2/5} b^{2/5}}}\right )}{5 a^{4/5} \sqrt {a^{2/5}-(-1)^{2/5} b^{2/5}}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt [5]{a} \tan \left (\frac {x}{2}\right )+(-1)^{3/5} \sqrt [5]{b}}{\sqrt {a^{2/5}+\sqrt [5]{-1} b^{2/5}}}\right )}{5 a^{4/5} \sqrt {a^{2/5}+\sqrt [5]{-1} b^{2/5}}} \]

[Out]

-2/5*arctan((b^(1/5)-a^(1/5)*tan(1/2*x))/(a^(2/5)-b^(2/5))^(1/2))/a^(4/5)/(a^(2/5)-b^(2/5))^(1/2)+2/5*arctan((
(-1)^(3/5)*b^(1/5)+a^(1/5)*tan(1/2*x))/(a^(2/5)+(-1)^(1/5)*b^(2/5))^(1/2))/a^(4/5)/(a^(2/5)+(-1)^(1/5)*b^(2/5)
)^(1/2)+2/5*arctan(((-1)^(1/5)*b^(1/5)+a^(1/5)*tan(1/2*x))/(a^(2/5)-(-1)^(2/5)*b^(2/5))^(1/2))/a^(4/5)/(a^(2/5
)-(-1)^(2/5)*b^(2/5))^(1/2)-2/5*arctan(((-1)^(4/5)*b^(1/5)-a^(1/5)*tan(1/2*x))/(a^(2/5)+(-1)^(3/5)*b^(2/5))^(1
/2))/a^(4/5)/(a^(2/5)+(-1)^(3/5)*b^(2/5))^(1/2)-2/5*arctan(((-1)^(2/5)*b^(1/5)-a^(1/5)*tan(1/2*x))/(a^(2/5)-(-
1)^(4/5)*b^(2/5))^(1/2))/a^(4/5)/(a^(2/5)-(-1)^(4/5)*b^(2/5))^(1/2)

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Rubi [A]  time = 0.48, antiderivative size = 379, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3213, 2660, 618, 204} \[ -\frac {2 \tan ^{-1}\left (\frac {\sqrt [5]{b}-\sqrt [5]{a} \tan \left (\frac {x}{2}\right )}{\sqrt {a^{2/5}-b^{2/5}}}\right )}{5 a^{4/5} \sqrt {a^{2/5}-b^{2/5}}}-\frac {2 \tan ^{-1}\left (\frac {(-1)^{2/5} \sqrt [5]{b}-\sqrt [5]{a} \tan \left (\frac {x}{2}\right )}{\sqrt {a^{2/5}-(-1)^{4/5} b^{2/5}}}\right )}{5 a^{4/5} \sqrt {a^{2/5}-(-1)^{4/5} b^{2/5}}}-\frac {2 \tan ^{-1}\left (\frac {(-1)^{4/5} \sqrt [5]{b}-\sqrt [5]{a} \tan \left (\frac {x}{2}\right )}{\sqrt {a^{2/5}+(-1)^{3/5} b^{2/5}}}\right )}{5 a^{4/5} \sqrt {a^{2/5}+(-1)^{3/5} b^{2/5}}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt [5]{a} \tan \left (\frac {x}{2}\right )+\sqrt [5]{-1} \sqrt [5]{b}}{\sqrt {a^{2/5}-(-1)^{2/5} b^{2/5}}}\right )}{5 a^{4/5} \sqrt {a^{2/5}-(-1)^{2/5} b^{2/5}}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt [5]{a} \tan \left (\frac {x}{2}\right )+(-1)^{3/5} \sqrt [5]{b}}{\sqrt {a^{2/5}+\sqrt [5]{-1} b^{2/5}}}\right )}{5 a^{4/5} \sqrt {a^{2/5}+\sqrt [5]{-1} b^{2/5}}} \]

Antiderivative was successfully verified.

[In]

Int[(a - b*Sin[x]^5)^(-1),x]

[Out]

(-2*ArcTan[(b^(1/5) - a^(1/5)*Tan[x/2])/Sqrt[a^(2/5) - b^(2/5)]])/(5*a^(4/5)*Sqrt[a^(2/5) - b^(2/5)]) - (2*Arc
Tan[((-1)^(2/5)*b^(1/5) - a^(1/5)*Tan[x/2])/Sqrt[a^(2/5) - (-1)^(4/5)*b^(2/5)]])/(5*a^(4/5)*Sqrt[a^(2/5) - (-1
)^(4/5)*b^(2/5)]) - (2*ArcTan[((-1)^(4/5)*b^(1/5) - a^(1/5)*Tan[x/2])/Sqrt[a^(2/5) + (-1)^(3/5)*b^(2/5)]])/(5*
a^(4/5)*Sqrt[a^(2/5) + (-1)^(3/5)*b^(2/5)]) + (2*ArcTan[((-1)^(1/5)*b^(1/5) + a^(1/5)*Tan[x/2])/Sqrt[a^(2/5) -
(-1)^(2/5)*b^(2/5)]])/(5*a^(4/5)*Sqrt[a^(2/5) - (-1)^(2/5)*b^(2/5)]) + (2*ArcTan[((-1)^(3/5)*b^(1/5) + a^(1/5
)*Tan[x/2])/Sqrt[a^(2/5) + (-1)^(1/5)*b^(2/5)]])/(5*a^(4/5)*Sqrt[a^(2/5) + (-1)^(1/5)*b^(2/5)])

Rule 204

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

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 2660

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[(2*e)/d, Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[a^2 - b^2, 0]

Rule 3213

Int[((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Int[ExpandTrig[(a + b*(c*sin[e + f*
x])^n)^p, x], x] /; FreeQ[{a, b, c, e, f, n}, x] && (IGtQ[p, 0] || (EqQ[p, -1] && IntegerQ[n]))

Rubi steps

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

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Mathematica [C]  time = 0.19, size = 149, normalized size = 0.39 \[ -\frac {8}{5} i \text {RootSum}\left [i \text {$\#$1}^{10} b-5 i \text {$\#$1}^8 b+10 i \text {$\#$1}^6 b+32 \text {$\#$1}^5 a-10 i \text {$\#$1}^4 b+5 i \text {$\#$1}^2 b-i b\& ,\frac {2 \text {$\#$1}^3 \tan ^{-1}\left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right )-i \text {$\#$1}^3 \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (x)+1\right )}{\text {$\#$1}^8 b-4 \text {$\#$1}^6 b+6 \text {$\#$1}^4 b-16 i \text {$\#$1}^3 a-4 \text {$\#$1}^2 b+b}\& \right ] \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a - b*Sin[x]^5)^(-1),x]

[Out]

((-8*I)/5)*RootSum[(-I)*b + (5*I)*b*#1^2 - (10*I)*b*#1^4 + 32*a*#1^5 + (10*I)*b*#1^6 - (5*I)*b*#1^8 + I*b*#1^1
0 & , (2*ArcTan[Sin[x]/(Cos[x] - #1)]*#1^3 - I*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^3)/(b - 4*b*#1^2 - (16*I)*a*#1^3
+ 6*b*#1^4 - 4*b*#1^6 + b*#1^8) & ]

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-b*sin(x)^5),x, algorithm="fricas")

[Out]

Exception raised: RuntimeError >> no explicit roots found

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {1}{b \sin \relax (x)^{5} - a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-b*sin(x)^5),x, algorithm="giac")

[Out]

integrate(-1/(b*sin(x)^5 - a), x)

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maple [C]  time = 0.25, size = 109, normalized size = 0.29 \[ \frac {\left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{10}+5 a \,\textit {\_Z}^{8}+10 a \,\textit {\_Z}^{6}-32 b \,\textit {\_Z}^{5}+10 a \,\textit {\_Z}^{4}+5 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{8}+4 \textit {\_R}^{6}+6 \textit {\_R}^{4}+4 \textit {\_R}^{2}+1\right ) \ln \left (\tan \left (\frac {x}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{9} a +4 \textit {\_R}^{7} a +6 \textit {\_R}^{5} a -16 \textit {\_R}^{4} b +4 \textit {\_R}^{3} a +\textit {\_R} a}\right )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a-b*sin(x)^5),x)

[Out]

1/5*sum((_R^8+4*_R^6+6*_R^4+4*_R^2+1)/(_R^9*a+4*_R^7*a+6*_R^5*a-16*_R^4*b+4*_R^3*a+_R*a)*ln(tan(1/2*x)-_R),_R=
RootOf(_Z^10*a+5*_Z^8*a+10*_Z^6*a-32*_Z^5*b+10*_Z^4*a+5*_Z^2*a+a))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {1}{b \sin \relax (x)^{5} - a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-b*sin(x)^5),x, algorithm="maxima")

[Out]

-integrate(1/(b*sin(x)^5 - a), x)

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mupad [B]  time = 20.14, size = 1515, normalized size = 4.00 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a - b*sin(x)^5),x)

[Out]

symsum(log(10995116277760*a*b^7*(16*tan(x/2) + 56*root(9765625*a^8*b^2*d^10 - 9765625*a^10*d^10 - 1953125*a^8*
d^8 - 156250*a^6*d^6 - 6250*a^4*d^4 - 125*a^2*d^2 - 1, d, k)*a + 5425*root(9765625*a^8*b^2*d^10 - 9765625*a^10
*d^10 - 1953125*a^8*d^8 - 156250*a^6*d^6 - 6250*a^4*d^4 - 125*a^2*d^2 - 1, d, k)^3*a^3 + 196875*root(9765625*a
^8*b^2*d^10 - 9765625*a^10*d^10 - 1953125*a^8*d^8 - 156250*a^6*d^6 - 6250*a^4*d^4 - 125*a^2*d^2 - 1, d, k)^5*a
^5 + 3171875*root(9765625*a^8*b^2*d^10 - 9765625*a^10*d^10 - 1953125*a^8*d^8 - 156250*a^6*d^6 - 6250*a^4*d^4 -
125*a^2*d^2 - 1, d, k)^7*a^7 + 19140625*root(9765625*a^8*b^2*d^10 - 9765625*a^10*d^10 - 1953125*a^8*d^8 - 156
250*a^6*d^6 - 6250*a^4*d^4 - 125*a^2*d^2 - 1, d, k)^9*a^9 + 1560*root(9765625*a^8*b^2*d^10 - 9765625*a^10*d^10
- 1953125*a^8*d^8 - 156250*a^6*d^6 - 6250*a^4*d^4 - 125*a^2*d^2 - 1, d, k)^2*a^2*tan(x/2) + 57000*root(976562
5*a^8*b^2*d^10 - 9765625*a^10*d^10 - 1953125*a^8*d^8 - 156250*a^6*d^6 - 6250*a^4*d^4 - 125*a^2*d^2 - 1, d, k)^
4*a^4*tan(x/2) + 925000*root(9765625*a^8*b^2*d^10 - 9765625*a^10*d^10 - 1953125*a^8*d^8 - 156250*a^6*d^6 - 625
0*a^4*d^4 - 125*a^2*d^2 - 1, d, k)^6*a^6*tan(x/2) + 5625000*root(9765625*a^8*b^2*d^10 - 9765625*a^10*d^10 - 19
53125*a^8*d^8 - 156250*a^6*d^6 - 6250*a^4*d^4 - 125*a^2*d^2 - 1, d, k)^8*a^8*tan(x/2) - 14000*root(9765625*a^8
*b^2*d^10 - 9765625*a^10*d^10 - 1953125*a^8*d^8 - 156250*a^6*d^6 - 6250*a^4*d^4 - 125*a^2*d^2 - 1, d, k)^4*a^3
*b - 175000*root(9765625*a^8*b^2*d^10 - 9765625*a^10*d^10 - 1953125*a^8*d^8 - 156250*a^6*d^6 - 6250*a^4*d^4 -
125*a^2*d^2 - 1, d, k)^6*a^5*b - 546875*root(9765625*a^8*b^2*d^10 - 9765625*a^10*d^10 - 1953125*a^8*d^8 - 1562
50*a^6*d^6 - 6250*a^4*d^4 - 125*a^2*d^2 - 1, d, k)^8*a^7*b - 128*root(9765625*a^8*b^2*d^10 - 9765625*a^10*d^10
- 1953125*a^8*d^8 - 156250*a^6*d^6 - 6250*a^4*d^4 - 125*a^2*d^2 - 1, d, k)*b*tan(x/2) + 1000000*root(9765625*
a^8*b^2*d^10 - 9765625*a^10*d^10 - 1953125*a^8*d^8 - 156250*a^6*d^6 - 6250*a^4*d^4 - 125*a^2*d^2 - 1, d, k)^7*
a^5*b^2 - 18750000*root(9765625*a^8*b^2*d^10 - 9765625*a^10*d^10 - 1953125*a^8*d^8 - 156250*a^6*d^6 - 6250*a^4
*d^4 - 125*a^2*d^2 - 1, d, k)^9*a^7*b^2 - 320*root(9765625*a^8*b^2*d^10 - 9765625*a^10*d^10 - 1953125*a^8*d^8
- 156250*a^6*d^6 - 6250*a^4*d^4 - 125*a^2*d^2 - 1, d, k)^2*a*b - 6400*root(9765625*a^8*b^2*d^10 - 9765625*a^10
*d^10 - 1953125*a^8*d^8 - 156250*a^6*d^6 - 6250*a^4*d^4 - 125*a^2*d^2 - 1, d, k)^3*a^2*b*tan(x/2) - 100000*roo
t(9765625*a^8*b^2*d^10 - 9765625*a^10*d^10 - 1953125*a^8*d^8 - 156250*a^6*d^6 - 6250*a^4*d^4 - 125*a^2*d^2 - 1
, d, k)^5*a^4*b*tan(x/2) - 500000*root(9765625*a^8*b^2*d^10 - 9765625*a^10*d^10 - 1953125*a^8*d^8 - 156250*a^6
*d^6 - 6250*a^4*d^4 - 125*a^2*d^2 - 1, d, k)^7*a^6*b*tan(x/2) - 390625*root(9765625*a^8*b^2*d^10 - 9765625*a^1
0*d^10 - 1953125*a^8*d^8 - 156250*a^6*d^6 - 6250*a^4*d^4 - 125*a^2*d^2 - 1, d, k)^9*a^8*b*tan(x/2) + 400000*ro
ot(9765625*a^8*b^2*d^10 - 9765625*a^10*d^10 - 1953125*a^8*d^8 - 156250*a^6*d^6 - 6250*a^4*d^4 - 125*a^2*d^2 -
1, d, k)^6*a^4*b^2*tan(x/2) - 5000000*root(9765625*a^8*b^2*d^10 - 9765625*a^10*d^10 - 1953125*a^8*d^8 - 156250
*a^6*d^6 - 6250*a^4*d^4 - 125*a^2*d^2 - 1, d, k)^8*a^6*b^2*tan(x/2)))*root(9765625*a^8*b^2*d^10 - 9765625*a^10
*d^10 - 1953125*a^8*d^8 - 156250*a^6*d^6 - 6250*a^4*d^4 - 125*a^2*d^2 - 1, d, k), k, 1, 10)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{a - b \sin ^{5}{\relax (x )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-b*sin(x)**5),x)

[Out]

Integral(1/(a - b*sin(x)**5), x)

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