3.3.52 \(\int \frac {1}{a-b \sin ^5(x)} \, dx\) [252]

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

[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 = 0.34, 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 = {3292, 2739, 632, 210} \begin {gather*} -\frac {2 \text {ArcTan}\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 \text {ArcTan}\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 \text {ArcTan}\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 \text {ArcTan}\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 \text {ArcTan}\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}}} \end {gather*}

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 210

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

Rule 632

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 2739

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 3292

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 \text {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 \text {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 \text {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 \text {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 \text {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 \text {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 \text {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 \text {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 \text {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 \text {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] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.13, size = 149, normalized size = 0.39 \begin {gather*} -\frac {8}{5} i \text {RootSum}\left [-i b+5 i b \text {$\#$1}^2-10 i b \text {$\#$1}^4+32 a \text {$\#$1}^5+10 i b \text {$\#$1}^6-5 i b \text {$\#$1}^8+i b \text {$\#$1}^{10}\&,\frac {2 \tan ^{-1}\left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^3-i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3}{b-4 b \text {$\#$1}^2-16 i a \text {$\#$1}^3+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8}\&\right ] \end {gather*}

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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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.64, size = 109, normalized size = 0.29

method	result	size
default	\(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{10}+5 a \,\textit {\_Z}^{8}+10 a \,\textit {\_Z}^{6}-32 b \,\textit {\_Z}^{5}+10 a \,\textit {\_Z}^{4}+5 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{8}+4 \textit {\_R}^{6}+6 \textit {\_R}^{4}+4 \textit {\_R}^{2}+1\right ) \ln \left (\tan \left (\frac {x}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{9} a +4 \textit {\_R}^{7} a +6 \textit {\_R}^{5} a -16 \textit {\_R}^{4} b +4 \textit {\_R}^{3} a +\textit {\_R} a}\right )}{5}\)	\(109\)
risch	\(\munderset {\textit {\_R} =\RootOf \left (1+\left (9765625 a^{10}-9765625 a^{8} b^{2}\right ) \textit {\_Z}^{10}+1953125 a^{8} \textit {\_Z}^{8}+156250 a^{6} \textit {\_Z}^{6}+6250 a^{4} \textit {\_Z}^{4}+125 a^{2} \textit {\_Z}^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{i x}+\left (-\frac {11718750 a^{10}}{b}[phone] a^{8} b \right ) \textit {\_R}^{9}+\left (-\frac {1171875 i a^{9}}{b}+1171875 i a^{7} b \right ) \textit {\_R}^{8}+\left (-\frac {2109375 a^{8}}{b}-234375 a^{6} b \right ) \textit {\_R}^{7}+\left (-\frac {218750 i a^{7}}{b}-15625 i a^{5} b \right ) \textit {\_R}^{6}+\left (-\frac {143750 a^{6}}{b}+3125 a^{4} b \right ) \textit {\_R}^{5}-\frac {15625 i a^{5} \textit {\_R}^{4}}{b}-\frac {4375 a^{4} \textit {\_R}^{3}}{b}-\frac {500 i a^{3} \textit {\_R}^{2}}{b}-\frac {50 a^{2} \textit {\_R}}{b}-\frac {6 i a}{b}\right )\)	\(216\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 \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)^5),x, algorithm="maxima")

[Out]

-integrate(1/(b*sin(x)^5 - a), x)

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{a - b \sin ^{5}{\left (x \right )}}\, dx \end {gather*}

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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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)^5),x, algorithm="giac")

[Out]

integrate(-1/(b*sin(x)^5 - a), x)

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Mupad [B]
time = 20.14, size = 1515, normalized size = 4.00 \begin {gather*} \sum _{k=1}^{10}\ln \left (a\,b^7\,\left (16\,\mathrm {tan}\left (\frac {x}{2}\right )+\mathrm {root}\left (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\right )\,a\,56+{\mathrm {root}\left (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\right )}^3\,a^3\,5425+{\mathrm {root}\left (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\right )}^5\,a^5\,196875+{\mathrm {root}\left (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\right )}^7\,a^7\,3171875+{\mathrm {root}\left (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\right )}^9\,a^9\,19140625+{\mathrm {root}\left (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\right )}^2\,a^2\,\mathrm {tan}\left (\frac {x}{2}\right )\,1560+{\mathrm {root}\left (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\right )}^4\,a^4\,\mathrm {tan}\left (\frac {x}{2}\right )\,57000+{\mathrm {root}\left (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\right )}^6\,a^6\,\mathrm {tan}\left (\frac {x}{2}\right )\,925000+{\mathrm {root}\left (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\right )}^8\,a^8\,\mathrm {tan}\left (\frac {x}{2}\right )\,5625000-{\mathrm {root}\left (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\right )}^4\,a^3\,b\,14000-{\mathrm {root}\left (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\right )}^6\,a^5\,b\,175000-{\mathrm {root}\left (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\right )}^8\,a^7\,b\,546875-\mathrm {root}\left (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\right )\,b\,\mathrm {tan}\left (\frac {x}{2}\right )\,128+{\mathrm {root}\left (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\right )}^7\,a^5\,b^2\,1000000-{\mathrm {root}\left (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\right )}^9\,a^7\,b^2\,18750000-{\mathrm {root}\left (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\right )}^2\,a\,b\,320-{\mathrm {root}\left (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\right )}^3\,a^2\,b\,\mathrm {tan}\left (\frac {x}{2}\right )\,6400-{\mathrm {root}\left (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\right )}^5\,a^4\,b\,\mathrm {tan}\left (\frac {x}{2}\right )\,100000-{\mathrm {root}\left (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\right )}^7\,a^6\,b\,\mathrm {tan}\left (\frac {x}{2}\right )\,500000-{\mathrm {root}\left (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\right )}^9\,a^8\,b\,\mathrm {tan}\left (\frac {x}{2}\right )\,390625+{\mathrm {root}\left (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\right )}^6\,a^4\,b^2\,\mathrm {tan}\left (\frac {x}{2}\right )\,400000-{\mathrm {root}\left (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\right )}^8\,a^6\,b^2\,\mathrm {tan}\left (\frac {x}{2}\right )\,5000000\right )\,10995116277760\right )\,\mathrm {root}\left (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\right ) \end {gather*}

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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