3.1.82 \(\int \frac {(e x)^{-1+n}}{(a+b \csc (c+d x^n))^2} \, dx\) [82]

Optimal. Leaf size=156 \[ \frac {(e x)^n}{a^2 e n}+\frac {2 b \left (2 a^2-b^2\right ) x^{-n} (e x)^n \tanh ^{-1}\left (\frac {a+b \tan \left (\frac {1}{2} \left (c+d x^n\right )\right )}{\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right )^{3/2} d e n}-\frac {b^2 x^{-n} (e x)^n \cot \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (a+b \csc \left (c+d x^n\right )\right )} \]

[Out]

(e*x)^n/a^2/e/n+2*b*(2*a^2-b^2)*(e*x)^n*arctanh((a+b*tan(1/2*c+1/2*d*x^n))/(a^2-b^2)^(1/2))/a^2/(a^2-b^2)^(3/2
)/d/e/n/(x^n)-b^2*(e*x)^n*cot(c+d*x^n)/a/(a^2-b^2)/d/e/n/(x^n)/(a+b*csc(c+d*x^n))

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Rubi [A]
time = 0.20, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {4294, 4290, 3870, 4004, 3916, 2739, 632, 212} \begin {gather*} \frac {2 b \left (2 a^2-b^2\right ) x^{-n} (e x)^n \tanh ^{-1}\left (\frac {a+b \tan \left (\frac {1}{2} \left (c+d x^n\right )\right )}{\sqrt {a^2-b^2}}\right )}{a^2 d e n \left (a^2-b^2\right )^{3/2}}-\frac {b^2 x^{-n} (e x)^n \cot \left (c+d x^n\right )}{a d e n \left (a^2-b^2\right ) \left (a+b \csc \left (c+d x^n\right )\right )}+\frac {(e x)^n}{a^2 e n} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(e*x)^(-1 + n)/(a + b*Csc[c + d*x^n])^2,x]

[Out]

(e*x)^n/(a^2*e*n) + (2*b*(2*a^2 - b^2)*(e*x)^n*ArcTanh[(a + b*Tan[(c + d*x^n)/2])/Sqrt[a^2 - b^2]])/(a^2*(a^2
- b^2)^(3/2)*d*e*n*x^n) - (b^2*(e*x)^n*Cot[c + d*x^n])/(a*(a^2 - b^2)*d*e*n*x^n*(a + b*Csc[c + d*x^n]))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[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 3870

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Simp[b^2*Cot[c + d*x]*((a + b*Csc[c + d*x])^(n +
1)/(a*d*(n + 1)*(a^2 - b^2))), x] + Dist[1/(a*(n + 1)*(a^2 - b^2)), Int[(a + b*Csc[c + d*x])^(n + 1)*Simp[(a^
2 - b^2)*(n + 1) - a*b*(n + 1)*Csc[c + d*x] + b^2*(n + 2)*Csc[c + d*x]^2, x], x], x] /; FreeQ[{a, b, c, d}, x]
&& NeQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]

Rule 3916

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[1/b, Int[1/(1 + (a/b)*Si
n[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 4004

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[c*(x/a),
x] - Dist[(b*c - a*d)/a, Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[
b*c - a*d, 0]

Rule 4290

Int[((a_.) + Csc[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Csc[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[
(m + 1)/n], 0] && IntegerQ[p]

Rule 4294

Int[((a_.) + Csc[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*((e_)*(x_))^(m_.), x_Symbol] :> Dist[e^IntPart[m]*((e*x
)^FracPart[m]/x^FracPart[m]), Int[x^m*(a + b*Csc[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x]

Rubi steps

\begin {align*} \int \frac {(e x)^{-1+n}}{\left (a+b \csc \left (c+d x^n\right )\right )^2} \, dx &=\frac {\left (x^{-n} (e x)^n\right ) \int \frac {x^{-1+n}}{\left (a+b \csc \left (c+d x^n\right )\right )^2} \, dx}{e}\\ &=\frac {\left (x^{-n} (e x)^n\right ) \text {Subst}\left (\int \frac {1}{(a+b \csc (c+d x))^2} \, dx,x,x^n\right )}{e n}\\ &=-\frac {b^2 x^{-n} (e x)^n \cot \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (a+b \csc \left (c+d x^n\right )\right )}-\frac {\left (x^{-n} (e x)^n\right ) \text {Subst}\left (\int \frac {-a^2+b^2+a b \csc (c+d x)}{a+b \csc (c+d x)} \, dx,x,x^n\right )}{a \left (a^2-b^2\right ) e n}\\ &=\frac {(e x)^n}{a^2 e n}-\frac {b^2 x^{-n} (e x)^n \cot \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (a+b \csc \left (c+d x^n\right )\right )}+\frac {\left (\left (-a^2 b+b \left (-a^2+b^2\right )\right ) x^{-n} (e x)^n\right ) \text {Subst}\left (\int \frac {\csc (c+d x)}{a+b \csc (c+d x)} \, dx,x,x^n\right )}{a^2 \left (a^2-b^2\right ) e n}\\ &=\frac {(e x)^n}{a^2 e n}-\frac {b^2 x^{-n} (e x)^n \cot \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (a+b \csc \left (c+d x^n\right )\right )}+\frac {\left (\left (-a^2 b+b \left (-a^2+b^2\right )\right ) x^{-n} (e x)^n\right ) \text {Subst}\left (\int \frac {1}{1+\frac {a \sin (c+d x)}{b}} \, dx,x,x^n\right )}{a^2 b \left (a^2-b^2\right ) e n}\\ &=\frac {(e x)^n}{a^2 e n}-\frac {b^2 x^{-n} (e x)^n \cot \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (a+b \csc \left (c+d x^n\right )\right )}+\frac {\left (2 \left (-a^2 b+b \left (-a^2+b^2\right )\right ) x^{-n} (e x)^n\right ) \text {Subst}\left (\int \frac {1}{1+\frac {2 a x}{b}+x^2} \, dx,x,\tan \left (\frac {1}{2} \left (c+d x^n\right )\right )\right )}{a^2 b \left (a^2-b^2\right ) d e n}\\ &=\frac {(e x)^n}{a^2 e n}-\frac {b^2 x^{-n} (e x)^n \cot \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (a+b \csc \left (c+d x^n\right )\right )}-\frac {\left (4 \left (-a^2 b+b \left (-a^2+b^2\right )\right ) x^{-n} (e x)^n\right ) \text {Subst}\left (\int \frac {1}{-4 \left (1-\frac {a^2}{b^2}\right )-x^2} \, dx,x,\frac {2 a}{b}+2 \tan \left (\frac {1}{2} \left (c+d x^n\right )\right )\right )}{a^2 b \left (a^2-b^2\right ) d e n}\\ &=\frac {(e x)^n}{a^2 e n}+\frac {2 b \left (2 a^2-b^2\right ) x^{-n} (e x)^n \tanh ^{-1}\left (\frac {b \left (\frac {a}{b}+\tan \left (\frac {1}{2} \left (c+d x^n\right )\right )\right )}{\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right )^{3/2} d e n}-\frac {b^2 x^{-n} (e x)^n \cot \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (a+b \csc \left (c+d x^n\right )\right )}\\ \end {align*}

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Mathematica [A]
time = 1.06, size = 176, normalized size = 1.13 \begin {gather*} \frac {x^{-n} (e x)^n \left (2 b \left (-2 a^2+b^2\right ) \text {ArcTan}\left (\frac {a+b \tan \left (\frac {1}{2} \left (c+d x^n\right )\right )}{\sqrt {-a^2+b^2}}\right ) \left (a+b \csc \left (c+d x^n\right )\right )+\sqrt {-a^2+b^2} \left (-a b^2 \cot \left (c+d x^n\right )+\left (a^2-b^2\right ) \left (c+d x^n\right ) \left (a+b \csc \left (c+d x^n\right )\right )\right )\right )}{a^2 (a-b) (a+b) \sqrt {-a^2+b^2} d e n \left (a+b \csc \left (c+d x^n\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(e*x)^(-1 + n)/(a + b*Csc[c + d*x^n])^2,x]

[Out]

((e*x)^n*(2*b*(-2*a^2 + b^2)*ArcTan[(a + b*Tan[(c + d*x^n)/2])/Sqrt[-a^2 + b^2]]*(a + b*Csc[c + d*x^n]) + Sqrt
[-a^2 + b^2]*(-(a*b^2*Cot[c + d*x^n]) + (a^2 - b^2)*(c + d*x^n)*(a + b*Csc[c + d*x^n]))))/(a^2*(a - b)*(a + b)
*Sqrt[-a^2 + b^2]*d*e*n*x^n*(a + b*Csc[c + d*x^n]))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.29, size = 706, normalized size = 4.53

method	result	size
risch	\(\frac {x \,{\mathrm e}^{\frac {\left (-1+n \right ) \left (-i \mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right ) \pi +i \mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e x \right )^{2} \pi +i \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )^{2} \pi -i \mathrm {csgn}\left (i e x \right )^{3} \pi +2 \ln \left (e \right )+2 \ln \left (x \right )\right )}{2}}}{a^{2} n}-\frac {2 i b^{2} x \,x^{-n} \left (\frac {i a \,x^{n} e^{n} \left (-1\right )^{\frac {\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )}{2}} {\mathrm e}^{\frac {i \pi \,\mathrm {csgn}\left (i e x \right ) \left (-n \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i x \right )+n \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e x \right )+n \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )-n \mathrm {csgn}\left (i e x \right )^{2}-\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e x \right )-\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )+\mathrm {csgn}\left (i e x \right )^{2}\right )}{2}}}{x e}+\frac {b \,x^{n} e^{n} \left (-1\right )^{\frac {\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )}{2}} {\mathrm e}^{-\frac {i \mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e x \right )^{2} \pi }{2}} {\mathrm e}^{-\frac {i \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )^{2} \pi }{2}} {\mathrm e}^{\frac {i \mathrm {csgn}\left (i e x \right )^{3} \pi }{2}} {\mathrm e}^{-\frac {i \pi n \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )}{2}} {\mathrm e}^{\frac {i \pi n \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e x \right )^{2}}{2}} {\mathrm e}^{\frac {i \pi n \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )^{2}}{2}} {\mathrm e}^{-\frac {i \pi n \mathrm {csgn}\left (i e x \right )^{3}}{2}} {\mathrm e}^{i c} {\mathrm e}^{i x^{n} d}}{x e}\right )}{a^{2} \left (-a^{2}+b^{2}\right ) d n \left (2 b \,{\mathrm e}^{i \left (c +d \,x^{n}\right )}-i a \,{\mathrm e}^{2 i \left (c +d \,x^{n}\right )}+i a \right )}-\frac {2 i \arctan \left (\frac {2 i a \,{\mathrm e}^{i \left (d \,x^{n}+2 c \right )}-2 b \,{\mathrm e}^{i c}}{2 \sqrt {a^{2} {\mathrm e}^{2 i c}-{\mathrm e}^{2 i c} b^{2}}}\right ) e^{n} \left (-2 a^{2}+b^{2}\right ) b \,{\mathrm e}^{\frac {i \left (-\pi n \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )+\pi n \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e x \right )^{2}+\pi n \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )^{2}-\pi n \mathrm {csgn}\left (i e x \right )^{3}+\pi \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )-\pi \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e x \right )^{2}-\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )^{2}+\pi \mathrm {csgn}\left (i e x \right )^{3}+2 c \right )}{2}}}{\sqrt {a^{2} {\mathrm e}^{2 i c}-{\mathrm e}^{2 i c} b^{2}}\, d e n \,a^{2} \left (-a^{2}+b^{2}\right )}\)	\(706\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^(-1+n)/(a+b*csc(c+d*x^n))^2,x,method=_RETURNVERBOSE)

[Out]

1/a^2/n*x*exp(1/2*(-1+n)*(-I*csgn(I*e)*csgn(I*x)*csgn(I*e*x)*Pi+I*csgn(I*e)*csgn(I*e*x)^2*Pi+I*csgn(I*x)*csgn(
I*e*x)^2*Pi-I*csgn(I*e*x)^3*Pi+2*ln(e)+2*ln(x)))-2*I*b^2*x/a^2/(-a^2+b^2)/d/n/(x^n)/(2*b*exp(I*(c+d*x^n))-I*a*
exp(2*I*(c+d*x^n))+I*a)*(I*a*x^n*e^n/x/e*(-1)^(1/2*csgn(I*e)*csgn(I*x)*csgn(I*e*x))*exp(1/2*I*Pi*csgn(I*e*x)*(
-n*csgn(I*e)*csgn(I*x)+n*csgn(I*e)*csgn(I*e*x)+n*csgn(I*x)*csgn(I*e*x)-n*csgn(I*e*x)^2-csgn(I*e)*csgn(I*e*x)-c
sgn(I*x)*csgn(I*e*x)+csgn(I*e*x)^2))+b*x^n*e^n/x/e*(-1)^(1/2*csgn(I*e)*csgn(I*x)*csgn(I*e*x))*exp(-1/2*I*Pi*cs
gn(I*e)*csgn(I*e*x)^2)*exp(-1/2*I*Pi*csgn(I*x)*csgn(I*e*x)^2)*exp(1/2*I*Pi*csgn(I*e*x)^3)*exp(-1/2*I*Pi*n*csgn
(I*e)*csgn(I*x)*csgn(I*e*x))*exp(1/2*I*Pi*n*csgn(I*e)*csgn(I*e*x)^2)*exp(1/2*I*Pi*n*csgn(I*x)*csgn(I*e*x)^2)*e
xp(-1/2*I*Pi*n*csgn(I*e*x)^3)*exp(I*c)*exp(I*x^n*d))-2*I*arctan(1/2*(2*I*a*exp(I*(d*x^n+2*c))-2*b*exp(I*c))/(a
^2*exp(2*I*c)-exp(2*I*c)*b^2)^(1/2))/(a^2*exp(2*I*c)-exp(2*I*c)*b^2)^(1/2)/d/e*e^n/n/a^2/(-a^2+b^2)*(-2*a^2+b^
2)*b*exp(1/2*I*(-Pi*n*csgn(I*e)*csgn(I*x)*csgn(I*e*x)+Pi*n*csgn(I*e)*csgn(I*e*x)^2+Pi*n*csgn(I*x)*csgn(I*e*x)^
2-Pi*n*csgn(I*e*x)^3+Pi*csgn(I*e)*csgn(I*x)*csgn(I*e*x)-Pi*csgn(I*e)*csgn(I*e*x)^2-Pi*csgn(I*x)*csgn(I*e*x)^2+
Pi*csgn(I*e*x)^3+2*c))

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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((e*x)^(-1+n)/(a+b*csc(c+d*x^n))^2,x, algorithm="maxima")

[Out]

-(2*a*b^3*cos(d*x^n + c)*e^n - (a^4 - a^2*b^2)*d*cos(2*d*x^n + 2*c)^2*e^(n*log(x) + n) - 4*(a^2*b^2 - b^4)*d*c
os(d*x^n + c)^2*e^(n*log(x) + n) - (a^4 - a^2*b^2)*d*e^(n*log(x) + n)*sin(2*d*x^n + 2*c)^2 - 4*(a^2*b^2 - b^4)
*d*e^(n*log(x) + n)*sin(d*x^n + c)^2 - 4*(a^3*b - a*b^3)*d*e^(n*log(x) + n)*sin(d*x^n + c) - (a^4 - a^2*b^2)*d
*e^(n*log(x) + n) + 2*(a*b^3*cos(d*x^n + c)*e^n + 2*(a^3*b - a*b^3)*d*e^(n*log(x) + n)*sin(d*x^n + c) + (a^4 -
a^2*b^2)*d*e^(n*log(x) + n))*cos(2*d*x^n + 2*c) - 2*((2*a^8*b*e - 3*a^6*b^3*e + a^4*b^5*e)*d*n*cos(2*d*x^n +
2*c)^2*e^n*sin(c) + 4*(2*a^6*b^3*e - 3*a^4*b^5*e + a^2*b^7*e)*d*n*cos(d*x^n + c)^2*e^n*sin(c) + 4*(2*a^7*b^2*e
- 3*a^5*b^4*e + a^3*b^6*e)*d*n*cos(d*x^n + c)*e^n*sin(2*d*x^n + 2*c)*sin(c) + (2*a^8*b*e - 3*a^6*b^3*e + a^4*
b^5*e)*d*n*e^n*sin(2*d*x^n + 2*c)^2*sin(c) + 4*(2*a^6*b^3*e - 3*a^4*b^5*e + a^2*b^7*e)*d*n*e^n*sin(d*x^n + c)^
2*sin(c) + 4*(2*a^7*b^2*e - 3*a^5*b^4*e + a^3*b^6*e)*d*n*e^n*sin(d*x^n + c)*sin(c) + (2*a^8*b*e - 3*a^6*b^3*e
+ a^4*b^5*e)*d*n*e^n*sin(c) - 2*(2*(2*a^7*b^2*e - 3*a^5*b^4*e + a^3*b^6*e)*d*n*e^n*sin(d*x^n + c)*sin(c) + (2*
a^8*b*e - 3*a^6*b^3*e + a^4*b^5*e)*d*n*e^n*sin(c))*cos(2*d*x^n + 2*c))*integrate((a^3*x^n*cos(2*d*x^n + 2*c)*c
os(d*x^n) + a^3*x^n*sin(2*d*x^n + 2*c)*sin(d*x^n) - 2*(a^2*b - b^3)*x^n*cos(d*x^n)^2*sin(c) - 2*(a^2*b - b^3)*
x^n*sin(d*x^n)^2*sin(c) - (a^3 - a*b^2)*x^n*cos(d*x^n) - (a*b^2*x^n*cos(d*x^n)*cos(2*c) + a*b^2*x^n*sin(d*x^n)
*sin(2*c))*cos(2*d*x^n) - (a*b^2*x^n*cos(2*c)*sin(d*x^n) - a*b^2*x^n*cos(d*x^n)*sin(2*c))*sin(2*d*x^n))/(a^8*x
*cos(2*d*x^n + 2*c)^2*e + a^8*x*e*sin(2*d*x^n + 2*c)^2 + (a^4*b^4*cos(2*c)^2*e + a^4*b^4*e*sin(2*c)^2)*x*cos(2
*d*x^n)^2 + 4*((a^6*b^2*e - 2*a^4*b^4*e + a^2*b^6*e)*cos(c)^2 + (a^6*b^2*e - 2*a^4*b^4*e + a^2*b^6*e)*sin(c)^2
)*x*cos(d*x^n)^2 + (a^4*b^4*cos(2*c)^2*e + a^4*b^4*e*sin(2*c)^2)*x*sin(2*d*x^n)^2 + 4*(a^7*b*e - 2*a^5*b^3*e +
a^3*b^5*e)*x*cos(c)*sin(d*x^n) + 4*((a^6*b^2*e - 2*a^4*b^4*e + a^2*b^6*e)*cos(c)^2 + (a^6*b^2*e - 2*a^4*b^4*e
+ a^2*b^6*e)*sin(c)^2)*x*sin(d*x^n)^2 + 4*(a^7*b*e - 2*a^5*b^3*e + a^3*b^5*e)*x*cos(d*x^n)*sin(c) + (a^8*e -
2*a^6*b^2*e + a^4*b^4*e)*x - 2*(2*((a^5*b^3*e - a^3*b^5*e)*cos(c)*sin(2*c) - (a^5*b^3*e - a^3*b^5*e)*cos(2*c)*
sin(c))*x*cos(d*x^n) - (a^6*b^2*e - a^4*b^4*e)*x*cos(2*c) - 2*((a^5*b^3*e - a^3*b^5*e)*cos(2*c)*cos(c) + (a^5*
b^3*e - a^3*b^5*e)*sin(2*c)*sin(c))*x*sin(d*x^n))*cos(2*d*x^n) - 2*(a^6*b^2*x*cos(2*d*x^n)*cos(2*c)*e - a^6*b^
2*x*e*sin(2*d*x^n)*sin(2*c) + 2*(a^7*b*e - a^5*b^3*e)*x*cos(c)*sin(d*x^n) + 2*(a^7*b*e - a^5*b^3*e)*x*cos(d*x^
n)*sin(c) + (a^8*e - a^6*b^2*e)*x)*cos(2*d*x^n + 2*c) - 2*(2*((a^5*b^3*e - a^3*b^5*e)*cos(2*c)*cos(c) + (a^5*b
^3*e - a^3*b^5*e)*sin(2*c)*sin(c))*x*cos(d*x^n) + 2*((a^5*b^3*e - a^3*b^5*e)*cos(c)*sin(2*c) - (a^5*b^3*e - a^
3*b^5*e)*cos(2*c)*sin(c))*x*sin(d*x^n) + (a^6*b^2*e - a^4*b^4*e)*x*sin(2*c))*sin(2*d*x^n) - 2*(a^6*b^2*x*cos(2
*c)*e*sin(2*d*x^n) + a^6*b^2*x*cos(2*d*x^n)*e*sin(2*c) - 2*(a^7*b*e - a^5*b^3*e)*x*cos(d*x^n)*cos(c) + 2*(a^7*
b*e - a^5*b^3*e)*x*sin(d*x^n)*sin(c))*sin(2*d*x^n + 2*c)), x) + 2*((2*a^8*b*e - 3*a^6*b^3*e + a^4*b^5*e)*d*n*c
os(2*d*x^n + 2*c)^2*cos(c)*e^n + 4*(2*a^6*b^3*e - 3*a^4*b^5*e + a^2*b^7*e)*d*n*cos(d*x^n + c)^2*cos(c)*e^n + 4
*(2*a^7*b^2*e - 3*a^5*b^4*e + a^3*b^6*e)*d*n*cos(d*x^n + c)*cos(c)*e^n*sin(2*d*x^n + 2*c) + (2*a^8*b*e - 3*a^6
*b^3*e + a^4*b^5*e)*d*n*cos(c)*e^n*sin(2*d*x^n + 2*c)^2 + 4*(2*a^6*b^3*e - 3*a^4*b^5*e + a^2*b^7*e)*d*n*cos(c)
*e^n*sin(d*x^n + c)^2 + 4*(2*a^7*b^2*e - 3*a^5*b^4*e + a^3*b^6*e)*d*n*cos(c)*e^n*sin(d*x^n + c) + (2*a^8*b*e -
3*a^6*b^3*e + a^4*b^5*e)*d*n*cos(c)*e^n - 2*(2*(2*a^7*b^2*e - 3*a^5*b^4*e + a^3*b^6*e)*d*n*cos(c)*e^n*sin(d*x
^n + c) + (2*a^8*b*e - 3*a^6*b^3*e + a^4*b^5*e)*d*n*cos(c)*e^n)*cos(2*d*x^n + 2*c))*integrate((a^3*x^n*cos(d*x
^n)*sin(2*d*x^n + 2*c) - a^3*x^n*cos(2*d*x^n + 2*c)*sin(d*x^n) + 2*(a^2*b - b^3)*x^n*cos(d*x^n)^2*cos(c) + 2*(
a^2*b - b^3)*x^n*cos(c)*sin(d*x^n)^2 + (a^3 - a*b^2)*x^n*sin(d*x^n) + (a*b^2*x^n*cos(2*c)*sin(d*x^n) - a*b^2*x
^n*cos(d*x^n)*sin(2*c))*cos(2*d*x^n) - (a*b^2*x^n*cos(d*x^n)*cos(2*c) + a*b^2*x^n*sin(d*x^n)*sin(2*c))*sin(2*d
*x^n))/(a^8*x*cos(2*d*x^n + 2*c)^2*e + a^8*x*e*sin(2*d*x^n + 2*c)^2 + (a^4*b^4*cos(2*c)^2*e + a^4*b^4*e*sin(2*
c)^2)*x*cos(2*d*x^n)^2 + 4*((a^6*b^2*e - 2*a^4*b^4*e + a^2*b^6*e)*cos(c)^2 + (a^6*b^2*e - 2*a^4*b^4*e + a^2*b^
6*e)*sin(c)^2)*x*cos(d*x^n)^2 + (a^4*b^4*cos(2*c)^2*e + a^4*b^4*e*sin(2*c)^2)*x*sin(2*d*x^n)^2 + 4*(a^7*b*e -
2*a^5*b^3*e + a^3*b^5*e)*x*cos(c)*sin(d*x^n) + 4*((a^6*b^2*e - 2*a^4*b^4*e + a^2*b^6*e)*cos(c)^2 + (a^6*b^2*e
- 2*a^4*b^4*e + a^2*b^6*e)*sin(c)^2)*x*sin(d*x^n)^2 + 4*(a^7*b*e - 2*a^5*b^3*e + a^3*b^5*e)*x*cos(d*x^n)*sin(c
) + (a^8*e - 2*a^6*b^2*e + a^4*b^4*e)*x - 2*(2*((a^5*b^3*e - a^3*b^5*e)*cos(c)*sin(2*c) - (a^5*b^3*e - a^3*b^5
*e)*cos(2*c)*sin(c))*x*cos(d*x^n) - (a^6*b^2*e - a^4*b^4*e)*x*cos(2*c) - 2*((a^5*b^3*e - a^3*b^5*e)*cos(2*c)*c
os(c) + (a^5*b^3*e - a^3*b^5*e)*sin(2*c)*sin(c))*x*sin(d*x^n))*cos(2*d*x^n) - 2*(a^6*b^2*x*cos(2*d*x^n)*cos(2*
c)*e - a^6*b^2*x*e*sin(2*d*x^n)*sin(2*c) + 2*(a...

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Fricas [A]
time = 3.53, size = 620, normalized size = 3.97 \begin {gather*} \left [\frac {2 \, {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d x^{n} e^{\left (n - 1\right )} \sin \left (d x^{n} + c\right ) + 2 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d x^{n} e^{\left (n - 1\right )} - 2 \, {\left (a^{3} b^{2} - a b^{4}\right )} \cos \left (d x^{n} + c\right ) e^{\left (n - 1\right )} + {\left ({\left (2 \, a^{3} b - a b^{3}\right )} \sqrt {a^{2} - b^{2}} e^{\left (n - 1\right )} \sin \left (d x^{n} + c\right ) + {\left (2 \, a^{2} b^{2} - b^{4}\right )} \sqrt {a^{2} - b^{2}} e^{\left (n - 1\right )}\right )} \log \left (\frac {{\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x^{n} + c\right )^{2} + 2 \, \sqrt {a^{2} - b^{2}} a \cos \left (d x^{n} + c\right ) + a^{2} + b^{2} + 2 \, {\left (\sqrt {a^{2} - b^{2}} b \cos \left (d x^{n} + c\right ) + a b\right )} \sin \left (d x^{n} + c\right )}{a^{2} \cos \left (d x^{n} + c\right )^{2} - 2 \, a b \sin \left (d x^{n} + c\right ) - a^{2} - b^{2}}\right )}{2 \, {\left ({\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d n \sin \left (d x^{n} + c\right ) + {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d n\right )}}, \frac {{\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d x^{n} e^{\left (n - 1\right )} \sin \left (d x^{n} + c\right ) + {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d x^{n} e^{\left (n - 1\right )} - {\left (a^{3} b^{2} - a b^{4}\right )} \cos \left (d x^{n} + c\right ) e^{\left (n - 1\right )} + {\left ({\left (2 \, a^{3} b - a b^{3}\right )} \sqrt {-a^{2} + b^{2}} e^{\left (n - 1\right )} \sin \left (d x^{n} + c\right ) + {\left (2 \, a^{2} b^{2} - b^{4}\right )} \sqrt {-a^{2} + b^{2}} e^{\left (n - 1\right )}\right )} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} b \sin \left (d x^{n} + c\right ) + \sqrt {-a^{2} + b^{2}} a}{{\left (a^{2} - b^{2}\right )} \cos \left (d x^{n} + c\right )}\right )}{{\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d n \sin \left (d x^{n} + c\right ) + {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d n}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^(-1+n)/(a+b*csc(c+d*x^n))^2,x, algorithm="fricas")

[Out]

[1/2*(2*(a^5 - 2*a^3*b^2 + a*b^4)*d*x^n*e^(n - 1)*sin(d*x^n + c) + 2*(a^4*b - 2*a^2*b^3 + b^5)*d*x^n*e^(n - 1)
- 2*(a^3*b^2 - a*b^4)*cos(d*x^n + c)*e^(n - 1) + ((2*a^3*b - a*b^3)*sqrt(a^2 - b^2)*e^(n - 1)*sin(d*x^n + c)
+ (2*a^2*b^2 - b^4)*sqrt(a^2 - b^2)*e^(n - 1))*log(((a^2 - 2*b^2)*cos(d*x^n + c)^2 + 2*sqrt(a^2 - b^2)*a*cos(d
*x^n + c) + a^2 + b^2 + 2*(sqrt(a^2 - b^2)*b*cos(d*x^n + c) + a*b)*sin(d*x^n + c))/(a^2*cos(d*x^n + c)^2 - 2*a
*b*sin(d*x^n + c) - a^2 - b^2)))/((a^7 - 2*a^5*b^2 + a^3*b^4)*d*n*sin(d*x^n + c) + (a^6*b - 2*a^4*b^3 + a^2*b^
5)*d*n), ((a^5 - 2*a^3*b^2 + a*b^4)*d*x^n*e^(n - 1)*sin(d*x^n + c) + (a^4*b - 2*a^2*b^3 + b^5)*d*x^n*e^(n - 1)
- (a^3*b^2 - a*b^4)*cos(d*x^n + c)*e^(n - 1) + ((2*a^3*b - a*b^3)*sqrt(-a^2 + b^2)*e^(n - 1)*sin(d*x^n + c) +
(2*a^2*b^2 - b^4)*sqrt(-a^2 + b^2)*e^(n - 1))*arctan(-(sqrt(-a^2 + b^2)*b*sin(d*x^n + c) + sqrt(-a^2 + b^2)*a
)/((a^2 - b^2)*cos(d*x^n + c))))/((a^7 - 2*a^5*b^2 + a^3*b^4)*d*n*sin(d*x^n + c) + (a^6*b - 2*a^4*b^3 + a^2*b^
5)*d*n)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (e x\right )^{n - 1}}{\left (a + b \csc {\left (c + d x^{n} \right )}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**(-1+n)/(a+b*csc(c+d*x**n))**2,x)

[Out]

Integral((e*x)**(n - 1)/(a + b*csc(c + d*x**n))**2, 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((e*x)^(-1+n)/(a+b*csc(c+d*x^n))^2,x, algorithm="giac")

[Out]

integrate((e*x)^(n - 1)/(b*csc(d*x^n + c) + a)^2, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (e\,x\right )}^{n-1}}{{\left (a+\frac {b}{\sin \left (c+d\,x^n\right )}\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^(n - 1)/(a + b/sin(c + d*x^n))^2,x)

[Out]

int((e*x)^(n - 1)/(a + b/sin(c + d*x^n))^2, x)

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