3.1.83 \(\int \frac {(e x)^{-1+2 n}}{(a+b \csc (c+d x^n))^2} \, dx\) [83]
Optimal. Leaf size=778 \[ \frac {(e x)^{2 n}}{2 a^2 e n}-\frac {i b^3 x^{-n} (e x)^{2 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}+\frac {2 i b x^{-n} (e x)^{2 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d e n}+\frac {i b^3 x^{-n} (e x)^{2 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}-\frac {2 i b x^{-n} (e x)^{2 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d e n}+\frac {b^2 x^{-2 n} (e x)^{2 n} \log \left (b+a \sin \left (c+d x^n\right )\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}-\frac {b^3 x^{-2 n} (e x)^{2 n} \text {PolyLog}\left (2,\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2 e n}+\frac {2 b x^{-2 n} (e x)^{2 n} \text {PolyLog}\left (2,\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2 e n}+\frac {b^3 x^{-2 n} (e x)^{2 n} \text {PolyLog}\left (2,\frac {i a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2 e n}-\frac {2 b x^{-2 n} (e x)^{2 n} \text {PolyLog}\left (2,\frac {i a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2 e n}-\frac {b^2 x^{-n} (e x)^{2 n} \cos \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \sin \left (c+d x^n\right )\right )} \]
[Out]
1/2*(e*x)^(2*n)/a^2/e/n+b^2*(e*x)^(2*n)*ln(b+a*sin(c+d*x^n))/a^2/(a^2-b^2)/d^2/e/n/(x^(2*n))-I*b^3*(e*x)^(2*n)
*ln(1-I*a*exp(I*(c+d*x^n))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d/e/n/(x^n)+I*b^3*(e*x)^(2*n)*ln(1-I*a*e
xp(I*(c+d*x^n))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d/e/n/(x^n)-b^3*(e*x)^(2*n)*polylog(2,I*a*exp(I*(c+
d*x^n))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^2/e/n/(x^(2*n))+b^3*(e*x)^(2*n)*polylog(2,I*a*exp(I*(c+d*
x^n))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^2/e/n/(x^(2*n))-b^2*(e*x)^(2*n)*cos(c+d*x^n)/a/(a^2-b^2)/d/
e/n/(x^n)/(b+a*sin(c+d*x^n))+2*I*b*(e*x)^(2*n)*ln(1-I*a*exp(I*(c+d*x^n))/(b-(-a^2+b^2)^(1/2)))/a^2/d/e/n/(x^n)
/(-a^2+b^2)^(1/2)-2*I*b*(e*x)^(2*n)*ln(1-I*a*exp(I*(c+d*x^n))/(b+(-a^2+b^2)^(1/2)))/a^2/d/e/n/(x^n)/(-a^2+b^2)
^(1/2)+2*b*(e*x)^(2*n)*polylog(2,I*a*exp(I*(c+d*x^n))/(b-(-a^2+b^2)^(1/2)))/a^2/d^2/e/n/(x^(2*n))/(-a^2+b^2)^(
1/2)-2*b*(e*x)^(2*n)*polylog(2,I*a*exp(I*(c+d*x^n))/(b+(-a^2+b^2)^(1/2)))/a^2/d^2/e/n/(x^(2*n))/(-a^2+b^2)^(1/
2)
________________________________________________________________________________________
Rubi [A]
time = 0.94, antiderivative size = 778, normalized size of antiderivative = 1.00, number of steps
used = 23, number of rules used = 11, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {4294, 4290,
4276, 3405, 3404, 2296, 2221, 2317, 2438, 2747, 31} \begin {gather*} \frac {2 b x^{-2 n} (e x)^{2 n} \text {Li}_2\left (\frac {i a e^{i \left (d x^n+c\right )}}{b-\sqrt {b^2-a^2}}\right )}{a^2 d^2 e n \sqrt {b^2-a^2}}-\frac {2 b x^{-2 n} (e x)^{2 n} \text {Li}_2\left (\frac {i a e^{i \left (d x^n+c\right )}}{b+\sqrt {b^2-a^2}}\right )}{a^2 d^2 e n \sqrt {b^2-a^2}}+\frac {b^2 x^{-2 n} (e x)^{2 n} \log \left (a \sin \left (c+d x^n\right )+b\right )}{a^2 d^2 e n \left (a^2-b^2\right )}+\frac {2 i b x^{-n} (e x)^{2 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {b^2-a^2}}\right )}{a^2 d e n \sqrt {b^2-a^2}}-\frac {2 i b x^{-n} (e x)^{2 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{\sqrt {b^2-a^2}+b}\right )}{a^2 d e n \sqrt {b^2-a^2}}-\frac {b^2 x^{-n} (e x)^{2 n} \cos \left (c+d x^n\right )}{a d e n \left (a^2-b^2\right ) \left (a \sin \left (c+d x^n\right )+b\right )}-\frac {b^3 x^{-2 n} (e x)^{2 n} \text {Li}_2\left (\frac {i a e^{i \left (d x^n+c\right )}}{b-\sqrt {b^2-a^2}}\right )}{a^2 d^2 e n \left (b^2-a^2\right )^{3/2}}+\frac {b^3 x^{-2 n} (e x)^{2 n} \text {Li}_2\left (\frac {i a e^{i \left (d x^n+c\right )}}{b+\sqrt {b^2-a^2}}\right )}{a^2 d^2 e n \left (b^2-a^2\right )^{3/2}}-\frac {i b^3 x^{-n} (e x)^{2 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {b^2-a^2}}\right )}{a^2 d e n \left (b^2-a^2\right )^{3/2}}+\frac {i b^3 x^{-n} (e x)^{2 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{\sqrt {b^2-a^2}+b}\right )}{a^2 d e n \left (b^2-a^2\right )^{3/2}}+\frac {(e x)^{2 n}}{2 a^2 e n} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(e*x)^(-1 + 2*n)/(a + b*Csc[c + d*x^n])^2,x]
[Out]
(e*x)^(2*n)/(2*a^2*e*n) - (I*b^3*(e*x)^(2*n)*Log[1 - (I*a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a
^2 + b^2)^(3/2)*d*e*n*x^n) + ((2*I)*b*(e*x)^(2*n)*Log[1 - (I*a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2])])/(a^
2*Sqrt[-a^2 + b^2]*d*e*n*x^n) + (I*b^3*(e*x)^(2*n)*Log[1 - (I*a*E^(I*(c + d*x^n)))/(b + Sqrt[-a^2 + b^2])])/(a
^2*(-a^2 + b^2)^(3/2)*d*e*n*x^n) - ((2*I)*b*(e*x)^(2*n)*Log[1 - (I*a*E^(I*(c + d*x^n)))/(b + Sqrt[-a^2 + b^2])
])/(a^2*Sqrt[-a^2 + b^2]*d*e*n*x^n) + (b^2*(e*x)^(2*n)*Log[b + a*Sin[c + d*x^n]])/(a^2*(a^2 - b^2)*d^2*e*n*x^(
2*n)) - (b^3*(e*x)^(2*n)*PolyLog[2, (I*a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d
^2*e*n*x^(2*n)) + (2*b*(e*x)^(2*n)*PolyLog[2, (I*a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2
+ b^2]*d^2*e*n*x^(2*n)) + (b^3*(e*x)^(2*n)*PolyLog[2, (I*a*E^(I*(c + d*x^n)))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-
a^2 + b^2)^(3/2)*d^2*e*n*x^(2*n)) - (2*b*(e*x)^(2*n)*PolyLog[2, (I*a*E^(I*(c + d*x^n)))/(b + Sqrt[-a^2 + b^2])
])/(a^2*Sqrt[-a^2 + b^2]*d^2*e*n*x^(2*n)) - (b^2*(e*x)^(2*n)*Cos[c + d*x^n])/(a*(a^2 - b^2)*d*e*n*x^n*(b + a*S
in[c + d*x^n]))
Rule 31
Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]
Rule 2221
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Rule 2296
Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[2*(c/q), Int[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Dist[2*(c/q), Int[(f + g
*x)^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]
Rule 2317
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Rule 2438
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]
Rule 2747
Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]
Rule 3404
Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[2, Int[(c + d*x)^m*(E
^(I*(e + f*x))/(I*b + 2*a*E^(I*(e + f*x)) - I*b*E^(2*I*(e + f*x)))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
Rule 3405
Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[b*(c + d*x)^m*(Cos[
e + f*x]/(f*(a^2 - b^2)*(a + b*Sin[e + f*x]))), x] + (Dist[a/(a^2 - b^2), Int[(c + d*x)^m/(a + b*Sin[e + f*x])
, x], x] - Dist[b*d*(m/(f*(a^2 - b^2))), Int[(c + d*x)^(m - 1)*(Cos[e + f*x]/(a + b*Sin[e + f*x])), x], x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
Rule 4276
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, 1/(Sin[e + f*x]^n/(b + a*Sin[e + f*x])^n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[n, 0] &
& IGtQ[m, 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+2 n}}{\left (a+b \csc \left (c+d x^n\right )\right )^2} \, dx &=\frac {\left (x^{-2 n} (e x)^{2 n}\right ) \int \frac {x^{-1+2 n}}{\left (a+b \csc \left (c+d x^n\right )\right )^2} \, dx}{e}\\ &=\frac {\left (x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {x}{(a+b \csc (c+d x))^2} \, dx,x,x^n\right )}{e n}\\ &=\frac {\left (x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \left (\frac {x}{a^2}+\frac {b^2 x}{a^2 (b+a \sin (c+d x))^2}-\frac {2 b x}{a^2 (b+a \sin (c+d x))}\right ) \, dx,x,x^n\right )}{e n}\\ &=\frac {(e x)^{2 n}}{2 a^2 e n}-\frac {\left (2 b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {x}{b+a \sin (c+d x)} \, dx,x,x^n\right )}{a^2 e n}+\frac {\left (b^2 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {x}{(b+a \sin (c+d x))^2} \, dx,x,x^n\right )}{a^2 e n}\\ &=\frac {(e x)^{2 n}}{2 a^2 e n}-\frac {b^2 x^{-n} (e x)^{2 n} \cos \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \sin \left (c+d x^n\right )\right )}-\frac {\left (4 b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {e^{i (c+d x)} x}{i a+2 b e^{i (c+d x)}-i a e^{2 i (c+d x)}} \, dx,x,x^n\right )}{a^2 e n}-\frac {\left (b^3 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {x}{b+a \sin (c+d x)} \, dx,x,x^n\right )}{a^2 \left (a^2-b^2\right ) e n}+\frac {\left (b^2 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {\cos (c+d x)}{b+a \sin (c+d x)} \, dx,x,x^n\right )}{a \left (a^2-b^2\right ) d e n}\\ &=\frac {(e x)^{2 n}}{2 a^2 e n}-\frac {b^2 x^{-n} (e x)^{2 n} \cos \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \sin \left (c+d x^n\right )\right )}-\frac {\left (2 b^3 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {e^{i (c+d x)} x}{i a+2 b e^{i (c+d x)}-i a e^{2 i (c+d x)}} \, dx,x,x^n\right )}{a^2 \left (a^2-b^2\right ) e n}+\frac {\left (4 i b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {e^{i (c+d x)} x}{2 b-2 \sqrt {-a^2+b^2}-2 i a e^{i (c+d x)}} \, dx,x,x^n\right )}{a \sqrt {-a^2+b^2} e n}-\frac {\left (4 i b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {e^{i (c+d x)} x}{2 b+2 \sqrt {-a^2+b^2}-2 i a e^{i (c+d x)}} \, dx,x,x^n\right )}{a \sqrt {-a^2+b^2} e n}+\frac {\left (b^2 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {1}{b+x} \, dx,x,a \sin \left (c+d x^n\right )\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}\\ &=\frac {(e x)^{2 n}}{2 a^2 e n}+\frac {2 i b x^{-n} (e x)^{2 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d e n}-\frac {2 i b x^{-n} (e x)^{2 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d e n}+\frac {b^2 x^{-2 n} (e x)^{2 n} \log \left (b+a \sin \left (c+d x^n\right )\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}-\frac {b^2 x^{-n} (e x)^{2 n} \cos \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \sin \left (c+d x^n\right )\right )}+\frac {\left (2 i b^3 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {e^{i (c+d x)} x}{2 b-2 \sqrt {-a^2+b^2}-2 i a e^{i (c+d x)}} \, dx,x,x^n\right )}{a \left (a^2-b^2\right ) \sqrt {-a^2+b^2} e n}-\frac {\left (2 i b^3 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {e^{i (c+d x)} x}{2 b+2 \sqrt {-a^2+b^2}-2 i a e^{i (c+d x)}} \, dx,x,x^n\right )}{a \left (a^2-b^2\right ) \sqrt {-a^2+b^2} e n}-\frac {\left (2 i b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \log \left (1-\frac {2 i a e^{i (c+d x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,x^n\right )}{a^2 \sqrt {-a^2+b^2} d e n}+\frac {\left (2 i b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \log \left (1-\frac {2 i a e^{i (c+d x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,x^n\right )}{a^2 \sqrt {-a^2+b^2} d e n}\\ &=\frac {(e x)^{2 n}}{2 a^2 e n}-\frac {i b^3 x^{-n} (e x)^{2 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}+\frac {2 i b x^{-n} (e x)^{2 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d e n}+\frac {i b^3 x^{-n} (e x)^{2 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}-\frac {2 i b x^{-n} (e x)^{2 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d e n}+\frac {b^2 x^{-2 n} (e x)^{2 n} \log \left (b+a \sin \left (c+d x^n\right )\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}-\frac {b^2 x^{-n} (e x)^{2 n} \cos \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \sin \left (c+d x^n\right )\right )}-\frac {\left (2 b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {2 i a x}{2 b-2 \sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^n\right )}\right )}{a^2 \sqrt {-a^2+b^2} d^2 e n}+\frac {\left (2 b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {2 i a x}{2 b+2 \sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^n\right )}\right )}{a^2 \sqrt {-a^2+b^2} d^2 e n}-\frac {\left (i b^3 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \log \left (1-\frac {2 i a e^{i (c+d x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,x^n\right )}{a^2 \left (a^2-b^2\right ) \sqrt {-a^2+b^2} d e n}+\frac {\left (i b^3 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \log \left (1-\frac {2 i a e^{i (c+d x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,x^n\right )}{a^2 \left (a^2-b^2\right ) \sqrt {-a^2+b^2} d e n}\\ &=\frac {(e x)^{2 n}}{2 a^2 e n}-\frac {i b^3 x^{-n} (e x)^{2 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}+\frac {2 i b x^{-n} (e x)^{2 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d e n}+\frac {i b^3 x^{-n} (e x)^{2 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}-\frac {2 i b x^{-n} (e x)^{2 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d e n}+\frac {b^2 x^{-2 n} (e x)^{2 n} \log \left (b+a \sin \left (c+d x^n\right )\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}+\frac {2 b x^{-2 n} (e x)^{2 n} \text {Li}_2\left (\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2 e n}-\frac {2 b x^{-2 n} (e x)^{2 n} \text {Li}_2\left (\frac {i a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2 e n}-\frac {b^2 x^{-n} (e x)^{2 n} \cos \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \sin \left (c+d x^n\right )\right )}-\frac {\left (b^3 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {2 i a x}{2 b-2 \sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^n\right )}\right )}{a^2 \left (a^2-b^2\right ) \sqrt {-a^2+b^2} d^2 e n}+\frac {\left (b^3 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {2 i a x}{2 b+2 \sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^n\right )}\right )}{a^2 \left (a^2-b^2\right ) \sqrt {-a^2+b^2} d^2 e n}\\ &=\frac {(e x)^{2 n}}{2 a^2 e n}-\frac {i b^3 x^{-n} (e x)^{2 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}+\frac {2 i b x^{-n} (e x)^{2 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d e n}+\frac {i b^3 x^{-n} (e x)^{2 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}-\frac {2 i b x^{-n} (e x)^{2 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d e n}+\frac {b^2 x^{-2 n} (e x)^{2 n} \log \left (b+a \sin \left (c+d x^n\right )\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}-\frac {b^3 x^{-2 n} (e x)^{2 n} \text {Li}_2\left (\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2 e n}+\frac {2 b x^{-2 n} (e x)^{2 n} \text {Li}_2\left (\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2 e n}+\frac {b^3 x^{-2 n} (e x)^{2 n} \text {Li}_2\left (\frac {i a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2 e n}-\frac {2 b x^{-2 n} (e x)^{2 n} \text {Li}_2\left (\frac {i a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2 e n}-\frac {b^2 x^{-n} (e x)^{2 n} \cos \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \sin \left (c+d x^n\right )\right )}\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(2839\) vs. \(2(778)=1556\).
time = 10.52, size = 2839, normalized size = 3.65 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Integrate[(e*x)^(-1 + 2*n)/(a + b*Csc[c + d*x^n])^2,x]
[Out]
-1/2*(b^2*x^(1 - n)*(e*x)^(-1 + 2*n)*Csc[c/2]*Csc[c + d*x^n]^2*Sec[c/2]*(b*Cos[c] + a*Sin[d*x^n])*(b + a*Sin[c
+ d*x^n]))/(a^2*(-a + b)*(a + b)*d*n*(a + b*Csc[c + d*x^n])^2) - (b^2*x^(1 - n)*(e*x)^(-1 + 2*n)*Cot[c]*Csc[c
+ d*x^n]^2*(b + a*Sin[c + d*x^n])^2)/(a^2*(-a^2 + b^2)*d*n*(a + b*Csc[c + d*x^n])^2) + (2*b^3*x^(1 - 2*n)*(e*
x)^(-1 + 2*n)*ArcTanh[(a*Cos[c + d*x^n] + I*(b + a*Sin[c + d*x^n]))/Sqrt[a^2 - b^2]]*Cot[c]*Csc[c + d*x^n]^2*(
b + a*Sin[c + d*x^n])^2)/(a^2*(a^2 - b^2)^(3/2)*d^2*n*(a + b*Csc[c + d*x^n])^2) - (2*b*x^(1 - 2*n)*(e*x)^(-1 +
2*n)*Csc[c + d*x^n]^2*((Pi*ArcTan[(a + b*Tan[(c + d*x^n)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + (2*(-c + P
i/2 - d*x^n)*ArcTanh[((a + b)*Cot[(-c + Pi/2 - d*x^n)/2])/Sqrt[a^2 - b^2]] - 2*(-c + ArcCos[-(b/a)])*ArcTanh[(
(a - b)*Tan[(-c + Pi/2 - d*x^n)/2])/Sqrt[a^2 - b^2]] + (ArcCos[-(b/a)] - (2*I)*(ArcTanh[((a + b)*Cot[(-c + Pi/
2 - d*x^n)/2])/Sqrt[a^2 - b^2]] - ArcTanh[((a - b)*Tan[(-c + Pi/2 - d*x^n)/2])/Sqrt[a^2 - b^2]]))*Log[Sqrt[a^2
- b^2]/(Sqrt[2]*Sqrt[a]*E^((I/2)*(-c + Pi/2 - d*x^n))*Sqrt[b + a*Sin[c + d*x^n]])] + (ArcCos[-(b/a)] + (2*I)*
(ArcTanh[((a + b)*Cot[(-c + Pi/2 - d*x^n)/2])/Sqrt[a^2 - b^2]] - ArcTanh[((a - b)*Tan[(-c + Pi/2 - d*x^n)/2])/
Sqrt[a^2 - b^2]]))*Log[(Sqrt[a^2 - b^2]*E^((I/2)*(-c + Pi/2 - d*x^n)))/(Sqrt[2]*Sqrt[a]*Sqrt[b + a*Sin[c + d*x
^n]])] - (ArcCos[-(b/a)] + (2*I)*ArcTanh[((a - b)*Tan[(-c + Pi/2 - d*x^n)/2])/Sqrt[a^2 - b^2]])*Log[1 - ((b -
I*Sqrt[a^2 - b^2])*(a + b - Sqrt[a^2 - b^2]*Tan[(-c + Pi/2 - d*x^n)/2]))/(a*(a + b + Sqrt[a^2 - b^2]*Tan[(-c +
Pi/2 - d*x^n)/2]))] + (-ArcCos[-(b/a)] + (2*I)*ArcTanh[((a - b)*Tan[(-c + Pi/2 - d*x^n)/2])/Sqrt[a^2 - b^2]])
*Log[1 - ((b + I*Sqrt[a^2 - b^2])*(a + b - Sqrt[a^2 - b^2]*Tan[(-c + Pi/2 - d*x^n)/2]))/(a*(a + b + Sqrt[a^2 -
b^2]*Tan[(-c + Pi/2 - d*x^n)/2]))] + I*(PolyLog[2, ((b - I*Sqrt[a^2 - b^2])*(a + b - Sqrt[a^2 - b^2]*Tan[(-c
+ Pi/2 - d*x^n)/2]))/(a*(a + b + Sqrt[a^2 - b^2]*Tan[(-c + Pi/2 - d*x^n)/2]))] - PolyLog[2, ((b + I*Sqrt[a^2 -
b^2])*(a + b - Sqrt[a^2 - b^2]*Tan[(-c + Pi/2 - d*x^n)/2]))/(a*(a + b + Sqrt[a^2 - b^2]*Tan[(-c + Pi/2 - d*x^
n)/2]))]))/Sqrt[a^2 - b^2])*(b + a*Sin[c + d*x^n])^2)/((a^2 - b^2)*d^2*n*(a + b*Csc[c + d*x^n])^2) + (b^3*x^(1
- 2*n)*(e*x)^(-1 + 2*n)*Csc[c + d*x^n]^2*((Pi*ArcTan[(a + b*Tan[(c + d*x^n)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2
+ b^2] + (2*(-c + Pi/2 - d*x^n)*ArcTanh[((a + b)*Cot[(-c + Pi/2 - d*x^n)/2])/Sqrt[a^2 - b^2]] - 2*(-c + ArcCos
[-(b/a)])*ArcTanh[((a - b)*Tan[(-c + Pi/2 - d*x^n)/2])/Sqrt[a^2 - b^2]] + (ArcCos[-(b/a)] - (2*I)*(ArcTanh[((a
+ b)*Cot[(-c + Pi/2 - d*x^n)/2])/Sqrt[a^2 - b^2]] - ArcTanh[((a - b)*Tan[(-c + Pi/2 - d*x^n)/2])/Sqrt[a^2 - b
^2]]))*Log[Sqrt[a^2 - b^2]/(Sqrt[2]*Sqrt[a]*E^((I/2)*(-c + Pi/2 - d*x^n))*Sqrt[b + a*Sin[c + d*x^n]])] + (ArcC
os[-(b/a)] + (2*I)*(ArcTanh[((a + b)*Cot[(-c + Pi/2 - d*x^n)/2])/Sqrt[a^2 - b^2]] - ArcTanh[((a - b)*Tan[(-c +
Pi/2 - d*x^n)/2])/Sqrt[a^2 - b^2]]))*Log[(Sqrt[a^2 - b^2]*E^((I/2)*(-c + Pi/2 - d*x^n)))/(Sqrt[2]*Sqrt[a]*Sqr
t[b + a*Sin[c + d*x^n]])] - (ArcCos[-(b/a)] + (2*I)*ArcTanh[((a - b)*Tan[(-c + Pi/2 - d*x^n)/2])/Sqrt[a^2 - b^
2]])*Log[1 - ((b - I*Sqrt[a^2 - b^2])*(a + b - Sqrt[a^2 - b^2]*Tan[(-c + Pi/2 - d*x^n)/2]))/(a*(a + b + Sqrt[a
^2 - b^2]*Tan[(-c + Pi/2 - d*x^n)/2]))] + (-ArcCos[-(b/a)] + (2*I)*ArcTanh[((a - b)*Tan[(-c + Pi/2 - d*x^n)/2]
)/Sqrt[a^2 - b^2]])*Log[1 - ((b + I*Sqrt[a^2 - b^2])*(a + b - Sqrt[a^2 - b^2]*Tan[(-c + Pi/2 - d*x^n)/2]))/(a*
(a + b + Sqrt[a^2 - b^2]*Tan[(-c + Pi/2 - d*x^n)/2]))] + I*(PolyLog[2, ((b - I*Sqrt[a^2 - b^2])*(a + b - Sqrt[
a^2 - b^2]*Tan[(-c + Pi/2 - d*x^n)/2]))/(a*(a + b + Sqrt[a^2 - b^2]*Tan[(-c + Pi/2 - d*x^n)/2]))] - PolyLog[2,
((b + I*Sqrt[a^2 - b^2])*(a + b - Sqrt[a^2 - b^2]*Tan[(-c + Pi/2 - d*x^n)/2]))/(a*(a + b + Sqrt[a^2 - b^2]*Ta
n[(-c + Pi/2 - d*x^n)/2]))]))/Sqrt[a^2 - b^2])*(b + a*Sin[c + d*x^n])^2)/(a^2*(a^2 - b^2)*d^2*n*(a + b*Csc[c +
d*x^n])^2) + (x^(1 - n)*(e*x)^(-1 + 2*n)*Csc[c/2]*Csc[c + d*x^n]^2*Sec[c/2]*(-2*b^2*Cos[c] + a^2*d*x^n*Sin[c]
- b^2*d*x^n*Sin[c])*(b + a*Sin[c + d*x^n])^2)/(4*a^2*(a - b)*(a + b)*d*n*(a + b*Csc[c + d*x^n])^2) + (b^2*x^(
1 - 2*n)*(e*x)^(-1 + 2*n)*Csc[c]*Csc[c + d*x^n]^2*(-(a*d*x^n*Cos[c]) + a*Log[b + a*Cos[d*x^n]*Sin[c] + a*Cos[c
]*Sin[d*x^n]]*Sin[c] + ((2*I)*a*b*ArcTan[(I*a*Cos[c] - I*(-b + a*Sin[c])*Tan[(d*x^n)/2])/Sqrt[-b^2 + a^2*Cos[c
]^2 + a^2*Sin[c]^2]]*Cos[c])/Sqrt[-b^2 + a^2*Cos[c]^2 + a^2*Sin[c]^2])*(b + a*Sin[c + d*x^n])^2)/(a*(a^2 - b^2
)*d^2*n*(a + b*Csc[c + d*x^n])^2*(a^2*Cos[c]^2 + a^2*Sin[c]^2))
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.39, size = 2847, normalized size = 3.66
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| method |
result |
size |
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| risch |
\(\text {Expression too large to display}\) |
\(2847\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x)^(-1+2*n)/(a+b*csc(c+d*x^n))^2,x,method=_RETURNVERBOSE)
[Out]
1/2/a^2/n*x*exp(1/2*(-1+2*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)*c
sgn(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)^2*(e^n)^2/x/e*(-1)^(1/2*csgn(I*e)*csgn(I*x)*csgn(I*e*x))*exp(1/2*I*Pi*c
sgn(I*e*x)*(-2*n*csgn(I*e*x)^2+2*n*csgn(I*e)*csgn(I*e*x)+2*n*csgn(I*x)*csgn(I*e*x)-2*n*csgn(I*e)*csgn(I*x)+csg
n(I*e*x)^2-csgn(I*e)*csgn(I*e*x)-csgn(I*x)*csgn(I*e*x)))+b*(x^n)^2*(e^n)^2/x/e*(-1)^(1/2*csgn(I*e)*csgn(I*x)*c
sgn(I*e*x))*exp(-1/2*I*Pi*csgn(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(-I*Pi*n*csgn(I*e)*csgn(I*x)*csgn(I*e*x))*exp(I*Pi*n*csgn(I*e)*csgn(I*e*x)^2)*exp(I*Pi*n*csgn(I*x)*
csgn(I*e*x)^2)*exp(-I*Pi*n*csgn(I*e*x)^3)*exp(I*x^n*d)*exp(I*c))+2*b/(a^2-b^2)/d*ln((I*exp(I*c)*b+a*exp(I*(d*x
^n+2*c))+(a^2*exp(2*I*c)-exp(2*I*c)*b^2)^(1/2))/(I*exp(I*c)*b+(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)*x^n/n/e*(e^n)^2*exp(1/2*I*(-2*Pi*n*csgn(I*e)*csgn(I*x)*csgn(I*e*x)+2*Pi*n*csgn(I
*e)*csgn(I*e*x)^2+2*Pi*n*csgn(I*x)*csgn(I*e*x)^2-2*Pi*n*csgn(I*e*x)^3+Pi*csgn(I*e)*csgn(I*x)*csgn(I*e*x)-Pi*cs
gn(I*e)*csgn(I*e*x)^2-Pi*csgn(I*x)*csgn(I*e*x)^2+Pi*csgn(I*e*x)^3+2*c))-2*I*b/(a^2-b^2)/d^2*dilog((I*exp(I*c)*
b+a*exp(I*(d*x^n+2*c))+(a^2*exp(2*I*c)-exp(2*I*c)*b^2)^(1/2))/(I*exp(I*c)*b+(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)/n/e*(e^n)^2*exp(1/2*I*(-2*Pi*n*csgn(I*e)*csgn(I*x)*csgn(I*e*x)+2*P
i*n*csgn(I*e)*csgn(I*e*x)^2+2*Pi*n*csgn(I*x)*csgn(I*e*x)^2-2*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))+b^3/(a^2-b^2)/d/a^2*ln((-I*e
xp(I*c)*b-a*exp(I*(d*x^n+2*c))+(a^2*exp(2*I*c)-exp(2*I*c)*b^2)^(1/2))/(-I*exp(I*c)*b+(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)*x^n/n/e*(e^n)^2*exp(1/2*I*(-2*Pi*n*csgn(I*e)*csgn(I*x)*cs
gn(I*e*x)+2*Pi*n*csgn(I*e)*csgn(I*e*x)^2+2*Pi*n*csgn(I*x)*csgn(I*e*x)^2-2*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))-b^3/(a^2-b^2)/d
/a^2*ln((I*exp(I*c)*b+a*exp(I*(d*x^n+2*c))+(a^2*exp(2*I*c)-exp(2*I*c)*b^2)^(1/2))/(I*exp(I*c)*b+(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)*x^n/n/e*(e^n)^2*exp(1/2*I*(-2*Pi*n*csgn(I*e)*c
sgn(I*x)*csgn(I*e*x)+2*Pi*n*csgn(I*e)*csgn(I*e*x)^2+2*Pi*n*csgn(I*x)*csgn(I*e*x)^2-2*Pi*n*csgn(I*e*x)^3+Pi*csg
n(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))-2*b/
(a^2-b^2)/d*ln((-I*exp(I*c)*b-a*exp(I*(d*x^n+2*c))+(a^2*exp(2*I*c)-exp(2*I*c)*b^2)^(1/2))/(-I*exp(I*c)*b+(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)*x^n/n/e*(e^n)^2*exp(1/2*I*(-2*Pi*n*cs
gn(I*e)*csgn(I*x)*csgn(I*e*x)+2*Pi*n*csgn(I*e)*csgn(I*e*x)^2+2*Pi*n*csgn(I*x)*csgn(I*e*x)^2-2*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))-2*b^2/(a^2-b^2)/d^2/a^2*ln(exp(I*x^n*d))/n/e*(e^n)^2*exp(1/2*I*csgn(I*e*x)*Pi*(-1+2*n)*(csgn(I*e*x)-csgn(
I*x))*(-csgn(I*e*x)+csgn(I*e)))+I*b^3/(a^2-b^2)/d^2/a^2*dilog((I*exp(I*c)*b+a*exp(I*(d*x^n+2*c))+(a^2*exp(2*I*
c)-exp(2*I*c)*b^2)^(1/2))/(I*exp(I*c)*b+(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)/n/e*(e^n)^2*exp(1/2*I*(-2*Pi*n*csgn(I*e)*csgn(I*x)*csgn(I*e*x)+2*Pi*n*csgn(I*e)*csgn(I*e*x)^2+2*Pi*n*c
sgn(I*x)*csgn(I*e*x)^2-2*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*c
sgn(I*x)*csgn(I*e*x)^2+Pi*csgn(I*e*x)^3+2*c))+2*I*b/(a^2-b^2)/d^2*dilog((-I*exp(I*c)*b-a*exp(I*(d*x^n+2*c))+(a
^2*exp(2*I*c)-exp(2*I*c)*b^2)^(1/2))/(-I*exp(I*c)*b+(a^2*exp(2*I*c)-exp(2*I*c)*b^2)^(1/2)))/(a^2*exp(2*I*c)-ex
p(2*I*c)*b^2)^(1/2)/n/e*(e^n)^2*exp(1/2*I*(-2*Pi*n*csgn(I*e)*csgn(I*x)*csgn(I*e*x)+2*Pi*n*csgn(I*e)*csgn(I*e*x
)^2+2*Pi*n*csgn(I*x)*csgn(I*e*x)^2-2*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))-I*b^3/(a^2-b^2)/d^2/a^2*dilog((-I*exp(I*c)*b-a*exp(I
*(d*x^n+2*c))+(a^2*exp(2*I*c)-exp(2*I*c)*b^2)^(1/2))/(-I*exp(I*c)*b+(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)/n/e*(e^n)^2*exp(1/2*I*(-2*Pi*n*csgn(I*e)*csgn(I*x)*csgn(I*e*x)+2*Pi*n*csgn
(I*e)*csgn(I*e*x)^2+2*Pi*n*csgn(I*x)*csgn(I*e*x)^2-2*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))+b^2/(a^2-b^2)/d^2/a^2*ln(I*a*exp(2*I
*(c+d*x^n))-2*b*exp(I*(c+d*x^n))-I*a)/n/e*(e^n)^2*exp(1/2*I*csgn(I*e*x)*Pi*(-1+2*n)*(csgn(I*e*x)-csgn(I*x))*(-
csgn(I*e*x)+csgn(I*e)))
________________________________________________________________________________________
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+2*n)/(a+b*csc(c+d*x^n))^2,x, algorithm="maxima")
[Out]
-1/2*(4*a*b^3*cos(d*x^n + c)*e^(n*log(x) + 2*n) - (a^4 - a^2*b^2)*d*cos(2*d*x^n + 2*c)^2*e^(2*n*log(x) + 2*n)
- 4*(a^2*b^2 - b^4)*d*cos(d*x^n + c)^2*e^(2*n*log(x) + 2*n) - (a^4 - a^2*b^2)*d*e^(2*n*log(x) + 2*n)*sin(2*d*x
^n + 2*c)^2 - 4*(a^2*b^2 - b^4)*d*e^(2*n*log(x) + 2*n)*sin(d*x^n + c)^2 - 4*(a^3*b - a*b^3)*d*e^(2*n*log(x) +
2*n)*sin(d*x^n + c) - (a^4 - a^2*b^2)*d*e^(2*n*log(x) + 2*n) + 2*(2*a*b^3*cos(d*x^n + c)*e^(n*log(x) + 2*n) +
2*(a^3*b - a*b^3)*d*e^(2*n*log(x) + 2*n)*sin(d*x^n + c) + (a^4 - a^2*b^2)*d*e^(2*n*log(x) + 2*n))*cos(2*d*x^n
+ 2*c) - 2*((a^6*e - a^4*b^2*e)*d*n*cos(2*d*x^n + 2*c)^2 + 4*(a^4*b^2*e - a^2*b^4*e)*d*n*cos(d*x^n + c)^2 + 4*
(a^5*b*e - a^3*b^3*e)*d*n*cos(d*x^n + c)*sin(2*d*x^n + 2*c) + (a^6*e - a^4*b^2*e)*d*n*sin(2*d*x^n + 2*c)^2 + 4
*(a^4*b^2*e - a^2*b^4*e)*d*n*sin(d*x^n + c)^2 + 4*(a^5*b*e - a^3*b^3*e)*d*n*sin(d*x^n + c) + (a^6*e - a^4*b^2*
e)*d*n - 2*(2*(a^5*b*e - a^3*b^3*e)*d*n*sin(d*x^n + c) + (a^6*e - a^4*b^2*e)*d*n)*cos(2*d*x^n + 2*c))*integrat
e(-2*(a^2*b^4*cos(2*c)*e^(n*log(x) + 2*n)*sin(2*d*x^n) + a^2*b^4*cos(2*d*x^n)*e^(n*log(x) + 2*n)*sin(2*c) - 2*
(a^3*b^3 - a*b^5)*cos(d*x^n)*cos(c)*e^(n*log(x) + 2*n) + 2*(a^3*b^3 - a*b^5)*e^(n*log(x) + 2*n)*sin(d*x^n)*sin
(c) - (a^3*b^3*cos(d*x^n + c)*e^(n*log(x) + 2*n) + (2*a^5*b - a^3*b^3)*d*e^(2*n*log(x) + 2*n)*sin(d*x^n + c))*
cos(2*d*x^n + 2*c) + ((a*b^5*cos(2*c)*e^(n*log(x) + 2*n) - (2*a^3*b^3 - a*b^5)*d*e^(2*n*log(x) + 2*n)*sin(2*c)
)*cos(2*d*x^n) + 2*((2*a^4*b^2 - 3*a^2*b^4 + b^6)*d*cos(c)*e^(2*n*log(x) + 2*n) + (a^2*b^4 - b^6)*e^(n*log(x)
+ 2*n)*sin(c))*cos(d*x^n) + (a^3*b^3 - a*b^5)*e^(n*log(x) + 2*n) - (a*b^5*e^(n*log(x) + 2*n)*sin(2*c) + (2*a^3
*b^3 - a*b^5)*d*cos(2*c)*e^(2*n*log(x) + 2*n))*sin(2*d*x^n) - 2*((2*a^4*b^2 - 3*a^2*b^4 + b^6)*d*e^(2*n*log(x)
+ 2*n)*sin(c) - (a^2*b^4 - b^6)*cos(c)*e^(n*log(x) + 2*n))*sin(d*x^n))*cos(d*x^n + c) - (a^3*b^3*e^(n*log(x)
+ 2*n)*sin(d*x^n + c) + a^4*b^2*e^(n*log(x) + 2*n) - (2*a^5*b - a^3*b^3)*d*cos(d*x^n + c)*e^(2*n*log(x) + 2*n)
)*sin(2*d*x^n + 2*c) + ((2*a^5*b - 3*a^3*b^3 + a*b^5)*d*e^(2*n*log(x) + 2*n) + (a*b^5*e^(n*log(x) + 2*n)*sin(2
*c) + (2*a^3*b^3 - a*b^5)*d*cos(2*c)*e^(2*n*log(x) + 2*n))*cos(2*d*x^n) + 2*((2*a^4*b^2 - 3*a^2*b^4 + b^6)*d*e
^(2*n*log(x) + 2*n)*sin(c) - (a^2*b^4 - b^6)*cos(c)*e^(n*log(x) + 2*n))*cos(d*x^n) + (a*b^5*cos(2*c)*e^(n*log(
x) + 2*n) - (2*a^3*b^3 - a*b^5)*d*e^(2*n*log(x) + 2*n)*sin(2*c))*sin(2*d*x^n) + 2*((2*a^4*b^2 - 3*a^2*b^4 + b^
6)*d*cos(c)*e^(2*n*log(x) + 2*n) + (a^2*b^4 - b^6)*e^(n*log(x) + 2*n)*sin(c))*sin(d*x^n))*sin(d*x^n + c))/(a^8
*d*x*cos(2*d*x^n + 2*c)^2*e + a^8*d*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)*d
*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)*d*x*cos(d*x^n)^2 + (a^4*b^4*cos(2*c)^2*e + a^4*b^4*e*sin(2*c)^2)*d*x*sin(2*d*x^n)^2 + 4*(a^7*b*e - 2
*a^5*b^3*e + a^3*b^5*e)*d*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)*d*x*sin(d*x^n)^2 + 4*(a^7*b*e - 2*a^5*b^3*e + a^3*b^5*e)*d*x*cos(d*x^n)*
sin(c) + (a^8*e - 2*a^6*b^2*e + a^4*b^4*e)*d*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))*d*x*cos(d*x^n) - (a^6*b^2*e - a^4*b^4*e)*d*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))*d*x*sin(d*x^n))*cos(2*d*x^n) - 2*(a^6*b^2*d*x*cos(
2*d*x^n)*cos(2*c)*e - a^6*b^2*d*x*e*sin(2*d*x^n)*sin(2*c) + 2*(a^7*b*e - a^5*b^3*e)*d*x*cos(c)*sin(d*x^n) + 2*
(a^7*b*e - a^5*b^3*e)*d*x*cos(d*x^n)*sin(c) + (a^8*e - a^6*b^2*e)*d*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))*d*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))*d*x*sin(d*x^n) + (a^6*b^2*e - a^4*b^4*e)*d*x*s
in(2*c))*sin(2*d*x^n) - 2*(a^6*b^2*d*x*cos(2*c)*e*sin(2*d*x^n) + a^6*b^2*d*x*cos(2*d*x^n)*e*sin(2*c) - 2*(a^7*
b*e - a^5*b^3*e)*d*x*cos(d*x^n)*cos(c) + 2*(a^7*b*e - a^5*b^3*e)*d*x*sin(d*x^n)*sin(c))*sin(2*d*x^n + 2*c)), x
) + 4*(a*b^3*e^(n*log(x) + 2*n)*sin(d*x^n + c) + a^2*b^2*e^(n*log(x) + 2*n) - (a^3*b - a*b^3)*d*cos(d*x^n + c)
*e^(2*n*log(x) + 2*n))*sin(2*d*x^n + 2*c))/((a^6*e - a^4*b^2*e)*d*n*cos(2*d*x^n + 2*c)^2 + 4*(a^4*b^2*e - a^2*
b^4*e)*d*n*cos(d*x^n + c)^2 + 4*(a^5*b*e - a^3*b^3*e)*d*n*cos(d*x^n + c)*sin(2*d*x^n + 2*c) + (a^6*e - a^4*b^2
*e)*d*n*sin(2*d*x^n + 2*c)^2 + 4*(a^4*b^2*e - a^2*b^4*e)*d*n*sin(d*x^n + c)^2 + 4*(a^5*b*e - a^3*b^3*e)*d*n*si
n(d*x^n + c) + (a^6*e - a^4*b^2*e)*d*n - 2*(2*(a^5*b*e - a^3*b^3*e)*d*n*sin(d*x^n + c) + (a^6*e - a^4*b^2*e)*d
*n)*cos(2*d*x^n + 2*c))
________________________________________________________________________________________
Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 2458 vs. \(2 (710) = 1420\).
time = 5.34, size = 2458, normalized size = 3.16 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^(-1+2*n)/(a+b*csc(c+d*x^n))^2,x, algorithm="fricas")
[Out]
1/2*((a^5 - 2*a^3*b^2 + a*b^4)*d^2*x^(2*n)*e^(2*n - 1)*sin(d*x^n + c) + (a^4*b - 2*a^2*b^3 + b^5)*d^2*x^(2*n)*
e^(2*n - 1) - 2*(a^3*b^2 - a*b^4)*d*x^n*cos(d*x^n + c)*e^(2*n - 1) + ((2*I*a^4*b - I*a^2*b^3)*sqrt((a^2 - b^2)
/a^2)*e^(2*n - 1)*sin(d*x^n + c) + (2*I*a^3*b^2 - I*a*b^4)*sqrt((a^2 - b^2)/a^2)*e^(2*n - 1))*dilog(((a*sqrt((
a^2 - b^2)/a^2) + I*b)*cos(d*x^n + c) + (I*a*sqrt((a^2 - b^2)/a^2) - b)*sin(d*x^n + c) - a)/a + 1) + ((2*I*a^4
*b - I*a^2*b^3)*sqrt((a^2 - b^2)/a^2)*e^(2*n - 1)*sin(d*x^n + c) + (2*I*a^3*b^2 - I*a*b^4)*sqrt((a^2 - b^2)/a^
2)*e^(2*n - 1))*dilog(-((a*sqrt((a^2 - b^2)/a^2) + I*b)*cos(d*x^n + c) - (I*a*sqrt((a^2 - b^2)/a^2) - b)*sin(d
*x^n + c) + a)/a + 1) + ((-2*I*a^4*b + I*a^2*b^3)*sqrt((a^2 - b^2)/a^2)*e^(2*n - 1)*sin(d*x^n + c) + (-2*I*a^3
*b^2 + I*a*b^4)*sqrt((a^2 - b^2)/a^2)*e^(2*n - 1))*dilog(((a*sqrt((a^2 - b^2)/a^2) - I*b)*cos(d*x^n + c) + (-I
*a*sqrt((a^2 - b^2)/a^2) - b)*sin(d*x^n + c) - a)/a + 1) + ((-2*I*a^4*b + I*a^2*b^3)*sqrt((a^2 - b^2)/a^2)*e^(
2*n - 1)*sin(d*x^n + c) + (-2*I*a^3*b^2 + I*a*b^4)*sqrt((a^2 - b^2)/a^2)*e^(2*n - 1))*dilog(-((a*sqrt((a^2 - b
^2)/a^2) - I*b)*cos(d*x^n + c) - (-I*a*sqrt((a^2 - b^2)/a^2) - b)*sin(d*x^n + c) + a)/a + 1) - ((2*a^3*b^2 - a
*b^4)*c*sqrt((a^2 - b^2)/a^2)*e^(2*n - 1) - (a^2*b^3 - b^5)*e^(2*n - 1) + ((2*a^4*b - a^2*b^3)*c*sqrt((a^2 - b
^2)/a^2)*e^(2*n - 1) - (a^3*b^2 - a*b^4)*e^(2*n - 1))*sin(d*x^n + c))*log(2*a*cos(d*x^n + c) + 2*I*a*sin(d*x^n
+ c) + 2*a*sqrt((a^2 - b^2)/a^2) + 2*I*b) - ((2*a^3*b^2 - a*b^4)*c*sqrt((a^2 - b^2)/a^2)*e^(2*n - 1) - (a^2*b
^3 - b^5)*e^(2*n - 1) + ((2*a^4*b - a^2*b^3)*c*sqrt((a^2 - b^2)/a^2)*e^(2*n - 1) - (a^3*b^2 - a*b^4)*e^(2*n -
1))*sin(d*x^n + c))*log(2*a*cos(d*x^n + c) - 2*I*a*sin(d*x^n + c) + 2*a*sqrt((a^2 - b^2)/a^2) - 2*I*b) + ((2*a
^3*b^2 - a*b^4)*c*sqrt((a^2 - b^2)/a^2)*e^(2*n - 1) + (a^2*b^3 - b^5)*e^(2*n - 1) + ((2*a^4*b - a^2*b^3)*c*sqr
t((a^2 - b^2)/a^2)*e^(2*n - 1) + (a^3*b^2 - a*b^4)*e^(2*n - 1))*sin(d*x^n + c))*log(-2*a*cos(d*x^n + c) + 2*I*
a*sin(d*x^n + c) + 2*a*sqrt((a^2 - b^2)/a^2) + 2*I*b) + ((2*a^3*b^2 - a*b^4)*c*sqrt((a^2 - b^2)/a^2)*e^(2*n -
1) + (a^2*b^3 - b^5)*e^(2*n - 1) + ((2*a^4*b - a^2*b^3)*c*sqrt((a^2 - b^2)/a^2)*e^(2*n - 1) + (a^3*b^2 - a*b^4
)*e^(2*n - 1))*sin(d*x^n + c))*log(-2*a*cos(d*x^n + c) - 2*I*a*sin(d*x^n + c) + 2*a*sqrt((a^2 - b^2)/a^2) - 2*
I*b) - ((2*a^3*b^2 - a*b^4)*d*x^n*sqrt((a^2 - b^2)/a^2)*e^(2*n - 1) + (2*a^3*b^2 - a*b^4)*c*sqrt((a^2 - b^2)/a
^2)*e^(2*n - 1) + ((2*a^4*b - a^2*b^3)*d*x^n*sqrt((a^2 - b^2)/a^2)*e^(2*n - 1) + (2*a^4*b - a^2*b^3)*c*sqrt((a
^2 - b^2)/a^2)*e^(2*n - 1))*sin(d*x^n + c))*log(-((a*sqrt((a^2 - b^2)/a^2) + I*b)*cos(d*x^n + c) + (I*a*sqrt((
a^2 - b^2)/a^2) - b)*sin(d*x^n + c) - a)/a) + ((2*a^3*b^2 - a*b^4)*d*x^n*sqrt((a^2 - b^2)/a^2)*e^(2*n - 1) + (
2*a^3*b^2 - a*b^4)*c*sqrt((a^2 - b^2)/a^2)*e^(2*n - 1) + ((2*a^4*b - a^2*b^3)*d*x^n*sqrt((a^2 - b^2)/a^2)*e^(2
*n - 1) + (2*a^4*b - a^2*b^3)*c*sqrt((a^2 - b^2)/a^2)*e^(2*n - 1))*sin(d*x^n + c))*log(((a*sqrt((a^2 - b^2)/a^
2) + I*b)*cos(d*x^n + c) - (I*a*sqrt((a^2 - b^2)/a^2) - b)*sin(d*x^n + c) + a)/a) - ((2*a^3*b^2 - a*b^4)*d*x^n
*sqrt((a^2 - b^2)/a^2)*e^(2*n - 1) + (2*a^3*b^2 - a*b^4)*c*sqrt((a^2 - b^2)/a^2)*e^(2*n - 1) + ((2*a^4*b - a^2
*b^3)*d*x^n*sqrt((a^2 - b^2)/a^2)*e^(2*n - 1) + (2*a^4*b - a^2*b^3)*c*sqrt((a^2 - b^2)/a^2)*e^(2*n - 1))*sin(d
*x^n + c))*log(-((a*sqrt((a^2 - b^2)/a^2) - I*b)*cos(d*x^n + c) + (-I*a*sqrt((a^2 - b^2)/a^2) - b)*sin(d*x^n +
c) - a)/a) + ((2*a^3*b^2 - a*b^4)*d*x^n*sqrt((a^2 - b^2)/a^2)*e^(2*n - 1) + (2*a^3*b^2 - a*b^4)*c*sqrt((a^2 -
b^2)/a^2)*e^(2*n - 1) + ((2*a^4*b - a^2*b^3)*d*x^n*sqrt((a^2 - b^2)/a^2)*e^(2*n - 1) + (2*a^4*b - a^2*b^3)*c*
sqrt((a^2 - b^2)/a^2)*e^(2*n - 1))*sin(d*x^n + c))*log(((a*sqrt((a^2 - b^2)/a^2) - I*b)*cos(d*x^n + c) - (-I*a
*sqrt((a^2 - b^2)/a^2) - b)*sin(d*x^n + c) + a)/a))/((a^7 - 2*a^5*b^2 + a^3*b^4)*d^2*n*sin(d*x^n + c) + (a^6*b
- 2*a^4*b^3 + a^2*b^5)*d^2*n)
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (e x\right )^{2 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+2*n)/(a+b*csc(c+d*x**n))**2,x)
[Out]
Integral((e*x)**(2*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+2*n)/(a+b*csc(c+d*x^n))^2,x, algorithm="giac")
[Out]
integrate((e*x)^(2*n - 1)/(b*csc(d*x^n + c) + a)^2, x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (e\,x\right )}^{2\,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)^(2*n - 1)/(a + b/sin(c + d*x^n))^2,x)
[Out]
int((e*x)^(2*n - 1)/(a + b/sin(c + d*x^n))^2, x)
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