\(\int \frac {1}{(a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac {5}{2}}(d+e x)} \, dx\) [471]
Optimal result
Integrand size = 33, antiderivative size = 492 \[
\int \frac {1}{(a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac {5}{2}}(d+e x)} \, dx=\frac {8 b E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(c,a)\right )|\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right ) (b+c \cos (d+e x)+a \sin (d+e x))^3}{3 \left (a^2-b^2+c^2\right )^2 e (a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac {5}{2}}(d+e x) \sqrt {\frac {b+c \cos (d+e x)+a \sin (d+e x)}{b+\sqrt {a^2+c^2}}}}+\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(c,a)\right ),\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right ) (b+c \cos (d+e x)+a \sin (d+e x))^2 \sqrt {\frac {b+c \cos (d+e x)+a \sin (d+e x)}{b+\sqrt {a^2+c^2}}}}{3 \left (a^2-b^2+c^2\right ) e (a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac {5}{2}}(d+e x)}-\frac {2 (b+c \cos (d+e x)+a \sin (d+e x)) (a \cos (d+e x)-c \sin (d+e x))}{3 \left (a^2-b^2+c^2\right ) e (a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac {5}{2}}(d+e x)}+\frac {8 (b+c \cos (d+e x)+a \sin (d+e x))^2 (a b \cos (d+e x)-b c \sin (d+e x))}{3 \left (a^2-b^2+c^2\right )^2 e (a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac {5}{2}}(d+e x)}
\]
[Out]
-2/3*(b+c*cos(e*x+d)+a*sin(e*x+d))*(a*cos(e*x+d)-c*sin(e*x+d))/(a^2-b^2+c^2)/e/(a+c*cot(e*x+d)+b*csc(e*x+d))^(
5/2)/sin(e*x+d)^(5/2)+8/3*(b+c*cos(e*x+d)+a*sin(e*x+d))^2*(a*b*cos(e*x+d)-b*c*sin(e*x+d))/(a^2-b^2+c^2)^2/e/(a
+c*cot(e*x+d)+b*csc(e*x+d))^(5/2)/sin(e*x+d)^(5/2)+8/3*b*(cos(1/2*d+1/2*e*x-1/2*arctan(c,a))^2)^(1/2)/cos(1/2*
d+1/2*e*x-1/2*arctan(c,a))*EllipticE(sin(1/2*d+1/2*e*x-1/2*arctan(c,a)),2^(1/2)*((a^2+c^2)^(1/2)/(b+(a^2+c^2)^
(1/2)))^(1/2))*(b+c*cos(e*x+d)+a*sin(e*x+d))^3/(a^2-b^2+c^2)^2/e/(a+c*cot(e*x+d)+b*csc(e*x+d))^(5/2)/sin(e*x+d
)^(5/2)/((b+c*cos(e*x+d)+a*sin(e*x+d))/(b+(a^2+c^2)^(1/2)))^(1/2)+2/3*(cos(1/2*d+1/2*e*x-1/2*arctan(c,a))^2)^(
1/2)/cos(1/2*d+1/2*e*x-1/2*arctan(c,a))*EllipticF(sin(1/2*d+1/2*e*x-1/2*arctan(c,a)),2^(1/2)*((a^2+c^2)^(1/2)/
(b+(a^2+c^2)^(1/2)))^(1/2))*(b+c*cos(e*x+d)+a*sin(e*x+d))^2*((b+c*cos(e*x+d)+a*sin(e*x+d))/(b+(a^2+c^2)^(1/2))
)^(1/2)/(a^2-b^2+c^2)/e/(a+c*cot(e*x+d)+b*csc(e*x+d))^(5/2)/sin(e*x+d)^(5/2)
Rubi [A] (verified)
Time = 0.61 (sec) , antiderivative size = 492, normalized size of antiderivative = 1.00, number of
steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {3243, 3208, 3235, 3228, 3198,
2732, 3206, 2740} \[
\int \frac {1}{(a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac {5}{2}}(d+e x)} \, dx=\frac {2 \sqrt {\frac {a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt {a^2+c^2}+b}} (a \sin (d+e x)+b+c \cos (d+e x))^2 \operatorname {EllipticF}\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(c,a)\right ),\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right )}{3 e \left (a^2-b^2+c^2\right ) \sin ^{\frac {5}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}+\frac {8 b (a \sin (d+e x)+b+c \cos (d+e x))^3 E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(c,a)\right )|\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right )}{3 e \left (a^2-b^2+c^2\right )^2 \sin ^{\frac {5}{2}}(d+e x) \sqrt {\frac {a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt {a^2+c^2}+b}} (a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}+\frac {8 (a b \cos (d+e x)-b c \sin (d+e x)) (a \sin (d+e x)+b+c \cos (d+e x))^2}{3 e \left (a^2-b^2+c^2\right )^2 \sin ^{\frac {5}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}-\frac {2 (a \cos (d+e x)-c \sin (d+e x)) (a \sin (d+e x)+b+c \cos (d+e x))}{3 e \left (a^2-b^2+c^2\right ) \sin ^{\frac {5}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}
\]
[In]
Int[1/((a + c*Cot[d + e*x] + b*Csc[d + e*x])^(5/2)*Sin[d + e*x]^(5/2)),x]
[Out]
(8*b*EllipticE[(d + e*x - ArcTan[c, a])/2, (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*(b + c*Cos[d + e*x] + a*
Sin[d + e*x])^3)/(3*(a^2 - b^2 + c^2)^2*e*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(5/2)*Sin[d + e*x]^(5/2)*Sqrt[
(b + c*Cos[d + e*x] + a*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])]) + (2*EllipticF[(d + e*x - ArcTan[c, a])/2, (2*Sq
rt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*(b + c*Cos[d + e*x] + a*Sin[d + e*x])^2*Sqrt[(b + c*Cos[d + e*x] + a*Sin
[d + e*x])/(b + Sqrt[a^2 + c^2])])/(3*(a^2 - b^2 + c^2)*e*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(5/2)*Sin[d +
e*x]^(5/2)) - (2*(b + c*Cos[d + e*x] + a*Sin[d + e*x])*(a*Cos[d + e*x] - c*Sin[d + e*x]))/(3*(a^2 - b^2 + c^2)
*e*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(5/2)*Sin[d + e*x]^(5/2)) + (8*(b + c*Cos[d + e*x] + a*Sin[d + e*x])^
2*(a*b*Cos[d + e*x] - b*c*Sin[d + e*x]))/(3*(a^2 - b^2 + c^2)^2*e*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(5/2)*
Sin[d + e*x]^(5/2))
Rule 2732
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2
+ d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Rule 2740
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*Sqrt[a + b]))*EllipticF[(1/2)*(c - P
i/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Rule 3198
Int[Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*C
os[d + e*x] + c*Sin[d + e*x]]/Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])], Int[Sqrt[a/(a
+ Sqrt[b^2 + c^2]) + (Sqrt[b^2 + c^2]/(a + Sqrt[b^2 + c^2]))*Cos[d + e*x - ArcTan[b, c]]], x], x] /; FreeQ[{a
, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[b^2 + c^2, 0] && !GtQ[a + Sqrt[b^2 + c^2], 0]
Rule 3206
Int[1/Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a +
b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])]/Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]], Int[1/Sqrt[
a/(a + Sqrt[b^2 + c^2]) + (Sqrt[b^2 + c^2]/(a + Sqrt[b^2 + c^2]))*Cos[d + e*x - ArcTan[b, c]]], x], x] /; Free
Q[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[b^2 + c^2, 0] && !GtQ[a + Sqrt[b^2 + c^2], 0]
Rule 3208
Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^(n_), x_Symbol] :> Simp[((-c)*Cos[d
+ e*x] + b*Sin[d + e*x])*((a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1)/(e*(n + 1)*(a^2 - b^2 - c^2))), x] +
Dist[1/((n + 1)*(a^2 - b^2 - c^2)), Int[(a*(n + 1) - b*(n + 2)*Cos[d + e*x] - c*(n + 2)*Sin[d + e*x])*(a + b*C
os[d + e*x] + c*Sin[d + e*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && LtQ[n
, -1] && NeQ[n, -3/2]
Rule 3228
Int[((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_)])/Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.)
+ (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]], x_Symbol] :> Dist[B/b, Int[Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]
, x], x] + Dist[(A*b - a*B)/b, Int[1/Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]], x], x] /; FreeQ[{a, b, c, d, e
, A, B, C}, x] && EqQ[B*c - b*C, 0] && NeQ[A*b - a*B, 0]
Rule 3235
Int[((a_.) + cos[(d_.) + (e_.)*(x_)]*(b_.) + (c_.)*sin[(d_.) + (e_.)*(x_)])^(n_)*((A_.) + cos[(d_.) + (e_.)*(x
_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_)]), x_Symbol] :> Simp[(-(c*B - b*C - (a*C - c*A)*Cos[d + e*x] + (a*B -
b*A)*Sin[d + e*x]))*((a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1)/(e*(n + 1)*(a^2 - b^2 - c^2))), x] + Dist[
1/((n + 1)*(a^2 - b^2 - c^2)), Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1)*Simp[(n + 1)*(a*A - b*B - c*C
) + (n + 2)*(a*B - b*A)*Cos[d + e*x] + (n + 2)*(a*C - c*A)*Sin[d + e*x], x], x], x] /; FreeQ[{a, b, c, d, e, A
, B, C}, x] && LtQ[n, -1] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[n, -2]
Rule 3243
Int[((a_.) + csc[(d_.) + (e_.)*(x_)]*(b_.) + cot[(d_.) + (e_.)*(x_)]*(c_.))^(n_)*sin[(d_.) + (e_.)*(x_)]^(n_),
x_Symbol] :> Dist[Sin[d + e*x]^n*((a + b*Csc[d + e*x] + c*Cot[d + e*x])^n/(b + a*Sin[d + e*x] + c*Cos[d + e*x
])^n), Int[(b + a*Sin[d + e*x] + c*Cos[d + e*x])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && !IntegerQ[n]
Rubi steps \begin{align*}
\text {integral}& = \frac {(b+c \cos (d+e x)+a \sin (d+e x))^{5/2} \int \frac {1}{(b+c \cos (d+e x)+a \sin (d+e x))^{5/2}} \, dx}{(a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac {5}{2}}(d+e x)} \\ & = -\frac {2 (b+c \cos (d+e x)+a \sin (d+e x)) (a \cos (d+e x)-c \sin (d+e x))}{3 \left (a^2-b^2+c^2\right ) e (a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac {5}{2}}(d+e x)}+\frac {\left (2 (b+c \cos (d+e x)+a \sin (d+e x))^{5/2}\right ) \int \frac {-\frac {3 b}{2}+\frac {1}{2} c \cos (d+e x)+\frac {1}{2} a \sin (d+e x)}{(b+c \cos (d+e x)+a \sin (d+e x))^{3/2}} \, dx}{3 \left (a^2-b^2+c^2\right ) (a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac {5}{2}}(d+e x)} \\ & = -\frac {2 (b+c \cos (d+e x)+a \sin (d+e x)) (a \cos (d+e x)-c \sin (d+e x))}{3 \left (a^2-b^2+c^2\right ) e (a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac {5}{2}}(d+e x)}+\frac {8 (b+c \cos (d+e x)+a \sin (d+e x))^2 (a b \cos (d+e x)-b c \sin (d+e x))}{3 \left (a^2-b^2+c^2\right )^2 e (a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac {5}{2}}(d+e x)}+\frac {\left (4 (b+c \cos (d+e x)+a \sin (d+e x))^{5/2}\right ) \int \frac {\frac {1}{4} \left (a^2+3 b^2+c^2\right )+b c \cos (d+e x)+a b \sin (d+e x)}{\sqrt {b+c \cos (d+e x)+a \sin (d+e x)}} \, dx}{3 \left (a^2-b^2+c^2\right )^2 (a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac {5}{2}}(d+e x)} \\ & = -\frac {2 (b+c \cos (d+e x)+a \sin (d+e x)) (a \cos (d+e x)-c \sin (d+e x))}{3 \left (a^2-b^2+c^2\right ) e (a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac {5}{2}}(d+e x)}+\frac {8 (b+c \cos (d+e x)+a \sin (d+e x))^2 (a b \cos (d+e x)-b c \sin (d+e x))}{3 \left (a^2-b^2+c^2\right )^2 e (a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac {5}{2}}(d+e x)}+\frac {\left (4 b (b+c \cos (d+e x)+a \sin (d+e x))^{5/2}\right ) \int \sqrt {b+c \cos (d+e x)+a \sin (d+e x)} \, dx}{3 \left (a^2-b^2+c^2\right )^2 (a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac {5}{2}}(d+e x)}+\frac {(b+c \cos (d+e x)+a \sin (d+e x))^{5/2} \int \frac {1}{\sqrt {b+c \cos (d+e x)+a \sin (d+e x)}} \, dx}{3 \left (a^2-b^2+c^2\right ) (a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac {5}{2}}(d+e x)} \\ & = -\frac {2 (b+c \cos (d+e x)+a \sin (d+e x)) (a \cos (d+e x)-c \sin (d+e x))}{3 \left (a^2-b^2+c^2\right ) e (a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac {5}{2}}(d+e x)}+\frac {8 (b+c \cos (d+e x)+a \sin (d+e x))^2 (a b \cos (d+e x)-b c \sin (d+e x))}{3 \left (a^2-b^2+c^2\right )^2 e (a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac {5}{2}}(d+e x)}+\frac {\left (4 b (b+c \cos (d+e x)+a \sin (d+e x))^3\right ) \int \sqrt {\frac {b}{b+\sqrt {a^2+c^2}}+\frac {\sqrt {a^2+c^2} \cos \left (d+e x-\tan ^{-1}(c,a)\right )}{b+\sqrt {a^2+c^2}}} \, dx}{3 \left (a^2-b^2+c^2\right )^2 (a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac {5}{2}}(d+e x) \sqrt {\frac {b+c \cos (d+e x)+a \sin (d+e x)}{b+\sqrt {a^2+c^2}}}}+\frac {\left ((b+c \cos (d+e x)+a \sin (d+e x))^2 \sqrt {\frac {b+c \cos (d+e x)+a \sin (d+e x)}{b+\sqrt {a^2+c^2}}}\right ) \int \frac {1}{\sqrt {\frac {b}{b+\sqrt {a^2+c^2}}+\frac {\sqrt {a^2+c^2} \cos \left (d+e x-\tan ^{-1}(c,a)\right )}{b+\sqrt {a^2+c^2}}}} \, dx}{3 \left (a^2-b^2+c^2\right ) (a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac {5}{2}}(d+e x)} \\ & = \frac {8 b E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(c,a)\right )|\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right ) (b+c \cos (d+e x)+a \sin (d+e x))^3}{3 \left (a^2-b^2+c^2\right )^2 e (a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac {5}{2}}(d+e x) \sqrt {\frac {b+c \cos (d+e x)+a \sin (d+e x)}{b+\sqrt {a^2+c^2}}}}+\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(c,a)\right ),\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right ) (b+c \cos (d+e x)+a \sin (d+e x))^2 \sqrt {\frac {b+c \cos (d+e x)+a \sin (d+e x)}{b+\sqrt {a^2+c^2}}}}{3 \left (a^2-b^2+c^2\right ) e (a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac {5}{2}}(d+e x)}-\frac {2 (b+c \cos (d+e x)+a \sin (d+e x)) (a \cos (d+e x)-c \sin (d+e x))}{3 \left (a^2-b^2+c^2\right ) e (a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac {5}{2}}(d+e x)}+\frac {8 (b+c \cos (d+e x)+a \sin (d+e x))^2 (a b \cos (d+e x)-b c \sin (d+e x))}{3 \left (a^2-b^2+c^2\right )^2 e (a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac {5}{2}}(d+e x)} \\
\end{align*}
Mathematica [C] (warning: unable to verify)
Result contains complex when optimal does not.
Time = 18.63 (sec) , antiderivative size = 6066, normalized size of antiderivative = 12.33
\[
\int \frac {1}{(a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac {5}{2}}(d+e x)} \, dx=\text {Result too large to show}
\]
[In]
Integrate[1/((a + c*Cot[d + e*x] + b*Csc[d + e*x])^(5/2)*Sin[d + e*x]^(5/2)),x]
[Out]
Result too large to show
Maple [C] (warning: unable to verify)
Result contains complex when optimal does not.
Time = 28.20 (sec) , antiderivative size = 60004, normalized size of antiderivative =
121.96
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method | result | size |
| | |
default |
\(\text {Expression too large to display}\) |
\(60004\) |
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|
|
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[In]
int(1/(a+c*cot(e*x+d)+b*csc(e*x+d))^(5/2)/sin(e*x+d)^(5/2),x,method=_RETURNVERBOSE)
[Out]
result too large to display
Fricas [C] (verification not implemented)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.25 (sec) , antiderivative size = 2798, normalized size of antiderivative = 5.69
\[
\int \frac {1}{(a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac {5}{2}}(d+e x)} \, dx=\text {Too large to display}
\]
[In]
integrate(1/(a+c*cot(e*x+d)+b*csc(e*x+d))^(5/2)/sin(e*x+d)^(5/2),x, algorithm="fricas")
[Out]
1/9*((sqrt(2)*(3*a^5 + a^3*b^2 - a*b^2*c^2 - I*b^2*c^3 - 3*a*c^4 - 3*I*c^5 + I*(3*a^4 + a^2*b^2)*c)*cos(e*x +
d)^2 - 2*sqrt(2)*(3*a*b*c^3 + 3*I*b*c^4 + I*(3*a^2*b + b^3)*c^2 + (3*a^3*b + a*b^3)*c)*cos(e*x + d) - 2*(sqrt(
2)*(3*a^2*c^3 + 3*I*a*c^4 + I*(3*a^3 + a*b^2)*c^2 + (3*a^4 + a^2*b^2)*c)*cos(e*x + d) + sqrt(2)*(3*a^4*b + a^2
*b^3 + 3*a^2*b*c^2 + 3*I*a*b*c^3 + I*(3*a^3*b + a*b^3)*c))*sin(e*x + d) - sqrt(2)*(3*a^5 + 4*a^3*b^2 + a*b^4 +
3*I*(a^2 + b^2)*c^3 + 3*(a^3 + a*b^2)*c^2 + I*(3*a^4 + 4*a^2*b^2 + b^4)*c))*sqrt(I*a + c)*weierstrassPInverse
(4/3*(3*a^4 - 4*a^2*b^2 + 4*b^2*c^2 + 6*I*a*c^3 - 3*c^4 + 2*I*(3*a^3 - 4*a*b^2)*c)/(a^4 + 2*a^2*c^2 + c^4), -8
/27*(-9*I*a^5*b + 8*I*a^3*b^3 + 27*I*a*b*c^4 - 9*b*c^5 + 2*(9*a^2*b + 4*b^3)*c^3 + 6*I*(3*a^3*b - 4*a*b^3)*c^2
+ 3*(9*a^4*b - 8*a^2*b^3)*c)/(a^6 + 3*a^4*c^2 + 3*a^2*c^4 + c^6), 1/3*(-2*I*a*b + 2*b*c + 3*(a^2 + c^2)*cos(e
*x + d) - 3*(I*a^2 + I*c^2)*sin(e*x + d))/(a^2 + c^2)) + (sqrt(2)*(3*a^5 + a^3*b^2 - a*b^2*c^2 + I*b^2*c^3 - 3
*a*c^4 + 3*I*c^5 - I*(3*a^4 + a^2*b^2)*c)*cos(e*x + d)^2 - 2*sqrt(2)*(3*a*b*c^3 - 3*I*b*c^4 - I*(3*a^2*b + b^3
)*c^2 + (3*a^3*b + a*b^3)*c)*cos(e*x + d) - 2*(sqrt(2)*(3*a^2*c^3 - 3*I*a*c^4 - I*(3*a^3 + a*b^2)*c^2 + (3*a^4
+ a^2*b^2)*c)*cos(e*x + d) + sqrt(2)*(3*a^4*b + a^2*b^3 + 3*a^2*b*c^2 - 3*I*a*b*c^3 - I*(3*a^3*b + a*b^3)*c))
*sin(e*x + d) - sqrt(2)*(3*a^5 + 4*a^3*b^2 + a*b^4 - 3*I*(a^2 + b^2)*c^3 + 3*(a^3 + a*b^2)*c^2 - I*(3*a^4 + 4*
a^2*b^2 + b^4)*c))*sqrt(-I*a + c)*weierstrassPInverse(4/3*(3*a^4 - 4*a^2*b^2 + 4*b^2*c^2 - 6*I*a*c^3 - 3*c^4 -
2*I*(3*a^3 - 4*a*b^2)*c)/(a^4 + 2*a^2*c^2 + c^4), -8/27*(9*I*a^5*b - 8*I*a^3*b^3 - 27*I*a*b*c^4 - 9*b*c^5 + 2
*(9*a^2*b + 4*b^3)*c^3 - 6*I*(3*a^3*b - 4*a*b^3)*c^2 + 3*(9*a^4*b - 8*a^2*b^3)*c)/(a^6 + 3*a^4*c^2 + 3*a^2*c^4
+ c^6), 1/3*(2*I*a*b + 2*b*c + 3*(a^2 + c^2)*cos(e*x + d) - 3*(-I*a^2 - I*c^2)*sin(e*x + d))/(a^2 + c^2)) + 1
2*(sqrt(2)*(-I*a^4*b + I*b*c^4)*cos(e*x + d)^2 + 2*sqrt(2)*(I*a^2*b^2*c + I*b^2*c^3)*cos(e*x + d) + 2*(sqrt(2)
*(I*a^3*b*c + I*a*b*c^3)*cos(e*x + d) + sqrt(2)*(I*a^3*b^2 + I*a*b^2*c^2))*sin(e*x + d) + sqrt(2)*(I*a^4*b + I
*a^2*b^3 + I*(a^2*b + b^3)*c^2))*sqrt(I*a + c)*weierstrassZeta(4/3*(3*a^4 - 4*a^2*b^2 + 4*b^2*c^2 + 6*I*a*c^3
- 3*c^4 + 2*I*(3*a^3 - 4*a*b^2)*c)/(a^4 + 2*a^2*c^2 + c^4), -8/27*(-9*I*a^5*b + 8*I*a^3*b^3 + 27*I*a*b*c^4 - 9
*b*c^5 + 2*(9*a^2*b + 4*b^3)*c^3 + 6*I*(3*a^3*b - 4*a*b^3)*c^2 + 3*(9*a^4*b - 8*a^2*b^3)*c)/(a^6 + 3*a^4*c^2 +
3*a^2*c^4 + c^6), weierstrassPInverse(4/3*(3*a^4 - 4*a^2*b^2 + 4*b^2*c^2 + 6*I*a*c^3 - 3*c^4 + 2*I*(3*a^3 - 4
*a*b^2)*c)/(a^4 + 2*a^2*c^2 + c^4), -8/27*(-9*I*a^5*b + 8*I*a^3*b^3 + 27*I*a*b*c^4 - 9*b*c^5 + 2*(9*a^2*b + 4*
b^3)*c^3 + 6*I*(3*a^3*b - 4*a*b^3)*c^2 + 3*(9*a^4*b - 8*a^2*b^3)*c)/(a^6 + 3*a^4*c^2 + 3*a^2*c^4 + c^6), 1/3*(
-2*I*a*b + 2*b*c + 3*(a^2 + c^2)*cos(e*x + d) - 3*(I*a^2 + I*c^2)*sin(e*x + d))/(a^2 + c^2))) + 12*(sqrt(2)*(I
*a^4*b - I*b*c^4)*cos(e*x + d)^2 + 2*sqrt(2)*(-I*a^2*b^2*c - I*b^2*c^3)*cos(e*x + d) + 2*(sqrt(2)*(-I*a^3*b*c
- I*a*b*c^3)*cos(e*x + d) + sqrt(2)*(-I*a^3*b^2 - I*a*b^2*c^2))*sin(e*x + d) + sqrt(2)*(-I*a^4*b - I*a^2*b^3 -
I*(a^2*b + b^3)*c^2))*sqrt(-I*a + c)*weierstrassZeta(4/3*(3*a^4 - 4*a^2*b^2 + 4*b^2*c^2 - 6*I*a*c^3 - 3*c^4 -
2*I*(3*a^3 - 4*a*b^2)*c)/(a^4 + 2*a^2*c^2 + c^4), -8/27*(9*I*a^5*b - 8*I*a^3*b^3 - 27*I*a*b*c^4 - 9*b*c^5 + 2
*(9*a^2*b + 4*b^3)*c^3 - 6*I*(3*a^3*b - 4*a*b^3)*c^2 + 3*(9*a^4*b - 8*a^2*b^3)*c)/(a^6 + 3*a^4*c^2 + 3*a^2*c^4
+ c^6), weierstrassPInverse(4/3*(3*a^4 - 4*a^2*b^2 + 4*b^2*c^2 - 6*I*a*c^3 - 3*c^4 - 2*I*(3*a^3 - 4*a*b^2)*c)
/(a^4 + 2*a^2*c^2 + c^4), -8/27*(9*I*a^5*b - 8*I*a^3*b^3 - 27*I*a*b*c^4 - 9*b*c^5 + 2*(9*a^2*b + 4*b^3)*c^3 -
6*I*(3*a^3*b - 4*a*b^3)*c^2 + 3*(9*a^4*b - 8*a^2*b^3)*c)/(a^6 + 3*a^4*c^2 + 3*a^2*c^4 + c^6), 1/3*(2*I*a*b + 2
*b*c + 3*(a^2 + c^2)*cos(e*x + d) - 3*(-I*a^2 - I*c^2)*sin(e*x + d))/(a^2 + c^2))) + 6*(4*a^3*b*c + 4*a*b*c^3
- 8*(a^3*b*c + a*b*c^3)*cos(e*x + d)^2 + (a^5 - 5*a^3*b^2 + a*c^4 + (2*a^3 - 5*a*b^2)*c^2)*cos(e*x + d) - (c^5
+ (2*a^2 - 5*b^2)*c^3 + (a^4 - 5*a^2*b^2)*c + 4*(a^4*b - b*c^4)*cos(e*x + d))*sin(e*x + d))*sqrt((c*cos(e*x +
d) + a*sin(e*x + d) + b)/sin(e*x + d))*sqrt(sin(e*x + d)))/((a^8 - 2*a^6*b^2 + a^4*b^4 - c^8 - 2*(a^2 - b^2)*
c^6 + (2*a^2*b^2 - b^4)*c^4 + 2*(a^6 - a^4*b^2)*c^2)*e*cos(e*x + d)^2 - 2*(b*c^7 + (3*a^2*b - 2*b^3)*c^5 + (3*
a^4*b - 4*a^2*b^3 + b^5)*c^3 + (a^6*b - 2*a^4*b^3 + a^2*b^5)*c)*e*cos(e*x + d) - (a^8 - a^6*b^2 - a^4*b^4 + a^
2*b^6 + (a^2 + b^2)*c^6 + (3*a^4 + a^2*b^2 - 2*b^4)*c^4 + (3*a^6 - a^4*b^2 - 3*a^2*b^4 + b^6)*c^2)*e - 2*((a*c
^7 + (3*a^3 - 2*a*b^2)*c^5 + (3*a^5 - 4*a^3*b^2 + a*b^4)*c^3 + (a^7 - 2*a^5*b^2 + a^3*b^4)*c)*e*cos(e*x + d) +
(a^7*b - 2*a^5*b^3 + a^3*b^5 + a*b*c^6 + (3*a^3*b - 2*a*b^3)*c^4 + (3*a^5*b - 4*a^3*b^3 + a*b^5)*c^2)*e)*sin(
e*x + d))
Sympy [F(-1)]
Timed out. \[
\int \frac {1}{(a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac {5}{2}}(d+e x)} \, dx=\text {Timed out}
\]
[In]
integrate(1/(a+c*cot(e*x+d)+b*csc(e*x+d))**(5/2)/sin(e*x+d)**(5/2),x)
[Out]
Timed out
Maxima [F]
\[
\int \frac {1}{(a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac {5}{2}}(d+e x)} \, dx=\int { \frac {1}{{\left (c \cot \left (e x + d\right ) + b \csc \left (e x + d\right ) + a\right )}^{\frac {5}{2}} \sin \left (e x + d\right )^{\frac {5}{2}}} \,d x }
\]
[In]
integrate(1/(a+c*cot(e*x+d)+b*csc(e*x+d))^(5/2)/sin(e*x+d)^(5/2),x, algorithm="maxima")
[Out]
integrate(1/((c*cot(e*x + d) + b*csc(e*x + d) + a)^(5/2)*sin(e*x + d)^(5/2)), x)
Giac [F(-1)]
Timed out. \[
\int \frac {1}{(a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac {5}{2}}(d+e x)} \, dx=\text {Timed out}
\]
[In]
integrate(1/(a+c*cot(e*x+d)+b*csc(e*x+d))^(5/2)/sin(e*x+d)^(5/2),x, algorithm="giac")
[Out]
Timed out
Mupad [F(-1)]
Timed out. \[
\int \frac {1}{(a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac {5}{2}}(d+e x)} \, dx=\int \frac {1}{{\sin \left (d+e\,x\right )}^{5/2}\,{\left (a+c\,\mathrm {cot}\left (d+e\,x\right )+\frac {b}{\sin \left (d+e\,x\right )}\right )}^{5/2}} \,d x
\]
[In]
int(1/(sin(d + e*x)^(5/2)*(a + c*cot(d + e*x) + b/sin(d + e*x))^(5/2)),x)
[Out]
int(1/(sin(d + e*x)^(5/2)*(a + c*cot(d + e*x) + b/sin(d + e*x))^(5/2)), x)