3.1.76 \(\int \frac {(e \cot (c+d x))^{5/2}}{(a+b \cot (c+d x))^2} \, dx\) [76]

Optimal. Leaf size=393 \[ -\frac {a^{3/2} \left (a^2+5 b^2\right ) e^{5/2} \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{b^{3/2} \left (a^2+b^2\right )^2 d}-\frac {\left (a^2+2 a b-b^2\right ) e^{5/2} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {\left (a^2+2 a b-b^2\right ) e^{5/2} \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {a^2 e^2 \sqrt {e \cot (c+d x)}}{b \left (a^2+b^2\right ) d (a+b \cot (c+d x))}+\frac {\left (a^2-2 a b-b^2\right ) e^{5/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\left (a^2-2 a b-b^2\right ) e^{5/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d} \]

[Out]

-a^(3/2)*(a^2+5*b^2)*e^(5/2)*arctan(b^(1/2)*(e*cot(d*x+c))^(1/2)/a^(1/2)/e^(1/2))/b^(3/2)/(a^2+b^2)^2/d-1/2*(a
^2+2*a*b-b^2)*e^(5/2)*arctan(1-2^(1/2)*(e*cot(d*x+c))^(1/2)/e^(1/2))/(a^2+b^2)^2/d*2^(1/2)+1/2*(a^2+2*a*b-b^2)
*e^(5/2)*arctan(1+2^(1/2)*(e*cot(d*x+c))^(1/2)/e^(1/2))/(a^2+b^2)^2/d*2^(1/2)+1/4*(a^2-2*a*b-b^2)*e^(5/2)*ln(e
^(1/2)+cot(d*x+c)*e^(1/2)-2^(1/2)*(e*cot(d*x+c))^(1/2))/(a^2+b^2)^2/d*2^(1/2)-1/4*(a^2-2*a*b-b^2)*e^(5/2)*ln(e
^(1/2)+cot(d*x+c)*e^(1/2)+2^(1/2)*(e*cot(d*x+c))^(1/2))/(a^2+b^2)^2/d*2^(1/2)+a^2*e^2*(e*cot(d*x+c))^(1/2)/b/(
a^2+b^2)/d/(a+b*cot(d*x+c))

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Rubi [A]
time = 0.50, antiderivative size = 393, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {3646, 3734, 3615, 1182, 1176, 631, 210, 1179, 642, 3715, 65, 211} \begin {gather*} -\frac {e^{5/2} \left (a^2+2 a b-b^2\right ) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d \left (a^2+b^2\right )^2}+\frac {e^{5/2} \left (a^2+2 a b-b^2\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^2}+\frac {e^{5/2} \left (a^2-2 a b-b^2\right ) \log \left (\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^2}-\frac {e^{5/2} \left (a^2-2 a b-b^2\right ) \log \left (\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^2}+\frac {a^2 e^2 \sqrt {e \cot (c+d x)}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}-\frac {a^{3/2} e^{5/2} \left (a^2+5 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{b^{3/2} d \left (a^2+b^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(e*Cot[c + d*x])^(5/2)/(a + b*Cot[c + d*x])^2,x]

[Out]

-((a^(3/2)*(a^2 + 5*b^2)*e^(5/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cot[c + d*x]])/(Sqrt[a]*Sqrt[e])])/(b^(3/2)*(a^2 + b^2
)^2*d)) - ((a^2 + 2*a*b - b^2)*e^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2
)^2*d) + ((a^2 + 2*a*b - b^2)*e^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)
^2*d) + (a^2*e^2*Sqrt[e*Cot[c + d*x]])/(b*(a^2 + b^2)*d*(a + b*Cot[c + d*x])) + ((a^2 - 2*a*b - b^2)*e^(5/2)*L
og[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) - ((a^2 - 2*a*b
- b^2)*e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^2*d
)

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

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 211

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

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 1182

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a*c, 2]}, Dist[(d*q + a*e)/(2*a*c),
Int[(q + c*x^2)/(a + c*x^4), x], x] + Dist[(d*q - a*e)/(2*a*c), Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ
[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(-a)*c]

Rule 3615

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[2/f, Subst[I
nt[(b*c + d*x^2)/(b^2 + x^4), x], x, Sqrt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2,
0] && NeQ[c^2 + d^2, 0]

Rule 3646

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(b*c - a*d)^2*(a + b*Tan[e + f*x])^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - D
ist[1/(d*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a^2*d*(b*d*(
m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c*(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3
*a*b^2*d)*Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*(n + 1)))*Tan[e + f*x]^2, x
], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && Gt
Q[m, 2] && LtQ[n, -1] && IntegerQ[2*m]

Rule 3715

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*
tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[A/f, Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x]
/; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]

Rule 3734

Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (
f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/(a^2 + b^2), Int[(c + d*Tan[e + f*
x])^n*Simp[b*B + a*(A - C) + (a*B - b*(A - C))*Tan[e + f*x], x], x], x] + Dist[(A*b^2 - a*b*B + a^2*C)/(a^2 +
b^2), Int[(c + d*Tan[e + f*x])^n*((1 + Tan[e + f*x]^2)/(a + b*Tan[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e,
f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] &&  !GtQ[n, 0] &&  !LeQ[n, -
1]

Rubi steps

\begin {align*} \int \frac {(e \cot (c+d x))^{5/2}}{(a+b \cot (c+d x))^2} \, dx &=\frac {a^2 e^2 \sqrt {e \cot (c+d x)}}{b \left (a^2+b^2\right ) d (a+b \cot (c+d x))}-\frac {\int \frac {-\frac {1}{2} a^2 e^3+a b e^3 \cot (c+d x)-\frac {1}{2} \left (a^2+2 b^2\right ) e^3 \cot ^2(c+d x)}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))} \, dx}{b \left (a^2+b^2\right )}\\ &=\frac {a^2 e^2 \sqrt {e \cot (c+d x)}}{b \left (a^2+b^2\right ) d (a+b \cot (c+d x))}-\frac {\int \frac {2 a b^2 e^3+b \left (a^2-b^2\right ) e^3 \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx}{b \left (a^2+b^2\right )^2}+\frac {\left (a^2 \left (a^2+5 b^2\right ) e^3\right ) \int \frac {1+\cot ^2(c+d x)}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))} \, dx}{2 b \left (a^2+b^2\right )^2}\\ &=\frac {a^2 e^2 \sqrt {e \cot (c+d x)}}{b \left (a^2+b^2\right ) d (a+b \cot (c+d x))}-\frac {2 \text {Subst}\left (\int \frac {-2 a b^2 e^4-b \left (a^2-b^2\right ) e^3 x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{b \left (a^2+b^2\right )^2 d}+\frac {\left (a^2 \left (a^2+5 b^2\right ) e^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-e x} (a-b x)} \, dx,x,-\cot (c+d x)\right )}{2 b \left (a^2+b^2\right )^2 d}\\ &=\frac {a^2 e^2 \sqrt {e \cot (c+d x)}}{b \left (a^2+b^2\right ) d (a+b \cot (c+d x))}-\frac {\left (a^2 \left (a^2+5 b^2\right ) e^2\right ) \text {Subst}\left (\int \frac {1}{a+\frac {b x^2}{e}} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{b \left (a^2+b^2\right )^2 d}-\frac {\left (\left (a^2-2 a b-b^2\right ) e^3\right ) \text {Subst}\left (\int \frac {e-x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}+\frac {\left (\left (a^2+2 a b-b^2\right ) e^3\right ) \text {Subst}\left (\int \frac {e+x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}\\ &=-\frac {a^{3/2} \left (a^2+5 b^2\right ) e^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{b^{3/2} \left (a^2+b^2\right )^2 d}+\frac {a^2 e^2 \sqrt {e \cot (c+d x)}}{b \left (a^2+b^2\right ) d (a+b \cot (c+d x))}+\frac {\left (\left (a^2-2 a b-b^2\right ) e^{5/2}\right ) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {e}+2 x}{-e-\sqrt {2} \sqrt {e} x-x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {\left (\left (a^2-2 a b-b^2\right ) e^{5/2}\right ) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {e}-2 x}{-e+\sqrt {2} \sqrt {e} x-x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {\left (\left (a^2+2 a b-b^2\right ) e^3\right ) \text {Subst}\left (\int \frac {1}{e-\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d}+\frac {\left (\left (a^2+2 a b-b^2\right ) e^3\right ) \text {Subst}\left (\int \frac {1}{e+\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d}\\ &=-\frac {a^{3/2} \left (a^2+5 b^2\right ) e^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{b^{3/2} \left (a^2+b^2\right )^2 d}+\frac {a^2 e^2 \sqrt {e \cot (c+d x)}}{b \left (a^2+b^2\right ) d (a+b \cot (c+d x))}+\frac {\left (a^2-2 a b-b^2\right ) e^{5/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\left (a^2-2 a b-b^2\right ) e^{5/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {\left (\left (a^2+2 a b-b^2\right ) e^{5/2}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\left (\left (a^2+2 a b-b^2\right ) e^{5/2}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}\\ &=-\frac {a^{3/2} \left (a^2+5 b^2\right ) e^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{b^{3/2} \left (a^2+b^2\right )^2 d}-\frac {\left (a^2+2 a b-b^2\right ) e^{5/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {\left (a^2+2 a b-b^2\right ) e^{5/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {a^2 e^2 \sqrt {e \cot (c+d x)}}{b \left (a^2+b^2\right ) d (a+b \cot (c+d x))}+\frac {\left (a^2-2 a b-b^2\right ) e^{5/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\left (a^2-2 a b-b^2\right ) e^{5/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 2.81, size = 390, normalized size = 0.99 \begin {gather*} -\frac {(e \cot (c+d x))^{5/2} \left (-28 a^2 b^{3/2} \left (a^2-b^2\right ) \cot ^{\frac {3}{2}}(c+d x) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\cot ^2(c+d x)\right )+12 b^{7/2} \left (a^2+b^2\right ) \cot ^{\frac {7}{2}}(c+d x) \, _2F_1\left (2,\frac {7}{2};\frac {9}{2};-\frac {b \cot (c+d x)}{a}\right )-7 a^2 \left (-6 \sqrt {2} a b^{5/2} \text {ArcTan}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )+6 \sqrt {2} a b^{5/2} \text {ArcTan}\left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )+24 a^{7/2} \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {\cot (c+d x)}}{\sqrt {a}}\right )-24 a^3 \sqrt {b} \sqrt {\cot (c+d x)}-24 a b^{5/2} \sqrt {\cot (c+d x)}+4 a^2 b^{3/2} \cot ^{\frac {3}{2}}(c+d x)+4 b^{7/2} \cot ^{\frac {3}{2}}(c+d x)-3 \sqrt {2} a b^{5/2} \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )+3 \sqrt {2} a b^{5/2} \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )\right )}{42 a^2 b^{3/2} \left (a^2+b^2\right )^2 d \cot ^{\frac {5}{2}}(c+d x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(e*Cot[c + d*x])^(5/2)/(a + b*Cot[c + d*x])^2,x]

[Out]

-1/42*((e*Cot[c + d*x])^(5/2)*(-28*a^2*b^(3/2)*(a^2 - b^2)*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -
Cot[c + d*x]^2] + 12*b^(7/2)*(a^2 + b^2)*Cot[c + d*x]^(7/2)*Hypergeometric2F1[2, 7/2, 9/2, -((b*Cot[c + d*x])/
a)] - 7*a^2*(-6*Sqrt[2]*a*b^(5/2)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] + 6*Sqrt[2]*a*b^(5/2)*ArcTan[1 + Sqrt
[2]*Sqrt[Cot[c + d*x]]] + 24*a^(7/2)*ArcTan[(Sqrt[b]*Sqrt[Cot[c + d*x]])/Sqrt[a]] - 24*a^3*Sqrt[b]*Sqrt[Cot[c
+ d*x]] - 24*a*b^(5/2)*Sqrt[Cot[c + d*x]] + 4*a^2*b^(3/2)*Cot[c + d*x]^(3/2) + 4*b^(7/2)*Cot[c + d*x]^(3/2) -
3*Sqrt[2]*a*b^(5/2)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] + 3*Sqrt[2]*a*b^(5/2)*Log[1 + Sqrt[2]*S
qrt[Cot[c + d*x]] + Cot[c + d*x]])))/(a^2*b^(3/2)*(a^2 + b^2)^2*d*Cot[c + d*x]^(5/2))

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Maple [A]
time = 0.62, size = 387, normalized size = 0.98

method	result	size
derivativedivides	\(-\frac {2 e^{3} \left (\frac {a^{2} \left (-\frac {\left (a^{2}+b^{2}\right ) \sqrt {e \cot \left (d x +c \right )}}{2 b \left (e \cot \left (d x +c \right ) b +a e \right )}+\frac {\left (a^{2}+5 b^{2}\right ) \arctan \left (\frac {b \sqrt {e \cot \left (d x +c \right )}}{\sqrt {a e b}}\right )}{2 b \sqrt {a e b}}\right )}{\left (a^{2}+b^{2}\right )^{2}}+\frac {-\frac {a b \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 e}+\frac {\left (-a^{2}+b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}}{\left (a^{2}+b^{2}\right )^{2}}\right )}{d}\)	\(387\)
default	\(-\frac {2 e^{3} \left (\frac {a^{2} \left (-\frac {\left (a^{2}+b^{2}\right ) \sqrt {e \cot \left (d x +c \right )}}{2 b \left (e \cot \left (d x +c \right ) b +a e \right )}+\frac {\left (a^{2}+5 b^{2}\right ) \arctan \left (\frac {b \sqrt {e \cot \left (d x +c \right )}}{\sqrt {a e b}}\right )}{2 b \sqrt {a e b}}\right )}{\left (a^{2}+b^{2}\right )^{2}}+\frac {-\frac {a b \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 e}+\frac {\left (-a^{2}+b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}}{\left (a^{2}+b^{2}\right )^{2}}\right )}{d}\)	\(387\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*cot(d*x+c))^(5/2)/(a+b*cot(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

-2/d*e^3*(a^2/(a^2+b^2)^2*(-1/2*(a^2+b^2)/b*(e*cot(d*x+c))^(1/2)/(e*cot(d*x+c)*b+a*e)+1/2*(a^2+5*b^2)/b/(a*e*b
)^(1/2)*arctan(b*(e*cot(d*x+c))^(1/2)/(a*e*b)^(1/2)))+1/(a^2+b^2)^2*(-1/4*a/e*b*(e^2)^(1/4)*2^(1/2)*(ln((e*cot
(d*x+c)+(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2))/(e*cot(d*x+c)-(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2
^(1/2)+(e^2)^(1/2)))+2*arctan(2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)-2*arctan(-2^(1/2)/(e^2)^(1/4)*(e*cot
(d*x+c))^(1/2)+1))+1/8*(-a^2+b^2)/(e^2)^(1/4)*2^(1/2)*(ln((e*cot(d*x+c)-(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/
2)+(e^2)^(1/2))/(e*cot(d*x+c)+(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2)))+2*arctan(2^(1/2)/(e^2)^(1
/4)*(e*cot(d*x+c))^(1/2)+1)-2*arctan(-2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1))))

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Maxima [A]
time = 0.50, size = 286, normalized size = 0.73 \begin {gather*} -\frac {{\left (\frac {4 \, {\left (a^{4} + 5 \, a^{2} b^{2}\right )} \arctan \left (\frac {b}{\sqrt {a b} \sqrt {\tan \left (d x + c\right )}}\right )}{{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \sqrt {a b}} - \frac {4 \, a^{2}}{{\left (a^{3} b + a b^{3} + \frac {a^{2} b^{2} + b^{4}}{\tan \left (d x + c\right )}\right )} \sqrt {\tan \left (d x + c\right )}} - \frac {2 \, \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 2 \, \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}}\right )} e^{\frac {5}{2}}}{4 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*cot(d*x+c))^(5/2)/(a+b*cot(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/4*(4*(a^4 + 5*a^2*b^2)*arctan(b/(sqrt(a*b)*sqrt(tan(d*x + c))))/((a^4*b + 2*a^2*b^3 + b^5)*sqrt(a*b)) - 4*a
^2/((a^3*b + a*b^3 + (a^2*b^2 + b^4)/tan(d*x + c))*sqrt(tan(d*x + c))) - (2*sqrt(2)*(a^2 + 2*a*b - b^2)*arctan
(1/2*sqrt(2)*(sqrt(2) + 2/sqrt(tan(d*x + c)))) + 2*sqrt(2)*(a^2 + 2*a*b - b^2)*arctan(-1/2*sqrt(2)*(sqrt(2) -
2/sqrt(tan(d*x + c)))) - sqrt(2)*(a^2 - 2*a*b - b^2)*log(sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1) + sq
rt(2)*(a^2 - 2*a*b - b^2)*log(-sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1))/(a^4 + 2*a^2*b^2 + b^4))*e^(5
/2)/d

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*cot(d*x+c))^(5/2)/(a+b*cot(d*x+c))^2,x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (e \cot {\left (c + d x \right )}\right )^{\frac {5}{2}}}{\left (a + b \cot {\left (c + d x \right )}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*cot(d*x+c))**(5/2)/(a+b*cot(d*x+c))**2,x)

[Out]

Integral((e*cot(c + d*x))**(5/2)/(a + b*cot(c + d*x))**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*cot(d*x+c))^(5/2)/(a+b*cot(d*x+c))^2,x, algorithm="giac")

[Out]

integrate((e*cot(d*x + c))^(5/2)/(b*cot(d*x + c) + a)^2, x)

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Mupad [B]
time = 3.12, size = 2500, normalized size = 6.36 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*cot(c + d*x))^(5/2)/(a + b*cot(c + d*x))^2,x)

[Out]

atan(((((((8*(96*a^2*b^14*d^4*e^13 + 480*a^4*b^12*d^4*e^13 + 960*a^6*b^10*d^4*e^13 + 960*a^8*b^8*d^4*e^13 + 48
0*a^10*b^6*d^4*e^13 + 96*a^12*b^4*d^4*e^13))/(b^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^5 + 6*a^4*b^5*d^5 + 4*a^6*b^3*
d^5) - (16*(e*cot(c + d*x))^(1/2)*((e^5*1i)/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^
2)))^(1/2)*(32*b^18*d^4*e^10 + 160*a^2*b^16*d^4*e^10 + 288*a^4*b^14*d^4*e^10 + 160*a^6*b^12*d^4*e^10 - 160*a^8
*b^10*d^4*e^10 - 288*a^10*b^8*d^4*e^10 - 160*a^12*b^6*d^4*e^10 - 32*a^14*b^4*d^4*e^10))/(b^9*d^4 + a^8*b*d^4 +
4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4))*((e^5*1i)/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i
- 6*a^2*b^2*d^2)))^(1/2) + (16*(e*cot(c + d*x))^(1/2)*(60*a*b^13*d^2*e^15 + 8*a^13*b*d^2*e^15 + 52*a^3*b^11*d
^2*e^15 + 128*a^5*b^9*d^2*e^15 + 424*a^7*b^7*d^2*e^15 + 380*a^9*b^5*d^2*e^15 + 100*a^11*b^3*d^2*e^15))/(b^9*d^
4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4))*((e^5*1i)/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i
- a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2) + (8*(4*a*b^11*d^2*e^18 + 16*a^11*b*d^2*e^18 - 304*a^3*b^9*d^2*e^18 -
120*a^5*b^7*d^2*e^18 + 320*a^7*b^5*d^2*e^18 + 148*a^9*b^3*d^2*e^18))/(b^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^5 + 6
*a^4*b^5*d^5 + 4*a^6*b^3*d^5))*((e^5*1i)/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2))
)^(1/2) + (16*(e*cot(c + d*x))^(1/2)*(a^10*e^20 - 2*b^10*e^20 - 4*a^2*b^8*e^20 - 27*a^4*b^6*e^20 + 15*a^6*b^4*
e^20 + 9*a^8*b^2*e^20))/(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4))*((e^5*1i)/(4*(a
^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*1i - (((((8*(96*a^2*b^14*d^4*e^13 + 48
0*a^4*b^12*d^4*e^13 + 960*a^6*b^10*d^4*e^13 + 960*a^8*b^8*d^4*e^13 + 480*a^10*b^6*d^4*e^13 + 96*a^12*b^4*d^4*e
^13))/(b^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^5 + 6*a^4*b^5*d^5 + 4*a^6*b^3*d^5) + (16*(e*cot(c + d*x))^(1/2)*((e^5
*1i)/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*(32*b^18*d^4*e^10 + 160*a^2*
b^16*d^4*e^10 + 288*a^4*b^14*d^4*e^10 + 160*a^6*b^12*d^4*e^10 - 160*a^8*b^10*d^4*e^10 - 288*a^10*b^8*d^4*e^10
- 160*a^12*b^6*d^4*e^10 - 32*a^14*b^4*d^4*e^10))/(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*
b^3*d^4))*((e^5*1i)/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2) - (16*(e*cot(
c + d*x))^(1/2)*(60*a*b^13*d^2*e^15 + 8*a^13*b*d^2*e^15 + 52*a^3*b^11*d^2*e^15 + 128*a^5*b^9*d^2*e^15 + 424*a^
7*b^7*d^2*e^15 + 380*a^9*b^5*d^2*e^15 + 100*a^11*b^3*d^2*e^15))/(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b
^5*d^4 + 4*a^6*b^3*d^4))*((e^5*1i)/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2
) + (8*(4*a*b^11*d^2*e^18 + 16*a^11*b*d^2*e^18 - 304*a^3*b^9*d^2*e^18 - 120*a^5*b^7*d^2*e^18 + 320*a^7*b^5*d^2
*e^18 + 148*a^9*b^3*d^2*e^18))/(b^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^5 + 6*a^4*b^5*d^5 + 4*a^6*b^3*d^5))*((e^5*1i
)/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2) - (16*(e*cot(c + d*x))^(1/2)*(a
^10*e^20 - 2*b^10*e^20 - 4*a^2*b^8*e^20 - 27*a^4*b^6*e^20 + 15*a^6*b^4*e^20 + 9*a^8*b^2*e^20))/(b^9*d^4 + a^8*
b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4))*((e^5*1i)/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b
*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*1i)/((16*(a^8*e^23 + 10*a^2*b^6*e^23 + 27*a^4*b^4*e^23 + 10*a^6*b^2*e^23))/(b
^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^5 + 6*a^4*b^5*d^5 + 4*a^6*b^3*d^5) + (((((8*(96*a^2*b^14*d^4*e^13 + 480*a^4*b
^12*d^4*e^13 + 960*a^6*b^10*d^4*e^13 + 960*a^8*b^8*d^4*e^13 + 480*a^10*b^6*d^4*e^13 + 96*a^12*b^4*d^4*e^13))/(
b^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^5 + 6*a^4*b^5*d^5 + 4*a^6*b^3*d^5) - (16*(e*cot(c + d*x))^(1/2)*((e^5*1i)/(4
*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*(32*b^18*d^4*e^10 + 160*a^2*b^16*d^
4*e^10 + 288*a^4*b^14*d^4*e^10 + 160*a^6*b^12*d^4*e^10 - 160*a^8*b^10*d^4*e^10 - 288*a^10*b^8*d^4*e^10 - 160*a
^12*b^6*d^4*e^10 - 32*a^14*b^4*d^4*e^10))/(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4
))*((e^5*1i)/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2) + (16*(e*cot(c + d*x
))^(1/2)*(60*a*b^13*d^2*e^15 + 8*a^13*b*d^2*e^15 + 52*a^3*b^11*d^2*e^15 + 128*a^5*b^9*d^2*e^15 + 424*a^7*b^7*d
^2*e^15 + 380*a^9*b^5*d^2*e^15 + 100*a^11*b^3*d^2*e^15))/(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4
+ 4*a^6*b^3*d^4))*((e^5*1i)/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2) + (8*
(4*a*b^11*d^2*e^18 + 16*a^11*b*d^2*e^18 - 304*a^3*b^9*d^2*e^18 - 120*a^5*b^7*d^2*e^18 + 320*a^7*b^5*d^2*e^18 +
148*a^9*b^3*d^2*e^18))/(b^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^5 + 6*a^4*b^5*d^5 + 4*a^6*b^3*d^5))*((e^5*1i)/(4*(a
^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2) + (16*(e*cot(c + d*x))^(1/2)*(a^10*e^2
0 - 2*b^10*e^20 - 4*a^2*b^8*e^20 - 27*a^4*b^6*e^20 + 15*a^6*b^4*e^20 + 9*a^8*b^2*e^20))/(b^9*d^4 + a^8*b*d^4 +
4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4))*((e^5*1i)/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i
- 6*a^2*b^2*d^2)))^(1/2) + (((((8*(96*a^2*b^14...

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