\(\int \frac {\cot ^{\frac {5}{2}}(c+d x)}{a+b \tan (c+d x)} \, dx\) [820]

Optimal result

Integrand size = 23, antiderivative size = 271 \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{a+b \tan (c+d x)} \, dx=-\frac {(a-b) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}+\frac {(a-b) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}-\frac {2 b^{7/2} \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{a^{5/2} \left (a^2+b^2\right ) d}+\frac {2 b \sqrt {\cot (c+d x)}}{a^2 d}-\frac {2 \cot ^{\frac {3}{2}}(c+d x)}{3 a d}+\frac {(a+b) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}-\frac {(a+b) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d} \]

[Out]

-2*b^(7/2)*arctan(a^(1/2)*cot(d*x+c)^(1/2)/b^(1/2))/a^(5/2)/(a^2+b^2)/d-2/3*cot(d*x+c)^(3/2)/a/d+1/2*(a-b)*arc
tan(-1+2^(1/2)*cot(d*x+c)^(1/2))/(a^2+b^2)/d*2^(1/2)+1/2*(a-b)*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))/(a^2+b^2)/d*
2^(1/2)+1/4*(a+b)*ln(1+cot(d*x+c)-2^(1/2)*cot(d*x+c)^(1/2))/(a^2+b^2)/d*2^(1/2)-1/4*(a+b)*ln(1+cot(d*x+c)+2^(1
/2)*cot(d*x+c)^(1/2))/(a^2+b^2)/d*2^(1/2)+2*b*cot(d*x+c)^(1/2)/a^2/d

Rubi [A] (verified)

Time = 0.79 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {3754, 3647, 3728, 3735, 3615, 1182, 1176, 631, 210, 1179, 642, 3715, 65, 211} \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{a+b \tan (c+d x)} \, dx=-\frac {(a-b) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )}+\frac {(a-b) \arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )}+\frac {(a+b) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )}-\frac {(a+b) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )}+\frac {2 b \sqrt {\cot (c+d x)}}{a^2 d}-\frac {2 b^{7/2} \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{a^{5/2} d \left (a^2+b^2\right )}-\frac {2 \cot ^{\frac {3}{2}}(c+d x)}{3 a d} \]

[In]

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

[Out]

-(((a - b)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d)) + ((a - b)*ArcTan[1 + Sqrt[2]*Sqrt
[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - (2*b^(7/2)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/(a^(5/2)*(
a^2 + b^2)*d) + (2*b*Sqrt[Cot[c + d*x]])/(a^2*d) - (2*Cot[c + d*x]^(3/2))/(3*a*d) + ((a + b)*Log[1 - Sqrt[2]*S
qrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) - ((a + b)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + C
ot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^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 3647

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[b^2*(a + b*Tan[e + f*x])^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(m + n - 1))), x] + Dist[1/(d*(m + n -
1)), Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^n*Simp[a^3*d*(m + n - 1) - b^2*(b*c*(m - 2) + a*d*(
1 + n)) + b*d*(m + n - 1)*(3*a^2 - b^2)*Tan[e + f*x] - b^2*(b*c*(m - 2) - a*d*(3*m + 2*n - 4))*Tan[e + f*x]^2,
x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
&& IntegerQ[2*m] && GtQ[m, 2] && (GeQ[n, -1] || IntegerQ[m]) &&  !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[c, 0]
&& NeQ[a, 0])))

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 3728

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*
tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*(a + b*Tan[e + f*x])^m*((c + d
*Tan[e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Dist[1/(d*(m + n + 1)), Int[(a + b*Tan[e + f*x])^(m - 1)*(c +
d*Tan[e + f*x])^n*Simp[a*A*d*(m + n + 1) - C*(b*c*m + a*d*(n + 1)) + d*(A*b + a*B - b*C)*(m + n + 1)*Tan[e + f
*x] - (C*m*(b*c - a*d) - b*B*d*(m + n + 1))*Tan[e + f*x]^2, 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[m, 0] &&  !(IGtQ[n, 0] && ( !Intege
rQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rule 3735

Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2))/((a_.) + (b_.)*ta
n[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/(a^2 + b^2), Int[(c + d*Tan[e + f*x])^n*Simp[a*(A - C) - (A*b - b*
C)*Tan[e + f*x], x], x], x] + Dist[(A*b^2 + 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, 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]

Rule 3754

Int[(cot[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^(n_.))^(p_.), x_Symbol] :> Dist
[d^(n*p), Int[(d*Cot[e + f*x])^(m - n*p)*(b + a*Cot[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, d, e, f, m, n, p}, x
] &&  !IntegerQ[m] && IntegersQ[n, p]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\cot ^{\frac {7}{2}}(c+d x)}{b+a \cot (c+d x)} \, dx \\ & = -\frac {2 \cot ^{\frac {3}{2}}(c+d x)}{3 a d}-\frac {2 \int \frac {\sqrt {\cot (c+d x)} \left (\frac {3 b}{2}+\frac {3}{2} a \cot (c+d x)+\frac {3}{2} b \cot ^2(c+d x)\right )}{b+a \cot (c+d x)} \, dx}{3 a} \\ & = \frac {2 b \sqrt {\cot (c+d x)}}{a^2 d}-\frac {2 \cot ^{\frac {3}{2}}(c+d x)}{3 a d}+\frac {4 \int \frac {\frac {3 b^2}{4}-\frac {3}{4} \left (a^2-b^2\right ) \cot ^2(c+d x)}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))} \, dx}{3 a^2} \\ & = \frac {2 b \sqrt {\cot (c+d x)}}{a^2 d}-\frac {2 \cot ^{\frac {3}{2}}(c+d x)}{3 a d}+\frac {4 \int \frac {\frac {3 a^2 b}{4}-\frac {3}{4} a^3 \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx}{3 a^2 \left (a^2+b^2\right )}+\frac {b^4 \int \frac {1+\cot ^2(c+d x)}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))} \, dx}{a^2 \left (a^2+b^2\right )} \\ & = \frac {2 b \sqrt {\cot (c+d x)}}{a^2 d}-\frac {2 \cot ^{\frac {3}{2}}(c+d x)}{3 a d}+\frac {8 \text {Subst}\left (\int \frac {-\frac {3 a^2 b}{4}+\frac {3 a^3 x^2}{4}}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{3 a^2 \left (a^2+b^2\right ) d}+\frac {b^4 \text {Subst}\left (\int \frac {1}{\sqrt {-x} (b-a x)} \, dx,x,-\cot (c+d x)\right )}{a^2 \left (a^2+b^2\right ) d} \\ & = \frac {2 b \sqrt {\cot (c+d x)}}{a^2 d}-\frac {2 \cot ^{\frac {3}{2}}(c+d x)}{3 a d}+\frac {(a-b) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{\left (a^2+b^2\right ) d}-\frac {\left (2 b^4\right ) \text {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d}-\frac {(a+b) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{\left (a^2+b^2\right ) d} \\ & = -\frac {2 b^{7/2} \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{a^{5/2} \left (a^2+b^2\right ) d}+\frac {2 b \sqrt {\cot (c+d x)}}{a^2 d}-\frac {2 \cot ^{\frac {3}{2}}(c+d x)}{3 a d}+\frac {(a-b) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 \left (a^2+b^2\right ) d}+\frac {(a-b) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 \left (a^2+b^2\right ) d}+\frac {(a+b) \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}+\frac {(a+b) \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d} \\ & = -\frac {2 b^{7/2} \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{a^{5/2} \left (a^2+b^2\right ) d}+\frac {2 b \sqrt {\cot (c+d x)}}{a^2 d}-\frac {2 \cot ^{\frac {3}{2}}(c+d x)}{3 a d}+\frac {(a+b) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}-\frac {(a+b) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}+\frac {(a-b) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}-\frac {(a-b) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d} \\ & = -\frac {(a-b) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}+\frac {(a-b) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}-\frac {2 b^{7/2} \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{a^{5/2} \left (a^2+b^2\right ) d}+\frac {2 b \sqrt {\cot (c+d x)}}{a^2 d}-\frac {2 \cot ^{\frac {3}{2}}(c+d x)}{3 a d}+\frac {(a+b) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}-\frac {(a+b) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.42 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.12 \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{a+b \tan (c+d x)} \, dx=\frac {6 \sqrt {2} a^{5/2} b \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-6 \sqrt {2} a^{5/2} b \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )-24 b^{7/2} \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )+24 a^{5/2} b \sqrt {\cot (c+d x)}+24 \sqrt {a} b^3 \sqrt {\cot (c+d x)}-8 a^{7/2} \cot ^{\frac {3}{2}}(c+d x)-8 a^{3/2} b^2 \cot ^{\frac {3}{2}}(c+d x)+8 a^{7/2} \cot ^{\frac {3}{2}}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\cot ^2(c+d x)\right )+3 \sqrt {2} a^{5/2} b \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-3 \sqrt {2} a^{5/2} b \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{12 a^{5/2} \left (a^2+b^2\right ) d} \]

[In]

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

[Out]

(6*Sqrt[2]*a^(5/2)*b*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 6*Sqrt[2]*a^(5/2)*b*ArcTan[1 + Sqrt[2]*Sqrt[Cot[
c + d*x]]] - 24*b^(7/2)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]] + 24*a^(5/2)*b*Sqrt[Cot[c + d*x]] + 24*Sq
rt[a]*b^3*Sqrt[Cot[c + d*x]] - 8*a^(7/2)*Cot[c + d*x]^(3/2) - 8*a^(3/2)*b^2*Cot[c + d*x]^(3/2) + 8*a^(7/2)*Cot
[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2] + 3*Sqrt[2]*a^(5/2)*b*Log[1 - Sqrt[2]*Sqrt[Cot
[c + d*x]] + Cot[c + d*x]] - 3*Sqrt[2]*a^(5/2)*b*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(12*a^(5/
2)*(a^2 + b^2)*d)

Maple [A] (verified)

Time = 1.40 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.50

method	result	size
derivativedivides	\(-\frac {\left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {5}{2}} \tan \left (d x +c \right ) \left (3 \sqrt {a b}\, \ln \left (-\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )-\tan \left (d x +c \right )-1}\right ) \sqrt {2}\, \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right ) a^{3}+6 \sqrt {a b}\, \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}\, \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right ) a^{3}-6 \sqrt {a b}\, \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}\, \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right ) a^{2} b +6 \sqrt {a b}\, \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}\, \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right ) a^{3}-6 \sqrt {a b}\, \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}\, \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right ) a^{2} b -3 \sqrt {a b}\, \ln \left (-\frac {\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )-\tan \left (d x +c \right )-1}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right ) \sqrt {2}\, \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right ) a^{2} b -24 b^{4} \arctan \left (\frac {b \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a b}}\right ) \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )-24 \sqrt {a b}\, \tan \left (d x +c \right ) a^{2} b -24 \sqrt {a b}\, \tan \left (d x +c \right ) b^{3}+8 \sqrt {a b}\, a^{3}+8 \sqrt {a b}\, a \,b^{2}\right )}{12 d \,a^{2} \left (a^{2}+b^{2}\right ) \sqrt {a b}}\)	\(406\)
default	\(-\frac {\left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {5}{2}} \tan \left (d x +c \right ) \left (3 \sqrt {a b}\, \ln \left (-\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )-\tan \left (d x +c \right )-1}\right ) \sqrt {2}\, \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right ) a^{3}+6 \sqrt {a b}\, \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}\, \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right ) a^{3}-6 \sqrt {a b}\, \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}\, \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right ) a^{2} b +6 \sqrt {a b}\, \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}\, \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right ) a^{3}-6 \sqrt {a b}\, \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}\, \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right ) a^{2} b -3 \sqrt {a b}\, \ln \left (-\frac {\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )-\tan \left (d x +c \right )-1}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right ) \sqrt {2}\, \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right ) a^{2} b -24 b^{4} \arctan \left (\frac {b \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a b}}\right ) \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )-24 \sqrt {a b}\, \tan \left (d x +c \right ) a^{2} b -24 \sqrt {a b}\, \tan \left (d x +c \right ) b^{3}+8 \sqrt {a b}\, a^{3}+8 \sqrt {a b}\, a \,b^{2}\right )}{12 d \,a^{2} \left (a^{2}+b^{2}\right ) \sqrt {a b}}\)	\(406\)

[In]

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

[Out]

-1/12/d*(1/tan(d*x+c))^(5/2)*tan(d*x+c)*(3*(a*b)^(1/2)*ln(-(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(2^(1/2)*ta
n(d*x+c)^(1/2)-tan(d*x+c)-1))*2^(1/2)*tan(d*x+c)^(3/2)*a^3+6*(a*b)^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*2^
(1/2)*tan(d*x+c)^(3/2)*a^3-6*(a*b)^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*tan(d*x+c)^(3/2)*a^2*b+6*(
a*b)^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*tan(d*x+c)^(3/2)*a^3-6*(a*b)^(1/2)*arctan(-1+2^(1/2)*ta
n(d*x+c)^(1/2))*2^(1/2)*tan(d*x+c)^(3/2)*a^2*b-3*(a*b)^(1/2)*ln(-(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1)/(1+2^
(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))*2^(1/2)*tan(d*x+c)^(3/2)*a^2*b-24*b^4*arctan(b*tan(d*x+c)^(1/2)/(a*b)^(1/2
))*tan(d*x+c)^(3/2)-24*(a*b)^(1/2)*tan(d*x+c)*a^2*b-24*(a*b)^(1/2)*tan(d*x+c)*b^3+8*(a*b)^(1/2)*a^3+8*(a*b)^(1
/2)*a*b^2)/a^2/(a^2+b^2)/(a*b)^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1446 vs. \(2 (227) = 454\).

Time = 0.39 (sec) , antiderivative size = 2922, normalized size of antiderivative = 10.78 \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{a+b \tan (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/6*(6*b^3*sqrt(-b/a)*log((2*a*sqrt(-b/a)*sqrt(tan(d*x + c)) + b*tan(d*x + c) - a)/(b*tan(d*x + c) + a))*tan(
d*x + c) + 3*(a^4 + a^2*b^2)*d*sqrt(((a^4 + 2*a^2*b^2 + b^4)*d^2*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b
^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) + 2*a*b)/((a^4 + 2*a^2*b^2 + b^4)*d^2))*log(((a^4*b + 2*a^2*b^3 + b^5)
*d^3*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) + (a^3 - a*b^2)*d)*s
qrt(((a^4 + 2*a^2*b^2 + b^4)*d^2*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8
)*d^4)) + 2*a*b)/((a^4 + 2*a^2*b^2 + b^4)*d^2)) - (a^2 - b^2)*sqrt(tan(d*x + c)))*tan(d*x + c) - 3*(a^4 + a^2*
b^2)*d*sqrt(((a^4 + 2*a^2*b^2 + b^4)*d^2*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b
^6 + b^8)*d^4)) + 2*a*b)/((a^4 + 2*a^2*b^2 + b^4)*d^2))*log(-((a^4*b + 2*a^2*b^3 + b^5)*d^3*sqrt(-(a^4 - 2*a^2
*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) + (a^3 - a*b^2)*d)*sqrt(((a^4 + 2*a^2*b^2 +
b^4)*d^2*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) + 2*a*b)/((a^4
+ 2*a^2*b^2 + b^4)*d^2)) - (a^2 - b^2)*sqrt(tan(d*x + c)))*tan(d*x + c) - 3*(a^4 + a^2*b^2)*d*sqrt(-((a^4 + 2*
a^2*b^2 + b^4)*d^2*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) - 2*a*
b)/((a^4 + 2*a^2*b^2 + b^4)*d^2))*log(((a^4*b + 2*a^2*b^3 + b^5)*d^3*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a
^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) - (a^3 - a*b^2)*d)*sqrt(-((a^4 + 2*a^2*b^2 + b^4)*d^2*sqrt(-(a^4 -
2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) - 2*a*b)/((a^4 + 2*a^2*b^2 + b^4)*d^2
)) - (a^2 - b^2)*sqrt(tan(d*x + c)))*tan(d*x + c) + 3*(a^4 + a^2*b^2)*d*sqrt(-((a^4 + 2*a^2*b^2 + b^4)*d^2*sqr
t(-(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) - 2*a*b)/((a^4 + 2*a^2*b^2 +
b^4)*d^2))*log(-((a^4*b + 2*a^2*b^3 + b^5)*d^3*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 +
4*a^2*b^6 + b^8)*d^4)) - (a^3 - a*b^2)*d)*sqrt(-((a^4 + 2*a^2*b^2 + b^4)*d^2*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/((a
^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) - 2*a*b)/((a^4 + 2*a^2*b^2 + b^4)*d^2)) - (a^2 - b^2)*sqrt
(tan(d*x + c)))*tan(d*x + c) - 4*(a^3 + a*b^2 - 3*(a^2*b + b^3)*tan(d*x + c))/sqrt(tan(d*x + c)))/((a^4 + a^2*
b^2)*d*tan(d*x + c)), 1/6*(12*b^3*sqrt(b/a)*arctan(sqrt(b/a)*sqrt(tan(d*x + c)))*tan(d*x + c) + 3*(a^4 + a^2*b
^2)*d*sqrt(((a^4 + 2*a^2*b^2 + b^4)*d^2*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^
6 + b^8)*d^4)) + 2*a*b)/((a^4 + 2*a^2*b^2 + b^4)*d^2))*log(((a^4*b + 2*a^2*b^3 + b^5)*d^3*sqrt(-(a^4 - 2*a^2*b
^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) + (a^3 - a*b^2)*d)*sqrt(((a^4 + 2*a^2*b^2 + b
^4)*d^2*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) + 2*a*b)/((a^4 +
2*a^2*b^2 + b^4)*d^2)) - (a^2 - b^2)*sqrt(tan(d*x + c)))*tan(d*x + c) - 3*(a^4 + a^2*b^2)*d*sqrt(((a^4 + 2*a^2
*b^2 + b^4)*d^2*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) + 2*a*b)/
((a^4 + 2*a^2*b^2 + b^4)*d^2))*log(-((a^4*b + 2*a^2*b^3 + b^5)*d^3*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6
*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) + (a^3 - a*b^2)*d)*sqrt(((a^4 + 2*a^2*b^2 + b^4)*d^2*sqrt(-(a^4 - 2*
a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) + 2*a*b)/((a^4 + 2*a^2*b^2 + b^4)*d^2))
- (a^2 - b^2)*sqrt(tan(d*x + c)))*tan(d*x + c) - 3*(a^4 + a^2*b^2)*d*sqrt(-((a^4 + 2*a^2*b^2 + b^4)*d^2*sqrt(-
(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) - 2*a*b)/((a^4 + 2*a^2*b^2 + b^
4)*d^2))*log(((a^4*b + 2*a^2*b^3 + b^5)*d^3*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^
2*b^6 + b^8)*d^4)) - (a^3 - a*b^2)*d)*sqrt(-((a^4 + 2*a^2*b^2 + b^4)*d^2*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/((a^8 +
4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) - 2*a*b)/((a^4 + 2*a^2*b^2 + b^4)*d^2)) - (a^2 - b^2)*sqrt(tan
(d*x + c)))*tan(d*x + c) + 3*(a^4 + a^2*b^2)*d*sqrt(-((a^4 + 2*a^2*b^2 + b^4)*d^2*sqrt(-(a^4 - 2*a^2*b^2 + b^4
)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) - 2*a*b)/((a^4 + 2*a^2*b^2 + b^4)*d^2))*log(-((a^4*b
+ 2*a^2*b^3 + b^5)*d^3*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) -
(a^3 - a*b^2)*d)*sqrt(-((a^4 + 2*a^2*b^2 + b^4)*d^2*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^
4 + 4*a^2*b^6 + b^8)*d^4)) - 2*a*b)/((a^4 + 2*a^2*b^2 + b^4)*d^2)) - (a^2 - b^2)*sqrt(tan(d*x + c)))*tan(d*x +
c) - 4*(a^3 + a*b^2 - 3*(a^2*b + b^3)*tan(d*x + c))/sqrt(tan(d*x + c)))/((a^4 + a^2*b^2)*d*tan(d*x + c))]

Sympy [F]

\[ \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{a+b \tan (c+d x)} \, dx=\int \frac {\cot ^{\frac {5}{2}}{\left (c + d x \right )}}{a + b \tan {\left (c + d x \right )}}\, dx \]

[In]

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

[Out]

Integral(cot(c + d*x)**(5/2)/(a + b*tan(c + d*x)), x)

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.76 \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{a+b \tan (c+d x)} \, dx=-\frac {\frac {24 \, b^{4} \arctan \left (\frac {a}{\sqrt {a b} \sqrt {\tan \left (d x + c\right )}}\right )}{{\left (a^{4} + a^{2} b^{2}\right )} \sqrt {a b}} - \frac {3 \, {\left (2 \, \sqrt {2} {\left (a - b\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 2 \, \sqrt {2} {\left (a - b\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - \sqrt {2} {\left (a + b\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + \sqrt {2} {\left (a + b\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )\right )}}{a^{2} + b^{2}} - \frac {8 \, {\left (\frac {3 \, b}{\sqrt {\tan \left (d x + c\right )}} - \frac {a}{\tan \left (d x + c\right )^{\frac {3}{2}}}\right )}}{a^{2}}}{12 \, d} \]

[In]

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

[Out]

-1/12*(24*b^4*arctan(a/(sqrt(a*b)*sqrt(tan(d*x + c))))/((a^4 + a^2*b^2)*sqrt(a*b)) - 3*(2*sqrt(2)*(a - b)*arct
an(1/2*sqrt(2)*(sqrt(2) + 2/sqrt(tan(d*x + c)))) + 2*sqrt(2)*(a - b)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2/sqrt(tan
(d*x + c)))) - sqrt(2)*(a + b)*log(sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1) + sqrt(2)*(a + b)*log(-sqr
t(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1))/(a^2 + b^2) - 8*(3*b/sqrt(tan(d*x + c)) - a/tan(d*x + c)^(3/2))
/a^2)/d

Giac [F]

\[ \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{a+b \tan (c+d x)} \, dx=\int { \frac {\cot \left (d x + c\right )^{\frac {5}{2}}}{b \tan \left (d x + c\right ) + a} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{a+b \tan (c+d x)} \, dx=\int \frac {{\mathrm {cot}\left (c+d\,x\right )}^{5/2}}{a+b\,\mathrm {tan}\left (c+d\,x\right )} \,d x \]

[In]

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

[Out]

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