\(\int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx\) [830]

Optimal result

Integrand size = 23, antiderivative size = 313 \[ \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=-\frac {\left (a^2-2 a b-b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {\left (a^2-2 a b-b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\sqrt {a} \left (a^2-3 b^2\right ) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{\sqrt {b} \left (a^2+b^2\right )^2 d}+\frac {a \sqrt {\cot (c+d x)}}{\left (a^2+b^2\right ) d (b+a \cot (c+d x))}+\frac {\left (a^2+2 a b-b^2\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\left (a^2+2 a b-b^2\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {3754, 3649, 3734, 3615, 1182, 1176, 631, 210, 1179, 642, 3715, 65, 211} \[ \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=-\frac {\sqrt {a} \left (a^2-3 b^2\right ) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{\sqrt {b} d \left (a^2+b^2\right )^2}-\frac {\left (a^2-2 a b-b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^2}+\frac {\left (a^2-2 a b-b^2\right ) \arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^2}+\frac {a \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}+\frac {\left (a^2+2 a b-b^2\right ) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^2}-\frac {\left (a^2+2 a b-b^2\right ) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^2} \]

[In]

Int[1/(Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^2),x]

[Out]

-(((a^2 - 2*a*b - b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d)) + ((a^2 - 2*a*b - b^
2)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d) - (Sqrt[a]*(a^2 - 3*b^2)*ArcTan[(Sqrt[a]*
Sqrt[Cot[c + d*x]])/Sqrt[b]])/(Sqrt[b]*(a^2 + b^2)^2*d) + (a*Sqrt[Cot[c + d*x]])/((a^2 + b^2)*d*(b + a*Cot[c +
d*x])) + ((a^2 + 2*a*b - b^2)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d)
- ((a^2 + 2*a*b - b^2)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + 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 3649

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[b*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^n/(f*(m + 1)*(a^2 + b^2))), x] + Dist[1/((m + 1)*(a^2
+ b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 1)*Simp[a*c*(m + 1) - b*d*n - (b*c - a*d)*
(m + 1)*Tan[e + f*x] - b*d*(m + 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] && LtQ[m, -1] && GtQ[n, 0] && 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]

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

Mathematica [C] (verified)

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

Time = 6.11 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.21 \[ \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=-\frac {-\frac {4 \sqrt {a} b^{3/2} \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{\left (a^2+b^2\right )^2}+\frac {4 a b \sqrt {\cot (c+d x)}}{\left (a^2+b^2\right )^2}-\frac {\sqrt {a} \left (-b \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )+\sqrt {a} \sqrt {b} \sqrt {\cot (c+d x)}-a \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right ) \cot (c+d x)\right )}{\sqrt {b} \left (a^2+b^2\right ) (b+a \cot (c+d x))}-\frac {2 \left (a^2-b^2\right ) \cot ^{\frac {3}{2}}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\cot ^2(c+d x)\right )}{3 \left (a^2+b^2\right )^2}-\frac {a b \left (2 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-2 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )+8 \sqrt {\cot (c+d x)}+\sqrt {2} \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-\sqrt {2} \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )}{2 \left (a^2+b^2\right )^2}}{d} \]

[In]

Integrate[1/(Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^2),x]

[Out]

-(((-4*Sqrt[a]*b^(3/2)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/(a^2 + b^2)^2 + (4*a*b*Sqrt[Cot[c + d*x]]
)/(a^2 + b^2)^2 - (Sqrt[a]*(-(b*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]]) + Sqrt[a]*Sqrt[b]*Sqrt[Cot[c + d
*x]] - a*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]]*Cot[c + d*x]))/(Sqrt[b]*(a^2 + b^2)*(b + a*Cot[c + d*x])
) - (2*(a^2 - b^2)*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2])/(3*(a^2 + b^2)^2) - (a*
b*(2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + 8*Sqr
t[Cot[c + d*x]] + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Co
t[c + d*x]] + Cot[c + d*x]]))/(2*(a^2 + b^2)^2))/d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(948\) vs. \(2(273)=546\).

Time = 1.87 (sec) , antiderivative size = 949, normalized size of antiderivative = 3.03

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(949\)
default	\(\text {Expression too large to display}\)	\(949\)

[In]

int(1/cot(d*x+c)^(3/2)/(a+b*tan(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2633 vs. \(2 (273) = 546\).

Time = 0.44 (sec) , antiderivative size = 5291, normalized size of antiderivative = 16.90 \[ \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\text {Too large to display} \]

[In]

integrate(1/cot(d*x+c)^(3/2)/(a+b*tan(d*x+c))^2,x, algorithm="fricas")

[Out]

Too large to include

Sympy [F]

\[ \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\int \frac {1}{\left (a + b \tan {\left (c + d x \right )}\right )^{2} \cot ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \]

[In]

integrate(1/cot(d*x+c)**(3/2)/(a+b*tan(d*x+c))**2,x)

[Out]

Integral(1/((a + b*tan(c + d*x))**2*cot(c + d*x)**(3/2)), x)

Maxima [A] (verification not implemented)

none

Time = 0.40 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=-\frac {\frac {4 \, {\left (a^{3} - 3 \, a b^{2}\right )} \arctan \left (\frac {a}{\sqrt {a b} \sqrt {\tan \left (d x + c\right )}}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a b}} - \frac {2 \, \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 2 \, \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {4 \, a}{{\left (a^{2} b + b^{3} + \frac {a^{3} + a b^{2}}{\tan \left (d x + c\right )}\right )} \sqrt {\tan \left (d x + c\right )}}}{4 \, d} \]

[In]

integrate(1/cot(d*x+c)^(3/2)/(a+b*tan(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/4*(4*(a^3 - 3*a*b^2)*arctan(a/(sqrt(a*b)*sqrt(tan(d*x + c))))/((a^4 + 2*a^2*b^2 + b^4)*sqrt(a*b)) - (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)*a
rctan(-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) + sqrt(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) - 4*a/((a^2*b + b^3 + (a^3 + a*b^2)/tan(d*x + c))*sqrt(tan(d*x + c))))/d

Giac [F]

\[ \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\int { \frac {1}{{\left (b \tan \left (d x + c\right ) + a\right )}^{2} \cot \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]

[In]

integrate(1/cot(d*x+c)^(3/2)/(a+b*tan(d*x+c))^2,x, algorithm="giac")

[Out]

integrate(1/((b*tan(d*x + c) + a)^2*cot(d*x + c)^(3/2)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\int \frac {1}{{\mathrm {cot}\left (c+d\,x\right )}^{3/2}\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^2} \,d x \]

[In]

int(1/(cot(c + d*x)^(3/2)*(a + b*tan(c + d*x))^2),x)

[Out]

int(1/(cot(c + d*x)^(3/2)*(a + b*tan(c + d*x))^2), x)