\(\int \frac {\cot ^2(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx\) [535]

Optimal result

Integrand size = 23, antiderivative size = 461 \[ \int \frac {\cot ^2(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\frac {b \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}-\frac {b \text {arctanh}\left (\frac {\sqrt {a+\sqrt {a^2+b^2}}-\sqrt {2} \sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} \sqrt {a^2+b^2} \sqrt {a-\sqrt {a^2+b^2}} d}+\frac {b \text {arctanh}\left (\frac {\sqrt {a+\sqrt {a^2+b^2}}+\sqrt {2} \sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} \sqrt {a^2+b^2} \sqrt {a-\sqrt {a^2+b^2}} d}+\frac {b \log \left (a+\sqrt {a^2+b^2}+b \tan (c+d x)-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {a+\sqrt {a^2+b^2}} d}-\frac {b \log \left (a+\sqrt {a^2+b^2}+b \tan (c+d x)+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {a+\sqrt {a^2+b^2}} d}-\frac {\cot (c+d x) \sqrt {a+b \tan (c+d x)}}{a d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 461, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {3650, 3734, 12, 3566, 722, 1108, 648, 632, 212, 642, 3715, 65, 214} \[ \int \frac {\cot ^2(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\frac {b \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}-\frac {b \text {arctanh}\left (\frac {\sqrt {\sqrt {a^2+b^2}+a}-\sqrt {2} \sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} d \sqrt {a^2+b^2} \sqrt {a-\sqrt {a^2+b^2}}}+\frac {b \text {arctanh}\left (\frac {\sqrt {\sqrt {a^2+b^2}+a}+\sqrt {2} \sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} d \sqrt {a^2+b^2} \sqrt {a-\sqrt {a^2+b^2}}}+\frac {b \log \left (-\sqrt {2} \sqrt {\sqrt {a^2+b^2}+a} \sqrt {a+b \tan (c+d x)}+\sqrt {a^2+b^2}+a+b \tan (c+d x)\right )}{2 \sqrt {2} d \sqrt {a^2+b^2} \sqrt {\sqrt {a^2+b^2}+a}}-\frac {b \log \left (\sqrt {2} \sqrt {\sqrt {a^2+b^2}+a} \sqrt {a+b \tan (c+d x)}+\sqrt {a^2+b^2}+a+b \tan (c+d x)\right )}{2 \sqrt {2} d \sqrt {a^2+b^2} \sqrt {\sqrt {a^2+b^2}+a}}-\frac {\cot (c+d x) \sqrt {a+b \tan (c+d x)}}{a d} \]

[In]

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

[Out]

(b*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(a^(3/2)*d) - (b*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] - Sqrt[2]*Sq
rt[a + b*Tan[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^2]]])/(Sqrt[2]*Sqrt[a^2 + b^2]*Sqrt[a - Sqrt[a^2 + b^2]]*d) + (b
*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] + Sqrt[2]*Sqrt[a + b*Tan[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^2]]])/(Sqrt[2]*S
qrt[a^2 + b^2]*Sqrt[a - Sqrt[a^2 + b^2]]*d) + (b*Log[a + Sqrt[a^2 + b^2] + b*Tan[c + d*x] - Sqrt[2]*Sqrt[a + S
qrt[a^2 + b^2]]*Sqrt[a + b*Tan[c + d*x]]])/(2*Sqrt[2]*Sqrt[a^2 + b^2]*Sqrt[a + Sqrt[a^2 + b^2]]*d) - (b*Log[a
+ Sqrt[a^2 + b^2] + b*Tan[c + d*x] + Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sqrt[a + b*Tan[c + d*x]]])/(2*Sqrt[2]*S
qrt[a^2 + b^2]*Sqrt[a + Sqrt[a^2 + b^2]]*d) - (Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(a*d)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 214

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

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; 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 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 722

Int[1/(Sqrt[(d_) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2*e, Subst[Int[1/(c*d^2 + a*e^2 - 2*c
*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0]

Rule 1108

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}
, Dist[1/(2*c*q*r), Int[(r - x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(r + x)/(q + r*x + x^2), x], x
]]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[b^2 - 4*a*c]

Rule 3566

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[b/d, Subst[Int[(a + x)^n/(b^2 + x^2), x], x
, b*Tan[c + d*x]], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[a^2 + b^2, 0]

Rule 3650

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 + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(a^2 + b^2)*(b*c - a*d))), x] + D
ist[1/((m + 1)*(a^2 + b^2)*(b*c - a*d)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[a*(b*c -
a*d)*(m + 1) - b^2*d*(m + n + 2) - b*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b^2*d*(m + n + 2)*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] && I
ntegerQ[2*m] && LtQ[m, -1] && (LtQ[n, 0] || IntegerQ[m]) &&  !(ILtQ[n, -1] && ( !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 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*} \text {integral}& = -\frac {\cot (c+d x) \sqrt {a+b \tan (c+d x)}}{a d}-\frac {\int \frac {\cot (c+d x) \left (\frac {b}{2}+a \tan (c+d x)+\frac {1}{2} b \tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx}{a} \\ & = -\frac {\cot (c+d x) \sqrt {a+b \tan (c+d x)}}{a d}-\frac {\int \frac {a}{\sqrt {a+b \tan (c+d x)}} \, dx}{a}-\frac {b \int \frac {\cot (c+d x) \left (1+\tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx}{2 a} \\ & = -\frac {\cot (c+d x) \sqrt {a+b \tan (c+d x)}}{a d}-\frac {b \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 a d}-\int \frac {1}{\sqrt {a+b \tan (c+d x)}} \, dx \\ & = -\frac {\cot (c+d x) \sqrt {a+b \tan (c+d x)}}{a d}-\frac {\text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{a d}-\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {a+x} \left (b^2+x^2\right )} \, dx,x,b \tan (c+d x)\right )}{d} \\ & = \frac {b \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}-\frac {\cot (c+d x) \sqrt {a+b \tan (c+d x)}}{a d}-\frac {(2 b) \text {Subst}\left (\int \frac {1}{a^2+b^2-2 a x^2+x^4} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{d} \\ & = \frac {b \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}-\frac {\cot (c+d x) \sqrt {a+b \tan (c+d x)}}{a d}-\frac {b \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}}-x}{\sqrt {a^2+b^2}-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{\sqrt {2} \sqrt {a^2+b^2} \sqrt {a+\sqrt {a^2+b^2}} d}-\frac {b \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}}+x}{\sqrt {a^2+b^2}+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{\sqrt {2} \sqrt {a^2+b^2} \sqrt {a+\sqrt {a^2+b^2}} d} \\ & = \frac {b \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}-\frac {\cot (c+d x) \sqrt {a+b \tan (c+d x)}}{a d}-\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {a^2+b^2}-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {a^2+b^2} d}-\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {a^2+b^2}+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {a^2+b^2} d}+\frac {b \text {Subst}\left (\int \frac {-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}}+2 x}{\sqrt {a^2+b^2}-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {a+\sqrt {a^2+b^2}} d}-\frac {b \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}}+2 x}{\sqrt {a^2+b^2}+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {a+\sqrt {a^2+b^2}} d} \\ & = \frac {b \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}+\frac {b \log \left (a+\sqrt {a^2+b^2}+b \tan (c+d x)-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {a+\sqrt {a^2+b^2}} d}-\frac {b \log \left (a+\sqrt {a^2+b^2}+b \tan (c+d x)+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {a+\sqrt {a^2+b^2}} d}-\frac {\cot (c+d x) \sqrt {a+b \tan (c+d x)}}{a d}+\frac {b \text {Subst}\left (\int \frac {1}{2 \left (a-\sqrt {a^2+b^2}\right )-x^2} \, dx,x,-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}}+2 \sqrt {a+b \tan (c+d x)}\right )}{\sqrt {a^2+b^2} d}+\frac {b \text {Subst}\left (\int \frac {1}{2 \left (a-\sqrt {a^2+b^2}\right )-x^2} \, dx,x,\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}}+2 \sqrt {a+b \tan (c+d x)}\right )}{\sqrt {a^2+b^2} d} \\ & = \frac {b \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}-\frac {b \text {arctanh}\left (\frac {\sqrt {a+\sqrt {a^2+b^2}}-\sqrt {2} \sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} \sqrt {a^2+b^2} \sqrt {a-\sqrt {a^2+b^2}} d}+\frac {b \text {arctanh}\left (\frac {\sqrt {a+\sqrt {a^2+b^2}}+\sqrt {2} \sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} \sqrt {a^2+b^2} \sqrt {a-\sqrt {a^2+b^2}} d}+\frac {b \log \left (a+\sqrt {a^2+b^2}+b \tan (c+d x)-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {a+\sqrt {a^2+b^2}} d}-\frac {b \log \left (a+\sqrt {a^2+b^2}+b \tan (c+d x)+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {a+\sqrt {a^2+b^2}} d}-\frac {\cot (c+d x) \sqrt {a+b \tan (c+d x)}}{a d} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.84 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.31 \[ \int \frac {\cot ^2(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\frac {\frac {b \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2}}+\frac {i \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{\sqrt {a-i b}}-\frac {i \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{\sqrt {a+i b}}-\frac {\cot (c+d x) \sqrt {a+b \tan (c+d x)}}{a}}{d} \]

[In]

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

[Out]

((b*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/a^(3/2) + (I*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/S
qrt[a - I*b] - (I*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/Sqrt[a + I*b] - (Cot[c + d*x]*Sqrt[a + b*Ta
n[c + d*x]])/a)/d

Maple [F(-1)]

Timed out.

hanged

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 822 vs. \(2 (376) = 752\).

Time = 0.30 (sec) , antiderivative size = 1660, normalized size of antiderivative = 3.60 \[ \int \frac {\cot ^2(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[-1/2*(a^2*d*sqrt(-((a^2 + b^2)*d^2*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + a)/((a^2 + b^2)*d^2))*log(sqrt(
b*tan(d*x + c) + a)*b + ((a^3 + a*b^2)*d^3*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + b^2*d)*sqrt(-((a^2 + b^2
)*d^2*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + a)/((a^2 + b^2)*d^2)))*tan(d*x + c) - a^2*d*sqrt(-((a^2 + b^2
)*d^2*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + a)/((a^2 + b^2)*d^2))*log(sqrt(b*tan(d*x + c) + a)*b - ((a^3
+ a*b^2)*d^3*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + b^2*d)*sqrt(-((a^2 + b^2)*d^2*sqrt(-b^2/((a^4 + 2*a^2*
b^2 + b^4)*d^4)) + a)/((a^2 + b^2)*d^2)))*tan(d*x + c) - a^2*d*sqrt(((a^2 + b^2)*d^2*sqrt(-b^2/((a^4 + 2*a^2*b
^2 + b^4)*d^4)) - a)/((a^2 + b^2)*d^2))*log(sqrt(b*tan(d*x + c) + a)*b + ((a^3 + a*b^2)*d^3*sqrt(-b^2/((a^4 +
2*a^2*b^2 + b^4)*d^4)) - b^2*d)*sqrt(((a^2 + b^2)*d^2*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - a)/((a^2 + b^
2)*d^2)))*tan(d*x + c) + a^2*d*sqrt(((a^2 + b^2)*d^2*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - a)/((a^2 + b^2
)*d^2))*log(sqrt(b*tan(d*x + c) + a)*b - ((a^3 + a*b^2)*d^3*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - b^2*d)*
sqrt(((a^2 + b^2)*d^2*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - a)/((a^2 + b^2)*d^2)))*tan(d*x + c) - sqrt(a)
*b*log((b*tan(d*x + c) + 2*sqrt(b*tan(d*x + c) + a)*sqrt(a) + 2*a)/tan(d*x + c))*tan(d*x + c) + 2*sqrt(b*tan(d
*x + c) + a)*a)/(a^2*d*tan(d*x + c)), -1/2*(a^2*d*sqrt(-((a^2 + b^2)*d^2*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4)*d^
4)) + a)/((a^2 + b^2)*d^2))*log(sqrt(b*tan(d*x + c) + a)*b + ((a^3 + a*b^2)*d^3*sqrt(-b^2/((a^4 + 2*a^2*b^2 +
b^4)*d^4)) + b^2*d)*sqrt(-((a^2 + b^2)*d^2*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + a)/((a^2 + b^2)*d^2)))*t
an(d*x + c) - a^2*d*sqrt(-((a^2 + b^2)*d^2*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + a)/((a^2 + b^2)*d^2))*lo
g(sqrt(b*tan(d*x + c) + a)*b - ((a^3 + a*b^2)*d^3*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + b^2*d)*sqrt(-((a^
2 + b^2)*d^2*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + a)/((a^2 + b^2)*d^2)))*tan(d*x + c) - a^2*d*sqrt(((a^2
+ b^2)*d^2*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - a)/((a^2 + b^2)*d^2))*log(sqrt(b*tan(d*x + c) + a)*b +
((a^3 + a*b^2)*d^3*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - b^2*d)*sqrt(((a^2 + b^2)*d^2*sqrt(-b^2/((a^4 + 2
*a^2*b^2 + b^4)*d^4)) - a)/((a^2 + b^2)*d^2)))*tan(d*x + c) + a^2*d*sqrt(((a^2 + b^2)*d^2*sqrt(-b^2/((a^4 + 2*
a^2*b^2 + b^4)*d^4)) - a)/((a^2 + b^2)*d^2))*log(sqrt(b*tan(d*x + c) + a)*b - ((a^3 + a*b^2)*d^3*sqrt(-b^2/((a
^4 + 2*a^2*b^2 + b^4)*d^4)) - b^2*d)*sqrt(((a^2 + b^2)*d^2*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - a)/((a^2
+ b^2)*d^2)))*tan(d*x + c) + 2*sqrt(-a)*b*arctan(sqrt(b*tan(d*x + c) + a)*sqrt(-a)/a)*tan(d*x + c) + 2*sqrt(b
*tan(d*x + c) + a)*a)/(a^2*d*tan(d*x + c))]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-1)]

Timed out. \[ \int \frac {\cot ^2(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 8.02 (sec) , antiderivative size = 2145, normalized size of antiderivative = 4.65 \[ \int \frac {\cot ^2(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

atan((((-(a - b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*((((16*(16*a*b^11*d^4 + 8*a^3*b^9*d^4))/(a^2*d^5) - (16*(-(
a - b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*(32*a^2*b^10*d^4 + 48*a^4*b^8*d^4)*(a + b*tan(c + d*x))^(1/2))/(a^2*d
^4))*(-(a - b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2) + (16*(4*a*b^10*d^2 - 20*a^3*b^8*d^2)*(a + b*tan(c + d*x))^(1
/2))/(a^2*d^4))*(-(a - b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2) + (16*(2*b^11*d^2 + 8*a^2*b^9*d^2))/(a^2*d^5)) + (
16*(b^10 - 2*a^2*b^8)*(a + b*tan(c + d*x))^(1/2))/(a^2*d^4))*(-(a - b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*1i -
((-(a - b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*((((16*(16*a*b^11*d^4 + 8*a^3*b^9*d^4))/(a^2*d^5) + (16*(-(a - b*
1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*(32*a^2*b^10*d^4 + 48*a^4*b^8*d^4)*(a + b*tan(c + d*x))^(1/2))/(a^2*d^4))*(
-(a - b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2) - (16*(4*a*b^10*d^2 - 20*a^3*b^8*d^2)*(a + b*tan(c + d*x))^(1/2))/(
a^2*d^4))*(-(a - b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2) + (16*(2*b^11*d^2 + 8*a^2*b^9*d^2))/(a^2*d^5)) - (16*(b^
10 - 2*a^2*b^8)*(a + b*tan(c + d*x))^(1/2))/(a^2*d^4))*(-(a - b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*1i)/(((-(a
- b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*((((16*(16*a*b^11*d^4 + 8*a^3*b^9*d^4))/(a^2*d^5) - (16*(-(a - b*1i)/(4
*a^2*d^2 + 4*b^2*d^2))^(1/2)*(32*a^2*b^10*d^4 + 48*a^4*b^8*d^4)*(a + b*tan(c + d*x))^(1/2))/(a^2*d^4))*(-(a -
b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2) + (16*(4*a*b^10*d^2 - 20*a^3*b^8*d^2)*(a + b*tan(c + d*x))^(1/2))/(a^2*d^
4))*(-(a - b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2) + (16*(2*b^11*d^2 + 8*a^2*b^9*d^2))/(a^2*d^5)) + (16*(b^10 - 2
*a^2*b^8)*(a + b*tan(c + d*x))^(1/2))/(a^2*d^4))*(-(a - b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2) + ((-(a - b*1i)/(
4*a^2*d^2 + 4*b^2*d^2))^(1/2)*((((16*(16*a*b^11*d^4 + 8*a^3*b^9*d^4))/(a^2*d^5) + (16*(-(a - b*1i)/(4*a^2*d^2
+ 4*b^2*d^2))^(1/2)*(32*a^2*b^10*d^4 + 48*a^4*b^8*d^4)*(a + b*tan(c + d*x))^(1/2))/(a^2*d^4))*(-(a - b*1i)/(4*
a^2*d^2 + 4*b^2*d^2))^(1/2) - (16*(4*a*b^10*d^2 - 20*a^3*b^8*d^2)*(a + b*tan(c + d*x))^(1/2))/(a^2*d^4))*(-(a
- b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2) + (16*(2*b^11*d^2 + 8*a^2*b^9*d^2))/(a^2*d^5)) - (16*(b^10 - 2*a^2*b^8)
*(a + b*tan(c + d*x))^(1/2))/(a^2*d^4))*(-(a - b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2) + (32*b^9)/(a*d^5)))*(-(a
- b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*2i + (log(- ((-1/(d^2*(a - b*1i)))^(1/2)*(((-1/(d^2*(a - b*1i)))^(1/2)*
(((-1/(d^2*(a - b*1i)))^(1/2)*(((-1/(d^2*(a - b*1i)))^(1/2)*(128*b^8*(-1/(d^2*(a - b*1i)))^(1/2)*(3*a^2 + 2*b^
2)*(a + b*tan(c + d*x))^(1/2) + (128*b^9*(a^2 + 2*b^2))/(a*d)))/2 + (64*b^8*(5*a^2 - b^2)*(a + b*tan(c + d*x))
^(1/2))/(a*d^2)))/2 + (32*b^9*(4*a^2 + b^2))/(a^2*d^3)))/2 - (16*(b^10 - 2*a^2*b^8)*(a + b*tan(c + d*x))^(1/2)
)/(a^2*d^4)))/2 - (16*b^9)/(a*d^5))*(-1/(a*d^2 - b*d^2*1i))^(1/2))/2 - log(((-1/(d^2*(a - b*1i)))^(1/2)*(((-1/
(d^2*(a - b*1i)))^(1/2)*(((-1/(d^2*(a - b*1i)))^(1/2)*(((-1/(d^2*(a - b*1i)))^(1/2)*(128*b^8*(-1/(d^2*(a - b*1
i)))^(1/2)*(3*a^2 + 2*b^2)*(a + b*tan(c + d*x))^(1/2) - (128*b^9*(a^2 + 2*b^2))/(a*d)))/2 + (64*b^8*(5*a^2 - b
^2)*(a + b*tan(c + d*x))^(1/2))/(a*d^2)))/2 - (32*b^9*(4*a^2 + b^2))/(a^2*d^3)))/2 - (16*(b^10 - 2*a^2*b^8)*(a
+ b*tan(c + d*x))^(1/2))/(a^2*d^4)))/2 - (16*b^9)/(a*d^5))*(-1/(4*(a*d^2 - b*d^2*1i)))^(1/2) + (b*(a + b*tan(
c + d*x))^(1/2))/(a*(a*d - d*(a + b*tan(c + d*x)))) - (b*atan((b^11*(a + b*tan(c + d*x))^(1/2)*64i)/((a^3)^(1/
2)*(32*a*b^9 + (64*b^11)/a + (32*b^13)/a^3 + (32*b^15)/a^5)) + (b^13*(a + b*tan(c + d*x))^(1/2)*32i)/((a^3)^(1
/2)*(64*a*b^11 + 32*a^3*b^9 + (32*b^13)/a + (32*b^15)/a^3)) + (b^15*(a + b*tan(c + d*x))^(1/2)*32i)/((a^3)^(1/
2)*(32*a*b^13 + 64*a^3*b^11 + 32*a^5*b^9 + (32*b^15)/a)) + (b^9*(a + b*tan(c + d*x))^(1/2)*32i)/((a^3)^(1/2)*(
(32*b^9)/a + (64*b^11)/a^3 + (32*b^13)/a^5 + (32*b^15)/a^7)))*1i)/(d*(a^3)^(1/2))