\(\int \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)} \, dx\) [509]

Optimal result

Integrand size = 23, antiderivative size = 415 \[ \int \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)} \, dx=-\frac {b \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}-\frac {b \text {arctanh}\left (\frac {\sqrt {a+\sqrt {a^2+b^2}}-\sqrt {2} \sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} \sqrt {a-\sqrt {a^2+b^2}} d}+\frac {b \text {arctanh}\left (\frac {\sqrt {a+\sqrt {a^2+b^2}}+\sqrt {2} \sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} \sqrt {a-\sqrt {a^2+b^2}} d}-\frac {b \log \left (a+\sqrt {a^2+b^2}+b \tan (c+d x)-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} d}+\frac {b \log \left (a+\sqrt {a^2+b^2}+b \tan (c+d x)+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} d}-\frac {\cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 415, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {3649, 3734, 3566, 714, 1143, 648, 632, 212, 642, 3715, 65, 214} \[ \int \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)} \, dx=-\frac {b \text {arctanh}\left (\frac {\sqrt {\sqrt {a^2+b^2}+a}-\sqrt {2} \sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} d \sqrt {a-\sqrt {a^2+b^2}}}+\frac {b \text {arctanh}\left (\frac {\sqrt {\sqrt {a^2+b^2}+a}+\sqrt {2} \sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} d \sqrt {a-\sqrt {a^2+b^2}}}-\frac {b \log \left (-\sqrt {2} \sqrt {\sqrt {a^2+b^2}+a} \sqrt {a+b \tan (c+d x)}+\sqrt {a^2+b^2}+a+b \tan (c+d x)\right )}{2 \sqrt {2} d \sqrt {\sqrt {a^2+b^2}+a}}+\frac {b \log \left (\sqrt {2} \sqrt {\sqrt {a^2+b^2}+a} \sqrt {a+b \tan (c+d x)}+\sqrt {a^2+b^2}+a+b \tan (c+d x)\right )}{2 \sqrt {2} d \sqrt {\sqrt {a^2+b^2}+a}}-\frac {b \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}-\frac {\cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d} \]

[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]])/(Sqrt[a]*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]*Sqrt[a - Sqrt[a^2 + b^2]]*d) + (b*ArcTanh[(Sqr
t[a + Sqrt[a^2 + b^2]] + Sqrt[2]*Sqrt[a + b*Tan[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^2]]])/(Sqrt[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*Ta
n[c + d*x]]])/(2*Sqrt[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]*Sqrt[a + Sqrt[a^2 + b^2]]*d) - (Cot[c + d*x]*S
qrt[a + b*Tan[c + d*x]])/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 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 714

Int[Sqrt[(d_) + (e_.)*(x_)]/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[2*e, Subst[Int[x^2/(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 1143

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/
c, 2]}, Dist[1/(2*c*r), Int[x^(m - 1)/(q - r*x + x^2), x], x] - Dist[1/(2*c*r), Int[x^(m - 1)/(q + r*x + x^2),
x], x]]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && GeQ[m, 1] && LtQ[m, 3] && 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 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]

Rubi steps \begin{align*} \text {integral}& = -\frac {\cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}-\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 \\ & = -\frac {\cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}+\frac {1}{2} b \int \frac {\cot (c+d x) \left (1+\tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx-\int \sqrt {a+b \tan (c+d x)} \, dx \\ & = -\frac {\cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}+\frac {b \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d}-\frac {b \text {Subst}\left (\int \frac {\sqrt {a+x}}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{d} \\ & = -\frac {\cot (c+d x) \sqrt {a+b \tan (c+d x)}}{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 )}{d}-\frac {(2 b) \text {Subst}\left (\int \frac {x^2}{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 )}{\sqrt {a} d}-\frac {\cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}-\frac {b \text {Subst}\left (\int \frac {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+\sqrt {a^2+b^2}} d}+\frac {b \text {Subst}\left (\int \frac {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+\sqrt {a^2+b^2}} d} \\ & = -\frac {b \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}-\frac {\cot (c+d x) \sqrt {a+b \tan (c+d x)}}{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 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 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+\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+\sqrt {a^2+b^2}} d} \\ & = -\frac {b \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} 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+\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+\sqrt {a^2+b^2}} d}-\frac {\cot (c+d x) \sqrt {a+b \tan (c+d x)}}{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 )}{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 )}{d} \\ & = -\frac {b \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} 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-\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-\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+\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+\sqrt {a^2+b^2}} d}-\frac {\cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.81 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.33 \[ \int \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)} \, dx=-\frac {\frac {b \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a}}-i \sqrt {a-i b} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )+i \sqrt {a+i b} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )+\cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d} \]

[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]])/Sqrt[a] - I*Sqrt[a - I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sq
rt[a - I*b]] + I*Sqrt[a + I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + Cot[c + d*x]*Sqrt[a + b*Tan[c
+ d*x]])/d)

Maple [B] (warning: unable to verify)

result has leaf size over 500,000. Avoiding possible recursion issues.

Time = 34.82 (sec) , antiderivative size = 1278502, normalized size of antiderivative = 3080.73

\[\text {output too large to display}\]

[In]

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

[Out]

result too large to display

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 847, normalized size of antiderivative = 2.04 \[ \int \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)} \, dx=\left [\frac {a d \sqrt {-\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} + a}{d^{2}}} \log \left (d^{3} \sqrt {-\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} + a}{d^{2}}} \sqrt {-\frac {b^{2}}{d^{4}}} + \sqrt {b \tan \left (d x + c\right ) + a} b\right ) \tan \left (d x + c\right ) - a d \sqrt {-\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} + a}{d^{2}}} \log \left (-d^{3} \sqrt {-\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} + a}{d^{2}}} \sqrt {-\frac {b^{2}}{d^{4}}} + \sqrt {b \tan \left (d x + c\right ) + a} b\right ) \tan \left (d x + c\right ) - a d \sqrt {\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} - a}{d^{2}}} \log \left (d^{3} \sqrt {\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} - a}{d^{2}}} \sqrt {-\frac {b^{2}}{d^{4}}} + \sqrt {b \tan \left (d x + c\right ) + a} b\right ) \tan \left (d x + c\right ) + a d \sqrt {\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} - a}{d^{2}}} \log \left (-d^{3} \sqrt {\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} - a}{d^{2}}} \sqrt {-\frac {b^{2}}{d^{4}}} + \sqrt {b \tan \left (d x + c\right ) + a} b\right ) \tan \left (d x + c\right ) + \sqrt {a} b \log \left (\frac {b \tan \left (d x + c\right ) - 2 \, \sqrt {b \tan \left (d x + c\right ) + a} \sqrt {a} + 2 \, a}{\tan \left (d x + c\right )}\right ) \tan \left (d x + c\right ) - 2 \, \sqrt {b \tan \left (d x + c\right ) + a} a}{2 \, a d \tan \left (d x + c\right )}, \frac {a d \sqrt {-\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} + a}{d^{2}}} \log \left (d^{3} \sqrt {-\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} + a}{d^{2}}} \sqrt {-\frac {b^{2}}{d^{4}}} + \sqrt {b \tan \left (d x + c\right ) + a} b\right ) \tan \left (d x + c\right ) - a d \sqrt {-\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} + a}{d^{2}}} \log \left (-d^{3} \sqrt {-\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} + a}{d^{2}}} \sqrt {-\frac {b^{2}}{d^{4}}} + \sqrt {b \tan \left (d x + c\right ) + a} b\right ) \tan \left (d x + c\right ) - a d \sqrt {\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} - a}{d^{2}}} \log \left (d^{3} \sqrt {\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} - a}{d^{2}}} \sqrt {-\frac {b^{2}}{d^{4}}} + \sqrt {b \tan \left (d x + c\right ) + a} b\right ) \tan \left (d x + c\right ) + a d \sqrt {\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} - a}{d^{2}}} \log \left (-d^{3} \sqrt {\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} - a}{d^{2}}} \sqrt {-\frac {b^{2}}{d^{4}}} + \sqrt {b \tan \left (d x + c\right ) + a} b\right ) \tan \left (d x + c\right ) + 2 \, \sqrt {-a} b \arctan \left (\frac {\sqrt {b \tan \left (d x + c\right ) + a} \sqrt {-a}}{a}\right ) \tan \left (d x + c\right ) - 2 \, \sqrt {b \tan \left (d x + c\right ) + a} a}{2 \, a d \tan \left (d x + c\right )}\right ] \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-1)]

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

[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 = 4.90 (sec) , antiderivative size = 832, normalized size of antiderivative = 2.00 \[ \int \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)} \, dx=\mathrm {atan}\left (-\frac {b^{12}\,\sqrt {-\frac {a}{4\,d^2}-\frac {b\,1{}\mathrm {i}}{4\,d^2}}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}\,32{}\mathrm {i}}{\frac {16\,b^{13}}{d}+\frac {32\,a^2\,b^{11}}{d}+\frac {16\,a^4\,b^9}{d}+\frac {a\,b^{12}\,16{}\mathrm {i}}{d}+\frac {a^3\,b^{10}\,16{}\mathrm {i}}{d}}+\frac {64\,a\,b^{11}\,\sqrt {-\frac {a}{4\,d^2}-\frac {b\,1{}\mathrm {i}}{4\,d^2}}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{\frac {16\,b^{13}}{d}+\frac {32\,a^2\,b^{11}}{d}+\frac {16\,a^4\,b^9}{d}+\frac {a\,b^{12}\,16{}\mathrm {i}}{d}+\frac {a^3\,b^{10}\,16{}\mathrm {i}}{d}}+\frac {32\,a^3\,b^9\,\sqrt {-\frac {a}{4\,d^2}-\frac {b\,1{}\mathrm {i}}{4\,d^2}}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{\frac {16\,b^{13}}{d}+\frac {32\,a^2\,b^{11}}{d}+\frac {16\,a^4\,b^9}{d}+\frac {a\,b^{12}\,16{}\mathrm {i}}{d}+\frac {a^3\,b^{10}\,16{}\mathrm {i}}{d}}\right )\,\sqrt {-\frac {a+b\,1{}\mathrm {i}}{4\,d^2}}\,2{}\mathrm {i}-\mathrm {atan}\left (\frac {b^{12}\,\sqrt {-\frac {a}{4\,d^2}+\frac {b\,1{}\mathrm {i}}{4\,d^2}}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}\,32{}\mathrm {i}}{\frac {16\,b^{13}}{d}+\frac {32\,a^2\,b^{11}}{d}+\frac {16\,a^4\,b^9}{d}-\frac {a\,b^{12}\,16{}\mathrm {i}}{d}-\frac {a^3\,b^{10}\,16{}\mathrm {i}}{d}}+\frac {64\,a\,b^{11}\,\sqrt {-\frac {a}{4\,d^2}+\frac {b\,1{}\mathrm {i}}{4\,d^2}}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{\frac {16\,b^{13}}{d}+\frac {32\,a^2\,b^{11}}{d}+\frac {16\,a^4\,b^9}{d}-\frac {a\,b^{12}\,16{}\mathrm {i}}{d}-\frac {a^3\,b^{10}\,16{}\mathrm {i}}{d}}+\frac {32\,a^3\,b^9\,\sqrt {-\frac {a}{4\,d^2}+\frac {b\,1{}\mathrm {i}}{4\,d^2}}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{\frac {16\,b^{13}}{d}+\frac {32\,a^2\,b^{11}}{d}+\frac {16\,a^4\,b^9}{d}-\frac {a\,b^{12}\,16{}\mathrm {i}}{d}-\frac {a^3\,b^{10}\,16{}\mathrm {i}}{d}}\right )\,\sqrt {-\frac {a-b\,1{}\mathrm {i}}{4\,d^2}}\,2{}\mathrm {i}+\frac {b\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{a\,d-d\,\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}+\frac {b\,\mathrm {atan}\left (\frac {b^{13}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}\,128{}\mathrm {i}}{\sqrt {a}\,\left (128\,b^{13}+128\,a^2\,b^{11}+32\,a^4\,b^9+\frac {32\,b^{15}}{a^2}\right )}+\frac {a^{3/2}\,b^{11}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}\,128{}\mathrm {i}}{128\,b^{13}+128\,a^2\,b^{11}+32\,a^4\,b^9+\frac {32\,b^{15}}{a^2}}+\frac {a^{7/2}\,b^9\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}\,32{}\mathrm {i}}{128\,b^{13}+128\,a^2\,b^{11}+32\,a^4\,b^9+\frac {32\,b^{15}}{a^2}}+\frac {b^{15}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}\,32{}\mathrm {i}}{a^{5/2}\,\left (128\,b^{13}+128\,a^2\,b^{11}+32\,a^4\,b^9+\frac {32\,b^{15}}{a^2}\right )}\right )\,1{}\mathrm {i}}{\sqrt {a}\,d} \]

[In]

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

[Out]

atan((64*a*b^11*(- a/(4*d^2) - (b*1i)/(4*d^2))^(1/2)*(a + b*tan(c + d*x))^(1/2))/((16*b^13)/d + (a*b^12*16i)/d
+ (32*a^2*b^11)/d + (a^3*b^10*16i)/d + (16*a^4*b^9)/d) - (b^12*(- a/(4*d^2) - (b*1i)/(4*d^2))^(1/2)*(a + b*ta
n(c + d*x))^(1/2)*32i)/((16*b^13)/d + (a*b^12*16i)/d + (32*a^2*b^11)/d + (a^3*b^10*16i)/d + (16*a^4*b^9)/d) +
(32*a^3*b^9*(- a/(4*d^2) - (b*1i)/(4*d^2))^(1/2)*(a + b*tan(c + d*x))^(1/2))/((16*b^13)/d + (a*b^12*16i)/d + (
32*a^2*b^11)/d + (a^3*b^10*16i)/d + (16*a^4*b^9)/d))*(-(a + b*1i)/(4*d^2))^(1/2)*2i - atan((b^12*((b*1i)/(4*d^
2) - a/(4*d^2))^(1/2)*(a + b*tan(c + d*x))^(1/2)*32i)/((16*b^13)/d - (a*b^12*16i)/d + (32*a^2*b^11)/d - (a^3*b
^10*16i)/d + (16*a^4*b^9)/d) + (64*a*b^11*((b*1i)/(4*d^2) - a/(4*d^2))^(1/2)*(a + b*tan(c + d*x))^(1/2))/((16*
b^13)/d - (a*b^12*16i)/d + (32*a^2*b^11)/d - (a^3*b^10*16i)/d + (16*a^4*b^9)/d) + (32*a^3*b^9*((b*1i)/(4*d^2)
- a/(4*d^2))^(1/2)*(a + b*tan(c + d*x))^(1/2))/((16*b^13)/d - (a*b^12*16i)/d + (32*a^2*b^11)/d - (a^3*b^10*16i
)/d + (16*a^4*b^9)/d))*(-(a - b*1i)/(4*d^2))^(1/2)*2i + (b*(a + b*tan(c + d*x))^(1/2))/(a*d - d*(a + b*tan(c +
d*x))) + (b*atan((b^13*(a + b*tan(c + d*x))^(1/2)*128i)/(a^(1/2)*(128*b^13 + 128*a^2*b^11 + 32*a^4*b^9 + (32*
b^15)/a^2)) + (a^(3/2)*b^11*(a + b*tan(c + d*x))^(1/2)*128i)/(128*b^13 + 128*a^2*b^11 + 32*a^4*b^9 + (32*b^15)
/a^2) + (a^(7/2)*b^9*(a + b*tan(c + d*x))^(1/2)*32i)/(128*b^13 + 128*a^2*b^11 + 32*a^4*b^9 + (32*b^15)/a^2) +
(b^15*(a + b*tan(c + d*x))^(1/2)*32i)/(a^(5/2)*(128*b^13 + 128*a^2*b^11 + 32*a^4*b^9 + (32*b^15)/a^2)))*1i)/(a
^(1/2)*d)