\(\int \frac {\sqrt {\cot (c+d x)} (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^{3/2}} \, dx\) [653]

Optimal result

Integrand size = 38, antiderivative size = 151 \[ \int \frac {\sqrt {\cot (c+d x)} (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^{3/2}} \, dx=\frac {B \arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{\sqrt {i a-b} d}+\frac {B \text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{\sqrt {i a+b} d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.184, Rules used = {21, 4326, 3656, 926, 95, 211, 214} \[ \int \frac {\sqrt {\cot (c+d x)} (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^{3/2}} \, dx=\frac {B \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d \sqrt {-b+i a}}+\frac {B \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \text {arctanh}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d \sqrt {b+i a}} \]

[In]

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

[Out]

(B*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/
(Sqrt[I*a - b]*d) + (B*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]
*Sqrt[Tan[c + d*x]])/(Sqrt[I*a + b]*d)

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

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 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 926

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)^n, 1/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && NeQ[c*d^2 + a*e^2,
0] &&  !IntegerQ[m] &&  !IntegerQ[n]

Rule 3656

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Wit
h[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[(a + b*ff*x)^m*((c + d*ff*x)^n/(1 + ff^2*x^2)), x]
, x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] &&
NeQ[c^2 + d^2, 0]

Rule 4326

Int[(cot[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Dist[(c*Cot[a + b*x])^m*(c*Tan[a + b*x])^m, Int[A
ctivateTrig[u]/(c*Tan[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[m] && KnownTangentIntegrandQ
[u, x]

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt {\cot (c+d x)} (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^{3/2}} \, dx=\frac {(-1)^{3/4} B \left (-\frac {\arctan \left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{\sqrt {-a+i b}}-\frac {\arctan \left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{\sqrt {a+i b}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d} \]

[In]

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

[Out]

((-1)^(3/4)*B*(-(ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/Sqrt[-a + I*b
]) - ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/Sqrt[a + I*b])*Sqrt[Cot[c
+ d*x]]*Sqrt[Tan[c + d*x]])/d

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(761\) vs. \(2(123)=246\).

Time = 16.24 (sec) , antiderivative size = 762, normalized size of antiderivative = 5.05

method	result	size
default	\(\frac {B \sin \left (d x +c \right ) \left (\sqrt {b +\sqrt {a^{2}+b^{2}}}\, \ln \left (\frac {a \cot \left (d x +c \right ) \cos \left (d x +c \right )-2 a \cot \left (d x +c \right )-2 \sqrt {\left (\cot \left (d x +c \right )^{2} a -2 a \cot \left (d x +c \right ) \csc \left (d x +c \right )+a \csc \left (d x +c \right )^{2}-2 b \csc \left (d x +c \right )+2 \cot \left (d x +c \right ) b -a \right ) \left (\cos \left (d x +c \right )-1\right ) \csc \left (d x +c \right )}\, \sqrt {b +\sqrt {a^{2}+b^{2}}}\, \sin \left (d x +c \right )+a \csc \left (d x +c \right )+2 \sqrt {a^{2}+b^{2}}\, \cos \left (d x +c \right )+2 b \cos \left (d x +c \right )-a \sin \left (d x +c \right )-2 \sqrt {a^{2}+b^{2}}-2 b}{\cos \left (d x +c \right )-1}\right )-2 \sqrt {-b +\sqrt {a^{2}+b^{2}}}\, \arctan \left (\frac {\sqrt {b +\sqrt {a^{2}+b^{2}}}\, \cos \left (d x +c \right )-\sqrt {-\frac {2 \left (\cos \left (d x +c \right )^{2} b -\cos \left (d x +c \right ) \sin \left (d x +c \right ) a -b \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, \sin \left (d x +c \right )-\sqrt {b +\sqrt {a^{2}+b^{2}}}}{\sqrt {-b +\sqrt {a^{2}+b^{2}}}\, \left (\cos \left (d x +c \right )-1\right )}\right )-\sqrt {b +\sqrt {a^{2}+b^{2}}}\, \ln \left (\frac {a \cot \left (d x +c \right ) \cos \left (d x +c \right )-2 a \cot \left (d x +c \right )+2 \sqrt {\left (\cot \left (d x +c \right )^{2} a -2 a \cot \left (d x +c \right ) \csc \left (d x +c \right )+a \csc \left (d x +c \right )^{2}-2 b \csc \left (d x +c \right )+2 \cot \left (d x +c \right ) b -a \right ) \left (\cos \left (d x +c \right )-1\right ) \csc \left (d x +c \right )}\, \sqrt {b +\sqrt {a^{2}+b^{2}}}\, \sin \left (d x +c \right )+a \csc \left (d x +c \right )+2 \sqrt {a^{2}+b^{2}}\, \cos \left (d x +c \right )+2 b \cos \left (d x +c \right )-a \sin \left (d x +c \right )-2 \sqrt {a^{2}+b^{2}}-2 b}{\cos \left (d x +c \right )-1}\right )+2 \sqrt {-b +\sqrt {a^{2}+b^{2}}}\, \arctan \left (\frac {\sqrt {b +\sqrt {a^{2}+b^{2}}}\, \cos \left (d x +c \right )+\sqrt {-\frac {2 \left (\cos \left (d x +c \right )^{2} b -\cos \left (d x +c \right ) \sin \left (d x +c \right ) a -b \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, \sin \left (d x +c \right )-\sqrt {b +\sqrt {a^{2}+b^{2}}}}{\sqrt {-b +\sqrt {a^{2}+b^{2}}}\, \left (\cos \left (d x +c \right )-1\right )}\right )\right ) \sqrt {\cot \left (d x +c \right )}\, \sqrt {a +b \tan \left (d x +c \right )}}{2 d \sqrt {a^{2}+b^{2}}\, \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {2 \left (\cos \left (d x +c \right )^{2} b -\cos \left (d x +c \right ) \sin \left (d x +c \right ) a -b \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}}\)	\(762\)

[In]

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

[Out]

1/2*B/d/(a^2+b^2)^(1/2)*sin(d*x+c)*((b+(a^2+b^2)^(1/2))^(1/2)*ln((a*cot(d*x+c)*cos(d*x+c)-2*a*cot(d*x+c)-2*((c
ot(d*x+c)^2*a-2*a*cot(d*x+c)*csc(d*x+c)+a*csc(d*x+c)^2-2*b*csc(d*x+c)+2*cot(d*x+c)*b-a)*(cos(d*x+c)-1)*csc(d*x
+c))^(1/2)*(b+(a^2+b^2)^(1/2))^(1/2)*sin(d*x+c)+a*csc(d*x+c)+2*(a^2+b^2)^(1/2)*cos(d*x+c)+2*b*cos(d*x+c)-a*sin
(d*x+c)-2*(a^2+b^2)^(1/2)-2*b)/(cos(d*x+c)-1))-2*(-b+(a^2+b^2)^(1/2))^(1/2)*arctan(1/(-b+(a^2+b^2)^(1/2))^(1/2
)*((b+(a^2+b^2)^(1/2))^(1/2)*cos(d*x+c)-(-2*(cos(d*x+c)^2*b-cos(d*x+c)*sin(d*x+c)*a-b)/(cos(d*x+c)+1)^2)^(1/2)
*sin(d*x+c)-(b+(a^2+b^2)^(1/2))^(1/2))/(cos(d*x+c)-1))-(b+(a^2+b^2)^(1/2))^(1/2)*ln((a*cot(d*x+c)*cos(d*x+c)-2
*a*cot(d*x+c)+2*((cot(d*x+c)^2*a-2*a*cot(d*x+c)*csc(d*x+c)+a*csc(d*x+c)^2-2*b*csc(d*x+c)+2*cot(d*x+c)*b-a)*(co
s(d*x+c)-1)*csc(d*x+c))^(1/2)*(b+(a^2+b^2)^(1/2))^(1/2)*sin(d*x+c)+a*csc(d*x+c)+2*(a^2+b^2)^(1/2)*cos(d*x+c)+2
*b*cos(d*x+c)-a*sin(d*x+c)-2*(a^2+b^2)^(1/2)-2*b)/(cos(d*x+c)-1))+2*(-b+(a^2+b^2)^(1/2))^(1/2)*arctan(1/(-b+(a
^2+b^2)^(1/2))^(1/2)*((b+(a^2+b^2)^(1/2))^(1/2)*cos(d*x+c)+(-2*(cos(d*x+c)^2*b-cos(d*x+c)*sin(d*x+c)*a-b)/(cos
(d*x+c)+1)^2)^(1/2)*sin(d*x+c)-(b+(a^2+b^2)^(1/2))^(1/2))/(cos(d*x+c)-1)))*cot(d*x+c)^(1/2)*(a+b*tan(d*x+c))^(
1/2)/(cos(d*x+c)+1)/(-2*(cos(d*x+c)^2*b-cos(d*x+c)*sin(d*x+c)*a-b)/(cos(d*x+c)+1)^2)^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4609 vs. \(2 (119) = 238\).

Time = 0.79 (sec) , antiderivative size = 4609, normalized size of antiderivative = 30.52 \[ \int \frac {\sqrt {\cot (c+d x)} (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^{3/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/8*sqrt(((a^2 + b^2)*sqrt(-B^4*a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2 + B^2*b)/((a^2 + b^2)*d^2))*log(1/2*((2
*(B^2*a^3*b^3 + 4*B^2*a*b^5)*d*tan(d*x + c)^2 + 2*(B^2*a^6 + 5*B^2*a^4*b^2 + 8*B^2*a^2*b^4)*d*tan(d*x + c) + 2
*(B^2*a^5*b + 2*B^2*a^3*b^3)*d - ((a^7 + 8*a^5*b^2 + 19*a^3*b^4 + 12*a*b^6)*d^3*tan(d*x + c)^2 + 2*(a^6*b + 2*
a^4*b^3 - 3*a^2*b^5 - 4*b^7)*d^3*tan(d*x + c) - (a^7 + 4*a^5*b^2 + 7*a^3*b^4 + 4*a*b^6)*d^3)*sqrt(-B^4*a^2/((a
^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt(((a^2 + b^2)*sqrt(-B^4*a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2 + B^2*b)/((a^2
+ b^2)*d^2)) + 2*((B^3*a^5 + 3*B^3*a^3*b^2 + 4*B^3*a*b^4)*tan(d*x + c)^2 + 2*(B^3*a^4*b + 2*B^3*a^2*b^3)*tan(
d*x + c) - (2*(B*a^5*b + 3*B*a^3*b^3 + 2*B*a*b^5)*d^2*tan(d*x + c)^2 - (B*a^6 + 4*B*a^4*b^2 + 7*B*a^2*b^4 + 4*
B*b^6)*d^2*tan(d*x + c))*sqrt(-B^4*a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt(b*tan(d*x + c) + a)/sqrt(tan(d*x +
c)))/(tan(d*x + c)^2 + 1)) + 1/8*sqrt(((a^2 + b^2)*sqrt(-B^4*a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2 + B^2*b)/
((a^2 + b^2)*d^2))*log(-1/2*((2*(B^2*a^3*b^3 + 4*B^2*a*b^5)*d*tan(d*x + c)^2 + 2*(B^2*a^6 + 5*B^2*a^4*b^2 + 8*
B^2*a^2*b^4)*d*tan(d*x + c) + 2*(B^2*a^5*b + 2*B^2*a^3*b^3)*d - ((a^7 + 8*a^5*b^2 + 19*a^3*b^4 + 12*a*b^6)*d^3
*tan(d*x + c)^2 + 2*(a^6*b + 2*a^4*b^3 - 3*a^2*b^5 - 4*b^7)*d^3*tan(d*x + c) - (a^7 + 4*a^5*b^2 + 7*a^3*b^4 +
4*a*b^6)*d^3)*sqrt(-B^4*a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt(((a^2 + b^2)*sqrt(-B^4*a^2/((a^4 + 2*a^2*b^2
+ b^4)*d^4))*d^2 + B^2*b)/((a^2 + b^2)*d^2)) + 2*((B^3*a^5 + 3*B^3*a^3*b^2 + 4*B^3*a*b^4)*tan(d*x + c)^2 + 2*(
B^3*a^4*b + 2*B^3*a^2*b^3)*tan(d*x + c) - (2*(B*a^5*b + 3*B*a^3*b^3 + 2*B*a*b^5)*d^2*tan(d*x + c)^2 - (B*a^6 +
4*B*a^4*b^2 + 7*B*a^2*b^4 + 4*B*b^6)*d^2*tan(d*x + c))*sqrt(-B^4*a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt(b*t
an(d*x + c) + a)/sqrt(tan(d*x + c)))/(tan(d*x + c)^2 + 1)) - 1/8*sqrt(((a^2 + b^2)*sqrt(-B^4*a^2/((a^4 + 2*a^2
*b^2 + b^4)*d^4))*d^2 + B^2*b)/((a^2 + b^2)*d^2))*log(1/2*((2*(B^2*a^3*b^3 + 4*B^2*a*b^5)*d*tan(d*x + c)^2 + 2
*(B^2*a^6 + 5*B^2*a^4*b^2 + 8*B^2*a^2*b^4)*d*tan(d*x + c) + 2*(B^2*a^5*b + 2*B^2*a^3*b^3)*d - ((a^7 + 8*a^5*b^
2 + 19*a^3*b^4 + 12*a*b^6)*d^3*tan(d*x + c)^2 + 2*(a^6*b + 2*a^4*b^3 - 3*a^2*b^5 - 4*b^7)*d^3*tan(d*x + c) - (
a^7 + 4*a^5*b^2 + 7*a^3*b^4 + 4*a*b^6)*d^3)*sqrt(-B^4*a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt(((a^2 + b^2)*sq
rt(-B^4*a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2 + B^2*b)/((a^2 + b^2)*d^2)) - 2*((B^3*a^5 + 3*B^3*a^3*b^2 + 4*B
^3*a*b^4)*tan(d*x + c)^2 + 2*(B^3*a^4*b + 2*B^3*a^2*b^3)*tan(d*x + c) - (2*(B*a^5*b + 3*B*a^3*b^3 + 2*B*a*b^5)
*d^2*tan(d*x + c)^2 - (B*a^6 + 4*B*a^4*b^2 + 7*B*a^2*b^4 + 4*B*b^6)*d^2*tan(d*x + c))*sqrt(-B^4*a^2/((a^4 + 2*
a^2*b^2 + b^4)*d^4)))*sqrt(b*tan(d*x + c) + a)/sqrt(tan(d*x + c)))/(tan(d*x + c)^2 + 1)) - 1/8*sqrt(((a^2 + b^
2)*sqrt(-B^4*a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2 + B^2*b)/((a^2 + b^2)*d^2))*log(-1/2*((2*(B^2*a^3*b^3 + 4*
B^2*a*b^5)*d*tan(d*x + c)^2 + 2*(B^2*a^6 + 5*B^2*a^4*b^2 + 8*B^2*a^2*b^4)*d*tan(d*x + c) + 2*(B^2*a^5*b + 2*B^
2*a^3*b^3)*d - ((a^7 + 8*a^5*b^2 + 19*a^3*b^4 + 12*a*b^6)*d^3*tan(d*x + c)^2 + 2*(a^6*b + 2*a^4*b^3 - 3*a^2*b^
5 - 4*b^7)*d^3*tan(d*x + c) - (a^7 + 4*a^5*b^2 + 7*a^3*b^4 + 4*a*b^6)*d^3)*sqrt(-B^4*a^2/((a^4 + 2*a^2*b^2 + b
^4)*d^4)))*sqrt(((a^2 + b^2)*sqrt(-B^4*a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2 + B^2*b)/((a^2 + b^2)*d^2)) - 2*
((B^3*a^5 + 3*B^3*a^3*b^2 + 4*B^3*a*b^4)*tan(d*x + c)^2 + 2*(B^3*a^4*b + 2*B^3*a^2*b^3)*tan(d*x + c) - (2*(B*a
^5*b + 3*B*a^3*b^3 + 2*B*a*b^5)*d^2*tan(d*x + c)^2 - (B*a^6 + 4*B*a^4*b^2 + 7*B*a^2*b^4 + 4*B*b^6)*d^2*tan(d*x
+ c))*sqrt(-B^4*a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt(b*tan(d*x + c) + a)/sqrt(tan(d*x + c)))/(tan(d*x + c
)^2 + 1)) + 1/8*sqrt(-((a^2 + b^2)*sqrt(-B^4*a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2 - B^2*b)/((a^2 + b^2)*d^2)
)*log(1/2*((2*(B^2*a^3*b^3 + 4*B^2*a*b^5)*d*tan(d*x + c)^2 + 2*(B^2*a^6 + 5*B^2*a^4*b^2 + 8*B^2*a^2*b^4)*d*tan
(d*x + c) + 2*(B^2*a^5*b + 2*B^2*a^3*b^3)*d + ((a^7 + 8*a^5*b^2 + 19*a^3*b^4 + 12*a*b^6)*d^3*tan(d*x + c)^2 +
2*(a^6*b + 2*a^4*b^3 - 3*a^2*b^5 - 4*b^7)*d^3*tan(d*x + c) - (a^7 + 4*a^5*b^2 + 7*a^3*b^4 + 4*a*b^6)*d^3)*sqrt
(-B^4*a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt(-((a^2 + b^2)*sqrt(-B^4*a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2
- B^2*b)/((a^2 + b^2)*d^2)) + 2*((B^3*a^5 + 3*B^3*a^3*b^2 + 4*B^3*a*b^4)*tan(d*x + c)^2 + 2*(B^3*a^4*b + 2*B^3
*a^2*b^3)*tan(d*x + c) + (2*(B*a^5*b + 3*B*a^3*b^3 + 2*B*a*b^5)*d^2*tan(d*x + c)^2 - (B*a^6 + 4*B*a^4*b^2 + 7*
B*a^2*b^4 + 4*B*b^6)*d^2*tan(d*x + c))*sqrt(-B^4*a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt(b*tan(d*x + c) + a)/
sqrt(tan(d*x + c)))/(tan(d*x + c)^2 + 1)) + 1/8*sqrt(-((a^2 + b^2)*sqrt(-B^4*a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)
)*d^2 - B^2*b)/((a^2 + b^2)*d^2))*log(-1/2*((2*(B^2*a^3*b^3 + 4*B^2*a*b^5)*d*tan(d*x + c)^2 + 2*(B^2*a^6 + 5*B
^2*a^4*b^2 + 8*B^2*a^2*b^4)*d*tan(d*x + c) + 2*(B^2*a^5*b + 2*B^2*a^3*b^3)*d + ((a^7 + 8*a^5*b^2 + 19*a^3*b^4
+ 12*a*b^6)*d^3*tan(d*x + c)^2 + 2*(a^6*b + 2*a^4*b^3 - 3*a^2*b^5 - 4*b^7)*d^3*tan(d*x + c) - (a^7 + 4*a^5*b^2
+ 7*a^3*b^4 + 4*a*b^6)*d^3)*sqrt(-B^4*a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt(-((a^2 + b^2)*sqrt(-B^4*a^2/((
a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2 - B^2*b)/((a^2 + b^2)*d^2)) + 2*((B^3*a^5 + 3*B^3*a^3*b^2 + 4*B^3*a*b^4)*tan(
d*x + c)^2 + 2*(B^3*a^4*b + 2*B^3*a^2*b^3)*tan(d*x + c) + (2*(B*a^5*b + 3*B*a^3*b^3 + 2*B*a*b^5)*d^2*tan(d*x +
c)^2 - (B*a^6 + 4*B*a^4*b^2 + 7*B*a^2*b^4 + 4*B*b^6)*d^2*tan(d*x + c))*sqrt(-B^4*a^2/((a^4 + 2*a^2*b^2 + b^4)
*d^4)))*sqrt(b*tan(d*x + c) + a)/sqrt(tan(d*x + c)))/(tan(d*x + c)^2 + 1)) - 1/8*sqrt(-((a^2 + b^2)*sqrt(-B^4*
a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2 - B^2*b)/((a^2 + b^2)*d^2))*log(1/2*((2*(B^2*a^3*b^3 + 4*B^2*a*b^5)*d*t
an(d*x + c)^2 + 2*(B^2*a^6 + 5*B^2*a^4*b^2 + 8*B^2*a^2*b^4)*d*tan(d*x + c) + 2*(B^2*a^5*b + 2*B^2*a^3*b^3)*d +
((a^7 + 8*a^5*b^2 + 19*a^3*b^4 + 12*a*b^6)*d^3*tan(d*x + c)^2 + 2*(a^6*b + 2*a^4*b^3 - 3*a^2*b^5 - 4*b^7)*d^3
*tan(d*x + c) - (a^7 + 4*a^5*b^2 + 7*a^3*b^4 + 4*a*b^6)*d^3)*sqrt(-B^4*a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqr
t(-((a^2 + b^2)*sqrt(-B^4*a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2 - B^2*b)/((a^2 + b^2)*d^2)) - 2*((B^3*a^5 + 3
*B^3*a^3*b^2 + 4*B^3*a*b^4)*tan(d*x + c)^2 + 2*(B^3*a^4*b + 2*B^3*a^2*b^3)*tan(d*x + c) + (2*(B*a^5*b + 3*B*a^
3*b^3 + 2*B*a*b^5)*d^2*tan(d*x + c)^2 - (B*a^6 + 4*B*a^4*b^2 + 7*B*a^2*b^4 + 4*B*b^6)*d^2*tan(d*x + c))*sqrt(-
B^4*a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt(b*tan(d*x + c) + a)/sqrt(tan(d*x + c)))/(tan(d*x + c)^2 + 1)) - 1
/8*sqrt(-((a^2 + b^2)*sqrt(-B^4*a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2 - B^2*b)/((a^2 + b^2)*d^2))*log(-1/2*((
2*(B^2*a^3*b^3 + 4*B^2*a*b^5)*d*tan(d*x + c)^2 + 2*(B^2*a^6 + 5*B^2*a^4*b^2 + 8*B^2*a^2*b^4)*d*tan(d*x + c) +
2*(B^2*a^5*b + 2*B^2*a^3*b^3)*d + ((a^7 + 8*a^5*b^2 + 19*a^3*b^4 + 12*a*b^6)*d^3*tan(d*x + c)^2 + 2*(a^6*b + 2
*a^4*b^3 - 3*a^2*b^5 - 4*b^7)*d^3*tan(d*x + c) - (a^7 + 4*a^5*b^2 + 7*a^3*b^4 + 4*a*b^6)*d^3)*sqrt(-B^4*a^2/((
a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt(-((a^2 + b^2)*sqrt(-B^4*a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2 - B^2*b)/((a
^2 + b^2)*d^2)) - 2*((B^3*a^5 + 3*B^3*a^3*b^2 + 4*B^3*a*b^4)*tan(d*x + c)^2 + 2*(B^3*a^4*b + 2*B^3*a^2*b^3)*ta
n(d*x + c) + (2*(B*a^5*b + 3*B*a^3*b^3 + 2*B*a*b^5)*d^2*tan(d*x + c)^2 - (B*a^6 + 4*B*a^4*b^2 + 7*B*a^2*b^4 +
4*B*b^6)*d^2*tan(d*x + c))*sqrt(-B^4*a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt(b*tan(d*x + c) + a)/sqrt(tan(d*x
+ c)))/(tan(d*x + c)^2 + 1))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt {\cot (c+d x)} (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^{3/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(c
onst gen &

Mupad [F(-1)]

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

[In]

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

[Out]

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