\(\int \frac {a B+b B \tan (c+d x)}{\sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}} \, dx\) [654]

Optimal result

Integrand size = 38, antiderivative size = 157 \[ \int \frac {a B+b B \tan (c+d x)}{\sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}} \, dx=\frac {i B \arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{\sqrt {i a-b} d}-\frac {i B \text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{\sqrt {i a+b} d} \]

[Out]

I*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)-I*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 = 157, 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, 924, 95, 211, 214} \[ \int \frac {a B+b B \tan (c+d x)}{\sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}} \, dx=\frac {i B \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d \sqrt {-b+i a}}-\frac {i B \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \text {arctanh}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d \sqrt {b+i a}} \]

[In]

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

[Out]

(I*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) - (I*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 924

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

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 {1}{\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 {\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 {\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 {1}{2 (i-x) \sqrt {x} \sqrt {a+b x}}+\frac {1}{2 \sqrt {x} (i+x) \sqrt {a+b x}}\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 \frac {1}{(i-x) \sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac {\left (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 (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 (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 {i 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 {i 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 = 0.11 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.92 \[ \int \frac {a B+b B \tan (c+d x)}{\sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}} \, dx=\frac {\sqrt [4]{-1} B \left (-\frac {\arctan \left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{\sqrt {-a+i b}}+\frac {\arctan \left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{\sqrt {a+i b}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d} \]

[In]

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

[Out]

((-1)^(1/4)*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. \(1301\) vs. \(2(127)=254\).

Time = 13.34 (sec) , antiderivative size = 1302, normalized size of antiderivative = 8.29

method	result	size
default	\(\text {Expression too large to display}\)	\(1302\)

[In]

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

[Out]

-1/4*B/d*2^(1/2)/a/(a^2+b^2)^(1/2)/(-b+(a^2+b^2)^(1/2))^(1/2)*((-cos(d*x+c)+1)^2*csc(d*x+c)^2-1)*((a*(-cos(d*x
+c)+1)^2*csc(d*x+c)^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)/((-cos(d*x+c)+1)^2*csc(d*x+c)^2-1))^(1/2)*((a^2+b^2)^(1/2
)*(b+(a^2+b^2)^(1/2))^(1/2)*(-b+(a^2+b^2)^(1/2))^(1/2)*ln(-1/(-cos(d*x+c)+1)*(-a*(-cos(d*x+c)+1)^2*csc(d*x+c)+
2*(a^2+b^2)^(1/2)*(-cos(d*x+c)+1)-2*(-(-cos(d*x+c)+1)*(a*(-cos(d*x+c)+1)^2*csc(d*x+c)^2-2*b*(csc(d*x+c)-cot(d*
x+c))-a)*csc(d*x+c))^(1/2)*(b+(a^2+b^2)^(1/2))^(1/2)*sin(d*x+c)+2*b*(-cos(d*x+c)+1)+a*sin(d*x+c)))-(a^2+b^2)^(
1/2)*(b+(a^2+b^2)^(1/2))^(1/2)*(-b+(a^2+b^2)^(1/2))^(1/2)*ln(1/(-cos(d*x+c)+1)*(-a*(-cos(d*x+c)+1)^2*csc(d*x+c
)+2*(a^2+b^2)^(1/2)*(-cos(d*x+c)+1)+2*(-(-cos(d*x+c)+1)*(a*(-cos(d*x+c)+1)^2*csc(d*x+c)^2-2*b*(csc(d*x+c)-cot(
d*x+c))-a)*csc(d*x+c))^(1/2)*(b+(a^2+b^2)^(1/2))^(1/2)*sin(d*x+c)+2*b*(-cos(d*x+c)+1)+a*sin(d*x+c)))-ln(-1/(-c
os(d*x+c)+1)*(-a*(-cos(d*x+c)+1)^2*csc(d*x+c)+2*(a^2+b^2)^(1/2)*(-cos(d*x+c)+1)-2*(-(-cos(d*x+c)+1)*(a*(-cos(d
*x+c)+1)^2*csc(d*x+c)^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*csc(d*x+c))^(1/2)*(b+(a^2+b^2)^(1/2))^(1/2)*sin(d*x+c)+
2*b*(-cos(d*x+c)+1)+a*sin(d*x+c)))*b*(b+(a^2+b^2)^(1/2))^(1/2)*(-b+(a^2+b^2)^(1/2))^(1/2)+ln(1/(-cos(d*x+c)+1)
*(-a*(-cos(d*x+c)+1)^2*csc(d*x+c)+2*(a^2+b^2)^(1/2)*(-cos(d*x+c)+1)+2*(-(-cos(d*x+c)+1)*(a*(-cos(d*x+c)+1)^2*c
sc(d*x+c)^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*csc(d*x+c))^(1/2)*(b+(a^2+b^2)^(1/2))^(1/2)*sin(d*x+c)+2*b*(-cos(d*
x+c)+1)+a*sin(d*x+c)))*b*(b+(a^2+b^2)^(1/2))^(1/2)*(-b+(a^2+b^2)^(1/2))^(1/2)-2*arctan((-(b+(a^2+b^2)^(1/2))^(
1/2)*(csc(d*x+c)-cot(d*x+c))+(-(-cos(d*x+c)+1)*(a*(-cos(d*x+c)+1)^2*csc(d*x+c)^2-2*b*(csc(d*x+c)-cot(d*x+c))-a
)*csc(d*x+c))^(1/2))/(-cos(d*x+c)+1)*sin(d*x+c)/(-b+(a^2+b^2)^(1/2))^(1/2))*a^2-2*arctan(((b+(a^2+b^2)^(1/2))^
(1/2)*(csc(d*x+c)-cot(d*x+c))+(-(-cos(d*x+c)+1)*(a*(-cos(d*x+c)+1)^2*csc(d*x+c)^2-2*b*(csc(d*x+c)-cot(d*x+c))-
a)*csc(d*x+c))^(1/2))/(-cos(d*x+c)+1)*sin(d*x+c)/(-b+(a^2+b^2)^(1/2))^(1/2))*a^2)/(-1/(-cos(d*x+c)+1)*((-cos(d
*x+c)+1)^2*csc(d*x+c)-sin(d*x+c)))^(1/2)/(-(-cos(d*x+c)+1)*(a*(-cos(d*x+c)+1)^2*csc(d*x+c)^2-2*b*(csc(d*x+c)-c
ot(d*x+c))-a)*csc(d*x+c))^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4649 vs. \(2 (121) = 242\).

Time = 0.76 (sec) , antiderivative size = 4649, normalized size of antiderivative = 29.61 \[ \int \frac {a B+b B \tan (c+d x)}{\sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}} \, dx=\text {Too large to display} \]

[In]

integrate((B*a+b*B*tan(d*x+c))/cot(d*x+c)^(1/2)/(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((((B^
2*a^6 + 7*B^2*a^4*b^2 + 12*B^2*a^2*b^4)*d*tan(d*x + c)^2 + 2*(B^2*a^5*b + B^2*a^3*b^3 - 4*B^2*a*b^5)*d*tan(d*x
+ c) - (B^2*a^6 + 3*B^2*a^4*b^2 + 4*B^2*a^2*b^4)*d + 2*((a^4*b^3 + 5*a^2*b^5 + 4*b^7)*d^3*tan(d*x + c)^2 + (a
^7 + 6*a^5*b^2 + 13*a^3*b^4 + 8*a*b^6)*d^3*tan(d*x + c) + (a^6*b + 3*a^4*b^3 + 2*a^2*b^5)*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)*t
an(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(-(((B^2*a^6 + 7*B^2*a^4*b^2 + 12*B^2*a^2*b^4)*d*tan(d*x + c)^2 + 2*(B^2*a^5*b + B^2
*a^3*b^3 - 4*B^2*a*b^5)*d*tan(d*x + c) - (B^2*a^6 + 3*B^2*a^4*b^2 + 4*B^2*a^2*b^4)*d + 2*((a^4*b^3 + 5*a^2*b^5
+ 4*b^7)*d^3*tan(d*x + c)^2 + (a^7 + 6*a^5*b^2 + 13*a^3*b^4 + 8*a*b^6)*d^3*tan(d*x + c) + (a^6*b + 3*a^4*b^3
+ 2*a^2*b^5)*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)))*sqr
t(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((((B^2*a^6 + 7*B^2*a^4*b^2 + 12*B^2*a^2*b^4)*d*tan
(d*x + c)^2 + 2*(B^2*a^5*b + B^2*a^3*b^3 - 4*B^2*a*b^5)*d*tan(d*x + c) - (B^2*a^6 + 3*B^2*a^4*b^2 + 4*B^2*a^2*
b^4)*d + 2*((a^4*b^3 + 5*a^2*b^5 + 4*b^7)*d^3*tan(d*x + c)^2 + (a^7 + 6*a^5*b^2 + 13*a^3*b^4 + 8*a*b^6)*d^3*ta
n(d*x + c) + (a^6*b + 3*a^4*b^3 + 2*a^2*b^5)*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(-(((B^2*a^6 + 7*B^
2*a^4*b^2 + 12*B^2*a^2*b^4)*d*tan(d*x + c)^2 + 2*(B^2*a^5*b + B^2*a^3*b^3 - 4*B^2*a*b^5)*d*tan(d*x + c) - (B^2
*a^6 + 3*B^2*a^4*b^2 + 4*B^2*a^2*b^4)*d + 2*((a^4*b^3 + 5*a^2*b^5 + 4*b^7)*d^3*tan(d*x + c)^2 + (a^7 + 6*a^5*b
^2 + 13*a^3*b^4 + 8*a*b^6)*d^3*tan(d*x + c) + (a^6*b + 3*a^4*b^3 + 2*a^2*b^5)*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)))/(ta
n(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((((B^2*a^6 + 7*B^2*a^4*b^2 + 12*B^2*a^2*b^4)*d*tan(d*x + c)^2 + 2*(B^2*a^5*b + B^2*a^3*b^3 - 4*B
^2*a*b^5)*d*tan(d*x + c) - (B^2*a^6 + 3*B^2*a^4*b^2 + 4*B^2*a^2*b^4)*d - 2*((a^4*b^3 + 5*a^2*b^5 + 4*b^7)*d^3*
tan(d*x + c)^2 + (a^7 + 6*a^5*b^2 + 13*a^3*b^4 + 8*a*b^6)*d^3*tan(d*x + c) + (a^6*b + 3*a^4*b^3 + 2*a^2*b^5)*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(-(((B^2*a^6 + 7*B^2*a^4*b^2 + 12*B^2*a^2*b^4)*d*tan(d*x + c)^2 + 2
*(B^2*a^5*b + B^2*a^3*b^3 - 4*B^2*a*b^5)*d*tan(d*x + c) - (B^2*a^6 + 3*B^2*a^4*b^2 + 4*B^2*a^2*b^4)*d - 2*((a^
4*b^3 + 5*a^2*b^5 + 4*b^7)*d^3*tan(d*x + c)^2 + (a^7 + 6*a^5*b^2 + 13*a^3*b^4 + 8*a*b^6)*d^3*tan(d*x + c) + (a
^6*b + 3*a^4*b^3 + 2*a^2*b^5)*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((((B^2*a^6 + 7*B^2*a^4*b^2 + 12*B^2
*a^2*b^4)*d*tan(d*x + c)^2 + 2*(B^2*a^5*b + B^2*a^3*b^3 - 4*B^2*a*b^5)*d*tan(d*x + c) - (B^2*a^6 + 3*B^2*a^4*b
^2 + 4*B^2*a^2*b^4)*d - 2*((a^4*b^3 + 5*a^2*b^5 + 4*b^7)*d^3*tan(d*x + c)^2 + (a^7 + 6*a^5*b^2 + 13*a^3*b^4 +
8*a*b^6)*d^3*tan(d*x + c) + (a^6*b + 3*a^4*b^3 + 2*a^2*b^5)*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))*sqr
t(-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(-(((B
^2*a^6 + 7*B^2*a^4*b^2 + 12*B^2*a^2*b^4)*d*tan(d*x + c)^2 + 2*(B^2*a^5*b + B^2*a^3*b^3 - 4*B^2*a*b^5)*d*tan(d*
x + c) - (B^2*a^6 + 3*B^2*a^4*b^2 + 4*B^2*a^2*b^4)*d - 2*((a^4*b^3 + 5*a^2*b^5 + 4*b^7)*d^3*tan(d*x + c)^2 + (
a^7 + 6*a^5*b^2 + 13*a^3*b^4 + 8*a*b^6)*d^3*tan(d*x + c) + (a^6*b + 3*a^4*b^3 + 2*a^2*b^5)*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)*t
an(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]

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

[In]

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

[Out]

B*Integral(1/(sqrt(a + b*tan(c + d*x))*sqrt(cot(c + d*x))), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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