Integrand size = 38, antiderivative size = 157 \[ \int \frac {a B+b B \tan (c+d x)}{\sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}} \, dx=\frac {i B \arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{\sqrt {i a-b} d}-\frac {i B \text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{\sqrt {i a+b} d} \]
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)
Time = 0.11 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.92 \[ \int \frac {a B+b B \tan (c+d x)}{\sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}} \, dx=\frac {\sqrt [4]{-1} B \left (-\frac {\arctan \left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{\sqrt {-a+i b}}+\frac {\arctan \left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{\sqrt {a+i b}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d} \]
((-1)^(1/4)*B*(-(ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqr t[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
Time = 0.47 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.85, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.237, Rules used = {2011, 3042, 4729, 3042, 4058, 613, 104, 218, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {a B+b B \tan (c+d x)}{\sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2011 |
| \(\displaystyle B \int \frac {1}{\sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle B \int \frac {1}{\sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}dx\) |
| \(\Big \downarrow \) 4729 |
| \(\displaystyle B \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle B \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}dx\) |
| \(\Big \downarrow \) 4058 |
| \(\displaystyle \frac {B \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)} \left (\tan ^2(c+d x)+1\right )}d\tan (c+d x)}{d}\) |
| \(\Big \downarrow \) 613 |
| \(\displaystyle \frac {B \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{2} \int \frac {1}{\sqrt {\tan (c+d x)} (\tan (c+d x)+i) \sqrt {a+b \tan (c+d x)}}d\tan (c+d x)-\frac {1}{2} \int \frac {1}{(i-\tan (c+d x)) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}d\tan (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {B \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\int \frac {1}{\frac {(a-i b) \tan (c+d x)}{a+b \tan (c+d x)}+i}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}-\int \frac {1}{i-\frac {(a+i b) \tan (c+d x)}{a+b \tan (c+d x)}}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {B \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\int \frac {1}{\frac {(a-i b) \tan (c+d x)}{a+b \tan (c+d x)}+i}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}+\frac {i \arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{\sqrt {-b+i a}}\right )}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {B \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {i \arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{\sqrt {-b+i a}}-\frac {i \text {arctanh}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{\sqrt {b+i a}}\right )}{d}\) |
(B*((I*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] )/Sqrt[I*a - b] - (I*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b *Tan[c + d*x]]])/Sqrt[I*a + b])*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d
3.7.54.3.1 Defintions of rubi rules used
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[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] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
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]
Int[Sqrt[(e_.)*(x_)]/(Sqrt[(c_) + (d_.)*(x_)]*((a_) + (b_.)*(x_)^2)), x_Sym bol] :> Simp[e/(2*b) Int[1/(Sqrt[e*x]*Sqrt[c + d*x]*(Rt[-a/b, 2] + x)), x ], x] - Simp[e/(2*b) Int[1/(Sqrt[e*x]*Sqrt[c + d*x]*(Rt[-a/b, 2] - x)), x ], x] /; FreeQ[{a, b, c, d, e}, x]
Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Simp[(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])
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, S imp[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]
Int[(cot[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Simp[(c*Cot[a + b*x])^m*(c*Tan[a + b*x])^m Int[ActivateTrig[u]/(c*Tan[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[m] && KnownTangentIntegrandQ[u, x]
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
-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/(-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)))*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*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...
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 \[ \int \frac {a B+b B \tan (c+d x)}{\sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}} \, dx=\text {Too large to display} \]
-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)*sqr t(-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...
\[ \int \frac {a B+b B \tan (c+d x)}{\sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}} \, dx=B \int \frac {1}{\sqrt {a + b \tan {\left (c + d x \right )}} \sqrt {\cot {\left (c + d x \right )}}}\, dx \]
\[ \int \frac {a B+b B \tan (c+d x)}{\sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}} \, dx=\int { \frac {B b \tan \left (d x + c\right ) + B a}{{\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sqrt {\cot \left (d x + c\right )}} \,d x } \]
Timed out. \[ \int \frac {a B+b B \tan (c+d x)}{\sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {a B+b B \tan (c+d x)}{\sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}} \, dx=\int \frac {B\,a+B\,b\,\mathrm {tan}\left (c+d\,x\right )}{\sqrt {\mathrm {cot}\left (c+d\,x\right )}\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{3/2}} \,d x \]