Integrand size = 25, antiderivative size = 187 \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=-\frac {i \arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{\sqrt {i a-b} d}+\frac {i \text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{\sqrt {i a+b} d}-\frac {2 \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{a d} \]
-I*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*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)-2*cot(d*x+c)^(1/2)*(a+b*tan(d*x+c))^(1/2)/a/d
Time = 0.65 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.95 \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\frac {\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} \left (\frac {\sqrt [4]{-1} \arctan \left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{\sqrt {-a+i b}}-\frac {\sqrt [4]{-1} \arctan \left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{\sqrt {a+i b}}-\frac {2 \sqrt {a+b \tan (c+d x)}}{a \sqrt {\tan (c+d x)}}\right )}{d} \]
(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(((-1)^(1/4)*ArcTan[((-1)^(1/4)*Sqr t[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[-a + I*b] - ((-1)^(1/4)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[a + I*b] - (2*Sqrt[a + b*Tan[c + d*x]])/(a*Sqrt[T an[c + d*x]])))/d
Time = 0.56 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.90, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {3042, 4729, 3042, 4052, 27, 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.
| \(\displaystyle \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cot (c+d x)^{3/2}}{\sqrt {a+b \tan (c+d x)}}dx\) |
| \(\Big \downarrow \) 4729 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \frac {1}{\tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \frac {1}{\tan (c+d x)^{3/2} \sqrt {a+b \tan (c+d x)}}dx\) |
| \(\Big \downarrow \) 4052 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {2 \int \frac {a \sqrt {\tan (c+d x)}}{2 \sqrt {a+b \tan (c+d x)}}dx}{a}-\frac {2 \sqrt {a+b \tan (c+d x)}}{a d \sqrt {\tan (c+d x)}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\int \frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}dx-\frac {2 \sqrt {a+b \tan (c+d x)}}{a d \sqrt {\tan (c+d x)}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\int \frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}dx-\frac {2 \sqrt {a+b \tan (c+d x)}}{a d \sqrt {\tan (c+d x)}}\right )\) |
| \(\Big \downarrow \) 4058 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {\int \frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)} \left (\tan ^2(c+d x)+1\right )}d\tan (c+d x)}{d}-\frac {2 \sqrt {a+b \tan (c+d x)}}{a d \sqrt {\tan (c+d x)}}\right )\) |
| \(\Big \downarrow \) 613 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {2 \sqrt {a+b \tan (c+d x)}}{a d \sqrt {\tan (c+d x)}}-\frac {\frac {1}{2} \int \frac {1}{\sqrt {\tan (c+d x)} (\tan (c+d x)+i) \sqrt {a+b \tan (c+d x)}}d\tan (c+d x)-\frac {1}{2} \int \frac {1}{(i-\tan (c+d x)) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}d\tan (c+d x)}{d}\right )\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {2 \sqrt {a+b \tan (c+d x)}}{a d \sqrt {\tan (c+d x)}}-\frac {\int \frac {1}{\frac {(a-i b) \tan (c+d x)}{a+b \tan (c+d x)}+i}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}-\int \frac {1}{i-\frac {(a+i b) \tan (c+d x)}{a+b \tan (c+d x)}}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}}{d}\right )\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {2 \sqrt {a+b \tan (c+d x)}}{a d \sqrt {\tan (c+d x)}}-\frac {\int \frac {1}{\frac {(a-i b) \tan (c+d x)}{a+b \tan (c+d x)}+i}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}+\frac {i \arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{\sqrt {-b+i a}}}{d}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {2 \sqrt {a+b \tan (c+d x)}}{a d \sqrt {\tan (c+d x)}}-\frac {\frac {i \arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{\sqrt {-b+i a}}-\frac {i \text {arctanh}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{\sqrt {b+i a}}}{d}\right )\) |
Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(-(((I*ArcTan[(Sqrt[I*a - b]*Sqrt[Ta n[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])/d) - (2*Sqrt[a + b*Tan[c + d*x]])/(a*d*Sqrt[Tan[c + d*x]]))
3.9.62.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[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] + Simp[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] && IntegerQ[2*m] && LtQ[m, -1] && (LtQ[n, 0] || Integ erQ[m]) && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
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. \(1495\) vs. \(2(153)=306\).
Time = 40.49 (sec) , antiderivative size = 1496, normalized size of antiderivative = 8.00
1/4/d*(-1/(1-cos(d*x+c))*(csc(d*x+c)*(1-cos(d*x+c))^2-sin(d*x+c)))^(3/2)*( (csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)/(csc(d*x+c )^2*(1-cos(d*x+c))^2-1))^(1/2)*(ln(1/(1-cos(d*x+c))*(-csc(d*x+c)*a*(1-cos( d*x+c))^2+2*(a^2+b^2)^(1/2)*(1-cos(d*x+c))+2*sin(d*x+c)*(-csc(d*x+c)*(csc( d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*(1-cos(d*x+c))) ^(1/2)*(b+(a^2+b^2)^(1/2))^(1/2)+2*b*(1-cos(d*x+c))+sin(d*x+c)*a))*b*(b+(a ^2+b^2)^(1/2))^(1/2)*(-b+(a^2+b^2)^(1/2))^(1/2)*(csc(d*x+c)-cot(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/(1 -cos(d*x+c))*(-csc(d*x+c)*a*(1-cos(d*x+c))^2+2*(a^2+b^2)^(1/2)*(1-cos(d*x+ c))+2*sin(d*x+c)*(-csc(d*x+c)*(csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d* x+c)-cot(d*x+c))-a)*(1-cos(d*x+c)))^(1/2)*(b+(a^2+b^2)^(1/2))^(1/2)+2*b*(1 -cos(d*x+c))+sin(d*x+c)*a))*(csc(d*x+c)-cot(d*x+c))-ln(1/(1-cos(d*x+c))*(c sc(d*x+c)*a*(1-cos(d*x+c))^2+2*sin(d*x+c)*(-csc(d*x+c)*(csc(d*x+c)^2*a*(1- cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*(1-cos(d*x+c)))^(1/2)*(b+(a^2 +b^2)^(1/2))^(1/2)-2*(a^2+b^2)^(1/2)*(1-cos(d*x+c))-2*b*(1-cos(d*x+c))-sin (d*x+c)*a))*b*(b+(a^2+b^2)^(1/2))^(1/2)*(-b+(a^2+b^2)^(1/2))^(1/2)*(csc(d* x+c)-cot(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/(1-cos(d*x+c))*(csc(d*x+c)*a*(1-cos(d*x+c))^2+2*sin(d*x+c )*(-csc(d*x+c)*(csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c) )-a)*(1-cos(d*x+c)))^(1/2)*(b+(a^2+b^2)^(1/2))^(1/2)-2*(a^2+b^2)^(1/2)*...
Leaf count of result is larger than twice the leaf count of optimal. 4115 vs. \(2 (147) = 294\).
Time = 0.68 (sec) , antiderivative size = 4115, normalized size of antiderivative = 22.01 \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\text {Too large to display} \]
1/8*(a*d*sqrt(-((a^2 + b^2)*d^2*sqrt(-a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + b)/((a^2 + b^2)*d^2))*log((((a^6 + 7*a^4*b^2 + 12*a^2*b^4)*d*tan(d*x + c) ^2 + 2*(a^5*b + a^3*b^3 - 4*a*b^5)*d*tan(d*x + c) - (a^6 + 3*a^4*b^2 + 4*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(-a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt(-((a^2 + b^2 )*d^2*sqrt(-a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + b)/((a^2 + b^2)*d^2)) + 2 *((a^5 + 3*a^3*b^2 + 4*a*b^4)*tan(d*x + c)^2 + 2*(a^4*b + 2*a^2*b^3)*tan(d *x + c) - (2*(a^5*b + 3*a^3*b^3 + 2*a*b^5)*d^2*tan(d*x + c)^2 - (a^6 + 4*a ^4*b^2 + 7*a^2*b^4 + 4*b^6)*d^2*tan(d*x + c))*sqrt(-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)) + a*d*sqrt(-((a^2 + b^2)*d^2*sqrt(-a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4 )) + b)/((a^2 + b^2)*d^2))*log(-(((a^6 + 7*a^4*b^2 + 12*a^2*b^4)*d*tan(d*x + c)^2 + 2*(a^5*b + a^3*b^3 - 4*a*b^5)*d*tan(d*x + c) - (a^6 + 3*a^4*b^2 + 4*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(-a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt(-((a^2 + b^2)*d^2*sqrt(-a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + b)/((a^2 + b^2)*d^2) ) + 2*((a^5 + 3*a^3*b^2 + 4*a*b^4)*tan(d*x + c)^2 + 2*(a^4*b + 2*a^2*b^3)* tan(d*x + c) - (2*(a^5*b + 3*a^3*b^3 + 2*a*b^5)*d^2*tan(d*x + c)^2 - (a...
\[ \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\int \frac {\cot ^{\frac {3}{2}}{\left (c + d x \right )}}{\sqrt {a + b \tan {\left (c + d x \right )}}}\, dx \]
\[ \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\int { \frac {\cot \left (d x + c\right )^{\frac {3}{2}}}{\sqrt {b \tan \left (d x + c\right ) + a}} \,d x } \]
\[ \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\int { \frac {\cot \left (d x + c\right )^{\frac {3}{2}}}{\sqrt {b \tan \left (d x + c\right ) + a}} \,d x } \]
Timed out. \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\int \frac {{\mathrm {cot}\left (c+d\,x\right )}^{3/2}}{\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}} \,d x \]