Integrand size = 25, antiderivative size = 246 \[ \int \frac {(a+b \tan (c+d x))^{3/2}}{\sqrt {\cot (c+d x)}} \, dx=-\frac {i (i a-b)^{3/2} \arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}+\frac {3 a \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}+\frac {i (i a+b)^{3/2} \text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}+\frac {b \sqrt {a+b \tan (c+d x)}}{d \sqrt {\cot (c+d x)}} \]
-I*(I*a-b)^(3/2)*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*(I*a+b)^(3/2)*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+3*a*arctanh(b^(1/2)*tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))^(1/2))*b^(1/ 2)*cot(d*x+c)^(1/2)*tan(d*x+c)^(1/2)/d+b*(a+b*tan(d*x+c))^(1/2)/d/cot(d*x+ c)^(1/2)
Time = 2.10 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.19 \[ \int \frac {(a+b \tan (c+d x))^{3/2}}{\sqrt {\cot (c+d x)}} \, dx=\frac {\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} \left (3 \sqrt {a} \sqrt {b} \text {arcsinh}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right ) (a+b \tan (c+d x))+\sqrt {1+\frac {b \tan (c+d x)}{a}} \left (-\sqrt [4]{-1} (-a+i b)^{3/2} \arctan \left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {a+b \tan (c+d x)}+\sqrt [4]{-1} (a+i b)^{3/2} \arctan \left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {a+b \tan (c+d x)}+b \sqrt {\tan (c+d x)} (a+b \tan (c+d x))\right )\right )}{d \sqrt {a+b \tan (c+d x)} \sqrt {1+\frac {b \tan (c+d x)}{a}}} \]
(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(3*Sqrt[a]*Sqrt[b]*ArcSinh[(Sqrt[b] *Sqrt[Tan[c + d*x]])/Sqrt[a]]*(a + b*Tan[c + d*x]) + Sqrt[1 + (b*Tan[c + d *x])/a]*(-((-1)^(1/4)*(-a + I*b)^(3/2)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*S qrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[a + b*Tan[c + d*x]]) + ( -1)^(1/4)*(a + I*b)^(3/2)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d* x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[a + b*Tan[c + d*x]] + b*Sqrt[Tan[c + d *x]]*(a + b*Tan[c + d*x]))))/(d*Sqrt[a + b*Tan[c + d*x]]*Sqrt[1 + (b*Tan[c + d*x])/a])
Time = 1.02 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.83, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3042, 4729, 3042, 4053, 27, 3042, 4138, 2035, 2257, 2009}
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.
| \(\displaystyle \int \frac {(a+b \tan (c+d x))^{3/2}}{\sqrt {\cot (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \tan (c+d x))^{3/2}}{\sqrt {\cot (c+d x)}}dx\) |
| \(\Big \downarrow \) 4729 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}dx\) |
| \(\Big \downarrow \) 4053 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\int -\frac {-3 a b \tan ^2(c+d x)-2 \left (a^2-b^2\right ) \tan (c+d x)+a b}{2 \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx+\frac {b \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {b \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{d}-\frac {1}{2} \int \frac {-3 a b \tan ^2(c+d x)-2 \left (a^2-b^2\right ) \tan (c+d x)+a b}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {b \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{d}-\frac {1}{2} \int \frac {-3 a b \tan (c+d x)^2-2 \left (a^2-b^2\right ) \tan (c+d x)+a b}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx\right )\) |
| \(\Big \downarrow \) 4138 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {b \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{d}-\frac {\int \frac {-3 a b \tan ^2(c+d x)-2 \left (a^2-b^2\right ) \tan (c+d x)+a b}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)} \left (\tan ^2(c+d x)+1\right )}d\tan (c+d x)}{2 d}\right )\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {b \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{d}-\frac {\int \frac {-3 a b \tan ^2(c+d x)-2 \left (a^2-b^2\right ) \tan (c+d x)+a b}{\sqrt {a+b \tan (c+d x)} \left (\tan ^2(c+d x)+1\right )}d\sqrt {\tan (c+d x)}}{d}\right )\) |
| \(\Big \downarrow \) 2257 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {b \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{d}-\frac {\int \left (\frac {2 \left (2 a b-\left (a^2-b^2\right ) \tan (c+d x)\right )}{\sqrt {a+b \tan (c+d x)} \left (\tan ^2(c+d x)+1\right )}-\frac {3 a b}{\sqrt {a+b \tan (c+d x)}}\right )d\sqrt {\tan (c+d x)}}{d}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {b \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{d}-\frac {i (-b+i a)^{3/2} \arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )-3 a \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )-i (b+i a)^{3/2} \text {arctanh}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}\right )\) |
Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(-((I*(I*a - b)^(3/2)*ArcTan[(Sqrt[I *a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] - 3*a*Sqrt[b]*ArcTan h[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] - I*(I*a + b)^(3/ 2)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d ) + (b*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/d)
3.9.51.3.1 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[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol ] :> Int[ExpandIntegrand[Px*(d + e*x^2)^q*(a + c*x^4)^p, x], x] /; FreeQ[{a , c, d, e, q}, x] && PolyQ[Px, x] && IntegerQ[p]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(a + b*Tan[e + f*x])^(m - 1)*((c + d*Tan[e + f*x])^n/(f*(m + n - 1))), x] + Simp[1/(m + n - 1) Int[(a + b*Ta n[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n - 1)*Simp[a^2*c*(m + n - 1) - b *(b*c*(m - 1) + a*d*n) + (2*a*b*c + a^2*d - b^2*d)*(m + n - 1)*Tan[e + f*x] + b*(b*c*n + a*d*(2*m + n - 2))*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] && GtQ[m, 1] && GtQ[n, 0] && IntegerQ[2*n]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), 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*((A + B*ff*x + C*ff^2*x^ 2)/(1 + ff^2*x^2)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, c, d, e, f , A, B, C, 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. \(2209\) vs. \(2(200)=400\).
Time = 33.34 (sec) , antiderivative size = 2210, normalized size of antiderivative = 8.98
-1/4/d*(-6*cos(d*x+c)*2^(1/2)*(-b+(a^2+b^2)^(1/2))^(1/2)*b^(1/2)*arctanh(( (a*cos(d*x+c)+b*sin(d*x+c))*sin(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(csc(d*x+c) +cot(d*x+c))/b^(1/2))*a^2+cos(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(-(-cos(d*x+c)*cot(d*x+c)*a+2*sin(d*x+c) *(csc(d*x+c)*(cot(d*x+c)^2*a-2*cot(d*x+c)*csc(d*x+c)*a+csc(d*x+c)^2*a-2*b* csc(d*x+c)+2*cot(d*x+c)*b-a)*(cos(d*x+c)-1))^(1/2)*(b+(a^2+b^2)^(1/2))^(1/ 2)+2*a*cot(d*x+c)+sin(d*x+c)*a-2*(a^2+b^2)^(1/2)*cos(d*x+c)-2*b*cos(d*x+c) -csc(d*x+c)*a+2*(a^2+b^2)^(1/2)+2*b)/(cos(d*x+c)-1))*b-cos(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(-(cos(d*x+ c)*cot(d*x+c)*a-2*a*cot(d*x+c)+2*sin(d*x+c)*(csc(d*x+c)*(cot(d*x+c)^2*a-2* cot(d*x+c)*csc(d*x+c)*a+csc(d*x+c)^2*a-2*b*csc(d*x+c)+2*cot(d*x+c)*b-a)*(c os(d*x+c)-1))^(1/2)*(b+(a^2+b^2)^(1/2))^(1/2)+2*(a^2+b^2)^(1/2)*cos(d*x+c) +2*b*cos(d*x+c)-sin(d*x+c)*a+csc(d*x+c)*a-2*(a^2+b^2)^(1/2)-2*b)/(cos(d*x+ c)-1))*b-2*2^(1/2)*(-b+(a^2+b^2)^(1/2))^(1/2)*((sin(d*x+c)*cos(d*x+c)*a-co s(d*x+c)^2*b+b)/(cos(d*x+c)+1)^2)^(1/2)*a*b*cos(d*x+c)+cos(d*x+c)*(b+(a^2+ b^2)^(1/2))^(1/2)*(-b+(a^2+b^2)^(1/2))^(1/2)*ln(-(-cos(d*x+c)*cot(d*x+c)*a +2*sin(d*x+c)*(csc(d*x+c)*(cot(d*x+c)^2*a-2*cot(d*x+c)*csc(d*x+c)*a+csc(d* x+c)^2*a-2*b*csc(d*x+c)+2*cot(d*x+c)*b-a)*(cos(d*x+c)-1))^(1/2)*(b+(a^2+b^ 2)^(1/2))^(1/2)+2*a*cot(d*x+c)+sin(d*x+c)*a-2*(a^2+b^2)^(1/2)*cos(d*x+c)-2 *b*cos(d*x+c)-csc(d*x+c)*a+2*(a^2+b^2)^(1/2)+2*b)/(cos(d*x+c)-1))*a^2-c...
Leaf count of result is larger than twice the leaf count of optimal. 3718 vs. \(2 (194) = 388\).
Time = 1.10 (sec) , antiderivative size = 7469, normalized size of antiderivative = 30.36 \[ \int \frac {(a+b \tan (c+d x))^{3/2}}{\sqrt {\cot (c+d x)}} \, dx=\text {Too large to display} \]
\[ \int \frac {(a+b \tan (c+d x))^{3/2}}{\sqrt {\cot (c+d x)}} \, dx=\int \frac {\left (a + b \tan {\left (c + d x \right )}\right )^{\frac {3}{2}}}{\sqrt {\cot {\left (c + d x \right )}}}\, dx \]
\[ \int \frac {(a+b \tan (c+d x))^{3/2}}{\sqrt {\cot (c+d x)}} \, dx=\int { \frac {{\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}{\sqrt {\cot \left (d x + c\right )}} \,d x } \]
Timed out. \[ \int \frac {(a+b \tan (c+d x))^{3/2}}{\sqrt {\cot (c+d x)}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(a+b \tan (c+d x))^{3/2}}{\sqrt {\cot (c+d x)}} \, dx=\int \frac {{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{3/2}}{\sqrt {\mathrm {cot}\left (c+d\,x\right )}} \,d x \]