Integrand size = 25, antiderivative size = 246 \[ \int \frac {(a+b \tan (c+d x))^{3/2}}{\sqrt {\cot (c+d x)}} \, dx=-\frac {i (i a-b)^{3/2} \arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}+\frac {3 a \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}+\frac {i (i a+b)^{3/2} \text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}+\frac {b \sqrt {a+b \tan (c+d x)}}{d \sqrt {\cot (c+d x)}} \] Output:
-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+3*a*b^(1/2)*arctanh(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+b*(a+b*tan(d*x+c))^(1/2)/d/cot(d*x+ c)^(1/2)
Time = 1.34 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.19 \[ \int \frac {(a+b \tan (c+d x))^{3/2}}{\sqrt {\cot (c+d x)}} \, dx=\frac {\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} \left (3 \sqrt {a} \sqrt {b} \text {arcsinh}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right ) (a+b \tan (c+d x))+\sqrt {1+\frac {b \tan (c+d x)}{a}} \left (-\sqrt [4]{-1} (-a+i b)^{3/2} \arctan \left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {a+b \tan (c+d x)}+\sqrt [4]{-1} (a+i b)^{3/2} \arctan \left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {a+b \tan (c+d x)}+b \sqrt {\tan (c+d x)} (a+b \tan (c+d x))\right )\right )}{d \sqrt {a+b \tan (c+d x)} \sqrt {1+\frac {b \tan (c+d x)}{a}}} \] Input:
Integrate[(a + b*Tan[c + d*x])^(3/2)/Sqrt[Cot[c + d*x]],x]
Output:
(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 = 1.03 (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 |
\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {b \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{d}-\frac {i (-b+i a)^{3/2} \arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )-3 a \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )-i (b+i a)^{3/2} \text {arctanh}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}\right )\) |
Input:
Int[(a + b*Tan[c + d*x])^(3/2)/Sqrt[Cot[c + d*x]],x]
Output:
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)
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. \(1916\) vs. \(2(200)=400\).
Time = 3.53 (sec) , antiderivative size = 1917, normalized size of antiderivative = 7.79
Input:
int((a+b*tan(d*x+c))^(3/2)/cot(d*x+c)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/4/d*2^(1/2)/(-b+(a^2+b^2)^(1/2))^(1/2)/a*(-6*arctanh(1/b^(1/2)*((a*cos(d *x+c)+b*sin(d*x+c))*sin(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*sin(d*x+c)/(-1+cos( d*x+c)))*cos(d*x+c)*2^(1/2)*(-b+(a^2+b^2)^(1/2))^(1/2)*b^(1/2)*a^2-ln(-(2* 2^(1/2)*((a*cos(d*x+c)+b*sin(d*x+c))*sin(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(b +(a^2+b^2)^(1/2))^(1/2)*sin(d*x+c)-a*cot(d*x+c)*cos(d*x+c)+2*a*cot(d*x+c)- 2*(a^2+b^2)^(1/2)*cos(d*x+c)-2*cos(d*x+c)*b+sin(d*x+c)*a-a*csc(d*x+c)+2*(a ^2+b^2)^(1/2)+2*b)/(-1+cos(d*x+c)))*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)*b+ln((a*cot(d*x+c)*cos(d*x+c)+2* 2^(1/2)*((a*cos(d*x+c)+b*sin(d*x+c))*sin(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(b +(a^2+b^2)^(1/2))^(1/2)*sin(d*x+c)-2*a*cot(d*x+c)+2*(a^2+b^2)^(1/2)*cos(d* x+c)+2*cos(d*x+c)*b-sin(d*x+c)*a+a*csc(d*x+c)-2*(a^2+b^2)^(1/2)-2*b)/(-1+c os(d*x+c)))*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)*b-ln(-(2*2^(1/2)*((a*cos(d*x+c)+b*sin(d*x+c))*sin(d*x+c) /(cos(d*x+c)+1)^2)^(1/2)*(b+(a^2+b^2)^(1/2))^(1/2)*sin(d*x+c)-a*cot(d*x+c) *cos(d*x+c)+2*a*cot(d*x+c)-2*(a^2+b^2)^(1/2)*cos(d*x+c)-2*cos(d*x+c)*b+sin (d*x+c)*a-a*csc(d*x+c)+2*(a^2+b^2)^(1/2)+2*b)/(-1+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)*a^2+ln(-(2*2^(1/2)*(( a*cos(d*x+c)+b*sin(d*x+c))*sin(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(b+(a^2+b^2) ^(1/2))^(1/2)*sin(d*x+c)-a*cot(d*x+c)*cos(d*x+c)+2*a*cot(d*x+c)-2*(a^2+b^2 )^(1/2)*cos(d*x+c)-2*cos(d*x+c)*b+sin(d*x+c)*a-a*csc(d*x+c)+2*(a^2+b^2)...
Leaf count of result is larger than twice the leaf count of optimal. 3718 vs. \(2 (194) = 388\).
Time = 1.01 (sec) , antiderivative size = 7469, normalized size of antiderivative = 30.36 \[ \int \frac {(a+b \tan (c+d x))^{3/2}}{\sqrt {\cot (c+d x)}} \, dx=\text {Too large to display} \] Input:
integrate((a+b*tan(d*x+c))^(3/2)/cot(d*x+c)^(1/2),x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {(a+b \tan (c+d x))^{3/2}}{\sqrt {\cot (c+d x)}} \, dx=\int \frac {\left (a + b \tan {\left (c + d x \right )}\right )^{\frac {3}{2}}}{\sqrt {\cot {\left (c + d x \right )}}}\, dx \] Input:
integrate((a+b*tan(d*x+c))**(3/2)/cot(d*x+c)**(1/2),x)
Output:
Integral((a + b*tan(c + d*x))**(3/2)/sqrt(cot(c + d*x)), x)
\[ \int \frac {(a+b \tan (c+d x))^{3/2}}{\sqrt {\cot (c+d x)}} \, dx=\int { \frac {{\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}{\sqrt {\cot \left (d x + c\right )}} \,d x } \] Input:
integrate((a+b*tan(d*x+c))^(3/2)/cot(d*x+c)^(1/2),x, algorithm="maxima")
Output:
integrate((b*tan(d*x + c) + a)^(3/2)/sqrt(cot(d*x + c)), x)
Exception generated. \[ \int \frac {(a+b \tan (c+d x))^{3/2}}{\sqrt {\cot (c+d x)}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+b*tan(d*x+c))^(3/2)/cot(d*x+c)^(1/2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{%%%{1,[0,9,3]%%%}+%%%{4,[0,7,3]%%%}+%%%{6,[0,5,3]%%%}+%%%{ 4,[0,3,3]
Timed out. \[ \int \frac {(a+b \tan (c+d x))^{3/2}}{\sqrt {\cot (c+d x)}} \, dx=\int \frac {{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{3/2}}{\sqrt {\mathrm {cot}\left (c+d\,x\right )}} \,d x \] Input:
int((a + b*tan(c + d*x))^(3/2)/cot(c + d*x)^(1/2),x)
Output:
int((a + b*tan(c + d*x))^(3/2)/cot(c + d*x)^(1/2), x)
\[ \int \frac {(a+b \tan (c+d x))^{3/2}}{\sqrt {\cot (c+d x)}} \, dx=\left (\int \frac {\sqrt {a +\tan \left (d x +c \right ) b}\, \sqrt {\cot \left (d x +c \right )}\, \tan \left (d x +c \right )}{\cot \left (d x +c \right )}d x \right ) b +\left (\int \frac {\sqrt {a +\tan \left (d x +c \right ) b}\, \sqrt {\cot \left (d x +c \right )}}{\cot \left (d x +c \right )}d x \right ) a \] Input:
int((a+b*tan(d*x+c))^(3/2)/cot(d*x+c)^(1/2),x)
Output:
int((sqrt(tan(c + d*x)*b + a)*sqrt(cot(c + d*x))*tan(c + d*x))/cot(c + d*x ),x)*b + int((sqrt(tan(c + d*x)*b + a)*sqrt(cot(c + d*x)))/cot(c + d*x),x) *a