Integrand size = 25, antiderivative size = 145 \[ \int \frac {(a+b \tan (c+d x))^{3/2}}{\tan ^{\frac {3}{2}}(c+d x)} \, dx=\frac {i (i a-b)^{3/2} \arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {i (i a+b)^{3/2} \text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {2 a \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}} \] 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))/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))/d-2*a*(a+b*tan(d*x+c))^(1/2)/d/tan(d*x+c)^(1/2)
Time = 0.19 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.21 \[ \int \frac {(a+b \tan (c+d x))^{3/2}}{\tan ^{\frac {3}{2}}(c+d x)} \, dx=\frac {\sqrt [4]{-1} (-a+i b)^{3/2} \arctan \left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\tan (c+d x)}-\sqrt [4]{-1} (a+i b)^{3/2} \arctan \left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\tan (c+d x)}-2 a \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}} \] Input:
Integrate[(a + b*Tan[c + d*x])^(3/2)/Tan[c + d*x]^(3/2),x]
Output:
((-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[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*T an[c + d*x]]]*Sqrt[Tan[c + d*x]] - 2*a*Sqrt[a + b*Tan[c + d*x]])/(d*Sqrt[T an[c + d*x]])
Time = 0.80 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.12, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3042, 4050, 27, 3042, 4099, 3042, 4098, 104, 216, 219}
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.
\(\displaystyle \int \frac {(a+b \tan (c+d x))^{3/2}}{\tan ^{\frac {3}{2}}(c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a+b \tan (c+d x))^{3/2}}{\tan (c+d x)^{3/2}}dx\) |
\(\Big \downarrow \) 4050 |
\(\displaystyle -2 \int -\frac {2 a b-\left (a^2-b^2\right ) \tan (c+d x)}{2 \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx-\frac {2 a \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {2 a b-\left (a^2-b^2\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx-\frac {2 a \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {2 a b-\left (a^2-b^2\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx-\frac {2 a \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}\) |
\(\Big \downarrow \) 4099 |
\(\displaystyle \frac {1}{2} i (a-i b)^2 \int \frac {i \tan (c+d x)+1}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx-\frac {1}{2} i (a+i b)^2 \int \frac {1-i \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx-\frac {2 a \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} i (a-i b)^2 \int \frac {i \tan (c+d x)+1}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx-\frac {1}{2} i (a+i b)^2 \int \frac {1-i \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx-\frac {2 a \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}\) |
\(\Big \downarrow \) 4098 |
\(\displaystyle \frac {i (a-i b)^2 \int \frac {1}{(1-i \tan (c+d x)) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}d\tan (c+d x)}{2 d}-\frac {i (a+i b)^2 \int \frac {1}{(i \tan (c+d x)+1) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}d\tan (c+d x)}{2 d}-\frac {2 a \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}\) |
\(\Big \downarrow \) 104 |
\(\displaystyle \frac {i (a-i b)^2 \int \frac {1}{1-\frac {(i a+b) \tan (c+d x)}{a+b \tan (c+d x)}}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}}{d}-\frac {i (a+i b)^2 \int \frac {1}{\frac {(i a-b) \tan (c+d x)}{a+b \tan (c+d x)}+1}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}}{d}-\frac {2 a \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {i (a-i b)^2 \int \frac {1}{1-\frac {(i a+b) \tan (c+d x)}{a+b \tan (c+d x)}}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}}{d}-\frac {i (a+i b)^2 \arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d \sqrt {-b+i a}}-\frac {2 a \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {i (a+i b)^2 \arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d \sqrt {-b+i a}}+\frac {i (a-i b)^2 \text {arctanh}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d \sqrt {b+i a}}-\frac {2 a \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}\) |
Input:
Int[(a + b*Tan[c + d*x])^(3/2)/Tan[c + d*x]^(3/2),x]
Output:
((-I)*(a + I*b)^2*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan [c + d*x]]])/(Sqrt[I*a - b]*d) + (I*(a - I*b)^2*ArcTanh[(Sqrt[I*a + b]*Sqr t[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(Sqrt[I*a + b]*d) - (2*a*Sqrt[ a + b*Tan[c + d*x]])/(d*Sqrt[Tan[c + d*x]])
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[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n - 1)/(f*(m + 1)*(a^2 + b^2))), x] + Simp[1/((m + 1)*(a^2 + b^2)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^ (n - 2)*Simp[a*c^2*(m + 1) + a*d^2*(n - 1) + b*c*d*(m - n + 2) - (b*c^2 - 2 *a*c*d - b*d^2)*(m + 1)*Tan[e + f*x] - d*(b*c - a*d)*(m + n)*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] && LtQ[m, -1] && LtQ[1, n, 2] && IntegerQ[ 2*m]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si mp[A^2/f Subst[Int[(a + b*x)^m*((c + d*x)^n/(A - B*x)), x], x, Tan[e + f* x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[A^2 + B^2, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si mp[(A + I*B)/2 Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n*(1 - I*T an[e + f*x]), x], x] + Simp[(A - I*B)/2 Int[(a + b*Tan[e + f*x])^m*(c + d *Tan[e + f*x])^n*(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A , B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[A^2 + B^2, 0]
result has leaf size over 500,000. Avoiding possible recursion issues.
Time = 0.71 (sec) , antiderivative size = 1344185, normalized size of antiderivative = 9270.24
\[\text {output too large to display}\]
Input:
int((a+b*tan(d*x+c))^(3/2)/tan(d*x+c)^(3/2),x)
Output:
result too large to display
Leaf count of result is larger than twice the leaf count of optimal. 3461 vs. \(2 (113) = 226\).
Time = 0.41 (sec) , antiderivative size = 3461, normalized size of antiderivative = 23.87 \[ \int \frac {(a+b \tan (c+d x))^{3/2}}{\tan ^{\frac {3}{2}}(c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((a+b*tan(d*x+c))^(3/2)/tan(d*x+c)^(3/2),x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {(a+b \tan (c+d x))^{3/2}}{\tan ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {\left (a + b \tan {\left (c + d x \right )}\right )^{\frac {3}{2}}}{\tan ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \] Input:
integrate((a+b*tan(d*x+c))**(3/2)/tan(d*x+c)**(3/2),x)
Output:
Integral((a + b*tan(c + d*x))**(3/2)/tan(c + d*x)**(3/2), x)
\[ \int \frac {(a+b \tan (c+d x))^{3/2}}{\tan ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}{\tan \left (d x + c\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((a+b*tan(d*x+c))^(3/2)/tan(d*x+c)^(3/2),x, algorithm="maxima")
Output:
integrate((b*tan(d*x + c) + a)^(3/2)/tan(d*x + c)^(3/2), x)
Exception generated. \[ \int \frac {(a+b \tan (c+d x))^{3/2}}{\tan ^{\frac {3}{2}}(c+d x)} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+b*tan(d*x+c))^(3/2)/tan(d*x+c)^(3/2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {(a+b \tan (c+d x))^{3/2}}{\tan ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{3/2}}{{\mathrm {tan}\left (c+d\,x\right )}^{3/2}} \,d x \] Input:
int((a + b*tan(c + d*x))^(3/2)/tan(c + d*x)^(3/2),x)
Output:
int((a + b*tan(c + d*x))^(3/2)/tan(c + d*x)^(3/2), x)
\[ \int \frac {(a+b \tan (c+d x))^{3/2}}{\tan ^{\frac {3}{2}}(c+d x)} \, dx=\frac {-2 \sqrt {\tan \left (d x +c \right )}\, \sqrt {a +\tan \left (d x +c \right ) b}\, b^{2}+\left (\int \frac {\sqrt {\tan \left (d x +c \right )}\, \sqrt {a +\tan \left (d x +c \right ) b}}{\tan \left (d x +c \right )^{3} b +\tan \left (d x +c \right )^{2} a}d x \right ) \tan \left (d x +c \right ) a^{3} d -\left (\int \frac {\sqrt {\tan \left (d x +c \right )}\, \sqrt {a +\tan \left (d x +c \right ) b}}{\tan \left (d x +c \right )^{3} b +\tan \left (d x +c \right )^{2} a}d x \right ) \tan \left (d x +c \right ) a \,b^{2} d +2 \left (\int \frac {\sqrt {\tan \left (d x +c \right )}\, \sqrt {a +\tan \left (d x +c \right ) b}}{\tan \left (d x +c \right )^{2} b +\tan \left (d x +c \right ) a}d x \right ) \tan \left (d x +c \right ) a^{2} b d}{\tan \left (d x +c \right ) a d} \] Input:
int((a+b*tan(d*x+c))^(3/2)/tan(d*x+c)^(3/2),x)
Output:
( - 2*sqrt(tan(c + d*x))*sqrt(tan(c + d*x)*b + a)*b**2 + int((sqrt(tan(c + d*x))*sqrt(tan(c + d*x)*b + a))/(tan(c + d*x)**3*b + tan(c + d*x)**2*a),x )*tan(c + d*x)*a**3*d - int((sqrt(tan(c + d*x))*sqrt(tan(c + d*x)*b + a))/ (tan(c + d*x)**3*b + tan(c + d*x)**2*a),x)*tan(c + d*x)*a*b**2*d + 2*int(( sqrt(tan(c + d*x))*sqrt(tan(c + d*x)*b + a))/(tan(c + d*x)**2*b + tan(c + d*x)*a),x)*tan(c + d*x)*a**2*b*d)/(tan(c + d*x)*a*d)