Integrand size = 24, antiderivative size = 98 \[ \int \frac {\tan (c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {1}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {1}{2 a d \sqrt {a+i a \tan (c+d x)}} \] Output:
-1/4*arctanh(1/2*(a+I*a*tan(d*x+c))^(1/2)*2^(1/2)/a^(1/2))*2^(1/2)/a^(3/2) /d-1/3/d/(a+I*a*tan(d*x+c))^(3/2)+1/2/a/d/(a+I*a*tan(d*x+c))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.26 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.64 \[ \int \frac {\tan (c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=\frac {-2+\operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {1}{2} (1+i \tan (c+d x))\right ) (3+3 i \tan (c+d x))}{6 d (a+i a \tan (c+d x))^{3/2}} \] Input:
Integrate[Tan[c + d*x]/(a + I*a*Tan[c + d*x])^(3/2),x]
Output:
(-2 + Hypergeometric2F1[-1/2, 1, 1/2, (1 + I*Tan[c + d*x])/2]*(3 + (3*I)*T an[c + d*x]))/(6*d*(a + I*a*Tan[c + d*x])^(3/2))
Time = 0.40 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.07, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {3042, 4009, 3042, 3960, 3042, 3961, 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.
\(\displaystyle \int \frac {\tan (c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\tan (c+d x)}{(a+i a \tan (c+d x))^{3/2}}dx\) |
\(\Big \downarrow \) 4009 |
\(\displaystyle -\frac {i \int \frac {1}{\sqrt {i \tan (c+d x) a+a}}dx}{2 a}-\frac {1}{3 d (a+i a \tan (c+d x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {i \int \frac {1}{\sqrt {i \tan (c+d x) a+a}}dx}{2 a}-\frac {1}{3 d (a+i a \tan (c+d x))^{3/2}}\) |
\(\Big \downarrow \) 3960 |
\(\displaystyle -\frac {i \left (\frac {\int \sqrt {i \tan (c+d x) a+a}dx}{2 a}+\frac {i}{d \sqrt {a+i a \tan (c+d x)}}\right )}{2 a}-\frac {1}{3 d (a+i a \tan (c+d x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {i \left (\frac {\int \sqrt {i \tan (c+d x) a+a}dx}{2 a}+\frac {i}{d \sqrt {a+i a \tan (c+d x)}}\right )}{2 a}-\frac {1}{3 d (a+i a \tan (c+d x))^{3/2}}\) |
\(\Big \downarrow \) 3961 |
\(\displaystyle -\frac {i \left (\frac {i}{d \sqrt {a+i a \tan (c+d x)}}-\frac {i \int \frac {1}{a-i a \tan (c+d x)}d\sqrt {i \tan (c+d x) a+a}}{d}\right )}{2 a}-\frac {1}{3 d (a+i a \tan (c+d x))^{3/2}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {i \left (\frac {i}{d \sqrt {a+i a \tan (c+d x)}}-\frac {i \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {2} \sqrt {a} d}\right )}{2 a}-\frac {1}{3 d (a+i a \tan (c+d x))^{3/2}}\) |
Input:
Int[Tan[c + d*x]/(a + I*a*Tan[c + d*x])^(3/2),x]
Output:
-1/3*1/(d*(a + I*a*Tan[c + d*x])^(3/2)) - ((I/2)*(((-I)*ArcTanh[Sqrt[a + I *a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(Sqrt[2]*Sqrt[a]*d) + I/(d*Sqrt[a + I *a*Tan[c + d*x]])))/a
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[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[a*((a + b*Tan[c + d*x])^n/(2*b*d*n)), x] + Simp[1/(2*a) Int[(a + b*Tan[c + d*x])^ (n + 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0] && LtQ[n, 0]
Int[Sqrt[(a_) + (b_.)*tan[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2*(b/d) Subst[Int[1/(2*a - x^2), x], x, Sqrt[a + b*Tan[c + d*x]]], x] /; FreeQ[{a , b, c, d}, x] && EqQ[a^2 + b^2, 0]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-(b*c - a*d))*((a + b*Tan[e + f*x])^m/(2*a *f*m)), x] + Simp[(b*c + a*d)/(2*a*b) Int[(a + b*Tan[e + f*x])^(m + 1), x ], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2 , 0] && LtQ[m, 0]
Time = 1.33 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.73
method | result | size |
derivativedivides | \(\frac {-\frac {1}{3 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {1}{2 a \sqrt {a +i a \tan \left (d x +c \right )}}-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{4 a^{\frac {3}{2}}}}{d}\) | \(72\) |
default | \(\frac {-\frac {1}{3 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {1}{2 a \sqrt {a +i a \tan \left (d x +c \right )}}-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{4 a^{\frac {3}{2}}}}{d}\) | \(72\) |
Input:
int(tan(d*x+c)/(a+I*a*tan(d*x+c))^(3/2),x,method=_RETURNVERBOSE)
Output:
1/d*(-1/3/(a+I*a*tan(d*x+c))^(3/2)+1/2/a/(a+I*a*tan(d*x+c))^(1/2)-1/4/a^(3 /2)*2^(1/2)*arctanh(1/2*(a+I*a*tan(d*x+c))^(1/2)*2^(1/2)/a^(1/2)))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 271 vs. \(2 (73) = 146\).
Time = 0.09 (sec) , antiderivative size = 271, normalized size of antiderivative = 2.77 \[ \int \frac {\tan (c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=-\frac {{\left (3 \, \sqrt {\frac {1}{2}} a^{2} d \sqrt {\frac {1}{a^{3} d^{2}}} e^{\left (3 i \, d x + 3 i \, c\right )} \log \left (4 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a^{3} d^{2}}} + a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) - 3 \, \sqrt {\frac {1}{2}} a^{2} d \sqrt {\frac {1}{a^{3} d^{2}}} e^{\left (3 i \, d x + 3 i \, c\right )} \log \left (-4 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a^{3} d^{2}}} - a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) - \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (2 \, e^{\left (4 i \, d x + 4 i \, c\right )} + e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )}\right )} e^{\left (-3 i \, d x - 3 i \, c\right )}}{12 \, a^{2} d} \] Input:
integrate(tan(d*x+c)/(a+I*a*tan(d*x+c))^(3/2),x, algorithm="fricas")
Output:
-1/12*(3*sqrt(1/2)*a^2*d*sqrt(1/(a^3*d^2))*e^(3*I*d*x + 3*I*c)*log(4*(sqrt (2)*sqrt(1/2)*(a^2*d*e^(2*I*d*x + 2*I*c) + a^2*d)*sqrt(a/(e^(2*I*d*x + 2*I *c) + 1))*sqrt(1/(a^3*d^2)) + a*e^(I*d*x + I*c))*e^(-I*d*x - I*c)) - 3*sqr t(1/2)*a^2*d*sqrt(1/(a^3*d^2))*e^(3*I*d*x + 3*I*c)*log(-4*(sqrt(2)*sqrt(1/ 2)*(a^2*d*e^(2*I*d*x + 2*I*c) + a^2*d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*s qrt(1/(a^3*d^2)) - a*e^(I*d*x + I*c))*e^(-I*d*x - I*c)) - sqrt(2)*sqrt(a/( e^(2*I*d*x + 2*I*c) + 1))*(2*e^(4*I*d*x + 4*I*c) + e^(2*I*d*x + 2*I*c) - 1 ))*e^(-3*I*d*x - 3*I*c)/(a^2*d)
\[ \int \frac {\tan (c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=\int \frac {\tan {\left (c + d x \right )}}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(tan(d*x+c)/(a+I*a*tan(d*x+c))**(3/2),x)
Output:
Integral(tan(c + d*x)/(I*a*(tan(c + d*x) - I))**(3/2), x)
Time = 0.16 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.03 \[ \int \frac {\tan (c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=\frac {3 \, \sqrt {2} \sqrt {a} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right ) + \frac {4 \, {\left (3 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} a - 2 \, a^{2}\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}}{24 \, a^{2} d} \] Input:
integrate(tan(d*x+c)/(a+I*a*tan(d*x+c))^(3/2),x, algorithm="maxima")
Output:
1/24*(3*sqrt(2)*sqrt(a)*log(-(sqrt(2)*sqrt(a) - sqrt(I*a*tan(d*x + c) + a) )/(sqrt(2)*sqrt(a) + sqrt(I*a*tan(d*x + c) + a))) + 4*(3*(I*a*tan(d*x + c) + a)*a - 2*a^2)/(I*a*tan(d*x + c) + a)^(3/2))/(a^2*d)
Exception generated. \[ \int \frac {\tan (c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(tan(d*x+c)/(a+I*a*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:Error: Bad Argument TypeError: Bad Argument TypeDone
Time = 0.98 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.73 \[ \int \frac {\tan (c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=\frac {\frac {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}{2\,a}-\frac {1}{3}}{d\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2}}-\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{2\,\sqrt {a}}\right )}{4\,a^{3/2}\,d} \] Input:
int(tan(c + d*x)/(a + a*tan(c + d*x)*1i)^(3/2),x)
Output:
((a + a*tan(c + d*x)*1i)/(2*a) - 1/3)/(d*(a + a*tan(c + d*x)*1i)^(3/2)) - (2^(1/2)*atanh((2^(1/2)*(a + a*tan(c + d*x)*1i)^(1/2))/(2*a^(1/2))))/(4*a^ (3/2)*d)
\[ \int \frac {\tan (c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=\frac {\sqrt {a}\, \left (-2 \sqrt {\tan \left (d x +c \right ) i +1}+\left (\int \frac {\sqrt {\tan \left (d x +c \right ) i +1}}{\tan \left (d x +c \right )^{3} i +\tan \left (d x +c \right )^{2}+\tan \left (d x +c \right ) i +1}d x \right ) \tan \left (d x +c \right )^{2} d i +\left (\int \frac {\sqrt {\tan \left (d x +c \right ) i +1}}{\tan \left (d x +c \right )^{3} i +\tan \left (d x +c \right )^{2}+\tan \left (d x +c \right ) i +1}d x \right ) d i +\left (\int \frac {\sqrt {\tan \left (d x +c \right ) i +1}\, \tan \left (d x +c \right )^{2}}{\tan \left (d x +c \right )^{3} i +\tan \left (d x +c \right )^{2}+\tan \left (d x +c \right ) i +1}d x \right ) \tan \left (d x +c \right )^{2} d i +\left (\int \frac {\sqrt {\tan \left (d x +c \right ) i +1}\, \tan \left (d x +c \right )^{2}}{\tan \left (d x +c \right )^{3} i +\tan \left (d x +c \right )^{2}+\tan \left (d x +c \right ) i +1}d x \right ) d i \right )}{2 a^{2} d \left (\tan \left (d x +c \right )^{2}+1\right )} \] Input:
int(tan(d*x+c)/(a+I*a*tan(d*x+c))^(3/2),x)
Output:
(sqrt(a)*( - 2*sqrt(tan(c + d*x)*i + 1) + int(sqrt(tan(c + d*x)*i + 1)/(ta n(c + d*x)**3*i + tan(c + d*x)**2 + tan(c + d*x)*i + 1),x)*tan(c + d*x)**2 *d*i + int(sqrt(tan(c + d*x)*i + 1)/(tan(c + d*x)**3*i + tan(c + d*x)**2 + tan(c + d*x)*i + 1),x)*d*i + int((sqrt(tan(c + d*x)*i + 1)*tan(c + d*x)** 2)/(tan(c + d*x)**3*i + tan(c + d*x)**2 + tan(c + d*x)*i + 1),x)*tan(c + d *x)**2*d*i + int((sqrt(tan(c + d*x)*i + 1)*tan(c + d*x)**2)/(tan(c + d*x)* *3*i + tan(c + d*x)**2 + tan(c + d*x)*i + 1),x)*d*i))/(2*a**2*d*(tan(c + d *x)**2 + 1))