\(\int \frac {1}{\sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}} \, dx\) [1029]

Optimal result

Integrand size = 35, antiderivative size = 137 \[ \int \frac {1}{\sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}} \, dx=\frac {i}{f \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}-\frac {2 i \sqrt {a+i a \tan (e+f x)}}{3 a f (c-i c \tan (e+f x))^{3/2}}-\frac {2 i \sqrt {a+i a \tan (e+f x)}}{3 a c f \sqrt {c-i c \tan (e+f x)}} \] Output:

I/f/(a+I*a*tan(f*x+e))^(1/2)/(c-I*c*tan(f*x+e))^(3/2)-2/3*I*(a+I*a*tan(f*x 
+e))^(1/2)/a/f/(c-I*c*tan(f*x+e))^(3/2)-2/3*I*(a+I*a*tan(f*x+e))^(1/2)/a/c 
/f/(c-I*c*tan(f*x+e))^(1/2)
 

Mathematica [A] (verified)

Time = 0.98 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.57 \[ \int \frac {1}{\sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}} \, dx=\frac {1+2 i \tan (e+f x)+2 \tan ^2(e+f x)}{3 c f (i+\tan (e+f x)) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}} \] Input:

Integrate[1/(Sqrt[a + I*a*Tan[e + f*x]]*(c - I*c*Tan[e + f*x])^(3/2)),x]
 

Output:

(1 + (2*I)*Tan[e + f*x] + 2*Tan[e + f*x]^2)/(3*c*f*(I + Tan[e + f*x])*Sqrt 
[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.09, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3042, 4006, 55, 55, 48}

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.

\(\displaystyle \int \frac {1}{\sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}dx\)

\(\Big \downarrow \) 4006

\(\displaystyle \frac {a c \int \frac {1}{(i \tan (e+f x) a+a)^{3/2} (c-i c \tan (e+f x))^{5/2}}d\tan (e+f x)}{f}\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {a c \left (\frac {2 \int \frac {1}{\sqrt {i \tan (e+f x) a+a} (c-i c \tan (e+f x))^{5/2}}d\tan (e+f x)}{a}+\frac {i}{a c \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}\right )}{f}\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {a c \left (\frac {2 \left (\frac {\int \frac {1}{\sqrt {i \tan (e+f x) a+a} (c-i c \tan (e+f x))^{3/2}}d\tan (e+f x)}{3 c}-\frac {i \sqrt {a+i a \tan (e+f x)}}{3 a c (c-i c \tan (e+f x))^{3/2}}\right )}{a}+\frac {i}{a c \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}\right )}{f}\)

\(\Big \downarrow \) 48

\(\displaystyle \frac {a c \left (\frac {2 \left (-\frac {i \sqrt {a+i a \tan (e+f x)}}{3 a c^2 \sqrt {c-i c \tan (e+f x)}}-\frac {i \sqrt {a+i a \tan (e+f x)}}{3 a c (c-i c \tan (e+f x))^{3/2}}\right )}{a}+\frac {i}{a c \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}\right )}{f}\)

Input:

Int[1/(Sqrt[a + I*a*Tan[e + f*x]]*(c - I*c*Tan[e + f*x])^(3/2)),x]
 

Output:

(a*c*(I/(a*c*Sqrt[a + I*a*Tan[e + f*x]]*(c - I*c*Tan[e + f*x])^(3/2)) + (2 
*(((-1/3*I)*Sqrt[a + I*a*Tan[e + f*x]])/(a*c*(c - I*c*Tan[e + f*x])^(3/2)) 
 - ((I/3)*Sqrt[a + I*a*Tan[e + f*x]])/(a*c^2*Sqrt[c - I*c*Tan[e + f*x]]))) 
/a))/f
 

Defintions of rubi rules used

rule 48

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp 
[(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ 
a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
 

rule 55

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S 
implify[m + n + 2]/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^Simplify[m + 1]*( 
c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 
 2], 0] && NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ 
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] ||  !SumSimp 
lerQ[n, 1])
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4006

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + ( 
f_.)*(x_)])^(n_), x_Symbol] :> Simp[a*(c/f)   Subst[Int[(a + b*x)^(m - 1)*( 
c + d*x)^(n - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n 
}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 + b^2, 0]
 

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.64

method	result	size
risch	\(-\frac {i \left ({\mathrm e}^{4 i \left (f x +e \right )}+6 \,{\mathrm e}^{2 i \left (f x +e \right )}-3\right )}{12 c \sqrt {\frac {a \,{\mathrm e}^{2 i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) \sqrt {\frac {c}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}\)	\(88\)
derivativedivides	\(\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \left (2 i \tan \left (f x +e \right )^{3}+2 \tan \left (f x +e \right )^{4}+2 i \tan \left (f x +e \right )+3 \tan \left (f x +e \right )^{2}+1\right )}{3 f a \,c^{2} \left (i+\tan \left (f x +e \right )\right )^{3} \left (-\tan \left (f x +e \right )+i\right )^{2}}\)	\(109\)
default	\(\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \left (2 i \tan \left (f x +e \right )^{3}+2 \tan \left (f x +e \right )^{4}+2 i \tan \left (f x +e \right )+3 \tan \left (f x +e \right )^{2}+1\right )}{3 f a \,c^{2} \left (i+\tan \left (f x +e \right )\right )^{3} \left (-\tan \left (f x +e \right )+i\right )^{2}}\)	\(109\)

Input:

int(1/(a+I*a*tan(f*x+e))^(1/2)/(c-I*c*tan(f*x+e))^(3/2),x,method=_RETURNVE 
RBOSE)
 

Output:

-1/12*I/c/(a*exp(2*I*(f*x+e))/(exp(2*I*(f*x+e))+1))^(1/2)/(exp(2*I*(f*x+e) 
)+1)/(c/(exp(2*I*(f*x+e))+1))^(1/2)*(exp(4*I*(f*x+e))+6*exp(2*I*(f*x+e))-3 
)/f
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.81 \[ \int \frac {1}{\sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}} \, dx=\frac {\sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} {\left (-i \, e^{\left (6 i \, f x + 6 i \, e\right )} - 7 i \, e^{\left (4 i \, f x + 4 i \, e\right )} + 4 i \, e^{\left (3 i \, f x + 3 i \, e\right )} - 3 i \, e^{\left (2 i \, f x + 2 i \, e\right )} + 4 i \, e^{\left (i \, f x + i \, e\right )} + 3 i\right )} e^{\left (-i \, f x - i \, e\right )}}{12 \, a c^{2} f} \] Input:

integrate(1/(a+I*a*tan(f*x+e))^(1/2)/(c-I*c*tan(f*x+e))^(3/2),x, algorithm 
="fricas")
 

Output:

1/12*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(c/(e^(2*I*f*x + 2*I*e) + 1))*( 
-I*e^(6*I*f*x + 6*I*e) - 7*I*e^(4*I*f*x + 4*I*e) + 4*I*e^(3*I*f*x + 3*I*e) 
 - 3*I*e^(2*I*f*x + 2*I*e) + 4*I*e^(I*f*x + I*e) + 3*I)*e^(-I*f*x - I*e)/( 
a*c^2*f)
 

Sympy [F]

\[ \int \frac {1}{\sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}} \, dx=\int \frac {1}{\sqrt {i a \left (\tan {\left (e + f x \right )} - i\right )} \left (- i c \left (\tan {\left (e + f x \right )} + i\right )\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(1/(a+I*a*tan(f*x+e))**(1/2)/(c-I*c*tan(f*x+e))**(3/2),x)
 

Output:

Integral(1/(sqrt(I*a*(tan(e + f*x) - I))*(-I*c*(tan(e + f*x) + I))**(3/2)) 
, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{\sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(1/(a+I*a*tan(f*x+e))^(1/2)/(c-I*c*tan(f*x+e))^(3/2),x, algorithm 
="maxima")
 

Output:

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un 
defined.
 

Giac [F]

\[ \int \frac {1}{\sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}} \, dx=\int { \frac {1}{\sqrt {i \, a \tan \left (f x + e\right ) + a} {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(a+I*a*tan(f*x+e))^(1/2)/(c-I*c*tan(f*x+e))^(3/2),x, algorithm 
="giac")
 

Output:

integrate(1/(sqrt(I*a*tan(f*x + e) + a)*(-I*c*tan(f*x + e) + c)^(3/2)), x)
 

Mupad [B] (verification not implemented)

Time = 0.39 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}} \, dx=\frac {\sqrt {\frac {a\,\left (\cos \left (2\,e+2\,f\,x\right )+1+\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}\,\left (\cos \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}+2\,\sin \left (2\,e+2\,f\,x\right )-3{}\mathrm {i}\right )}{6\,a\,c\,f\,\sqrt {\frac {c\,\left (\cos \left (2\,e+2\,f\,x\right )+1-\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}} \] Input:

int(1/((a + a*tan(e + f*x)*1i)^(1/2)*(c - c*tan(e + f*x)*1i)^(3/2)),x)
 

Output:

(((a*(cos(2*e + 2*f*x) + sin(2*e + 2*f*x)*1i + 1))/(cos(2*e + 2*f*x) + 1)) 
^(1/2)*(cos(2*e + 2*f*x)*1i + 2*sin(2*e + 2*f*x) - 3i))/(6*a*c*f*((c*(cos( 
2*e + 2*f*x) - sin(2*e + 2*f*x)*1i + 1))/(cos(2*e + 2*f*x) + 1))^(1/2))
 

Reduce [F]

\[ \int \frac {1}{\sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}} \, dx=-\frac {\sqrt {c}\, \sqrt {a}\, \left (\int \frac {\sqrt {\tan \left (f x +e \right ) i +1}\, \sqrt {-\tan \left (f x +e \right ) i +1}}{\tan \left (f x +e \right )^{3} i -\tan \left (f x +e \right )^{2}+\tan \left (f x +e \right ) i -1}d x \right )}{a \,c^{2}} \] Input:

int(1/(a+I*a*tan(f*x+e))^(1/2)/(c-I*c*tan(f*x+e))^(3/2),x)
 

Output:

( - sqrt(c)*sqrt(a)*int((sqrt(tan(e + f*x)*i + 1)*sqrt( - tan(e + f*x)*i + 
 1))/(tan(e + f*x)**3*i - tan(e + f*x)**2 + tan(e + f*x)*i - 1),x))/(a*c** 
2)