\(\int \frac {1}{(a+i a \tan (e+f x))^{7/2} \sqrt {c-i c \tan (e+f x)}} \, dx\) [1023]

Optimal result

Integrand size = 35, antiderivative size = 186 \[ \int \frac {1}{(a+i a \tan (e+f x))^{7/2} \sqrt {c-i c \tan (e+f x)}} \, dx=\frac {i}{7 f (a+i a \tan (e+f x))^{7/2} \sqrt {c-i c \tan (e+f x)}}+\frac {4 i}{35 a f (a+i a \tan (e+f x))^{5/2} \sqrt {c-i c \tan (e+f x)}}+\frac {4 i}{35 a^2 f (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}+\frac {8 \tan (e+f x)}{35 a^3 f \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}} \] Output:

1/7*I/f/(a+I*a*tan(f*x+e))^(7/2)/(c-I*c*tan(f*x+e))^(1/2)+4/35*I/a/f/(a+I* 
a*tan(f*x+e))^(5/2)/(c-I*c*tan(f*x+e))^(1/2)+4/35*I/a^2/f/(a+I*a*tan(f*x+e 
))^(3/2)/(c-I*c*tan(f*x+e))^(1/2)+8/35*tan(f*x+e)/a^3/f/(a+I*a*tan(f*x+e)) 
^(1/2)/(c-I*c*tan(f*x+e))^(1/2)
 

Mathematica [A] (verified)

Time = 1.54 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.54 \[ \int \frac {1}{(a+i a \tan (e+f x))^{7/2} \sqrt {c-i c \tan (e+f x)}} \, dx=\frac {-13-4 i \tan (e+f x)-20 \tan ^2(e+f x)-24 i \tan ^3(e+f x)+8 \tan ^4(e+f x)}{35 a^3 f (-i+\tan (e+f x))^3 \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}} \] Input:

Integrate[1/((a + I*a*Tan[e + f*x])^(7/2)*Sqrt[c - I*c*Tan[e + f*x]]),x]
 

Output:

(-13 - (4*I)*Tan[e + f*x] - 20*Tan[e + f*x]^2 - (24*I)*Tan[e + f*x]^3 + 8* 
Tan[e + f*x]^4)/(35*a^3*f*(-I + Tan[e + f*x])^3*Sqrt[a + I*a*Tan[e + f*x]] 
*Sqrt[c - I*c*Tan[e + f*x]])
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.13, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {3042, 4006, 55, 55, 55, 41}

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.

\(\displaystyle \int \frac {1}{(a+i a \tan (e+f x))^{7/2} \sqrt {c-i c \tan (e+f x)}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{(a+i a \tan (e+f x))^{7/2} \sqrt {c-i c \tan (e+f x)}}dx\)

\(\Big \downarrow \) 4006

\(\displaystyle \frac {a c \int \frac {1}{(i \tan (e+f x) a+a)^{9/2} (c-i c \tan (e+f x))^{3/2}}d\tan (e+f x)}{f}\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {a c \left (\frac {4 \int \frac {1}{(i \tan (e+f x) a+a)^{7/2} (c-i c \tan (e+f x))^{3/2}}d\tan (e+f x)}{7 a}+\frac {i}{7 a c (a+i a \tan (e+f x))^{7/2} \sqrt {c-i c \tan (e+f x)}}\right )}{f}\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {a c \left (\frac {4 \left (\frac {3 \int \frac {1}{(i \tan (e+f x) a+a)^{5/2} (c-i c \tan (e+f x))^{3/2}}d\tan (e+f x)}{5 a}+\frac {i}{5 a c (a+i a \tan (e+f x))^{5/2} \sqrt {c-i c \tan (e+f x)}}\right )}{7 a}+\frac {i}{7 a c (a+i a \tan (e+f x))^{7/2} \sqrt {c-i c \tan (e+f x)}}\right )}{f}\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {a c \left (\frac {4 \left (\frac {3 \left (\frac {2 \int \frac {1}{(i \tan (e+f x) a+a)^{3/2} (c-i c \tan (e+f x))^{3/2}}d\tan (e+f x)}{3 a}+\frac {i}{3 a c (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}\right )}{5 a}+\frac {i}{5 a c (a+i a \tan (e+f x))^{5/2} \sqrt {c-i c \tan (e+f x)}}\right )}{7 a}+\frac {i}{7 a c (a+i a \tan (e+f x))^{7/2} \sqrt {c-i c \tan (e+f x)}}\right )}{f}\)

\(\Big \downarrow \) 41

\(\displaystyle \frac {a c \left (\frac {4 \left (\frac {3 \left (\frac {2 \tan (e+f x)}{3 a^2 c \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}+\frac {i}{3 a c (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}\right )}{5 a}+\frac {i}{5 a c (a+i a \tan (e+f x))^{5/2} \sqrt {c-i c \tan (e+f x)}}\right )}{7 a}+\frac {i}{7 a c (a+i a \tan (e+f x))^{7/2} \sqrt {c-i c \tan (e+f x)}}\right )}{f}\)

Input:

Int[1/((a + I*a*Tan[e + f*x])^(7/2)*Sqrt[c - I*c*Tan[e + f*x]]),x]
 

Output:

(a*c*((I/7)/(a*c*(a + I*a*Tan[e + f*x])^(7/2)*Sqrt[c - I*c*Tan[e + f*x]]) 
+ (4*((I/5)/(a*c*(a + I*a*Tan[e + f*x])^(5/2)*Sqrt[c - I*c*Tan[e + f*x]]) 
+ (3*((I/3)/(a*c*(a + I*a*Tan[e + f*x])^(3/2)*Sqrt[c - I*c*Tan[e + f*x]]) 
+ (2*Tan[e + f*x])/(3*a^2*c*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e 
+ f*x]])))/(5*a)))/(7*a)))/f
 

Defintions of rubi rules used

rule 41

Int[1/(((a_) + (b_.)*(x_))^(3/2)*((c_) + (d_.)*(x_))^(3/2)), x_Symbol] :> S 
imp[x/(a*c*Sqrt[a + b*x]*Sqrt[c + d*x]), x] /; FreeQ[{a, b, c, d}, x] && Eq 
Q[b*c + a*d, 0]
 

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 = 0.36 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.70

method	result	size
derivativedivides	\(\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \left (24 i \tan \left (f x +e \right )^{5}-8 \tan \left (f x +e \right )^{6}+28 i \tan \left (f x +e \right )^{3}+12 \tan \left (f x +e \right )^{4}+4 i \tan \left (f x +e \right )+33 \tan \left (f x +e \right )^{2}+13\right )}{35 f \,a^{4} c \left (-\tan \left (f x +e \right )+i\right )^{5} \left (i+\tan \left (f x +e \right )\right )^{2}}\)	\(130\)
default	\(\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \left (24 i \tan \left (f x +e \right )^{5}-8 \tan \left (f x +e \right )^{6}+28 i \tan \left (f x +e \right )^{3}+12 \tan \left (f x +e \right )^{4}+4 i \tan \left (f x +e \right )+33 \tan \left (f x +e \right )^{2}+13\right )}{35 f \,a^{4} c \left (-\tan \left (f x +e \right )+i\right )^{5} \left (i+\tan \left (f x +e \right )\right )^{2}}\)	\(130\)

Input:

int(1/(a+I*a*tan(f*x+e))^(7/2)/(c-I*c*tan(f*x+e))^(1/2),x,method=_RETURNVE 
RBOSE)
 

Output:

1/35/f*(a*(1+I*tan(f*x+e)))^(1/2)*(-c*(I*tan(f*x+e)-1))^(1/2)/a^4/c*(24*I* 
tan(f*x+e)^5-8*tan(f*x+e)^6+28*I*tan(f*x+e)^3+12*tan(f*x+e)^4+4*I*tan(f*x+ 
e)+33*tan(f*x+e)^2+13)/(-tan(f*x+e)+I)^5/(I+tan(f*x+e))^2
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.72 \[ \int \frac {1}{(a+i a \tan (e+f x))^{7/2} \sqrt {c-i c \tan (e+f x)}} \, dx=\frac {\sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} {\left (-35 i \, e^{\left (10 i \, f x + 10 i \, e\right )} - 208 i \, e^{\left (9 i \, f x + 9 i \, e\right )} + 105 i \, e^{\left (8 i \, f x + 8 i \, e\right )} - 208 i \, e^{\left (7 i \, f x + 7 i \, e\right )} + 210 i \, e^{\left (6 i \, f x + 6 i \, e\right )} + 98 i \, e^{\left (4 i \, f x + 4 i \, e\right )} + 33 i \, e^{\left (2 i \, f x + 2 i \, e\right )} + 5 i\right )} e^{\left (-7 i \, f x - 7 i \, e\right )}}{560 \, a^{4} c f} \] Input:

integrate(1/(a+I*a*tan(f*x+e))^(7/2)/(c-I*c*tan(f*x+e))^(1/2),x, algorithm 
="fricas")
 

Output:

1/560*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(c/(e^(2*I*f*x + 2*I*e) + 1))* 
(-35*I*e^(10*I*f*x + 10*I*e) - 208*I*e^(9*I*f*x + 9*I*e) + 105*I*e^(8*I*f* 
x + 8*I*e) - 208*I*e^(7*I*f*x + 7*I*e) + 210*I*e^(6*I*f*x + 6*I*e) + 98*I* 
e^(4*I*f*x + 4*I*e) + 33*I*e^(2*I*f*x + 2*I*e) + 5*I)*e^(-7*I*f*x - 7*I*e) 
/(a^4*c*f)
 

Sympy [F]

\[ \int \frac {1}{(a+i a \tan (e+f x))^{7/2} \sqrt {c-i c \tan (e+f x)}} \, dx=\int \frac {1}{\left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{\frac {7}{2}} \sqrt {- i c \left (\tan {\left (e + f x \right )} + i\right )}}\, dx \] Input:

integrate(1/(a+I*a*tan(f*x+e))**(7/2)/(c-I*c*tan(f*x+e))**(1/2),x)
 

Output:

Integral(1/((I*a*(tan(e + f*x) - I))**(7/2)*sqrt(-I*c*(tan(e + f*x) + I))) 
, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(a+i a \tan (e+f x))^{7/2} \sqrt {c-i c \tan (e+f x)}} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(1/(a+I*a*tan(f*x+e))^(7/2)/(c-I*c*tan(f*x+e))^(1/2),x, algorithm 
="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {1}{(a+i a \tan (e+f x))^{7/2} \sqrt {c-i c \tan (e+f x)}} \, dx=\int { \frac {1}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac {7}{2}} \sqrt {-i \, c \tan \left (f x + e\right ) + c}} \,d x } \] Input:

integrate(1/(a+I*a*tan(f*x+e))^(7/2)/(c-I*c*tan(f*x+e))^(1/2),x, algorithm 
="giac")
 

Output:

integrate(1/((I*a*tan(f*x + e) + a)^(7/2)*sqrt(-I*c*tan(f*x + e) + c)), x)
 

Mupad [B] (verification not implemented)

Time = 3.52 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.98 \[ \int \frac {1}{(a+i a \tan (e+f x))^{7/2} \sqrt {c-i c \tan (e+f x)}} \, dx=\frac {\sqrt {\frac {a\,\left (\cos \left (2\,e+2\,f\,x\right )+1+\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}\,\left (\cos \left (2\,e+2\,f\,x\right )\,140{}\mathrm {i}+\cos \left (4\,e+4\,f\,x\right )\,70{}\mathrm {i}+\cos \left (6\,e+6\,f\,x\right )\,28{}\mathrm {i}+\cos \left (8\,e+8\,f\,x\right )\,5{}\mathrm {i}+140\,\sin \left (2\,e+2\,f\,x\right )+70\,\sin \left (4\,e+4\,f\,x\right )+28\,\sin \left (6\,e+6\,f\,x\right )+5\,\sin \left (8\,e+8\,f\,x\right )-35{}\mathrm {i}\right )}{560\,a^4\,f\,\sqrt {\frac {c\,\left (\cos \left (2\,e+2\,f\,x\right )+1-\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}} \] Input:

int(1/((a + a*tan(e + f*x)*1i)^(7/2)*(c - c*tan(e + f*x)*1i)^(1/2)),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)*140i + cos(4*e + 4*f*x)*70i + cos(6*e + 6*f*x)*28 
i + cos(8*e + 8*f*x)*5i + 140*sin(2*e + 2*f*x) + 70*sin(4*e + 4*f*x) + 28* 
sin(6*e + 6*f*x) + 5*sin(8*e + 8*f*x) - 35i))/(560*a^4*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]

\[ \int \frac {1}{(a+i a \tan (e+f x))^{7/2} \sqrt {c-i c \tan (e+f x)}} \, dx=-\frac {\sqrt {c}\, \sqrt {a}\, \left (\int \frac {\sqrt {\tan \left (f x +e \right ) i +1}\, \sqrt {-\tan \left (f x +e \right ) i +1}}{\tan \left (f x +e \right )^{5} i +3 \tan \left (f x +e \right )^{4}-2 \tan \left (f x +e \right )^{3} i +2 \tan \left (f x +e \right )^{2}-3 \tan \left (f x +e \right ) i -1}d x \right )}{a^{4} c} \] Input:

int(1/(a+I*a*tan(f*x+e))^(7/2)/(c-I*c*tan(f*x+e))^(1/2),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)**5*i + 3*tan(e + f*x)**4 - 2*tan(e + f*x)**3*i + 2*tan( 
e + f*x)**2 - 3*tan(e + f*x)*i - 1),x))/(a**4*c)