Integrand size = 35, antiderivative size = 94 \[ \int \frac {1}{(a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}} \, dx=\frac {i}{3 f (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}+\frac {2 \tan (e+f x)}{3 a f \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}} \] Output:
1/3*I/f/(a+I*a*tan(f*x+e))^(3/2)/(c-I*c*tan(f*x+e))^(1/2)+2/3*tan(f*x+e)/a /f/(a+I*a*tan(f*x+e))^(1/2)/(c-I*c*tan(f*x+e))^(1/2)
Time = 0.73 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.83 \[ \int \frac {1}{(a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}} \, dx=\frac {1-2 i \tan (e+f x)+2 \tan ^2(e+f x)}{3 a f (-i+\tan (e+f x)) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}} \] Input:
Integrate[1/((a + I*a*Tan[e + f*x])^(3/2)*Sqrt[c - I*c*Tan[e + f*x]]),x]
Output:
(1 - (2*I)*Tan[e + f*x] + 2*Tan[e + f*x]^2)/(3*a*f*(-I + Tan[e + f*x])*Sqr t[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])
Time = 0.30 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.10, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {3042, 4006, 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.
\(\displaystyle \int \frac {1}{(a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{(a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}dx\) |
\(\Big \downarrow \) 4006 |
\(\displaystyle \frac {a c \int \frac {1}{(i \tan (e+f x) a+a)^{5/2} (c-i c \tan (e+f x))^{3/2}}d\tan (e+f x)}{f}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {a c \left (\frac {2 \int \frac {1}{(i \tan (e+f x) a+a)^{3/2} (c-i c \tan (e+f x))^{3/2}}d\tan (e+f x)}{3 a}+\frac {i}{3 a c (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}\right )}{f}\) |
\(\Big \downarrow \) 41 |
\(\displaystyle \frac {a c \left (\frac {2 \tan (e+f x)}{3 a^2 c \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}+\frac {i}{3 a c (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}\right )}{f}\) |
Input:
Int[1/((a + I*a*Tan[e + f*x])^(3/2)*Sqrt[c - I*c*Tan[e + f*x]]),x]
Output:
(a*c*((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]])))/f
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]
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])
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]
Time = 0.36 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.16
method | result | size |
derivativedivides | \(\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \left (2 i \tan \left (f x +e \right )^{3}-2 \tan \left (f x +e \right )^{4}+2 i \tan \left (f x +e \right )-3 \tan \left (f x +e \right )^{2}-1\right )}{3 f \,a^{2} c \left (-\tan \left (f x +e \right )+i\right )^{3} \left (i+\tan \left (f x +e \right )\right )^{2}}\) | \(109\) |
default | \(\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \left (2 i \tan \left (f x +e \right )^{3}-2 \tan \left (f x +e \right )^{4}+2 i \tan \left (f x +e \right )-3 \tan \left (f x +e \right )^{2}-1\right )}{3 f \,a^{2} c \left (-\tan \left (f x +e \right )+i\right )^{3} \left (i+\tan \left (f x +e \right )\right )^{2}}\) | \(109\) |
Input:
int(1/(a+I*a*tan(f*x+e))^(3/2)/(c-I*c*tan(f*x+e))^(1/2),x,method=_RETURNVE RBOSE)
Output:
1/3/f*(a*(1+I*tan(f*x+e)))^(1/2)*(-c*(I*tan(f*x+e)-1))^(1/2)/a^2/c*(2*I*ta n(f*x+e)^3-2*tan(f*x+e)^4+2*I*tan(f*x+e)-3*tan(f*x+e)^2-1)/(-tan(f*x+e)+I) ^3/(I+tan(f*x+e))^2
Time = 0.09 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.18 \[ \int \frac {1}{(a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}} \, dx=\frac {\sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} {\left (-3 i \, e^{\left (6 i \, f x + 6 i \, e\right )} - 4 i \, e^{\left (5 i \, f x + 5 i \, e\right )} + 3 i \, e^{\left (4 i \, f x + 4 i \, e\right )} - 4 i \, e^{\left (3 i \, f x + 3 i \, e\right )} + 7 i \, e^{\left (2 i \, f x + 2 i \, e\right )} + i\right )} e^{\left (-3 i \, f x - 3 i \, e\right )}}{12 \, a^{2} c f} \] Input:
integrate(1/(a+I*a*tan(f*x+e))^(3/2)/(c-I*c*tan(f*x+e))^(1/2),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))*( -3*I*e^(6*I*f*x + 6*I*e) - 4*I*e^(5*I*f*x + 5*I*e) + 3*I*e^(4*I*f*x + 4*I* e) - 4*I*e^(3*I*f*x + 3*I*e) + 7*I*e^(2*I*f*x + 2*I*e) + I)*e^(-3*I*f*x - 3*I*e)/(a^2*c*f)
\[ \int \frac {1}{(a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}} \, dx=\int \frac {1}{\left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{\frac {3}{2}} \sqrt {- i c \left (\tan {\left (e + f x \right )} + i\right )}}\, dx \] Input:
integrate(1/(a+I*a*tan(f*x+e))**(3/2)/(c-I*c*tan(f*x+e))**(1/2),x)
Output:
Integral(1/((I*a*(tan(e + f*x) - I))**(3/2)*sqrt(-I*c*(tan(e + f*x) + I))) , x)
Exception generated. \[ \int \frac {1}{(a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(1/(a+I*a*tan(f*x+e))^(3/2)/(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.
\[ \int \frac {1}{(a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}} \, dx=\int { \frac {1}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac {3}{2}} \sqrt {-i \, c \tan \left (f x + e\right ) + c}} \,d x } \] Input:
integrate(1/(a+I*a*tan(f*x+e))^(3/2)/(c-I*c*tan(f*x+e))^(1/2),x, algorithm ="giac")
Output:
integrate(1/((I*a*tan(f*x + e) + a)^(3/2)*sqrt(-I*c*tan(f*x + e) + c)), x)
Time = 2.55 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.44 \[ \int \frac {1}{(a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}} \, dx=\frac {\sqrt {\frac {a\,\left (\cos \left (2\,e+2\,f\,x\right )+1+\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}\,\left (\cos \left (2\,e+2\,f\,x\right )\,6{}\mathrm {i}+\cos \left (4\,e+4\,f\,x\right )\,1{}\mathrm {i}+6\,\sin \left (2\,e+2\,f\,x\right )+\sin \left (4\,e+4\,f\,x\right )-3{}\mathrm {i}\right )}{12\,a^2\,f\,\sqrt {\frac {c\,\left (\cos \left (2\,e+2\,f\,x\right )+1-\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}} \] Input:
int(1/((a + a*tan(e + f*x)*1i)^(3/2)*(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)*6i + cos(4*e + 4*f*x)*1i + 6*sin(2*e + 2*f*x) + s in(4*e + 4*f*x) - 3i))/(12*a^2*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))
\[ \int \frac {1}{(a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}} \, dx=\frac {\sqrt {c}\, \sqrt {a}\, \left (\int \frac {\sqrt {\tan \left (f x +e \right ) i +1}\, \sqrt {-\tan \left (f x +e \right ) i +1}}{\tan \left (f x +e \right )^{3} i +\tan \left (f x +e \right )^{2}+\tan \left (f x +e \right ) i +1}d x \right )}{a^{2} c} \] Input:
int(1/(a+I*a*tan(f*x+e))^(3/2)/(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)**3*i + tan(e + f*x)**2 + tan(e + f*x)*i + 1),x))/(a**2*c)