\(\int \frac {(a+i a \tan (e+f x))^2}{(c-i c \tan (e+f x))^{3/2}} \, dx\) [981]

Optimal result

Integrand size = 33, antiderivative size = 60 \[ \int \frac {(a+i a \tan (e+f x))^2}{(c-i c \tan (e+f x))^{3/2}} \, dx=-\frac {4 i a^2}{3 f (c-i c \tan (e+f x))^{3/2}}+\frac {2 i a^2}{c f \sqrt {c-i c \tan (e+f x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3603, 3568, 45} \[ \int \frac {(a+i a \tan (e+f x))^2}{(c-i c \tan (e+f x))^{3/2}} \, dx=\frac {2 i a^2}{c f \sqrt {c-i c \tan (e+f x)}}-\frac {4 i a^2}{3 f (c-i c \tan (e+f x))^{3/2}} \]

[In]

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

[Out]

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

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 3568

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

Rule 3603

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Di
st[a^m*c^m, Int[Sec[e + f*x]^(2*m)*(c + d*Tan[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&
EqQ[b*c + a*d, 0] && EqQ[a^2 + b^2, 0] && IntegerQ[m] &&  !(IGtQ[n, 0] && (LtQ[m, 0] || GtQ[m, n]))

Rubi steps \begin{align*} \text {integral}& = \left (a^2 c^2\right ) \int \frac {\sec ^4(e+f x)}{(c-i c \tan (e+f x))^{7/2}} \, dx \\ & = \frac {\left (i a^2\right ) \text {Subst}\left (\int \frac {c-x}{(c+x)^{5/2}} \, dx,x,-i c \tan (e+f x)\right )}{c f} \\ & = \frac {\left (i a^2\right ) \text {Subst}\left (\int \left (\frac {2 c}{(c+x)^{5/2}}-\frac {1}{(c+x)^{3/2}}\right ) \, dx,x,-i c \tan (e+f x)\right )}{c f} \\ & = -\frac {4 i a^2}{3 f (c-i c \tan (e+f x))^{3/2}}+\frac {2 i a^2}{c f \sqrt {c-i c \tan (e+f x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.13 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.92 \[ \int \frac {(a+i a \tan (e+f x))^2}{(c-i c \tan (e+f x))^{3/2}} \, dx=\frac {i a^2 \left (-\frac {4 c}{3 (c-i c \tan (e+f x))^{3/2}}+\frac {2}{\sqrt {c-i c \tan (e+f x)}}\right )}{c f} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.73

method	result	size
risch	\(-\frac {i a^{2} \left ({\mathrm e}^{2 i \left (f x +e \right )}-2\right ) \sqrt {2}}{3 c \sqrt {\frac {c}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}\)	\(44\)
derivativedivides	\(\frac {2 i a^{2} \left (\frac {1}{\sqrt {c -i c \tan \left (f x +e \right )}}-\frac {2 c}{3 \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}\right )}{f c}\)	\(45\)
default	\(\frac {2 i a^{2} \left (\frac {1}{\sqrt {c -i c \tan \left (f x +e \right )}}-\frac {2 c}{3 \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}\right )}{f c}\)	\(45\)
parts	\(\frac {2 i a^{2} c \left (\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{8 c^{\frac {5}{2}}}-\frac {1}{4 c^{2} \sqrt {c -i c \tan \left (f x +e \right )}}-\frac {1}{6 c \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}\right )}{f}+\frac {2 i a^{2} \left (-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{4 c^{\frac {3}{2}}}-\frac {1}{3 \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}+\frac {1}{2 c \sqrt {c -i c \tan \left (f x +e \right )}}\right )}{f}-\frac {2 i a^{2} \left (-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{8 \sqrt {c}}-\frac {3}{4 \sqrt {c -i c \tan \left (f x +e \right )}}+\frac {c}{6 \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}\right )}{f c}\)	\(235\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.03 \[ \int \frac {(a+i a \tan (e+f x))^2}{(c-i c \tan (e+f x))^{3/2}} \, dx=\frac {\sqrt {2} {\left (-i \, a^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + i \, a^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 2 i \, a^{2}\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{3 \, c^{2} f} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.73 \[ \int \frac {(a+i a \tan (e+f x))^2}{(c-i c \tan (e+f x))^{3/2}} \, dx=\frac {2 i \, {\left (3 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )} a^{2} - 2 \, a^{2} c\right )}}{3 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} c f} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 6.41 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.63 \[ \int \frac {(a+i a \tan (e+f x))^2}{(c-i c \tan (e+f x))^{3/2}} \, dx=\frac {a^2\,\sqrt {\frac {c\,\left (\cos \left (2\,e+2\,f\,x\right )+1-\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}\,\left (\cos \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}-\cos \left (4\,e+4\,f\,x\right )\,1{}\mathrm {i}-\sin \left (2\,e+2\,f\,x\right )+\sin \left (4\,e+4\,f\,x\right )+2{}\mathrm {i}\right )}{3\,c^2\,f} \]

[In]

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

[Out]

(a^2*((c*(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 - co
s(4*e + 4*f*x)*1i - sin(2*e + 2*f*x) + sin(4*e + 4*f*x) + 2i))/(3*c^2*f)