\(\int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{3/2}}{a+i a \tan (e+f x)} \, dx\) [766]

Optimal result

Integrand size = 43, antiderivative size = 144 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{3/2}}{a+i a \tan (e+f x)} \, dx=-\frac {(i A-5 B) c^{3/2} \text {arctanh}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{\sqrt {2} a f}+\frac {(i A-5 B) c \sqrt {c-i c \tan (e+f x)}}{2 a f}+\frac {(i A-B) (c-i c \tan (e+f x))^{3/2}}{2 a f (1+i \tan (e+f x))} \]

[Out]

-1/2*(I*A-5*B)*c^(3/2)*arctanh(1/2*(c-I*c*tan(f*x+e))^(1/2)*2^(1/2)/c^(1/2))/a/f*2^(1/2)+1/2*(I*A-5*B)*c*(c-I*
c*tan(f*x+e))^(1/2)/a/f+1/2*(I*A-B)*(c-I*c*tan(f*x+e))^(3/2)/a/f/(1+I*tan(f*x+e))

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.116, Rules used = {3669, 79, 52, 65, 214} \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{3/2}}{a+i a \tan (e+f x)} \, dx=-\frac {c^{3/2} (-5 B+i A) \text {arctanh}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{\sqrt {2} a f}+\frac {c (-5 B+i A) \sqrt {c-i c \tan (e+f x)}}{2 a f}+\frac {(-B+i A) (c-i c \tan (e+f x))^{3/2}}{2 a f (1+i \tan (e+f x))} \]

[In]

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

[Out]

-(((I*A - 5*B)*c^(3/2)*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])])/(Sqrt[2]*a*f)) + ((I*A - 5*B)*c*
Sqrt[c - I*c*Tan[e + f*x]])/(2*a*f) + ((I*A - B)*(c - I*c*Tan[e + f*x])^(3/2))/(2*a*f*(1 + I*Tan[e + f*x]))

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 3669

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

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

Mathematica [A] (verified)

Time = 3.90 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.75 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{3/2}}{a+i a \tan (e+f x)} \, dx=\frac {c \left (\sqrt {2} (-i A+5 B) \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )+\frac {2 (A+3 i B-2 B \tan (e+f x)) \sqrt {c-i c \tan (e+f x)}}{-i+\tan (e+f x)}\right )}{2 a f} \]

[In]

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

[Out]

(c*(Sqrt[2]*((-I)*A + 5*B)*Sqrt[c]*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])] + (2*(A + (3*I)*B - 2
*B*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(-I + Tan[e + f*x])))/(2*a*f)

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.77

method	result	size
derivativedivides	\(\frac {2 i c \left (i \sqrt {c -i c \tan \left (f x +e \right )}\, B +c \left (\frac {\left (\frac {i B}{4}+\frac {A}{4}\right ) \sqrt {c -i c \tan \left (f x +e \right )}}{\frac {c}{2}+\frac {i c \tan \left (f x +e \right )}{2}}-\frac {\left (\frac {A}{2}+\frac {5 i B}{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{2 \sqrt {c}}\right )\right )}{f a}\)	\(111\)
default	\(\frac {2 i c \left (i \sqrt {c -i c \tan \left (f x +e \right )}\, B +c \left (\frac {\left (\frac {i B}{4}+\frac {A}{4}\right ) \sqrt {c -i c \tan \left (f x +e \right )}}{\frac {c}{2}+\frac {i c \tan \left (f x +e \right )}{2}}-\frac {\left (\frac {A}{2}+\frac {5 i B}{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{2 \sqrt {c}}\right )\right )}{f a}\)	\(111\)

[In]

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

[Out]

2*I/f/a*c*(I*(c-I*c*tan(f*x+e))^(1/2)*B+c*((1/4*I*B+1/4*A)*(c-I*c*tan(f*x+e))^(1/2)/(1/2*c+1/2*I*c*tan(f*x+e))
-1/2*(1/2*A+5/2*I*B)*2^(1/2)/c^(1/2)*arctanh(1/2*(c-I*c*tan(f*x+e))^(1/2)*2^(1/2)/c^(1/2))))

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 330 vs. \(2 (111) = 222\).

Time = 0.25 (sec) , antiderivative size = 330, normalized size of antiderivative = 2.29 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{3/2}}{a+i a \tan (e+f x)} \, dx=-\frac {{\left (\sqrt {2} a f \sqrt {-\frac {{\left (A^{2} + 10 i \, A B - 25 \, B^{2}\right )} c^{3}}{a^{2} f^{2}}} e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (-\frac {2 \, {\left ({\left (i \, A - 5 \, B\right )} c^{2} + {\left (a f e^{\left (2 i \, f x + 2 i \, e\right )} + a f\right )} \sqrt {-\frac {{\left (A^{2} + 10 i \, A B - 25 \, B^{2}\right )} c^{3}}{a^{2} f^{2}}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-i \, f x - i \, e\right )}}{a f}\right ) - \sqrt {2} a f \sqrt {-\frac {{\left (A^{2} + 10 i \, A B - 25 \, B^{2}\right )} c^{3}}{a^{2} f^{2}}} e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (-\frac {2 \, {\left ({\left (i \, A - 5 \, B\right )} c^{2} - {\left (a f e^{\left (2 i \, f x + 2 i \, e\right )} + a f\right )} \sqrt {-\frac {{\left (A^{2} + 10 i \, A B - 25 \, B^{2}\right )} c^{3}}{a^{2} f^{2}}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-i \, f x - i \, e\right )}}{a f}\right ) + 2 \, \sqrt {2} {\left ({\left (-i \, A + 5 \, B\right )} c e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (-i \, A + B\right )} c\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{4 \, a f} \]

[In]

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

[Out]

-1/4*(sqrt(2)*a*f*sqrt(-(A^2 + 10*I*A*B - 25*B^2)*c^3/(a^2*f^2))*e^(2*I*f*x + 2*I*e)*log(-2*((I*A - 5*B)*c^2 +
(a*f*e^(2*I*f*x + 2*I*e) + a*f)*sqrt(-(A^2 + 10*I*A*B - 25*B^2)*c^3/(a^2*f^2))*sqrt(c/(e^(2*I*f*x + 2*I*e) +
1)))*e^(-I*f*x - I*e)/(a*f)) - sqrt(2)*a*f*sqrt(-(A^2 + 10*I*A*B - 25*B^2)*c^3/(a^2*f^2))*e^(2*I*f*x + 2*I*e)*
log(-2*((I*A - 5*B)*c^2 - (a*f*e^(2*I*f*x + 2*I*e) + a*f)*sqrt(-(A^2 + 10*I*A*B - 25*B^2)*c^3/(a^2*f^2))*sqrt(
c/(e^(2*I*f*x + 2*I*e) + 1)))*e^(-I*f*x - I*e)/(a*f)) + 2*sqrt(2)*((-I*A + 5*B)*c*e^(2*I*f*x + 2*I*e) + (-I*A
+ B)*c)*sqrt(c/(e^(2*I*f*x + 2*I*e) + 1)))*e^(-2*I*f*x - 2*I*e)/(a*f)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.96 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{3/2}}{a+i a \tan (e+f x)} \, dx=\frac {i \, {\left (\frac {\sqrt {2} {\left (A + 5 i \, B\right )} c^{\frac {5}{2}} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-i \, c \tan \left (f x + e\right ) + c}}{\sqrt {2} \sqrt {c} + \sqrt {-i \, c \tan \left (f x + e\right ) + c}}\right )}{a} - \frac {4 \, \sqrt {-i \, c \tan \left (f x + e\right ) + c} {\left (A + i \, B\right )} c^{3}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )} a - 2 \, a c} + \frac {8 i \, \sqrt {-i \, c \tan \left (f x + e\right ) + c} B c^{2}}{a}\right )}}{4 \, c f} \]

[In]

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

[Out]

1/4*I*(sqrt(2)*(A + 5*I*B)*c^(5/2)*log(-(sqrt(2)*sqrt(c) - sqrt(-I*c*tan(f*x + e) + c))/(sqrt(2)*sqrt(c) + sqr
t(-I*c*tan(f*x + e) + c)))/a - 4*sqrt(-I*c*tan(f*x + e) + c)*(A + I*B)*c^3/((-I*c*tan(f*x + e) + c)*a - 2*a*c)
+ 8*I*sqrt(-I*c*tan(f*x + e) + c)*B*c^2/a)/(c*f)

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.20 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.31 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{3/2}}{a+i a \tan (e+f x)} \, dx=\frac {B\,c^2\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}}{a\,f\,\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )-2\,a\,c\,f}-\frac {2\,B\,c\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}}{a\,f}+\frac {\sqrt {2}\,A\,{\left (-c\right )}^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}}{2\,\sqrt {-c}}\right )\,1{}\mathrm {i}}{2\,a\,f}+\frac {5\,\sqrt {2}\,B\,c^{3/2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}}{2\,\sqrt {c}}\right )}{2\,a\,f}+\frac {A\,c^2\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}\,1{}\mathrm {i}}{a\,f\,\left (c+c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )} \]

[In]

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

[Out]

(B*c^2*(c - c*tan(e + f*x)*1i)^(1/2))/(a*f*(c - c*tan(e + f*x)*1i) - 2*a*c*f) - (2*B*c*(c - c*tan(e + f*x)*1i)
^(1/2))/(a*f) + (2^(1/2)*A*(-c)^(3/2)*atan((2^(1/2)*(c - c*tan(e + f*x)*1i)^(1/2))/(2*(-c)^(1/2)))*1i)/(2*a*f)
+ (5*2^(1/2)*B*c^(3/2)*atanh((2^(1/2)*(c - c*tan(e + f*x)*1i)^(1/2))/(2*c^(1/2))))/(2*a*f) + (A*c^2*(c - c*ta
n(e + f*x)*1i)^(1/2)*1i)/(a*f*(c + c*tan(e + f*x)*1i))