\(\int \frac {A+B \tan (c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx\) [93]

Optimal result

Integrand size = 28, antiderivative size = 82 \[ \int \frac {A+B \tan (c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=-\frac {(i A+B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {2} \sqrt {a} d}+\frac {i A-B}{d \sqrt {a+i a \tan (c+d x)}} \]

[Out]

-1/2*(I*A+B)*arctanh(1/2*(a+I*a*tan(d*x+c))^(1/2)*2^(1/2)/a^(1/2))/d*2^(1/2)/a^(1/2)+(I*A-B)/d/(a+I*a*tan(d*x+
c))^(1/2)

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {3607, 3561, 212} \[ \int \frac {A+B \tan (c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\frac {-B+i A}{d \sqrt {a+i a \tan (c+d x)}}-\frac {(B+i A) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {2} \sqrt {a} d} \]

[In]

Int[(A + B*Tan[c + d*x])/Sqrt[a + I*a*Tan[c + d*x]],x]

[Out]

-(((I*A + B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(Sqrt[2]*Sqrt[a]*d)) + (I*A - B)/(d*Sqrt[a
+ I*a*Tan[c + d*x]])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 3561

Int[Sqrt[(a_) + (b_.)*tan[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[-2*(b/d), Subst[Int[1/(2*a - x^2), x], x, Sq
rt[a + b*Tan[c + d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0]

Rule 3607

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

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

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00 \[ \int \frac {A+B \tan (c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=-\frac {(i A+B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {2} \sqrt {a} d}+\frac {i A-B}{d \sqrt {a+i a \tan (c+d x)}} \]

[In]

Integrate[(A + B*Tan[c + d*x])/Sqrt[a + I*a*Tan[c + d*x]],x]

[Out]

-(((I*A + B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(Sqrt[2]*Sqrt[a]*d)) + (I*A - B)/(d*Sqrt[a
+ I*a*Tan[c + d*x]])

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.87

method	result	size
derivativedivides	\(\frac {2 i \left (-\frac {\left (\frac {A}{2}-\frac {i B}{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{2 \sqrt {a}}-\frac {-\frac {A}{2}-\frac {i B}{2}}{\sqrt {a +i a \tan \left (d x +c \right )}}\right )}{d}\)	\(71\)
default	\(\frac {2 i \left (-\frac {\left (\frac {A}{2}-\frac {i B}{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{2 \sqrt {a}}-\frac {-\frac {A}{2}-\frac {i B}{2}}{\sqrt {a +i a \tan \left (d x +c \right )}}\right )}{d}\)	\(71\)
parts	\(\frac {2 i A a \left (-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{4 a^{\frac {3}{2}}}+\frac {1}{2 a \sqrt {a +i a \tan \left (d x +c \right )}}\right )}{d}+\frac {B \left (-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{2 \sqrt {a}}-\frac {1}{\sqrt {a +i a \tan \left (d x +c \right )}}\right )}{d}\)	\(114\)

[In]

int((A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^(1/2),x,method=_RETURNVERBOSE)

[Out]

2*I/d*(-1/2*(1/2*A-1/2*I*B)*2^(1/2)/a^(1/2)*arctanh(1/2*(a+I*a*tan(d*x+c))^(1/2)*2^(1/2)/a^(1/2))-(-1/2*A-1/2*
I*B)/(a+I*a*tan(d*x+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. 328 vs. \(2 (63) = 126\).

Time = 0.26 (sec) , antiderivative size = 328, normalized size of antiderivative = 4.00 \[ \int \frac {A+B \tan (c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\frac {{\left (\sqrt {2} a d \sqrt {-\frac {A^{2} - 2 i \, A B - B^{2}}{a d^{2}}} e^{\left (i \, d x + i \, c\right )} \log \left (-\frac {4 \, {\left ({\left (-i \, A - B\right )} a e^{\left (i \, d x + i \, c\right )} + {\left (a d e^{\left (2 i \, d x + 2 i \, c\right )} + a d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {-\frac {A^{2} - 2 i \, A B - B^{2}}{a d^{2}}}\right )} e^{\left (-i \, d x - i \, c\right )}}{i \, A + B}\right ) - \sqrt {2} a d \sqrt {-\frac {A^{2} - 2 i \, A B - B^{2}}{a d^{2}}} e^{\left (i \, d x + i \, c\right )} \log \left (-\frac {4 \, {\left ({\left (-i \, A - B\right )} a e^{\left (i \, d x + i \, c\right )} - {\left (a d e^{\left (2 i \, d x + 2 i \, c\right )} + a d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {-\frac {A^{2} - 2 i \, A B - B^{2}}{a d^{2}}}\right )} e^{\left (-i \, d x - i \, c\right )}}{i \, A + B}\right ) - 2 \, \sqrt {2} {\left ({\left (-i \, A + B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, A + B\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-i \, d x - i \, c\right )}}{4 \, a d} \]

[In]

integrate((A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

1/4*(sqrt(2)*a*d*sqrt(-(A^2 - 2*I*A*B - B^2)/(a*d^2))*e^(I*d*x + I*c)*log(-4*((-I*A - B)*a*e^(I*d*x + I*c) + (
a*d*e^(2*I*d*x + 2*I*c) + a*d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt(-(A^2 - 2*I*A*B - B^2)/(a*d^2)))*e^(-I*d
*x - I*c)/(I*A + B)) - sqrt(2)*a*d*sqrt(-(A^2 - 2*I*A*B - B^2)/(a*d^2))*e^(I*d*x + I*c)*log(-4*((-I*A - B)*a*e
^(I*d*x + I*c) - (a*d*e^(2*I*d*x + 2*I*c) + a*d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt(-(A^2 - 2*I*A*B - B^2)
/(a*d^2)))*e^(-I*d*x - I*c)/(I*A + B)) - 2*sqrt(2)*((-I*A + B)*e^(2*I*d*x + 2*I*c) - I*A + B)*sqrt(a/(e^(2*I*d
*x + 2*I*c) + 1)))*e^(-I*d*x - I*c)/(a*d)

Sympy [F]

\[ \int \frac {A+B \tan (c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int \frac {A + B \tan {\left (c + d x \right )}}{\sqrt {i a \left (\tan {\left (c + d x \right )} - i\right )}}\, dx \]

[In]

integrate((A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))**(1/2),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.11 \[ \int \frac {A+B \tan (c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\frac {i \, {\left (\sqrt {2} {\left (A - i \, B\right )} \sqrt {a} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right ) + \frac {4 \, {\left (A + i \, B\right )} a}{\sqrt {i \, a \tan \left (d x + c\right ) + a}}\right )}}{4 \, a d} \]

[In]

integrate((A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

1/4*I*(sqrt(2)*(A - I*B)*sqrt(a)*log(-(sqrt(2)*sqrt(a) - sqrt(I*a*tan(d*x + c) + a))/(sqrt(2)*sqrt(a) + sqrt(I
*a*tan(d*x + c) + a))) + 4*(A + I*B)*a/sqrt(I*a*tan(d*x + c) + a))/(a*d)

Giac [F]

\[ \int \frac {A+B \tan (c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int { \frac {B \tan \left (d x + c\right ) + A}{\sqrt {i \, a \tan \left (d x + c\right ) + a}} \,d x } \]

[In]

integrate((A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate((B*tan(d*x + c) + A)/sqrt(I*a*tan(d*x + c) + a), x)

Mupad [B] (verification not implemented)

Time = 0.89 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.43 \[ \int \frac {A+B \tan (c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\frac {A\,1{}\mathrm {i}}{d\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}-\frac {B}{d\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}+\frac {\sqrt {2}\,A\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{2\,\sqrt {-a}}\right )\,1{}\mathrm {i}}{2\,\sqrt {-a}\,d}-\frac {\sqrt {2}\,B\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{2\,\sqrt {a}}\right )}{2\,\sqrt {a}\,d} \]

[In]

int((A + B*tan(c + d*x))/(a + a*tan(c + d*x)*1i)^(1/2),x)

[Out]

(A*1i)/(d*(a + a*tan(c + d*x)*1i)^(1/2)) - B/(d*(a + a*tan(c + d*x)*1i)^(1/2)) + (2^(1/2)*A*atan((2^(1/2)*(a +
a*tan(c + d*x)*1i)^(1/2))/(2*(-a)^(1/2)))*1i)/(2*(-a)^(1/2)*d) - (2^(1/2)*B*atanh((2^(1/2)*(a + a*tan(c + d*x
)*1i)^(1/2))/(2*a^(1/2))))/(2*a^(1/2)*d)