3.373 \(\int \frac {1+i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx\)
Optimal. Leaf size=45 \[ -\frac {2 i \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d \sqrt {a-i b}} \]
[Out]
-2*I*arctanh((a+b*tan(d*x+c))^(1/2)/(a-I*b)^(1/2))/d/(a-I*b)^(1/2)
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Rubi [A] time = 0.05, antiderivative size = 45, normalized size of antiderivative = 1.00,
number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used =
{3537, 63, 208} \[ -\frac {2 i \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d \sqrt {a-i b}} \]
Antiderivative was successfully verified.
[In]
Int[(1 + I*Tan[c + d*x])/Sqrt[a + b*Tan[c + d*x]],x]
[Out]
((-2*I)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/(Sqrt[a - I*b]*d)
Rule 63
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 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rule 3537
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c*
d)/f, Subst[Int[(a + (b*x)/d)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] &&
NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]
Rubi steps
\begin {align*} \int \frac {1+i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx &=\frac {i \operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a-i b x}} \, dx,x,i \tan (c+d x)\right )}{d}\\ &=-\frac {2 \operatorname {Subst}\left (\int \frac {1}{-1-\frac {i a}{b}+\frac {i x^2}{b}} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{b d}\\ &=-\frac {2 i \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{\sqrt {a-i b} d}\\ \end {align*}
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Mathematica [A] time = 1.67, size = 70, normalized size = 1.56 \[ -\frac {2 i \tanh ^{-1}\left (\frac {\sqrt {a-\frac {i b \left (-1+e^{2 i (c+d x)}\right )}{1+e^{2 i (c+d x)}}}}{\sqrt {a-i b}}\right )}{d \sqrt {a-i b}} \]
Antiderivative was successfully verified.
[In]
Integrate[(1 + I*Tan[c + d*x])/Sqrt[a + b*Tan[c + d*x]],x]
[Out]
((-2*I)*ArcTanh[Sqrt[a - (I*b*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]/Sqrt[a - I*b]])/(Sqrt[a -
I*b]*d)
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fricas [B] time = 0.48, size = 249, normalized size = 5.53 \[ \frac {1}{4} \, \sqrt {-\frac {4 i}{{\left (i \, a + b\right )} d^{2}}} \log \left ({\left ({\left ({\left (i \, a + b\right )} d e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (i \, a + b\right )} d\right )} \sqrt {\frac {{\left (a - i \, b\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + a + i \, b}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {-\frac {4 i}{{\left (i \, a + b\right )} d^{2}}} + 2 \, {\left (a - i \, b\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + 2 \, a\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}\right ) - \frac {1}{4} \, \sqrt {-\frac {4 i}{{\left (i \, a + b\right )} d^{2}}} \log \left ({\left ({\left ({\left (-i \, a - b\right )} d e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (-i \, a - b\right )} d\right )} \sqrt {\frac {{\left (a - i \, b\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + a + i \, b}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {-\frac {4 i}{{\left (i \, a + b\right )} d^{2}}} + 2 \, {\left (a - i \, b\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + 2 \, a\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((1+I*tan(d*x+c))/(a+b*tan(d*x+c))^(1/2),x, algorithm="fricas")
[Out]
1/4*sqrt(-4*I/((I*a + b)*d^2))*log((((I*a + b)*d*e^(2*I*d*x + 2*I*c) + (I*a + b)*d)*sqrt(((a - I*b)*e^(2*I*d*x
+ 2*I*c) + a + I*b)/(e^(2*I*d*x + 2*I*c) + 1))*sqrt(-4*I/((I*a + b)*d^2)) + 2*(a - I*b)*e^(2*I*d*x + 2*I*c) +
2*a)*e^(-2*I*d*x - 2*I*c)) - 1/4*sqrt(-4*I/((I*a + b)*d^2))*log((((-I*a - b)*d*e^(2*I*d*x + 2*I*c) + (-I*a -
b)*d)*sqrt(((a - I*b)*e^(2*I*d*x + 2*I*c) + a + I*b)/(e^(2*I*d*x + 2*I*c) + 1))*sqrt(-4*I/((I*a + b)*d^2)) + 2
*(a - I*b)*e^(2*I*d*x + 2*I*c) + 2*a)*e^(-2*I*d*x - 2*I*c))
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((1+I*tan(d*x+c))/(a+b*tan(d*x+c))^(1/2),x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unab
le to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Warning, need to choose a
branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [d]=[-84
,-49]sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument ValueWarning, need
to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assum
ing [d]=[82,-6]sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(const ge
n & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(const gen & e,const index_m
& i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(const gen & e,const index_m & i,const vecteur
& l) Error: Bad Argument Valuesym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Arg
ument Valuesym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly
/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(const gen &
e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(const gen & e,const index_m & i
,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l)
Error: Bad Argument ValueWarning, integration of abs or sign assumes constant sign by intervals (correct if t
he argument is real):Check [abs(t_nostep^2-1)]Discontinuities at zeroes of t_nostep^2-1 were not checkedEvalua
tion time: 82.3Done
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maple [B] time = 0.23, size = 1624, normalized size = 36.09 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((1+I*tan(d*x+c))/(a+b*tan(d*x+c))^(1/2),x)
[Out]
-1/2*I/d/(2*(a^2+b^2)^(1/2)+2*a)^(1/2)/(a^2+b^2)^(1/2)*ln(b*tan(d*x+c)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(
1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*a-I/d/((a^2+b^2)^(1/2)*a+a^2+b^2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a
^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*b^2+1/2/d/(2*(a^2+b^2)^(1/2)
+2*a)^(1/2)/(a^2+b^2)^(1/2)*ln(b*tan(d*x+c)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(
1/2))*b-I/d/(a^2+b^2)^(1/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+
2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*a+I/d/(2*(a^2+b^2)^(1/2)+2*a)^(1/2)/(a^2+b^2)^(1/2)/((a^2+b^2)^(1/2
)*a+a^2+b^2)*ln((a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b^2)^(1/2))*a*b^2+1/d
*b/(a^2+b^2)^(1/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2
))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))+I/d/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b
^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))+I/d/(2*(a^2+b^2)^(1/2)+2*a)^(1/2)/((a^2+b^2)^(1/2)*a+a^2+
b^2)*ln((a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b^2)^(1/2))*a^2+I/d/(2*(a^2+b
^2)^(1/2)+2*a)^(1/2)/(a^2+b^2)^(1/2)/((a^2+b^2)^(1/2)*a+a^2+b^2)*ln((a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+
2*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b^2)^(1/2))*a^3-1/2*I/d/(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*ln(b*tan(d*x+c)+a+(a+b*ta
n(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))-1/2/d/(2*(a^2+b^2)^(1/2)+2*a)^(1/2)/((a^2+b^2)^
(1/2)*a+a^2+b^2)*ln((a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b^2)^(1/2))*a*b-1
/2/d/(2*(a^2+b^2)^(1/2)+2*a)^(1/2)/(a^2+b^2)^(1/2)/((a^2+b^2)^(1/2)*a+a^2+b^2)*ln((a+b*tan(d*x+c))^(1/2)*(2*(a
^2+b^2)^(1/2)+2*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b^2)^(1/2))*a^2*b-1/2/d/(2*(a^2+b^2)^(1/2)+2*a)^(1/2)/(a^2+b^2)^(
1/2)/((a^2+b^2)^(1/2)*a+a^2+b^2)*ln((a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b
^2)^(1/2))*b^3+1/2*I/d/(2*(a^2+b^2)^(1/2)+2*a)^(1/2)/((a^2+b^2)^(1/2)*a+a^2+b^2)*ln((a+b*tan(d*x+c))^(1/2)*(2*
(a^2+b^2)^(1/2)+2*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b^2)^(1/2))*b^2-1/d/((a^2+b^2)^(1/2)*a+a^2+b^2)/(2*(a^2+b^2)^(1
/2)-2*a)^(1/2)*arctan(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*
a*b-1/d/(a^2+b^2)^(1/2)/((a^2+b^2)^(1/2)*a+a^2+b^2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^2+b^2)^(1/2)+2
*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*a^2*b-1/d/(a^2+b^2)^(1/2)/((a^2+b^2)^(1/2)*
a+a^2+b^2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a
^2+b^2)^(1/2)-2*a)^(1/2))*b^3
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maxima [B] time = 1.36, size = 6480, normalized size = 144.00 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((1+I*tan(d*x+c))/(a+b*tan(d*x+c))^(1/2),x, algorithm="maxima")
[Out]
-1/4*(sqrt(2*a^2 + 2*b^2)*sqrt(a + sqrt(a^2 + b^2))*(2*arctan2((b^2*cos(2*d*x + 2*c) - a*b*sin(2*d*x + 2*c) +
((a^4 + 2*a^2*b^2 + b^4)*cos(2*d*x + 2*c)^4 + (a^4 + 2*a^2*b^2 + b^4)*sin(2*d*x + 2*c)^4 + a^4 + 2*a^2*b^2 + b
^4 + 4*(a^4 + a^2*b^2)*cos(2*d*x + 2*c)^3 + 4*(a^3*b + a*b^3)*sin(2*d*x + 2*c)^3 + 2*(3*a^4 + 2*a^2*b^2 - b^4)
*cos(2*d*x + 2*c)^2 + 2*(a^4 + 2*a^2*b^2 + b^4 + (a^4 + 2*a^2*b^2 + b^4)*cos(2*d*x + 2*c)^2 + 2*(a^4 + a^2*b^2
)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c)^2 + 4*(a^4 + a^2*b^2)*cos(2*d*x + 2*c) + 4*(a^3*b + a*b^3 + (a^3*b + a*b^
3)*cos(2*d*x + 2*c)^2 + 2*(a^3*b + a*b^3)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c))^(1/4)*b*sin(1/2*arctan2(-2*(a*b*
cos(2*d*x + 2*c)^2 - a*b*sin(2*d*x + 2*c)^2 + a*b*cos(2*d*x + 2*c) - (a^2 + (a^2 - b^2)*cos(2*d*x + 2*c))*sin(
2*d*x + 2*c))/b^2, (2*a^2*cos(2*d*x + 2*c) + (a^2 - b^2)*cos(2*d*x + 2*c)^2 - (a^2 - b^2)*sin(2*d*x + 2*c)^2 +
a^2 + b^2 + 2*(2*a*b*cos(2*d*x + 2*c) + a*b)*sin(2*d*x + 2*c))/b^2)))/b^2, -(a*cos(2*d*x + 2*c) + b*sin(2*d*x
+ 2*c) - ((a^4 + 2*a^2*b^2 + b^4)*cos(2*d*x + 2*c)^4 + (a^4 + 2*a^2*b^2 + b^4)*sin(2*d*x + 2*c)^4 + a^4 + 2*a
^2*b^2 + b^4 + 4*(a^4 + a^2*b^2)*cos(2*d*x + 2*c)^3 + 4*(a^3*b + a*b^3)*sin(2*d*x + 2*c)^3 + 2*(3*a^4 + 2*a^2*
b^2 - b^4)*cos(2*d*x + 2*c)^2 + 2*(a^4 + 2*a^2*b^2 + b^4 + (a^4 + 2*a^2*b^2 + b^4)*cos(2*d*x + 2*c)^2 + 2*(a^4
+ a^2*b^2)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c)^2 + 4*(a^4 + a^2*b^2)*cos(2*d*x + 2*c) + 4*(a^3*b + a*b^3 + (a^
3*b + a*b^3)*cos(2*d*x + 2*c)^2 + 2*(a^3*b + a*b^3)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c))^(1/4)*cos(1/2*arctan2(
-2*(a*b*cos(2*d*x + 2*c)^2 - a*b*sin(2*d*x + 2*c)^2 + a*b*cos(2*d*x + 2*c) - (a^2 + (a^2 - b^2)*cos(2*d*x + 2*
c))*sin(2*d*x + 2*c))/b^2, (2*a^2*cos(2*d*x + 2*c) + (a^2 - b^2)*cos(2*d*x + 2*c)^2 - (a^2 - b^2)*sin(2*d*x +
2*c)^2 + a^2 + b^2 + 2*(2*a*b*cos(2*d*x + 2*c) + a*b)*sin(2*d*x + 2*c))/b^2)) + a)/b) - I*log((2*a^2*cos(2*d*x
+ 2*c) + (a^2 + b^2)*cos(2*d*x + 2*c)^2 + 2*a*b*sin(2*d*x + 2*c) + (a^2 + b^2)*sin(2*d*x + 2*c)^2 + sqrt((a^4
+ 2*a^2*b^2 + b^4)*cos(2*d*x + 2*c)^4 + (a^4 + 2*a^2*b^2 + b^4)*sin(2*d*x + 2*c)^4 + a^4 + 2*a^2*b^2 + b^4 +
4*(a^4 + a^2*b^2)*cos(2*d*x + 2*c)^3 + 4*(a^3*b + a*b^3)*sin(2*d*x + 2*c)^3 + 2*(3*a^4 + 2*a^2*b^2 - b^4)*cos(
2*d*x + 2*c)^2 + 2*(a^4 + 2*a^2*b^2 + b^4 + (a^4 + 2*a^2*b^2 + b^4)*cos(2*d*x + 2*c)^2 + 2*(a^4 + a^2*b^2)*cos
(2*d*x + 2*c))*sin(2*d*x + 2*c)^2 + 4*(a^4 + a^2*b^2)*cos(2*d*x + 2*c) + 4*(a^3*b + a*b^3 + (a^3*b + a*b^3)*co
s(2*d*x + 2*c)^2 + 2*(a^3*b + a*b^3)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c))*cos(1/2*arctan2(-2*(a*b*cos(2*d*x + 2
*c)^2 - a*b*sin(2*d*x + 2*c)^2 + a*b*cos(2*d*x + 2*c) - (a^2 + (a^2 - b^2)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c))
/b^2, (2*a^2*cos(2*d*x + 2*c) + (a^2 - b^2)*cos(2*d*x + 2*c)^2 - (a^2 - b^2)*sin(2*d*x + 2*c)^2 + a^2 + b^2 +
2*(2*a*b*cos(2*d*x + 2*c) + a*b)*sin(2*d*x + 2*c))/b^2))^2 + sqrt((a^4 + 2*a^2*b^2 + b^4)*cos(2*d*x + 2*c)^4 +
(a^4 + 2*a^2*b^2 + b^4)*sin(2*d*x + 2*c)^4 + a^4 + 2*a^2*b^2 + b^4 + 4*(a^4 + a^2*b^2)*cos(2*d*x + 2*c)^3 + 4
*(a^3*b + a*b^3)*sin(2*d*x + 2*c)^3 + 2*(3*a^4 + 2*a^2*b^2 - b^4)*cos(2*d*x + 2*c)^2 + 2*(a^4 + 2*a^2*b^2 + b^
4 + (a^4 + 2*a^2*b^2 + b^4)*cos(2*d*x + 2*c)^2 + 2*(a^4 + a^2*b^2)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c)^2 + 4*(a
^4 + a^2*b^2)*cos(2*d*x + 2*c) + 4*(a^3*b + a*b^3 + (a^3*b + a*b^3)*cos(2*d*x + 2*c)^2 + 2*(a^3*b + a*b^3)*cos
(2*d*x + 2*c))*sin(2*d*x + 2*c))*sin(1/2*arctan2(-2*(a*b*cos(2*d*x + 2*c)^2 - a*b*sin(2*d*x + 2*c)^2 + a*b*cos
(2*d*x + 2*c) - (a^2 + (a^2 - b^2)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c))/b^2, (2*a^2*cos(2*d*x + 2*c) + (a^2 - b
^2)*cos(2*d*x + 2*c)^2 - (a^2 - b^2)*sin(2*d*x + 2*c)^2 + a^2 + b^2 + 2*(2*a*b*cos(2*d*x + 2*c) + a*b)*sin(2*d
*x + 2*c))/b^2))^2 + a^2 - 2*((a^4 + 2*a^2*b^2 + b^4)*cos(2*d*x + 2*c)^4 + (a^4 + 2*a^2*b^2 + b^4)*sin(2*d*x +
2*c)^4 + a^4 + 2*a^2*b^2 + b^4 + 4*(a^4 + a^2*b^2)*cos(2*d*x + 2*c)^3 + 4*(a^3*b + a*b^3)*sin(2*d*x + 2*c)^3
+ 2*(3*a^4 + 2*a^2*b^2 - b^4)*cos(2*d*x + 2*c)^2 + 2*(a^4 + 2*a^2*b^2 + b^4 + (a^4 + 2*a^2*b^2 + b^4)*cos(2*d*
x + 2*c)^2 + 2*(a^4 + a^2*b^2)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c)^2 + 4*(a^4 + a^2*b^2)*cos(2*d*x + 2*c) + 4*(
a^3*b + a*b^3 + (a^3*b + a*b^3)*cos(2*d*x + 2*c)^2 + 2*(a^3*b + a*b^3)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c))^(1/
4)*(a*b*cos(2*d*x + 2*c) + b^2*sin(2*d*x + 2*c) + a*b)*cos(1/2*arctan2(-2*(a*b*cos(2*d*x + 2*c)^2 - a*b*sin(2*
d*x + 2*c)^2 + a*b*cos(2*d*x + 2*c) - (a^2 + (a^2 - b^2)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c))/b^2, (2*a^2*cos(2
*d*x + 2*c) + (a^2 - b^2)*cos(2*d*x + 2*c)^2 - (a^2 - b^2)*sin(2*d*x + 2*c)^2 + a^2 + b^2 + 2*(2*a*b*cos(2*d*x
+ 2*c) + a*b)*sin(2*d*x + 2*c))/b^2))/b + 2*((a^4 + 2*a^2*b^2 + b^4)*cos(2*d*x + 2*c)^4 + (a^4 + 2*a^2*b^2 +
b^4)*sin(2*d*x + 2*c)^4 + a^4 + 2*a^2*b^2 + b^4 + 4*(a^4 + a^2*b^2)*cos(2*d*x + 2*c)^3 + 4*(a^3*b + a*b^3)*sin
(2*d*x + 2*c)^3 + 2*(3*a^4 + 2*a^2*b^2 - b^4)*cos(2*d*x + 2*c)^2 + 2*(a^4 + 2*a^2*b^2 + b^4 + (a^4 + 2*a^2*b^2
+ b^4)*cos(2*d*x + 2*c)^2 + 2*(a^4 + a^2*b^2)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c)^2 + 4*(a^4 + a^2*b^2)*cos(2*
d*x + 2*c) + 4*(a^3*b + a*b^3 + (a^3*b + a*b^3)*cos(2*d*x + 2*c)^2 + 2*(a^3*b + a*b^3)*cos(2*d*x + 2*c))*sin(2
*d*x + 2*c))^(1/4)*(b^2*cos(2*d*x + 2*c) - a*b*sin(2*d*x + 2*c))*sin(1/2*arctan2(-2*(a*b*cos(2*d*x + 2*c)^2 -
a*b*sin(2*d*x + 2*c)^2 + a*b*cos(2*d*x + 2*c) - (a^2 + (a^2 - b^2)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c))/b^2, (2
*a^2*cos(2*d*x + 2*c) + (a^2 - b^2)*cos(2*d*x + 2*c)^2 - (a^2 - b^2)*sin(2*d*x + 2*c)^2 + a^2 + b^2 + 2*(2*a*b
*cos(2*d*x + 2*c) + a*b)*sin(2*d*x + 2*c))/b^2))/b)/b^2)) - sqrt(2*a^2 + 2*b^2)*sqrt(-a + sqrt(a^2 + b^2))*(-2
*I*arctan2((b^2*cos(2*d*x + 2*c) - a*b*sin(2*d*x + 2*c) + ((a^4 + 2*a^2*b^2 + b^4)*cos(2*d*x + 2*c)^4 + (a^4 +
2*a^2*b^2 + b^4)*sin(2*d*x + 2*c)^4 + a^4 + 2*a^2*b^2 + b^4 + 4*(a^4 + a^2*b^2)*cos(2*d*x + 2*c)^3 + 4*(a^3*b
+ a*b^3)*sin(2*d*x + 2*c)^3 + 2*(3*a^4 + 2*a^2*b^2 - b^4)*cos(2*d*x + 2*c)^2 + 2*(a^4 + 2*a^2*b^2 + b^4 + (a^
4 + 2*a^2*b^2 + b^4)*cos(2*d*x + 2*c)^2 + 2*(a^4 + a^2*b^2)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c)^2 + 4*(a^4 + a^
2*b^2)*cos(2*d*x + 2*c) + 4*(a^3*b + a*b^3 + (a^3*b + a*b^3)*cos(2*d*x + 2*c)^2 + 2*(a^3*b + a*b^3)*cos(2*d*x
+ 2*c))*sin(2*d*x + 2*c))^(1/4)*b*sin(1/2*arctan2(-2*(a*b*cos(2*d*x + 2*c)^2 - a*b*sin(2*d*x + 2*c)^2 + a*b*co
s(2*d*x + 2*c) - (a^2 + (a^2 - b^2)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c))/b^2, (2*a^2*cos(2*d*x + 2*c) + (a^2 -
b^2)*cos(2*d*x + 2*c)^2 - (a^2 - b^2)*sin(2*d*x + 2*c)^2 + a^2 + b^2 + 2*(2*a*b*cos(2*d*x + 2*c) + a*b)*sin(2*
d*x + 2*c))/b^2)))/b^2, -(a*cos(2*d*x + 2*c) + b*sin(2*d*x + 2*c) - ((a^4 + 2*a^2*b^2 + b^4)*cos(2*d*x + 2*c)^
4 + (a^4 + 2*a^2*b^2 + b^4)*sin(2*d*x + 2*c)^4 + a^4 + 2*a^2*b^2 + b^4 + 4*(a^4 + a^2*b^2)*cos(2*d*x + 2*c)^3
+ 4*(a^3*b + a*b^3)*sin(2*d*x + 2*c)^3 + 2*(3*a^4 + 2*a^2*b^2 - b^4)*cos(2*d*x + 2*c)^2 + 2*(a^4 + 2*a^2*b^2 +
b^4 + (a^4 + 2*a^2*b^2 + b^4)*cos(2*d*x + 2*c)^2 + 2*(a^4 + a^2*b^2)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c)^2 + 4
*(a^4 + a^2*b^2)*cos(2*d*x + 2*c) + 4*(a^3*b + a*b^3 + (a^3*b + a*b^3)*cos(2*d*x + 2*c)^2 + 2*(a^3*b + a*b^3)*
cos(2*d*x + 2*c))*sin(2*d*x + 2*c))^(1/4)*cos(1/2*arctan2(-2*(a*b*cos(2*d*x + 2*c)^2 - a*b*sin(2*d*x + 2*c)^2
+ a*b*cos(2*d*x + 2*c) - (a^2 + (a^2 - b^2)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c))/b^2, (2*a^2*cos(2*d*x + 2*c) +
(a^2 - b^2)*cos(2*d*x + 2*c)^2 - (a^2 - b^2)*sin(2*d*x + 2*c)^2 + a^2 + b^2 + 2*(2*a*b*cos(2*d*x + 2*c) + a*b
)*sin(2*d*x + 2*c))/b^2)) + a)/b) - log((2*a^2*cos(2*d*x + 2*c) + (a^2 + b^2)*cos(2*d*x + 2*c)^2 + 2*a*b*sin(2
*d*x + 2*c) + (a^2 + b^2)*sin(2*d*x + 2*c)^2 + sqrt((a^4 + 2*a^2*b^2 + b^4)*cos(2*d*x + 2*c)^4 + (a^4 + 2*a^2*
b^2 + b^4)*sin(2*d*x + 2*c)^4 + a^4 + 2*a^2*b^2 + b^4 + 4*(a^4 + a^2*b^2)*cos(2*d*x + 2*c)^3 + 4*(a^3*b + a*b^
3)*sin(2*d*x + 2*c)^3 + 2*(3*a^4 + 2*a^2*b^2 - b^4)*cos(2*d*x + 2*c)^2 + 2*(a^4 + 2*a^2*b^2 + b^4 + (a^4 + 2*a
^2*b^2 + b^4)*cos(2*d*x + 2*c)^2 + 2*(a^4 + a^2*b^2)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c)^2 + 4*(a^4 + a^2*b^2)*
cos(2*d*x + 2*c) + 4*(a^3*b + a*b^3 + (a^3*b + a*b^3)*cos(2*d*x + 2*c)^2 + 2*(a^3*b + a*b^3)*cos(2*d*x + 2*c))
*sin(2*d*x + 2*c))*cos(1/2*arctan2(-2*(a*b*cos(2*d*x + 2*c)^2 - a*b*sin(2*d*x + 2*c)^2 + a*b*cos(2*d*x + 2*c)
- (a^2 + (a^2 - b^2)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c))/b^2, (2*a^2*cos(2*d*x + 2*c) + (a^2 - b^2)*cos(2*d*x
+ 2*c)^2 - (a^2 - b^2)*sin(2*d*x + 2*c)^2 + a^2 + b^2 + 2*(2*a*b*cos(2*d*x + 2*c) + a*b)*sin(2*d*x + 2*c))/b^2
))^2 + sqrt((a^4 + 2*a^2*b^2 + b^4)*cos(2*d*x + 2*c)^4 + (a^4 + 2*a^2*b^2 + b^4)*sin(2*d*x + 2*c)^4 + a^4 + 2*
a^2*b^2 + b^4 + 4*(a^4 + a^2*b^2)*cos(2*d*x + 2*c)^3 + 4*(a^3*b + a*b^3)*sin(2*d*x + 2*c)^3 + 2*(3*a^4 + 2*a^2
*b^2 - b^4)*cos(2*d*x + 2*c)^2 + 2*(a^4 + 2*a^2*b^2 + b^4 + (a^4 + 2*a^2*b^2 + b^4)*cos(2*d*x + 2*c)^2 + 2*(a^
4 + a^2*b^2)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c)^2 + 4*(a^4 + a^2*b^2)*cos(2*d*x + 2*c) + 4*(a^3*b + a*b^3 + (a
^3*b + a*b^3)*cos(2*d*x + 2*c)^2 + 2*(a^3*b + a*b^3)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c))*sin(1/2*arctan2(-2*(a
*b*cos(2*d*x + 2*c)^2 - a*b*sin(2*d*x + 2*c)^2 + a*b*cos(2*d*x + 2*c) - (a^2 + (a^2 - b^2)*cos(2*d*x + 2*c))*s
in(2*d*x + 2*c))/b^2, (2*a^2*cos(2*d*x + 2*c) + (a^2 - b^2)*cos(2*d*x + 2*c)^2 - (a^2 - b^2)*sin(2*d*x + 2*c)^
2 + a^2 + b^2 + 2*(2*a*b*cos(2*d*x + 2*c) + a*b)*sin(2*d*x + 2*c))/b^2))^2 + a^2 - 2*((a^4 + 2*a^2*b^2 + b^4)*
cos(2*d*x + 2*c)^4 + (a^4 + 2*a^2*b^2 + b^4)*sin(2*d*x + 2*c)^4 + a^4 + 2*a^2*b^2 + b^4 + 4*(a^4 + a^2*b^2)*co
s(2*d*x + 2*c)^3 + 4*(a^3*b + a*b^3)*sin(2*d*x + 2*c)^3 + 2*(3*a^4 + 2*a^2*b^2 - b^4)*cos(2*d*x + 2*c)^2 + 2*(
a^4 + 2*a^2*b^2 + b^4 + (a^4 + 2*a^2*b^2 + b^4)*cos(2*d*x + 2*c)^2 + 2*(a^4 + a^2*b^2)*cos(2*d*x + 2*c))*sin(2
*d*x + 2*c)^2 + 4*(a^4 + a^2*b^2)*cos(2*d*x + 2*c) + 4*(a^3*b + a*b^3 + (a^3*b + a*b^3)*cos(2*d*x + 2*c)^2 + 2
*(a^3*b + a*b^3)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c))^(1/4)*(a*b*cos(2*d*x + 2*c) + b^2*sin(2*d*x + 2*c) + a*b)
*cos(1/2*arctan2(-2*(a*b*cos(2*d*x + 2*c)^2 - a*b*sin(2*d*x + 2*c)^2 + a*b*cos(2*d*x + 2*c) - (a^2 + (a^2 - b^
2)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c))/b^2, (2*a^2*cos(2*d*x + 2*c) + (a^2 - b^2)*cos(2*d*x + 2*c)^2 - (a^2 -
b^2)*sin(2*d*x + 2*c)^2 + a^2 + b^2 + 2*(2*a*b*cos(2*d*x + 2*c) + a*b)*sin(2*d*x + 2*c))/b^2))/b + 2*((a^4 + 2
*a^2*b^2 + b^4)*cos(2*d*x + 2*c)^4 + (a^4 + 2*a^2*b^2 + b^4)*sin(2*d*x + 2*c)^4 + a^4 + 2*a^2*b^2 + b^4 + 4*(a
^4 + a^2*b^2)*cos(2*d*x + 2*c)^3 + 4*(a^3*b + a*b^3)*sin(2*d*x + 2*c)^3 + 2*(3*a^4 + 2*a^2*b^2 - b^4)*cos(2*d*
x + 2*c)^2 + 2*(a^4 + 2*a^2*b^2 + b^4 + (a^4 + 2*a^2*b^2 + b^4)*cos(2*d*x + 2*c)^2 + 2*(a^4 + a^2*b^2)*cos(2*d
*x + 2*c))*sin(2*d*x + 2*c)^2 + 4*(a^4 + a^2*b^2)*cos(2*d*x + 2*c) + 4*(a^3*b + a*b^3 + (a^3*b + a*b^3)*cos(2*
d*x + 2*c)^2 + 2*(a^3*b + a*b^3)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c))^(1/4)*(b^2*cos(2*d*x + 2*c) - a*b*sin(2*d
*x + 2*c))*sin(1/2*arctan2(-2*(a*b*cos(2*d*x + 2*c)^2 - a*b*sin(2*d*x + 2*c)^2 + a*b*cos(2*d*x + 2*c) - (a^2 +
(a^2 - b^2)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c))/b^2, (2*a^2*cos(2*d*x + 2*c) + (a^2 - b^2)*cos(2*d*x + 2*c)^2
- (a^2 - b^2)*sin(2*d*x + 2*c)^2 + a^2 + b^2 + 2*(2*a*b*cos(2*d*x + 2*c) + a*b)*sin(2*d*x + 2*c))/b^2))/b)/b^
2)))/((a^2 + b^2)*d)
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mupad [B] time = 8.70, size = 1410, normalized size = 31.33 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((tan(c + d*x)*1i + 1)/(a + b*tan(c + d*x))^(1/2),x)
[Out]
2*atanh((32*b^2*((b*1i)/(4*a^2*d^2 + 4*b^2*d^2) - a/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*(a + b*tan(c + d*x))^(1/2))
/((a^2*b^2*d^2*64i)/(4*a^2*d^3 + 4*b^2*d^3) - (b^2*16i)/d + (64*a*b^3*d^2)/(4*a^2*d^3 + 4*b^2*d^3)) - (128*a^2
*b^2*((b*1i)/(4*a^2*d^2 + 4*b^2*d^2) - a/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*(a + b*tan(c + d*x))^(1/2))/((a^2*b^4*
d^2*256i)/(4*a^2*d^3 + 4*b^2*d^3) - (a^2*b^2*64i)/d - (b^4*64i)/d + (256*a^3*b^3*d^2)/(4*a^2*d^3 + 4*b^2*d^3)
+ (a^4*b^2*d^2*256i)/(4*a^2*d^3 + 4*b^2*d^3) + (256*a*b^5*d^2)/(4*a^2*d^3 + 4*b^2*d^3)) + (a*b^3*((b*1i)/(4*a^
2*d^2 + 4*b^2*d^2) - a/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*(a + b*tan(c + d*x))^(1/2)*128i)/((a^2*b^4*d^2*256i)/(4*
a^2*d^3 + 4*b^2*d^3) - (a^2*b^2*64i)/d - (b^4*64i)/d + (256*a^3*b^3*d^2)/(4*a^2*d^3 + 4*b^2*d^3) + (a^4*b^2*d^
2*256i)/(4*a^2*d^3 + 4*b^2*d^3) + (256*a*b^5*d^2)/(4*a^2*d^3 + 4*b^2*d^3)))*(-(a - b*1i)/(4*a^2*d^2 + 4*b^2*d^
2))^(1/2) + (log(d*(-1/(d^2*(a - b*1i)))^(1/2)*(a + b*tan(c + d*x))^(1/2)*1i + 1)*(-1/(a*d^2 - b*d^2*1i))^(1/2
))/2 - log(d*(-1/(d^2*(a - b*1i)))^(1/2)*(a + b*tan(c + d*x))^(1/2) + 1i)*(-1/(4*(a*d^2 - b*d^2*1i)))^(1/2) +
(log(16*b^2*(a + b*tan(c + d*x))^(1/2) + 16*b^3*d*(-1/(d^2*(a - b*1i)))^(1/2) - (16*a*b^2*(a + b*tan(c + d*x))
^(1/2))/(a - b*1i))*(-1/(a*d^2 - b*d^2*1i))^(1/2))/2 - log(16*b^3*d*(-1/(d^2*(a - b*1i)))^(1/2) - 16*b^2*(a +
b*tan(c + d*x))^(1/2) + (16*a*b^2*(a + b*tan(c + d*x))^(1/2))/(a - b*1i))*(-1/(4*(a*d^2 - b*d^2*1i)))^(1/2) +
2*atanh((32*b^2*((b*1i)/(4*a^2*d^2 + 4*b^2*d^2) - a/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*(a + b*tan(c + d*x))^(1/2))
/((b^4*d^2*64i)/(4*a^2*d^3 + 4*b^2*d^3) - (64*a*b^3*d^2)/(4*a^2*d^3 + 4*b^2*d^3)) + (a*b^3*((b*1i)/(4*a^2*d^2
+ 4*b^2*d^2) - a/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*(a + b*tan(c + d*x))^(1/2)*128i)/((b^6*d^2*256i)/(4*a^2*d^3 +
4*b^2*d^3) + (a^2*b^4*d^2*256i)/(4*a^2*d^3 + 4*b^2*d^3) - (256*a^3*b^3*d^2)/(4*a^2*d^3 + 4*b^2*d^3) - (256*a*b
^5*d^2)/(4*a^2*d^3 + 4*b^2*d^3)) - (128*a^2*b^2*((b*1i)/(4*a^2*d^2 + 4*b^2*d^2) - a/(4*a^2*d^2 + 4*b^2*d^2))^(
1/2)*(a + b*tan(c + d*x))^(1/2))/((b^6*d^2*256i)/(4*a^2*d^3 + 4*b^2*d^3) + (a^2*b^4*d^2*256i)/(4*a^2*d^3 + 4*b
^2*d^3) - (256*a^3*b^3*d^2)/(4*a^2*d^3 + 4*b^2*d^3) - (256*a*b^5*d^2)/(4*a^2*d^3 + 4*b^2*d^3)))*(-(a - b*1i)/(
4*a^2*d^2 + 4*b^2*d^2))^(1/2)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ i \left (\int \left (- \frac {i}{\sqrt {a + b \tan {\left (c + d x \right )}}}\right )\, dx + \int \frac {\tan {\left (c + d x \right )}}{\sqrt {a + b \tan {\left (c + d x \right )}}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((1+I*tan(d*x+c))/(a+b*tan(d*x+c))**(1/2),x)
[Out]
I*(Integral(-I/sqrt(a + b*tan(c + d*x)), x) + Integral(tan(c + d*x)/sqrt(a + b*tan(c + d*x)), x))
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