∫ (a + ia tan(e + fx))∘ __________

3.958 \(\int (a+i a \tan (e+f x)) \sqrt {c-i c \tan (e+f x)} \, dx\)

Optimal. Leaf size=25 \[ \frac {2 i a \sqrt {c-i c \tan (e+f x)}}{f} \]

[Out]

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

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Rubi [A] time = 0.10, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {3522, 3487, 32} \[ \frac {2 i a \sqrt {c-i c \tan (e+f x)}}{f} \]

Antiderivative was successfully veriﬁed.

[In]

Int[(a + I*a*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]],x]

[Out]

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

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N

eQ[m, -1]

Rule 3487

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 3522

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*} \int (a+i a \tan (e+f x)) \sqrt {c-i c \tan (e+f x)} \, dx &=(a c) \int \frac {\sec ^2(e+f x)}{\sqrt {c-i c \tan (e+f x)}} \, dx\\ &=\frac {(i a) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c+x}} \, dx,x,-i c \tan (e+f x)\right )}{f}\\ &=\frac {2 i a \sqrt {c-i c \tan (e+f x)}}{f}\\ \end {align*}

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Mathematica [A] time = 0.88, size = 25, normalized size = 1.00 \[ \frac {2 i a \sqrt {c-i c \tan (e+f x)}}{f} \]

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + I*a*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]],x]

[Out]

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

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fricas [A] time = 0.46, size = 26, normalized size = 1.04 \[ \frac {2 i \, \sqrt {2} a \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{f} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-I*c*tan(f*x+e))^(1/2)*(a+I*a*tan(f*x+e)),x, algorithm="fricas")

[Out]

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

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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-I*c*tan(f*x+e))^(1/2)*(a+I*a*tan(f*x+e)),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (2*pi/x/2)>(-2*pi/

x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check si