3.10.58 \(\int (a+i a \tan (e+f x)) \sqrt {c-i c \tan (e+f x)} \, dx\) [958]

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 = 0.07, 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 = {3603, 3568, 32} \begin {gather*} \frac {2 i a \sqrt {c-i c \tan (e+f x)}}{f} \end {gather*}

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 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*} \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) \text {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 = 0.42, size = 25, normalized size = 1.00 \begin {gather*} \frac {2 i a \sqrt {c-i c \tan (e+f x)}}{f} \end {gather*}

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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Maple [A]
time = 0.19, size = 22, normalized size = 0.88

method	result	size
derivativedivides	\(\frac {2 i a \sqrt {c -i c \tan \left (f x +e \right )}}{f}\)	\(22\)
default	\(\frac {2 i a \sqrt {c -i c \tan \left (f x +e \right )}}{f}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c-I*c*tan(f*x+e))^(1/2)*(a+I*a*tan(f*x+e)),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 20, normalized size = 0.80 \begin {gather*} \frac {2 i \, \sqrt {-i \, c \tan \left (f x + e\right ) + c} a}{f} \end {gather*}

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="maxima")

[Out]

2*I*sqrt(-I*c*tan(f*x + e) + c)*a/f

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Fricas [A]
time = 0.92, size = 27, normalized size = 1.08 \begin {gather*} \frac {2 i \, \sqrt {2} a \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{f} \end {gather*}

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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Sympy [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (20) = 40\).
time = 1.69, size = 42, normalized size = 1.68 \begin {gather*} \begin {cases} \frac {2 i a \sqrt {- i c \tan {\left (e + f x \right )} + c}}{f} & \text {for}\: f \neq 0 \\x \left (i a \tan {\left (e \right )} + a\right ) \sqrt {- i c \tan {\left (e \right )} + c} & \text {otherwise} \end {cases} \end {gather*}

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)

[Out]

Piecewise((2*I*a*sqrt(-I*c*tan(e + f*x) + c)/f, Ne(f, 0)), (x*(I*a*tan(e) + a)*sqrt(-I*c*tan(e) + c), True))

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Giac [A]
time = 1.15, size = 20, normalized size = 0.80 \begin {gather*} \frac {2 i \, \sqrt {-i \, c \tan \left (f x + e\right ) + c} a}{f} \end {gather*}

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]

2*I*sqrt(-I*c*tan(f*x + e) + c)*a/f

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Mupad [B]
time = 0.46, size = 29, normalized size = 1.16 \begin {gather*} \frac {\sqrt {2}\,a\,\sqrt {\frac {c}{{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1}}\,2{}\mathrm {i}}{f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2^(1/2)*a*(c/(exp(e*2i + f*x*2i) + 1))^(1/2)*2i)/f

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