3.86 \(\int \tan ^2(c+d x) \sqrt{a+i a \tan (c+d x)} \, dx\)

Optimal. Leaf size=76 \[ \frac{i \sqrt{2} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{d}-\frac{2 i (a+i a \tan (c+d x))^{3/2}}{3 a d} \]

[Out]

(I*Sqrt[2]*Sqrt[a]*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d - (((2*I)/3)*(a + I*a*Tan[c + d*x]
)^(3/2))/(a*d)

________________________________________________________________________________________

Rubi [A]  time = 0.079921, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {3543, 3480, 206} \[ \frac{i \sqrt{2} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{d}-\frac{2 i (a+i a \tan (c+d x))^{3/2}}{3 a d} \]

Antiderivative was successfully verified.

[In]

Int[Tan[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]],x]

[Out]

(I*Sqrt[2]*Sqrt[a]*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d - (((2*I)/3)*(a + I*a*Tan[c + d*x]
)^(3/2))/(a*d)

Rule 3543

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

Rule 3480

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 206

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

Rubi steps

\begin{align*} \int \tan ^2(c+d x) \sqrt{a+i a \tan (c+d x)} \, dx &=-\frac{2 i (a+i a \tan (c+d x))^{3/2}}{3 a d}-\int \sqrt{a+i a \tan (c+d x)} \, dx\\ &=-\frac{2 i (a+i a \tan (c+d x))^{3/2}}{3 a d}+\frac{(2 i a) \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,\sqrt{a+i a \tan (c+d x)}\right )}{d}\\ &=\frac{i \sqrt{2} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{d}-\frac{2 i (a+i a \tan (c+d x))^{3/2}}{3 a d}\\ \end{align*}

Mathematica [A]  time = 0.633703, size = 97, normalized size = 1.28 \[ -\frac{i e^{-i (c+d x)} \left (4 e^{3 i (c+d x)}-3 \left (1+e^{2 i (c+d x)}\right )^{3/2} \sinh ^{-1}\left (e^{i (c+d x)}\right )\right ) \sqrt{a+i a \tan (c+d x)}}{3 d \left (1+e^{2 i (c+d x)}\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]],x]

[Out]

((-I/3)*(4*E^((3*I)*(c + d*x)) - 3*(1 + E^((2*I)*(c + d*x)))^(3/2)*ArcSinh[E^(I*(c + d*x))])*Sqrt[a + I*a*Tan[
c + d*x]])/(d*E^(I*(c + d*x))*(1 + E^((2*I)*(c + d*x))))

________________________________________________________________________________________

Maple [A]  time = 0.039, size = 58, normalized size = 0.8 \begin{align*}{\frac{-2\,i}{ad} \left ({\frac{1}{3} \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}}}-{\frac{\sqrt{2}}{2}{a}^{{\frac{3}{2}}}{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{a+ia\tan \left ( dx+c \right ) }{\frac{1}{\sqrt{a}}}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+I*a*tan(d*x+c))^(1/2)*tan(d*x+c)^2,x)

[Out]

-2*I/d/a*(1/3*(a+I*a*tan(d*x+c))^(3/2)-1/2*a^(3/2)*2^(1/2)*arctanh(1/2*(a+I*a*tan(d*x+c))^(1/2)*2^(1/2)/a^(1/2
)))

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 2.27472, size = 711, normalized size = 9.36 \begin{align*} -\frac{3 \, \sqrt{2}{\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt{-\frac{a}{d^{2}}} \log \left ({\left (i \, \sqrt{2} d \sqrt{-\frac{a}{d^{2}}} e^{\left (2 i \, d x + 2 i \, c\right )} + \sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}\right ) - 3 \, \sqrt{2}{\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt{-\frac{a}{d^{2}}} \log \left ({\left (-i \, \sqrt{2} d \sqrt{-\frac{a}{d^{2}}} e^{\left (2 i \, d x + 2 i \, c\right )} + \sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}\right ) + 8 i \, \sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (3 i \, d x + 3 i \, c\right )}}{6 \,{\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(3*sqrt(2)*(d*e^(2*I*d*x + 2*I*c) + d)*sqrt(-a/d^2)*log((I*sqrt(2)*d*sqrt(-a/d^2)*e^(2*I*d*x + 2*I*c) + s
qrt(2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*(e^(2*I*d*x + 2*I*c) + 1)*e^(I*d*x + I*c))*e^(-2*I*d*x - 2*I*c)) - 3*
sqrt(2)*(d*e^(2*I*d*x + 2*I*c) + d)*sqrt(-a/d^2)*log((-I*sqrt(2)*d*sqrt(-a/d^2)*e^(2*I*d*x + 2*I*c) + sqrt(2)*
sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*(e^(2*I*d*x + 2*I*c) + 1)*e^(I*d*x + I*c))*e^(-2*I*d*x - 2*I*c)) + 8*I*sqrt(
2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*e^(3*I*d*x + 3*I*c))/(d*e^(2*I*d*x + 2*I*c) + d)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \left (i \tan{\left (c + d x \right )} + 1\right )} \tan ^{2}{\left (c + d x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a*(I*tan(c + d*x) + 1))*tan(c + d*x)**2, x)

________________________________________________________________________________________

Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out