3.350 \(\int \frac {\sec ^2(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx\)

Optimal. Leaf size=27 \[ \frac {2 i}{a d \sqrt {a+i a \tan (c+d x)}} \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {3487, 32} \[ \frac {2 i}{a d \sqrt {a+i a \tan (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^2/(a + I*a*Tan[c + d*x])^(3/2),x]

[Out]

(2*I)/(a*d*Sqrt[a + I*a*Tan[c + d*x]])

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]

Rubi steps

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

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Mathematica [A]  time = 0.16, size = 27, normalized size = 1.00 \[ \frac {2 i}{a d \sqrt {a+i a \tan (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^2/(a + I*a*Tan[c + d*x])^(3/2),x]

[Out]

(2*I)/(a*d*Sqrt[a + I*a*Tan[c + d*x]])

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fricas [B]  time = 0.71, size = 49, normalized size = 1.81 \[ \frac {\sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )} e^{\left (-i \, d x - i \, c\right )}}{a^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^2/(a+I*a*tan(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec \left (d x + c\right )^{2}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^2/(a+I*a*tan(d*x+c))^(3/2),x, algorithm="giac")

[Out]

integrate(sec(d*x + c)^2/(I*a*tan(d*x + c) + a)^(3/2), x)

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maple [A]  time = 0.19, size = 24, normalized size = 0.89 \[ \frac {2 i}{a d \sqrt {a +i a \tan \left (d x +c \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^2/(a+I*a*tan(d*x+c))^(3/2),x)

[Out]

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

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maxima [A]  time = 0.45, size = 21, normalized size = 0.78 \[ \frac {2 i}{\sqrt {i \, a \tan \left (d x + c\right ) + a} a d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^2/(a+I*a*tan(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

2*I/(sqrt(I*a*tan(d*x + c) + a)*a*d)

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mupad [B]  time = 0.24, size = 67, normalized size = 2.48 \[ \frac {\left ({\cos \left (c+d\,x\right )}^2\,2{}\mathrm {i}+\sin \left (2\,c+2\,d\,x\right )\right )\,\sqrt {\frac {a\,\left (2\,{\cos \left (c+d\,x\right )}^2+\sin \left (2\,c+2\,d\,x\right )\,1{}\mathrm {i}\right )}{2\,{\cos \left (c+d\,x\right )}^2}}}{a^2\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(c + d*x)^2*(a + a*tan(c + d*x)*1i)^(3/2)),x)

[Out]

((sin(2*c + 2*d*x) + cos(c + d*x)^2*2i)*((a*(sin(2*c + 2*d*x)*1i + 2*cos(c + d*x)^2))/(2*cos(c + d*x)^2))^(1/2
))/(a^2*d)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^{2}{\left (c + d x \right )}}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**2/(a+I*a*tan(d*x+c))**(3/2),x)

[Out]

Integral(sec(c + d*x)**2/(I*a*(tan(c + d*x) - I))**(3/2), x)

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