3.153 \(\int \frac{\sec ^4(c+d x)}{(a+i a \tan (c+d x))^4} \, dx\)

Optimal. Leaf size=29 \[ \frac{\tan (c+d x)}{d \left (a^2+i a^2 \tan (c+d x)\right )^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0412489, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {3487, 34} \[ \frac{\tan (c+d x)}{d \left (a^2+i a^2 \tan (c+d x)\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^4/(a + I*a*Tan[c + d*x])^4,x]

[Out]

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

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 34

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

Rubi steps

\begin{align*} \int \frac{\sec ^4(c+d x)}{(a+i a \tan (c+d x))^4} \, dx &=-\frac{i \operatorname{Subst}\left (\int \frac{a-x}{(a+x)^3} \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=\frac{\tan (c+d x)}{d \left (a^2+i a^2 \tan (c+d x)\right )^2}\\ \end{align*}

Mathematica [A]  time = 0.0511495, size = 32, normalized size = 1.1 \[ \frac{i \sec ^4(c+d x)}{4 d (a+i a \tan (c+d x))^4} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^4/(a + I*a*Tan[c + d*x])^4,x]

[Out]

((I/4)*Sec[c + d*x]^4)/(d*(a + I*a*Tan[c + d*x])^4)

________________________________________________________________________________________

Maple [A]  time = 0.084, size = 36, normalized size = 1.2 \begin{align*}{\frac{1}{{a}^{4}d} \left ({\frac{-i}{ \left ( \tan \left ( dx+c \right ) -i \right ) ^{2}}}- \left ( \tan \left ( dx+c \right ) -i \right ) ^{-1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^4/(a+I*a*tan(d*x+c))^4,x)

[Out]

1/d/a^4*(-I/(tan(d*x+c)-I)^2-1/(tan(d*x+c)-I))

________________________________________________________________________________________

Maxima [B]  time = 1.00374, size = 90, normalized size = 3.1 \begin{align*} -\frac{3 \,{\left (\tan \left (d x + c\right )^{2} - i \, \tan \left (d x + c\right )\right )}}{{\left (3 \, a^{4} \tan \left (d x + c\right )^{3} - 9 i \, a^{4} \tan \left (d x + c\right )^{2} - 9 \, a^{4} \tan \left (d x + c\right ) + 3 i \, a^{4}\right )} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^4/(a+I*a*tan(d*x+c))^4,x, algorithm="maxima")

[Out]

-3*(tan(d*x + c)^2 - I*tan(d*x + c))/((3*a^4*tan(d*x + c)^3 - 9*I*a^4*tan(d*x + c)^2 - 9*a^4*tan(d*x + c) + 3*
I*a^4)*d)

________________________________________________________________________________________

Fricas [A]  time = 2.37237, size = 49, normalized size = 1.69 \begin{align*} \frac{i \, e^{\left (-4 i \, d x - 4 i \, c\right )}}{4 \, a^{4} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^4/(a+I*a*tan(d*x+c))^4,x, algorithm="fricas")

[Out]

1/4*I*e^(-4*I*d*x - 4*I*c)/(a^4*d)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**4/(a+I*a*tan(d*x+c))**4,x)

[Out]

Exception raised: AttributeError

________________________________________________________________________________________

Giac [A]  time = 1.17843, size = 59, normalized size = 2.03 \begin{align*} -\frac{2 \,{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{a^{4} d{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - i\right )}^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^4/(a+I*a*tan(d*x+c))^4,x, algorithm="giac")

[Out]

-2*(tan(1/2*d*x + 1/2*c)^3 - tan(1/2*d*x + 1/2*c))/(a^4*d*(tan(1/2*d*x + 1/2*c) - I)^4)