### 3.63 $$\int \frac{1}{(c+d x)^2 (a+b \tan (e+f x))^2} \, dx$$

Optimal. Leaf size=22 $\text{Unintegrable}\left (\frac{1}{(c+d x)^2 (a+b \tan (e+f x))^2},x\right )$

[Out]

Unintegrable[1/((c + d*x)^2*(a + b*Tan[e + f*x])^2), x]

________________________________________________________________________________________

Rubi [A]  time = 0.0586367, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0., Rules used = {} $\int \frac{1}{(c+d x)^2 (a+b \tan (e+f x))^2} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

Int[1/((c + d*x)^2*(a + b*Tan[e + f*x])^2),x]

[Out]

Defer[Int][1/((c + d*x)^2*(a + b*Tan[e + f*x])^2), x]

Rubi steps

\begin{align*} \int \frac{1}{(c+d x)^2 (a+b \tan (e+f x))^2} \, dx &=\int \frac{1}{(c+d x)^2 (a+b \tan (e+f x))^2} \, dx\\ \end{align*}

Mathematica [A]  time = 16.2071, size = 0, normalized size = 0. $\int \frac{1}{(c+d x)^2 (a+b \tan (e+f x))^2} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

Integrate[1/((c + d*x)^2*(a + b*Tan[e + f*x])^2),x]

[Out]

Integrate[1/((c + d*x)^2*(a + b*Tan[e + f*x])^2), x]

________________________________________________________________________________________

Maple [A]  time = 10.503, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( dx+c \right ) ^{2} \left ( a+b\tan \left ( fx+e \right ) \right ) ^{2}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [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(1/(d*x+c)^2/(a+b*tan(f*x+e))^2,x, algorithm="maxima")

[Out]

Timed out

________________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{a^{2} d^{2} x^{2} + 2 \, a^{2} c d x + a^{2} c^{2} +{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \tan \left (f x + e\right )^{2} + 2 \,{\left (a b d^{2} x^{2} + 2 \, a b c d x + a b c^{2}\right )} \tan \left (f x + e\right )}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/(a^2*d^2*x^2 + 2*a^2*c*d*x + a^2*c^2 + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*tan(f*x + e)^2 + 2*(a*
b*d^2*x^2 + 2*a*b*c*d*x + a*b*c^2)*tan(f*x + e)), x)

________________________________________________________________________________________

Sympy [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(1/(d*x+c)**2/(a+b*tan(f*x+e))**2,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (d x + c\right )}^{2}{\left (b \tan \left (f x + e\right ) + a\right )}^{2}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((d*x + c)^2*(b*tan(f*x + e) + a)^2), x)