### 3.62 $$\int \frac{1}{(c+d x) (a+b \tan (e+f x))^2} \, dx$$

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

[Out]

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

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Rubi [A]  time = 0.0611239, 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 = {} $\int \frac{1}{(c+d x) (a+b \tan (e+f x))^2} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 15.8466, size = 0, normalized size = 0. $\int \frac{1}{(c+d x) (a+b \tan (e+f x))^2} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 5.47, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( dx+c \right ) \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)/(a+b*tan(f*x+e))^2,x)

[Out]

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

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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)/(a+b*tan(f*x+e))^2,x, algorithm="maxima")

[Out]

Timed out

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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