### 3.57 $$\int \frac{1}{(c+d x) (a+b \tan (e+f x))} \, dx$$

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

[Out]

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

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Rubi [A]  time = 0.0619765, 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 = {} $\int \frac{1}{(c+d x) (a+b \tan (e+f x))} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.68867, size = 0, normalized size = 0. $\int \frac{1}{(c+d x) (a+b \tan (e+f x))} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.685, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( dx+c \right ) \left ( a+b\tan \left ( fx+e \right ) \right ) }}\, 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)),x)

[Out]

int(1/(d*x+c)/(a+b*tan(f*x+e)),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)),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 d x + a c +{\left (b d x + 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)),x, algorithm="fricas")

[Out]

integral(1/(a*d*x + a*c + (b*d*x + 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 ) \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)),x)

[Out]

Integral(1/((a + b*tan(e + f*x))*(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 )}}\,{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)),x, algorithm="giac")

[Out]

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