3.1.58 \(\int \frac {1}{(c+d x)^2 (a+b \tan (e+f x))} \, dx\) [58]

Optimal. Leaf size=23 \[ \text {Int}\left (\frac {1}{(c+d x)^2 (a+b \tan (e+f x))},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1}{(c+d x)^2 (a+b \tan (e+f x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 4.40, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(c+d x)^2 (a+b \tan (e+f x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.35, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (d x +c \right )^{2} \left (a +b \tan \left (f x +e \right )\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*((a^2*b + b^3)*d^2*x + (a^2*b + b^3)*c*d)*integrate((2*a*b*cos(2*f*x + 2*e) - (a^2 - b^2)*sin(2*f*x + 2*e))
/((a^4 + 2*a^2*b^2 + b^4)*d^2*x^2 + 2*(a^4 + 2*a^2*b^2 + b^4)*c*d*x + (a^4 + 2*a^2*b^2 + b^4)*c^2 + ((a^4 + 2*
a^2*b^2 + b^4)*d^2*x^2 + 2*(a^4 + 2*a^2*b^2 + b^4)*c*d*x + (a^4 + 2*a^2*b^2 + b^4)*c^2)*cos(2*f*x + 2*e)^2 + (
(a^4 + 2*a^2*b^2 + b^4)*d^2*x^2 + 2*(a^4 + 2*a^2*b^2 + b^4)*c*d*x + (a^4 + 2*a^2*b^2 + b^4)*c^2)*sin(2*f*x + 2
*e)^2 + 2*((a^4 - b^4)*d^2*x^2 + 2*(a^4 - b^4)*c*d*x + (a^4 - b^4)*c^2)*cos(2*f*x + 2*e) + 4*((a^3*b + a*b^3)*
d^2*x^2 + 2*(a^3*b + a*b^3)*c*d*x + (a^3*b + a*b^3)*c^2)*sin(2*f*x + 2*e)), x) - a)/((a^2 + b^2)*d^2*x + (a^2
+ b^2)*c*d)

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b \tan {\left (e + f x \right )}\right ) \left (c + d x\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {1}{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )\,{\left (c+d\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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