3.1.62 \(\int \frac {1}{(c+d x) (a+b \tan (e+f x))^2} \, dx\) [62]

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

[Out]

Unintegrable(1/(d*x+c)/(a+b*tan(f*x+e))^2,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) (a+b \tan (e+f x))^2} \, dx \end {gather*}

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*}

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Mathematica [A]
time = 16.29, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(c+d x) (a+b \tan (e+f x))^2} \, dx \end {gather*}

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 = 0.43, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (d x +c \right ) \left (a +b \tan \left (f x +e \right )\right )^{2}}\, dx\]

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

[Out]

(((a^4 - b^4)*d*f*x + (a^4 - b^4)*c*f)*cos(2*f*x + 2*e)^2*log(d*x + c) + ((a^4 - b^4)*d*f*x + (a^4 - b^4)*c*f)
*log(d*x + c)*sin(2*f*x + 2*e)^2 - 2*(2*a*b^3*d - ((a^4 - 2*a^2*b^2 + b^4)*d*f*x + (a^4 - 2*a^2*b^2 + b^4)*c*f
)*log(d*x + c))*cos(2*f*x + 2*e) + ((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*f*x + (a^6 + 3*a^4*b^2 + 3*a^2*b^4
+ b^6)*c*d*f + ((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*f*x + (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*c*d*f)*cos(
2*f*x + 2*e)^2 + ((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*f*x + (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*c*d*f)*sin
(2*f*x + 2*e)^2 + 2*((a^6 + a^4*b^2 - a^2*b^4 - b^6)*d^2*f*x + (a^6 + a^4*b^2 - a^2*b^4 - b^6)*c*d*f)*cos(2*f*
x + 2*e) + 4*((a^5*b + 2*a^3*b^3 + a*b^5)*d^2*f*x + (a^5*b + 2*a^3*b^3 + a*b^5)*c*d*f)*sin(2*f*x + 2*e))*integ
rate(2*(2*(2*a^2*b^2*d*f*x + 2*a^2*b^2*c*f - a*b^3*d)*cos(2*f*x + 2*e) - (2*(a^3*b - a*b^3)*d*f*x + 2*(a^3*b -
a*b^3)*c*f - (a^2*b^2 - b^4)*d)*sin(2*f*x + 2*e))/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*f*x^2 + 2*(a^6 + 3
*a^4*b^2 + 3*a^2*b^4 + b^6)*c*d*f*x + (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*c^2*f + ((a^6 + 3*a^4*b^2 + 3*a^2*b^
4 + b^6)*d^2*f*x^2 + 2*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*c*d*f*x + (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*c^2*f
)*cos(2*f*x + 2*e)^2 + ((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*f*x^2 + 2*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*
c*d*f*x + (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*c^2*f)*sin(2*f*x + 2*e)^2 + 2*((a^6 + a^4*b^2 - a^2*b^4 - b^6)*d
^2*f*x^2 + 2*(a^6 + a^4*b^2 - a^2*b^4 - b^6)*c*d*f*x + (a^6 + a^4*b^2 - a^2*b^4 - b^6)*c^2*f)*cos(2*f*x + 2*e)
+ 4*((a^5*b + 2*a^3*b^3 + a*b^5)*d^2*f*x^2 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*c*d*f*x + (a^5*b + 2*a^3*b^3 + a*b
^5)*c^2*f)*sin(2*f*x + 2*e)), x) + ((a^4 - b^4)*d*f*x + (a^4 - b^4)*c*f)*log(d*x + c) + 2*((a^2*b^2 - b^4)*d +
2*((a^3*b - a*b^3)*d*f*x + (a^3*b - a*b^3)*c*f)*log(d*x + c))*sin(2*f*x + 2*e))/((a^6 + 3*a^4*b^2 + 3*a^2*b^4
+ b^6)*d^2*f*x + (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*c*d*f + ((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*f*x + (
a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*c*d*f)*cos(2*f*x + 2*e)^2 + ((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*f*x +
(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*c*d*f)*sin(2*f*x + 2*e)^2 + 2*((a^6 + a^4*b^2 - a^2*b^4 - b^6)*d^2*f*x + (
a^6 + a^4*b^2 - a^2*b^4 - b^6)*c*d*f)*cos(2*f*x + 2*e) + 4*((a^5*b + 2*a^3*b^3 + a*b^5)*d^2*f*x + (a^5*b + 2*a
^3*b^3 + a*b^5)*c*d*f)*sin(2*f*x + 2*e))

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

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 = 0.00, 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)/(a+b*tan(f*x+e))^2,x, algorithm="giac")

[Out]

integrate(1/((d*x + c)*(b*tan(f*x + e) + a)^2), 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 )}^2\,\left (c+d\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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