\(\int \frac {(a+b \tan (e+f x))^3}{(c+d x)^2} \, dx\) [53]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {(a+b \tan (e+f x))^3}{(c+d x)^2} \, dx=\text {Int}\left (\frac {(a+b \tan (e+f x))^3}{(c+d x)^2},x\right ) \]

[Out]

Unintegrable((a+b*tan(f*x+e))^3/(d*x+c)^2,x)

Rubi [N/A]

Not integrable

Time = 0.06 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {(a+b \tan (e+f x))^3}{(c+d x)^2} \, dx=\int \frac {(a+b \tan (e+f x))^3}{(c+d x)^2} \, dx \]

[In]

Int[(a + b*Tan[e + f*x])^3/(c + d*x)^2,x]

[Out]

Defer[Int][(a + b*Tan[e + f*x])^3/(c + d*x)^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {(a+b \tan (e+f x))^3}{(c+d x)^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 17.93 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(a+b \tan (e+f x))^3}{(c+d x)^2} \, dx=\int \frac {(a+b \tan (e+f x))^3}{(c+d x)^2} \, dx \]

[In]

Integrate[(a + b*Tan[e + f*x])^3/(c + d*x)^2,x]

[Out]

Integrate[(a + b*Tan[e + f*x])^3/(c + d*x)^2, x]

Maple [N/A] (verified)

Not integrable

Time = 0.76 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

\[\int \frac {\left (a +b \tan \left (f x +e \right )\right )^{3}}{\left (d x +c \right )^{2}}d x\]

[In]

int((a+b*tan(f*x+e))^3/(d*x+c)^2,x)

[Out]

int((a+b*tan(f*x+e))^3/(d*x+c)^2,x)

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 63, normalized size of antiderivative = 3.15 \[ \int \frac {(a+b \tan (e+f x))^3}{(c+d x)^2} \, dx=\int { \frac {{\left (b \tan \left (f x + e\right ) + a\right )}^{3}}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

integrate((a+b*tan(f*x+e))^3/(d*x+c)^2,x, algorithm="fricas")

[Out]

integral((b^3*tan(f*x + e)^3 + 3*a*b^2*tan(f*x + e)^2 + 3*a^2*b*tan(f*x + e) + a^3)/(d^2*x^2 + 2*c*d*x + c^2),
x)

Sympy [N/A]

Not integrable

Time = 2.44 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {(a+b \tan (e+f x))^3}{(c+d x)^2} \, dx=\int \frac {\left (a + b \tan {\left (e + f x \right )}\right )^{3}}{\left (c + d x\right )^{2}}\, dx \]

[In]

integrate((a+b*tan(f*x+e))**3/(d*x+c)**2,x)

[Out]

Integral((a + b*tan(e + f*x))**3/(c + d*x)**2, x)

Maxima [N/A]

Not integrable

Time = 9.08 (sec) , antiderivative size = 2194, normalized size of antiderivative = 109.70 \[ \int \frac {(a+b \tan (e+f x))^3}{(c+d x)^2} \, dx=\int { \frac {{\left (b \tan \left (f x + e\right ) + a\right )}^{3}}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

integrate((a+b*tan(f*x+e))^3/(d*x+c)^2,x, algorithm="maxima")

[Out]

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

Giac [N/A]

Not integrable

Time = 18.06 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(a+b \tan (e+f x))^3}{(c+d x)^2} \, dx=\int { \frac {{\left (b \tan \left (f x + e\right ) + a\right )}^{3}}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

integrate((a+b*tan(f*x+e))^3/(d*x+c)^2,x, algorithm="giac")

[Out]

integrate((b*tan(f*x + e) + a)^3/(d*x + c)^2, x)

Mupad [N/A]

Not integrable

Time = 4.93 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(a+b \tan (e+f x))^3}{(c+d x)^2} \, dx=\int \frac {{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^3}{{\left (c+d\,x\right )}^2} \,d x \]

[In]

int((a + b*tan(e + f*x))^3/(c + d*x)^2,x)

[Out]

int((a + b*tan(e + f*x))^3/(c + d*x)^2, x)