### 3.53 $$\int \frac{(a+b \tan (e+f x))^3}{(c+d x)^2} \, dx$$

Optimal. Leaf size=22 $\text{Unintegrable}\left (\frac{(a+b \tan (e+f x))^3}{(c+d x)^2},x\right )$

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0504774, 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 = {} $\int \frac{(a+b \tan (e+f x))^3}{(c+d x)^2} \, dx$

Veriﬁcation is Not applicable to the result.

[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*} \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\\ \end{align*}

Mathematica [A]  time = 18.6984, size = 0, normalized size = 0. $\int \frac{(a+b \tan (e+f x))^3}{(c+d x)^2} \, dx$

Veriﬁcation is Not applicable to the result.

[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 [A]  time = 3.512, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\tan \left ( fx+e \right ) \right ) ^{3}}{ \left ( dx+c \right ) ^{2}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{3} \tan \left (f x + e\right )^{3} + 3 \, a b^{2} \tan \left (f x + e\right )^{2} + 3 \, a^{2} b \tan \left (f x + e\right ) + a^{3}}{d^{2} x^{2} + 2 \, c d x + c^{2}}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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 [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \tan{\left (e + f x \right )}\right )^{3}}{\left (c + d x\right )^{2}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \tan \left (f x + e\right ) + a\right )}^{3}}{{\left (d x + c\right )}^{2}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)