### 3.52 $$\int \frac{(a+b \tan (e+f x))^3}{c+d x} \, dx$$

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

[Out]

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

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Rubi [A]  time = 0.0534729, 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 = {} $\int \frac{(a+b \tan (e+f x))^3}{c+d x} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 13.5921, size = 0, normalized size = 0. $\int \frac{(a+b \tan (e+f x))^3}{c+d x} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 1.775, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\tan \left ( fx+e \right ) \right ) ^{3}}{dx+c}}\, 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),x)

[Out]

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

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

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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 x + c}, 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),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*x + c), x)

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

[Out]

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

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

[Out]

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