3.52 \(\int \frac {(a+b \tan (e+f x))^3}{c+d x} \, dx\)
Optimal. Leaf size=23 \[ \text {Int}\left (\frac {(a+b \tan (e+f x))^3}{c+d x},x\right ) \]
[Out]
Unintegrable((a+b*tan(f*x+e))^3/(d*x+c),x)
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Rubi [A] time = 0.05, 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 =
{} \[ \int \frac {(a+b \tan (e+f x))^3}{c+d x} \, dx \]
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*}
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Mathematica [A] time = 14.57, size = 0, normalized size = 0.00 \[ \int \frac {(a+b \tan (e+f x))^3}{c+d x} \, dx \]
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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fricas [A] time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{3} \tan \left (f x + e\right )^{3} + 3 \, a b^{2} \tan \left (f x + e\right )^{2} + 3 \, a^{2} b \tan \left (f x + e\right ) + a^{3}}{d x + c}, x\right ) \]
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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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \tan \left (f x + e\right ) + a\right )}^{3}}{d x + c}\,{d x} \]
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)
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maple [A] time = 4.50, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \tan \left (f x +e \right )\right )^{3}}{d x +c}\, dx \]
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 = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]
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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mupad [A] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^3}{c+d\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*tan(e + f*x))^3/(c + d*x),x)
[Out]
int((a + b*tan(e + f*x))^3/(c + d*x), x)
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \tan {\left (e + f x \right )}\right )^{3}}{c + d x}\, dx \]
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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