3.308 \(\int \frac{\tan ^3(a+b x)}{(c+d x)^2} \, dx\)
Optimal. Leaf size=18 \[ \text{Unintegrable}\left (\frac{\tan ^3(a+b x)}{(c+d x)^2},x\right ) \]
[Out]
Unintegrable[Tan[a + b*x]^3/(c + d*x)^2, x]
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Rubi [A] time = 0.0366005, 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 = {}
\[ \int \frac{\tan ^3(a+b x)}{(c+d x)^2} \, dx \]
Verification is Not applicable to the result.
[In]
Int[Tan[a + b*x]^3/(c + d*x)^2,x]
[Out]
Defer[Int][Tan[a + b*x]^3/(c + d*x)^2, x]
Rubi steps
\begin{align*} \int \frac{\tan ^3(a+b x)}{(c+d x)^2} \, dx &=\int \frac{\tan ^3(a+b x)}{(c+d x)^2} \, dx\\ \end{align*}
Mathematica [A] time = 7.03406, size = 0, normalized size = 0. \[ \int \frac{\tan ^3(a+b x)}{(c+d x)^2} \, dx \]
Verification is Not applicable to the result.
[In]
Integrate[Tan[a + b*x]^3/(c + d*x)^2,x]
[Out]
Integrate[Tan[a + b*x]^3/(c + d*x)^2, x]
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Maple [A] time = 2.576, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \tan \left ( bx+a \right ) \right ) ^{3}}{ \left ( dx+c \right ) ^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(tan(b*x+a)^3/(d*x+c)^2,x)
[Out]
int(tan(b*x+a)^3/(d*x+c)^2,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(tan(b*x+a)^3/(d*x+c)^2,x, algorithm="maxima")
[Out]
(4*(b*d*x + b*c)*cos(2*b*x + 2*a)^2 + 4*(b*d*x + b*c)*sin(2*b*x + 2*a)^2 + 2*((b*d*x + b*c)*cos(2*b*x + 2*a) -
d*sin(2*b*x + 2*a))*cos(4*b*x + 4*a) + 2*(b*d*x + b*c)*cos(2*b*x + 2*a) - (b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*
b^2*c^2*d*x + b^2*c^3 + (b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*cos(4*b*x + 4*a)^2 + 4*(b^2*
d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*cos(2*b*x + 2*a)^2 + (b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b
^2*c^2*d*x + b^2*c^3)*sin(4*b*x + 4*a)^2 + 4*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*sin(4*b
*x + 4*a)*sin(2*b*x + 2*a) + 4*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*sin(2*b*x + 2*a)^2 +
2*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3 + 2*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x
+ b^2*c^3)*cos(2*b*x + 2*a))*cos(4*b*x + 4*a) + 4*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*co
s(2*b*x + 2*a))*integrate(2*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 - 3*d^2)*sin(2*b*x + 2*a)/(b^2*d^4*x^4 + 4*b^
2*c*d^3*x^3 + 6*b^2*c^2*d^2*x^2 + 4*b^2*c^3*d*x + b^2*c^4 + (b^2*d^4*x^4 + 4*b^2*c*d^3*x^3 + 6*b^2*c^2*d^2*x^2
+ 4*b^2*c^3*d*x + b^2*c^4)*cos(2*b*x + 2*a)^2 + (b^2*d^4*x^4 + 4*b^2*c*d^3*x^3 + 6*b^2*c^2*d^2*x^2 + 4*b^2*c^
3*d*x + b^2*c^4)*sin(2*b*x + 2*a)^2 + 2*(b^2*d^4*x^4 + 4*b^2*c*d^3*x^3 + 6*b^2*c^2*d^2*x^2 + 4*b^2*c^3*d*x + b
^2*c^4)*cos(2*b*x + 2*a)), x) + 2*(d*cos(2*b*x + 2*a) + (b*d*x + b*c)*sin(2*b*x + 2*a) + d)*sin(4*b*x + 4*a) +
2*d*sin(2*b*x + 2*a))/(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3 + (b^2*d^3*x^3 + 3*b^2*c*d^2*x
^2 + 3*b^2*c^2*d*x + b^2*c^3)*cos(4*b*x + 4*a)^2 + 4*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)
*cos(2*b*x + 2*a)^2 + (b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*sin(4*b*x + 4*a)^2 + 4*(b^2*d^
3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + 4*(b^2*d^3*x^3 + 3*b^2*
c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*sin(2*b*x + 2*a)^2 + 2*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b
^2*c^3 + 2*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*cos(2*b*x + 2*a))*cos(4*b*x + 4*a) + 4*(b
^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*cos(2*b*x + 2*a))
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\tan \left (b x + a\right )^{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(tan(b*x+a)^3/(d*x+c)^2,x, algorithm="fricas")
[Out]
integral(tan(b*x + a)^3/(d^2*x^2 + 2*c*d*x + c^2), x)
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Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan ^{3}{\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tan(b*x+a)**3/(d*x+c)**2,x)
[Out]
Integral(tan(a + b*x)**3/(c + d*x)**2, x)
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan \left (b x + 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(tan(b*x+a)^3/(d*x+c)^2,x, algorithm="giac")
[Out]
integrate(tan(b*x + a)^3/(d*x + c)^2, x)