3.177 \(\int \csc ^2(e+f x) (b \tan (e+f x))^n \, dx\)
Optimal. Leaf size=25 \[ -\frac{b (b \tan (e+f x))^{n-1}}{f (1-n)} \]
[Out]
-((b*(b*Tan[e + f*x])^(-1 + n))/(f*(1 - n)))
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Rubi [A] time = 0.0424143, antiderivative size = 25, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used =
{2591, 30} \[ -\frac{b (b \tan (e+f x))^{n-1}}{f (1-n)} \]
Antiderivative was successfully verified.
[In]
Int[Csc[e + f*x]^2*(b*Tan[e + f*x])^n,x]
[Out]
-((b*(b*Tan[e + f*x])^(-1 + n))/(f*(1 - n)))
Rule 2591
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> With[{ff = FreeFactors[Ta
n[e + f*x], x]}, Dist[(b*ff)/f, Subst[Int[(ff*x)^(m + n)/(b^2 + ff^2*x^2)^(m/2 + 1), x], x, (b*Tan[e + f*x])/f
f], x]] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2]
Rule 30
Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]
Rubi steps
\begin{align*} \int \csc ^2(e+f x) (b \tan (e+f x))^n \, dx &=\frac{b \operatorname{Subst}\left (\int x^{-2+n} \, dx,x,b \tan (e+f x)\right )}{f}\\ &=-\frac{b (b \tan (e+f x))^{-1+n}}{f (1-n)}\\ \end{align*}
Mathematica [A] time = 0.0661165, size = 22, normalized size = 0.88 \[ \frac{b (b \tan (e+f x))^{n-1}}{f (n-1)} \]
Antiderivative was successfully verified.
[In]
Integrate[Csc[e + f*x]^2*(b*Tan[e + f*x])^n,x]
[Out]
(b*(b*Tan[e + f*x])^(-1 + n))/(f*(-1 + n))
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Maple [C] time = 1.787, size = 4284, normalized size = 171.4 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csc(f*x+e)^2*(b*tan(f*x+e))^n,x)
[Out]
I/(-1+n)/f/(exp(2*I*(f*x+e))-1)*(b^n/((exp(I*(f*x+e))+I)^n)*(exp(I*(f*x+e))-1)^n*(exp(I*(f*x+e))+1)^n/((exp(I*
(f*x+e))-I)^n)*exp(-1/2*I*Pi*n)*exp(-1/2*I*n*Pi*csgn(I*b/(exp(I*(f*x+e))+I)/(exp(I*(f*x+e))-I)*exp(2*I*(f*x+e)
)-I*b/(exp(I*(f*x+e))+I)/(exp(I*(f*x+e))-I))^3)*exp(1/2*I*n*Pi*csgn(b/(exp(I*(f*x+e))+I)/(exp(I*(f*x+e))-I)*ex
p(2*I*(f*x+e))-b/(exp(I*(f*x+e))+I)/(exp(I*(f*x+e))-I))^2)*exp(-1/2*I*n*Pi*csgn(b/(exp(I*(f*x+e))+I)/(exp(I*(f
*x+e))-I)*exp(2*I*(f*x+e))-b/(exp(I*(f*x+e))+I)/(exp(I*(f*x+e))-I))^3)*exp(-1/2*I*n*Pi*csgn(I/(exp(I*(f*x+e))-
I)/(exp(I*(f*x+e))+I)*exp(2*I*(f*x+e))-I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I))*csgn(I*b/(exp(I*(f*x+e))+I)/(e
xp(I*(f*x+e))-I)*exp(2*I*(f*x+e))-I*b/(exp(I*(f*x+e))+I)/(exp(I*(f*x+e))-I))*csgn(I*b))*exp(-1/2*I*n*Pi*csgn(I
/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I)*exp(I*(f*x+e))+I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I))^3)*exp(-1/2*I*n
*Pi*csgn(I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I)*exp(2*I*(f*x+e))-I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I))^3)*
exp(-1/2*I*n*Pi*csgn(I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I)*exp(I*(f*x+e))+I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e
))+I))*csgn(I*exp(I*(f*x+e))+I)*csgn(I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I)))*exp(-1/2*I*csgn(I/(exp(I*(f*x+e
))-I)/(exp(I*(f*x+e))+I))*Pi*csgn(I/(exp(I*(f*x+e))-I))*csgn(I/(exp(I*(f*x+e))+I))*n)*exp(-1/2*I*csgn(I/(exp(I
*(f*x+e))-I)/(exp(I*(f*x+e))+I))^3*Pi*n)*exp(-1/2*I*n*Pi*csgn(I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I)*exp(I*(f
*x+e))+I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I))*csgn(I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I)*exp(2*I*(f*x+e))-
I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I))*csgn(I*exp(I*(f*x+e))-I))*exp(1/2*I*n*Pi*csgn(I/(exp(I*(f*x+e))-I)/(e
xp(I*(f*x+e))+I)*exp(I*(f*x+e))+I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I))*csgn(I/(exp(I*(f*x+e))-I)/(exp(I*(f*x
+e))+I)*exp(2*I*(f*x+e))-I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I))^2)*exp(1/2*I*n*Pi*csgn(I/(exp(I*(f*x+e))-I)/
(exp(I*(f*x+e))+I)*exp(2*I*(f*x+e))-I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I))^2*csgn(I*exp(I*(f*x+e))-I))*exp(1
/2*I*n*Pi*csgn(I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I)*exp(2*I*(f*x+e))-I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I
))*csgn(I*b/(exp(I*(f*x+e))+I)/(exp(I*(f*x+e))-I)*exp(2*I*(f*x+e))-I*b/(exp(I*(f*x+e))+I)/(exp(I*(f*x+e))-I))^
2)*exp(1/2*I*csgn(I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I))^2*Pi*csgn(I/(exp(I*(f*x+e))+I))*n)*exp(1/2*I*csgn(I
/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I))^2*Pi*csgn(I/(exp(I*(f*x+e))-I))*n)*exp(1/2*I*n*Pi*csgn(I/(exp(I*(f*x+e
))-I)/(exp(I*(f*x+e))+I)*exp(I*(f*x+e))+I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I))^2*csgn(I*exp(I*(f*x+e))+I))*e
xp(-1/2*I*n*Pi*csgn(I*b/(exp(I*(f*x+e))+I)/(exp(I*(f*x+e))-I)*exp(2*I*(f*x+e))-I*b/(exp(I*(f*x+e))+I)/(exp(I*(
f*x+e))-I))*csgn(b/(exp(I*(f*x+e))+I)/(exp(I*(f*x+e))-I)*exp(2*I*(f*x+e))-b/(exp(I*(f*x+e))+I)/(exp(I*(f*x+e))
-I)))*exp(1/2*I*n*Pi*csgn(I*b/(exp(I*(f*x+e))+I)/(exp(I*(f*x+e))-I)*exp(2*I*(f*x+e))-I*b/(exp(I*(f*x+e))+I)/(e
xp(I*(f*x+e))-I))^2*csgn(I*b))*exp(1/2*I*n*Pi*csgn(I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I)*exp(I*(f*x+e))+I/(e
xp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I))^2*csgn(I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I)))*exp(1/2*I*n*Pi*csgn(I*b/
(exp(I*(f*x+e))+I)/(exp(I*(f*x+e))-I)*exp(2*I*(f*x+e))-I*b/(exp(I*(f*x+e))+I)/(exp(I*(f*x+e))-I))*csgn(b/(exp(
I*(f*x+e))+I)/(exp(I*(f*x+e))-I)*exp(2*I*(f*x+e))-b/(exp(I*(f*x+e))+I)/(exp(I*(f*x+e))-I))^2)*exp(2*I*f*x)*exp
(2*I*e)+b^n/((exp(I*(f*x+e))+I)^n)*(exp(I*(f*x+e))-1)^n*(exp(I*(f*x+e))+1)^n/((exp(I*(f*x+e))-I)^n)*exp(-1/2*I
*Pi*n*(csgn(I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I)*exp(I*(f*x+e))+I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I))*cs
gn(I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I)*exp(2*I*(f*x+e))-I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I))*csgn(I*ex
p(I*(f*x+e))-I)-csgn(I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I))^2*csgn(I/(exp(I*(f*x+e))+I))-csgn(I/(exp(I*(f*x+
e))-I)/(exp(I*(f*x+e))+I)*exp(2*I*(f*x+e))-I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I))^2*csgn(I*exp(I*(f*x+e))-I)
-csgn(I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I)*exp(I*(f*x+e))+I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I))^2*csgn(I
*exp(I*(f*x+e))+I)-csgn(I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I)*exp(I*(f*x+e))+I/(exp(I*(f*x+e))-I)/(exp(I*(f*
x+e))+I))*csgn(I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I)*exp(2*I*(f*x+e))-I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I
))^2+csgn(I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I)*exp(2*I*(f*x+e))-I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I))^3+
csgn(I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I)*exp(I*(f*x+e))+I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I))*csgn(I*ex
p(I*(f*x+e))+I)*csgn(I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I))+csgn(I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I)*exp
(I*(f*x+e))+I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I))^3+csgn(I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I))^3-csgn(I*
b/(exp(I*(f*x+e))+I)/(exp(I*(f*x+e))-I)*exp(2*I*(f*x+e))-I*b/(exp(I*(f*x+e))+I)/(exp(I*(f*x+e))-I))*csgn(b/(ex
p(I*(f*x+e))+I)/(exp(I*(f*x+e))-I)*exp(2*I*(f*x+e))-b/(exp(I*(f*x+e))+I)/(exp(I*(f*x+e))-I))^2+csgn(I*b/(exp(I
*(f*x+e))+I)/(exp(I*(f*x+e))-I)*exp(2*I*(f*x+e))-I*b/(exp(I*(f*x+e))+I)/(exp(I*(f*x+e))-I))*csgn(b/(exp(I*(f*x
+e))+I)/(exp(I*(f*x+e))-I)*exp(2*I*(f*x+e))-b/(exp(I*(f*x+e))+I)/(exp(I*(f*x+e))-I))-csgn(I/(exp(I*(f*x+e))-I)
/(exp(I*(f*x+e))+I))^2*csgn(I/(exp(I*(f*x+e))-I))+csgn(b/(exp(I*(f*x+e))+I)/(exp(I*(f*x+e))-I)*exp(2*I*(f*x+e)
)-b/(exp(I*(f*x+e))+I)/(exp(I*(f*x+e))-I))^3+csgn(I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I))*csgn(I/(exp(I*(f*x+
e))-I))*csgn(I/(exp(I*(f*x+e))+I))-csgn(I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I)*exp(I*(f*x+e))+I/(exp(I*(f*x+e
))-I)/(exp(I*(f*x+e))+I))^2*csgn(I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I))+csgn(I/(exp(I*(f*x+e))-I)/(exp(I*(f*
x+e))+I)*exp(2*I*(f*x+e))-I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I))*csgn(I*b/(exp(I*(f*x+e))+I)/(exp(I*(f*x+e))
-I)*exp(2*I*(f*x+e))-I*b/(exp(I*(f*x+e))+I)/(exp(I*(f*x+e))-I))*csgn(I*b)+csgn(I*b/(exp(I*(f*x+e))+I)/(exp(I*(
f*x+e))-I)*exp(2*I*(f*x+e))-I*b/(exp(I*(f*x+e))+I)/(exp(I*(f*x+e))-I))^3-csgn(I*b/(exp(I*(f*x+e))+I)/(exp(I*(f
*x+e))-I)*exp(2*I*(f*x+e))-I*b/(exp(I*(f*x+e))+I)/(exp(I*(f*x+e))-I))^2*csgn(I*b)-csgn(I/(exp(I*(f*x+e))-I)/(e
xp(I*(f*x+e))+I)*exp(2*I*(f*x+e))-I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I))*csgn(I*b/(exp(I*(f*x+e))+I)/(exp(I*
(f*x+e))-I)*exp(2*I*(f*x+e))-I*b/(exp(I*(f*x+e))+I)/(exp(I*(f*x+e))-I))^2-csgn(b/(exp(I*(f*x+e))+I)/(exp(I*(f*
x+e))-I)*exp(2*I*(f*x+e))-b/(exp(I*(f*x+e))+I)/(exp(I*(f*x+e))-I))^2+1)))
________________________________________________________________________________________
Maxima [A] time = 0.963509, size = 38, normalized size = 1.52 \begin{align*} \frac{b^{n} \tan \left (f x + e\right )^{n}}{f{\left (n - 1\right )} \tan \left (f x + e\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(f*x+e)^2*(b*tan(f*x+e))^n,x, algorithm="maxima")
[Out]
b^n*tan(f*x + e)^n/(f*(n - 1)*tan(f*x + e))
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Fricas [A] time = 1.63448, size = 96, normalized size = 3.84 \begin{align*} \frac{\left (\frac{b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}\right )^{n} \cos \left (f x + e\right )}{{\left (f n - f\right )} \sin \left (f x + e\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(f*x+e)^2*(b*tan(f*x+e))^n,x, algorithm="fricas")
[Out]
(b*sin(f*x + e)/cos(f*x + e))^n*cos(f*x + e)/((f*n - f)*sin(f*x + e))
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \tan{\left (e + f x \right )}\right )^{n} \csc ^{2}{\left (e + f x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(f*x+e)**2*(b*tan(f*x+e))**n,x)
[Out]
Integral((b*tan(e + f*x))**n*csc(e + f*x)**2, x)
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \tan \left (f x + e\right )\right )^{n} \csc \left (f x + e\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(f*x+e)^2*(b*tan(f*x+e))^n,x, algorithm="giac")
[Out]
integrate((b*tan(f*x + e))^n*csc(f*x + e)^2, x)