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(f*x+e))^(-1+n)/f/(1-n)

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Rubi [A]  time = 0.04, antiderivative size = 25, normalized size of antiderivative = 1.00, 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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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]

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*}

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Mathematica [A]  time = 0.07, 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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fricas [A]  time = 0.44, size = 42, normalized size = 1.68 \[ \frac {\left (\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}\right )^{n} \cos \left (f x + e\right )}{{\left (f n - f\right )} \sin \left (f x + e\right )} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \tan \left (f x + e\right )\right )^{n} \csc \left (f x + e\right )^{2}\,{d x} \]

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)

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maple [C]  time = 3.66, size = 4284, normalized size = 171.36 \[ \text {output too large to display} \]

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)*(1/((exp(I*(f*x+e))+I)^n)/((exp(I*(f*x+e))-I)^n)*(exp(I*(f*x+e))+1)^n*(exp(I*(
f*x+e))-1)^n*b^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)/(ex
p(I*(f*x+e))+I))*csgn(I*exp(I*(f*x+e))-I))*exp(1/2*I*n*Pi*csgn(1/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I)*b*exp(2
*I*(f*x+e))-1/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I)*b)^2)*exp(1/2*I*n*Pi*csgn(I/(exp(I*(f*x+e))-I)/(exp(I*(f*x
+e))+I)*b*exp(2*I*(f*x+e))-I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I)*b)^2*csgn(I*b))*exp(-1/2*I*n*Pi*csgn(I/(exp
(I*(f*x+e))-I)/(exp(I*(f*x+e))+I)*b*exp(2*I*(f*x+e))-I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I)*b)^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))^3)*exp(-1/2*I*n*Pi*csgn(1/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I)*b*exp(2*I*(f*x+e))-1/(exp(I*(f*x+e))-I
)/(exp(I*(f*x+e))+I)*b)^3)*exp(-1/2*I*Pi*n)*exp(-1/2*I*Pi*csgn(I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I))^3*n)*e
xp(1/2*I*n*Pi*csgn(I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I)*b*exp(2*I*(f*x+e))-I/(exp(I*(f*x+e))-I)/(exp(I*(f*x
+e))+I)*b)*csgn(1/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I)*b*exp(2*I*(f*x+e))-1/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e)
)+I)*b)^2)*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)/(ex
p(I*(f*x+e))+I))^2*csgn(I*exp(I*(f*x+e))+I))*exp(1/2*I*Pi*csgn(I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I))^2*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)*b*exp(2*I*(f*x+e))-I/(e
xp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I)*b)*csgn(1/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I)*b*exp(2*I*(f*x+e))-1/(exp(
I*(f*x+e))-I)/(exp(I*(f*x+e))+I)*b))*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/(exp(I*(f*x+e))-I)/(ex
p(I*(f*x+e))+I)*b*exp(2*I*(f*x+e))-I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I)*b)^2)*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))^2)*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)/(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(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*Pi*csgn(I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I))^2*csgn(I/(exp(I*(f*x+e))-I))*n)*exp(-1/2*I*Pi*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))*n)*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
/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I)*b*exp(2*I*(f*x+e))-I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I)*b)*csgn(I*b)
)*exp(2*I*f*x)*exp(2*I*e)+1/((exp(I*(f*x+e))+I)^n)/((exp(I*(f*x+e))-I)^n)*(exp(I*(f*x+e))+1)^n*(exp(I*(f*x+e))
-1)^n*b^n*exp(-1/2*I*Pi*n*(-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))^3+csgn(1/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I)*b*exp(2*I*(f*x+e))-1/(ex
p(I*(f*x+e))-I)/(exp(I*(f*x+e))+I)*b)^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/(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)-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)*b*exp(2*I*(f*x+e))-I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I)*b)^3-csgn(1/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+
I)*b*exp(2*I*(f*x+e))-1/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I)*b)^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))*csgn(I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I)*b*exp
(2*I*(f*x+e))-I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I)*b)*csgn(I*b)-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(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(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)*b*exp(2*I*(f*x+e))-I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e)
)+I)*b)*csgn(1/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I)*b*exp(2*I*(f*x+e))-1/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I
)*b)+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))*csgn(I/(exp(I*(f*x+e))-I))*csgn(I/(exp(I*(f*x+e))+I))+csgn(I/(ex
p(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))-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(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)-c
sgn(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(I*(f*x+e))+I)*b*exp(2*I*(f*x+e))-I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I)*b)^2-csgn(I/(e
xp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I)*b*exp(2*I*(f*x+e))-I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I)*b)^2*csgn(I*b)-
csgn(I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I)*b*exp(2*I*(f*x+e))-I/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I)*b)*csg
n(1/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I)*b*exp(2*I*(f*x+e))-1/(exp(I*(f*x+e))-I)/(exp(I*(f*x+e))+I)*b)^2+1)))

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maxima [A]  time = 0.42, size = 28, normalized size = 1.12 \[ \frac {b^{n} \tan \left (f x + e\right )^{n}}{f {\left (n - 1\right )} \tan \left (f x + e\right )} \]

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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mupad [B]  time = 2.62, size = 53, normalized size = 2.12 \[ -\frac {\sin \left (2\,e+2\,f\,x\right )\,{\left (\frac {b\,\sin \left (2\,e+2\,f\,x\right )}{2\,{\cos \left (e+f\,x\right )}^2}\right )}^n}{2\,f\,\left ({\cos \left (e+f\,x\right )}^2-1\right )\,\left (n-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*tan(e + f*x))^n/sin(e + f*x)^2,x)

[Out]

-(sin(2*e + 2*f*x)*((b*sin(2*e + 2*f*x))/(2*cos(e + f*x)^2))^n)/(2*f*(cos(e + f*x)^2 - 1)*(n - 1))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \tan {\left (e + f x \right )}\right )^{n} \csc ^{2}{\left (e + f x \right )}\, dx \]

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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