3.4.77 \(\int \cot ^3(e+f x) (b \csc (e+f x))^m \, dx\) [377]
Optimal. Leaf size=43 \[ \frac {(b \csc (e+f x))^m}{f m}-\frac {(b \csc (e+f x))^{2+m}}{b^2 f (2+m)} \]
[Out]
(b*csc(f*x+e))^m/f/m-(b*csc(f*x+e))^(2+m)/b^2/f/(2+m)
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Rubi [A]
time = 0.04, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2686, 14}
\begin {gather*} \frac {(b \csc (e+f x))^m}{f m}-\frac {(b \csc (e+f x))^{m+2}}{b^2 f (m+2)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Cot[e + f*x]^3*(b*Csc[e + f*x])^m,x]
[Out]
(b*Csc[e + f*x])^m/(f*m) - (b*Csc[e + f*x])^(2 + m)/(b^2*f*(2 + m))
Rule 14
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&& !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Rule 2686
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] && !(IntegerQ[m/2] && LtQ[0, m, n + 1])
Rubi steps
\begin {align*} \int \cot ^3(e+f x) (b \csc (e+f x))^m \, dx &=-\frac {b \text {Subst}\left (\int (b x)^{-1+m} \left (-1+x^2\right ) \, dx,x,\csc (e+f x)\right )}{f}\\ &=-\frac {b \text {Subst}\left (\int \left (-(b x)^{-1+m}+\frac {(b x)^{1+m}}{b^2}\right ) \, dx,x,\csc (e+f x)\right )}{f}\\ &=\frac {(b \csc (e+f x))^m}{f m}-\frac {(b \csc (e+f x))^{2+m}}{b^2 f (2+m)}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 36, normalized size = 0.84 \begin {gather*} \frac {(b \csc (e+f x))^m \left (2+m-m \csc ^2(e+f x)\right )}{f m (2+m)} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Cot[e + f*x]^3*(b*Csc[e + f*x])^m,x]
[Out]
((b*Csc[e + f*x])^m*(2 + m - m*Csc[e + f*x]^2))/(f*m*(2 + m))
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.47, size = 3826, normalized size = 88.98
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| method |
result |
size |
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| risch |
\(\text {Expression too large to display}\) |
\(3826\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cot(f*x+e)^3*(b*csc(f*x+e))^m,x,method=_RETURNVERBOSE)
[Out]
1/(2+m)/f/(exp(2*I*(f*x+e))-1)^2/m*(m/((exp(2*I*(f*x+e))-1)^m)*exp(I*(Re(f*x)+Re(e)))^m*b^m*2^m*exp(-m*Im(f*x)
-m*Im(e))*exp(1/2*I*Pi*csgn(I*exp(I*(f*x+e))/(exp(2*I*(f*x+e))-1))^2*csgn(I/(exp(2*I*(f*x+e))-1))*m)*exp(1/2*I
*Pi*csgn(I*exp(I*(f*x+e))/(exp(2*I*(f*x+e))-1))*csgn(I*b/(exp(2*I*(f*x+e))-1)*exp(I*(f*x+e)))^2*m)*exp(1/2*I*P
i*csgn(I*b/(exp(2*I*(f*x+e))-1)*exp(I*(f*x+e)))^2*csgn(I*b)*m)*exp(1/2*I*Pi*csgn(I*b/(exp(2*I*(f*x+e))-1)*exp(
I*(f*x+e)))*csgn(b/(exp(2*I*(f*x+e))-1)*exp(I*(f*x+e)))^2*m)*exp(-1/2*I*Pi*csgn(I*exp(I*(f*x+e))/(exp(2*I*(f*x
+e))-1))^3*m)*exp(-1/2*I*Pi*csgn(I*b/(exp(2*I*(f*x+e))-1)*exp(I*(f*x+e)))*csgn(b/(exp(2*I*(f*x+e))-1)*exp(I*(f
*x+e)))*m)*exp(-1/2*I*Pi*csgn(I*exp(I*(f*x+e))/(exp(2*I*(f*x+e))-1))*csgn(I*exp(I*(f*x+e)))*csgn(I/(exp(2*I*(f
*x+e))-1))*m)*exp(1/2*I*Pi*csgn(b/(exp(2*I*(f*x+e))-1)*exp(I*(f*x+e)))^3*m)*exp(-1/2*I*Pi*csgn(b/(exp(2*I*(f*x
+e))-1)*exp(I*(f*x+e)))^2*m)*exp(1/2*I*Pi*csgn(I*exp(I*(f*x+e))/(exp(2*I*(f*x+e))-1))^2*csgn(I*exp(I*(f*x+e)))
*m)*exp(1/2*I*Pi*m)*exp(-1/2*I*Pi*csgn(I*exp(I*(f*x+e))/(exp(2*I*(f*x+e))-1))*csgn(I*b/(exp(2*I*(f*x+e))-1)*ex
p(I*(f*x+e)))*csgn(I*b)*m)*exp(-1/2*I*Pi*csgn(I*b/(exp(2*I*(f*x+e))-1)*exp(I*(f*x+e)))^3*m)*exp(4*I*f*x)*exp(4
*I*e)+2/((exp(2*I*(f*x+e))-1)^m)*exp(I*(Re(f*x)+Re(e)))^m*b^m*2^m*exp(-m*Im(f*x)-m*Im(e))*exp(1/2*I*Pi*csgn(I*
exp(I*(f*x+e))/(exp(2*I*(f*x+e))-1))^2*csgn(I/(exp(2*I*(f*x+e))-1))*m)*exp(1/2*I*Pi*csgn(I*exp(I*(f*x+e))/(exp
(2*I*(f*x+e))-1))*csgn(I*b/(exp(2*I*(f*x+e))-1)*exp(I*(f*x+e)))^2*m)*exp(1/2*I*Pi*csgn(I*b/(exp(2*I*(f*x+e))-1
)*exp(I*(f*x+e)))^2*csgn(I*b)*m)*exp(1/2*I*Pi*csgn(I*b/(exp(2*I*(f*x+e))-1)*exp(I*(f*x+e)))*csgn(b/(exp(2*I*(f
*x+e))-1)*exp(I*(f*x+e)))^2*m)*exp(-1/2*I*Pi*csgn(I*exp(I*(f*x+e))/(exp(2*I*(f*x+e))-1))^3*m)*exp(-1/2*I*Pi*cs
gn(I*b/(exp(2*I*(f*x+e))-1)*exp(I*(f*x+e)))*csgn(b/(exp(2*I*(f*x+e))-1)*exp(I*(f*x+e)))*m)*exp(-1/2*I*Pi*csgn(
I*exp(I*(f*x+e))/(exp(2*I*(f*x+e))-1))*csgn(I*exp(I*(f*x+e)))*csgn(I/(exp(2*I*(f*x+e))-1))*m)*exp(1/2*I*Pi*csg
n(b/(exp(2*I*(f*x+e))-1)*exp(I*(f*x+e)))^3*m)*exp(-1/2*I*Pi*csgn(b/(exp(2*I*(f*x+e))-1)*exp(I*(f*x+e)))^2*m)*e
xp(1/2*I*Pi*csgn(I*exp(I*(f*x+e))/(exp(2*I*(f*x+e))-1))^2*csgn(I*exp(I*(f*x+e)))*m)*exp(1/2*I*Pi*m)*exp(-1/2*I
*Pi*csgn(I*exp(I*(f*x+e))/(exp(2*I*(f*x+e))-1))*csgn(I*b/(exp(2*I*(f*x+e))-1)*exp(I*(f*x+e)))*csgn(I*b)*m)*exp
(-1/2*I*Pi*csgn(I*b/(exp(2*I*(f*x+e))-1)*exp(I*(f*x+e)))^3*m)*exp(4*I*f*x)*exp(4*I*e)+2*m/((exp(2*I*(f*x+e))-1
)^m)*exp(I*(Re(f*x)+Re(e)))^m*b^m*2^m*exp(-m*Im(f*x)-m*Im(e))*exp(1/2*I*Pi*csgn(I*exp(I*(f*x+e))/(exp(2*I*(f*x
+e))-1))^2*csgn(I/(exp(2*I*(f*x+e))-1))*m)*exp(1/2*I*Pi*csgn(I*exp(I*(f*x+e))/(exp(2*I*(f*x+e))-1))*csgn(I*b/(
exp(2*I*(f*x+e))-1)*exp(I*(f*x+e)))^2*m)*exp(1/2*I*Pi*csgn(I*b/(exp(2*I*(f*x+e))-1)*exp(I*(f*x+e)))^2*csgn(I*b
)*m)*exp(1/2*I*Pi*csgn(I*b/(exp(2*I*(f*x+e))-1)*exp(I*(f*x+e)))*csgn(b/(exp(2*I*(f*x+e))-1)*exp(I*(f*x+e)))^2*
m)*exp(-1/2*I*Pi*csgn(I*exp(I*(f*x+e))/(exp(2*I*(f*x+e))-1))^3*m)*exp(-1/2*I*Pi*csgn(I*b/(exp(2*I*(f*x+e))-1)*
exp(I*(f*x+e)))*csgn(b/(exp(2*I*(f*x+e))-1)*exp(I*(f*x+e)))*m)*exp(-1/2*I*Pi*csgn(I*exp(I*(f*x+e))/(exp(2*I*(f
*x+e))-1))*csgn(I*exp(I*(f*x+e)))*csgn(I/(exp(2*I*(f*x+e))-1))*m)*exp(1/2*I*Pi*csgn(b/(exp(2*I*(f*x+e))-1)*exp
(I*(f*x+e)))^3*m)*exp(-1/2*I*Pi*csgn(b/(exp(2*I*(f*x+e))-1)*exp(I*(f*x+e)))^2*m)*exp(1/2*I*Pi*csgn(I*exp(I*(f*
x+e))/(exp(2*I*(f*x+e))-1))^2*csgn(I*exp(I*(f*x+e)))*m)*exp(1/2*I*Pi*m)*exp(-1/2*I*Pi*csgn(I*exp(I*(f*x+e))/(e
xp(2*I*(f*x+e))-1))*csgn(I*b/(exp(2*I*(f*x+e))-1)*exp(I*(f*x+e)))*csgn(I*b)*m)*exp(-1/2*I*Pi*csgn(I*b/(exp(2*I
*(f*x+e))-1)*exp(I*(f*x+e)))^3*m)*exp(2*I*f*x)*exp(2*I*e)-4/((exp(2*I*(f*x+e))-1)^m)*exp(I*(Re(f*x)+Re(e)))^m*
b^m*2^m*exp(-m*Im(f*x)-m*Im(e))*exp(1/2*I*Pi*csgn(I*exp(I*(f*x+e))/(exp(2*I*(f*x+e))-1))^2*csgn(I/(exp(2*I*(f*
x+e))-1))*m)*exp(1/2*I*Pi*csgn(I*exp(I*(f*x+e))/(exp(2*I*(f*x+e))-1))*csgn(I*b/(exp(2*I*(f*x+e))-1)*exp(I*(f*x
+e)))^2*m)*exp(1/2*I*Pi*csgn(I*b/(exp(2*I*(f*x+e))-1)*exp(I*(f*x+e)))^2*csgn(I*b)*m)*exp(1/2*I*Pi*csgn(I*b/(ex
p(2*I*(f*x+e))-1)*exp(I*(f*x+e)))*csgn(b/(exp(2*I*(f*x+e))-1)*exp(I*(f*x+e)))^2*m)*exp(-1/2*I*Pi*csgn(I*exp(I*
(f*x+e))/(exp(2*I*(f*x+e))-1))^3*m)*exp(-1/2*I*Pi*csgn(I*b/(exp(2*I*(f*x+e))-1)*exp(I*(f*x+e)))*csgn(b/(exp(2*
I*(f*x+e))-1)*exp(I*(f*x+e)))*m)*exp(-1/2*I*Pi*csgn(I*exp(I*(f*x+e))/(exp(2*I*(f*x+e))-1))*csgn(I*exp(I*(f*x+e
)))*csgn(I/(exp(2*I*(f*x+e))-1))*m)*exp(1/2*I*Pi*csgn(b/(exp(2*I*(f*x+e))-1)*exp(I*(f*x+e)))^3*m)*exp(-1/2*I*P
i*csgn(b/(exp(2*I*(f*x+e))-1)*exp(I*(f*x+e)))^2*m)*exp(1/2*I*Pi*csgn(I*exp(I*(f*x+e))/(exp(2*I*(f*x+e))-1))^2*
csgn(I*exp(I*(f*x+e)))*m)*exp(1/2*I*Pi*m)*exp(-1/2*I*Pi*csgn(I*exp(I*(f*x+e))/(exp(2*I*(f*x+e))-1))*csgn(I*b/(
exp(2*I*(f*x+e))-1)*exp(I*(f*x+e)))*csgn(I*b)*m)*exp(-1/2*I*Pi*csgn(I*b/(exp(2*I*(f*x+e))-1)*exp(I*(f*x+e)))^3
*m)*exp(2*I*f*x)*exp(2*I*e)+m/((exp(2*I*(f*x+e))-1)^m)*exp(I*(Re(f*x)+Re(e)))^m*b^m*2^m*exp(-1/2*m*(-I*Pi*csgn
(b/(exp(2*I*(f*x+e))-1)*exp(I*(f*x+e)))^3+I*Pi*csgn(I*b/(exp(2*I*(f*x+e))-1)*exp(I*(f*x+e)))^3-I*Pi*csgn(I*exp
(I*(f*x+e))/(exp(2*I*(f*x+e))-1))*csgn(I*b/(exp...
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Maxima [A]
time = 0.29, size = 53, normalized size = 1.23 \begin {gather*} \frac {\frac {b^{m} \sin \left (f x + e\right )^{-m}}{m} - \frac {b^{m} \sin \left (f x + e\right )^{-m}}{{\left (m + 2\right )} \sin \left (f x + e\right )^{2}}}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(f*x+e)^3*(b*csc(f*x+e))^m,x, algorithm="maxima")
[Out]
(b^m*sin(f*x + e)^(-m)/m - b^m*sin(f*x + e)^(-m)/((m + 2)*sin(f*x + e)^2))/f
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Fricas [A]
time = 0.37, size = 63, normalized size = 1.47 \begin {gather*} -\frac {{\left ({\left (m + 2\right )} \cos \left (f x + e\right )^{2} - 2\right )} \left (\frac {b}{\sin \left (f x + e\right )}\right )^{m}}{f m^{2} - {\left (f m^{2} + 2 \, f m\right )} \cos \left (f x + e\right )^{2} + 2 \, f m} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(f*x+e)^3*(b*csc(f*x+e))^m,x, algorithm="fricas")
[Out]
-((m + 2)*cos(f*x + e)^2 - 2)*(b/sin(f*x + e))^m/(f*m^2 - (f*m^2 + 2*f*m)*cos(f*x + e)^2 + 2*f*m)
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \begin {cases} x \left (b \csc {\left (e \right )}\right )^{m} \cot ^{3}{\left (e \right )} & \text {for}\: f = 0 \\\frac {\int \frac {\cot ^{3}{\left (e + f x \right )}}{\csc ^{2}{\left (e + f x \right )}}\, dx}{b^{2}} & \text {for}\: m = -2 \\\frac {\log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} - \frac {\log {\left (\tan {\left (e + f x \right )} \right )}}{f} - \frac {1}{2 f \tan ^{2}{\left (e + f x \right )}} & \text {for}\: m = 0 \\- \frac {m \left (b \csc {\left (e + f x \right )}\right )^{m} \cot ^{2}{\left (e + f x \right )}}{f m^{2} + 2 f m} + \frac {2 \left (b \csc {\left (e + f x \right )}\right )^{m}}{f m^{2} + 2 f m} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(f*x+e)**3*(b*csc(f*x+e))**m,x)
[Out]
Piecewise((x*(b*csc(e))**m*cot(e)**3, Eq(f, 0)), (Integral(cot(e + f*x)**3/csc(e + f*x)**2, x)/b**2, Eq(m, -2)
), (log(tan(e + f*x)**2 + 1)/(2*f) - log(tan(e + f*x))/f - 1/(2*f*tan(e + f*x)**2), Eq(m, 0)), (-m*(b*csc(e +
f*x))**m*cot(e + f*x)**2/(f*m**2 + 2*f*m) + 2*(b*csc(e + f*x))**m/(f*m**2 + 2*f*m), True))
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(f*x+e)^3*(b*csc(f*x+e))^m,x, algorithm="giac")
[Out]
integrate((b*csc(f*x + e))^m*cot(f*x + e)^3, x)
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Mupad [B]
time = 3.44, size = 92, normalized size = 2.14 \begin {gather*} \frac {{\left (\frac {b}{\sin \left (e+f\,x\right )}\right )}^m\,\left (m+4\,{\sin \left (2\,e+2\,f\,x\right )}^2+m\,\left (2\,{\sin \left (2\,e+2\,f\,x\right )}^2-1\right )-16\,{\sin \left (e+f\,x\right )}^2\right )}{f\,m\,\left (2\,{\sin \left (2\,e+2\,f\,x\right )}^2-8\,{\sin \left (e+f\,x\right )}^2\right )\,\left (m+2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cot(e + f*x)^3*(b/sin(e + f*x))^m,x)
[Out]
((b/sin(e + f*x))^m*(m + 4*sin(2*e + 2*f*x)^2 + m*(2*sin(2*e + 2*f*x)^2 - 1) - 16*sin(e + f*x)^2))/(f*m*(2*sin
(2*e + 2*f*x)^2 - 8*sin(e + f*x)^2)*(m + 2))
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