3.352 \(\int (b \sec (e+f x))^m \tan ^3(e+f x) \, dx\)
Optimal. Leaf size=43 \[ \frac{(b \sec (e+f x))^{m+2}}{b^2 f (m+2)}-\frac{(b \sec (e+f x))^m}{f m} \]
[Out]
-((b*Sec[e + f*x])^m/(f*m)) + (b*Sec[e + f*x])^(2 + m)/(b^2*f*(2 + m))
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Rubi [A] time = 0.0488089, antiderivative size = 43, normalized size of antiderivative = 1.,
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 =
{2606, 14} \[ \frac{(b \sec (e+f x))^{m+2}}{b^2 f (m+2)}-\frac{(b \sec (e+f x))^m}{f m} \]
Antiderivative was successfully verified.
[In]
Int[(b*Sec[e + f*x])^m*Tan[e + f*x]^3,x]
[Out]
-((b*Sec[e + f*x])^m/(f*m)) + (b*Sec[e + f*x])^(2 + m)/(b^2*f*(2 + m))
Rule 2606
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])
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]]
Rubi steps
\begin{align*} \int (b \sec (e+f x))^m \tan ^3(e+f x) \, dx &=\frac{b \operatorname{Subst}\left (\int (b x)^{-1+m} \left (-1+x^2\right ) \, dx,x,\sec (e+f x)\right )}{f}\\ &=\frac{b \operatorname{Subst}\left (\int \left (-(b x)^{-1+m}+\frac{(b x)^{1+m}}{b^2}\right ) \, dx,x,\sec (e+f x)\right )}{f}\\ &=-\frac{(b \sec (e+f x))^m}{f m}+\frac{(b \sec (e+f x))^{2+m}}{b^2 f (2+m)}\\ \end{align*}
Mathematica [A] time = 0.106865, size = 34, normalized size = 0.79 \[ \frac{\left (\frac{\sec ^2(e+f x)}{m+2}-\frac{1}{m}\right ) (b \sec (e+f x))^m}{f} \]
Antiderivative was successfully verified.
[In]
Integrate[(b*Sec[e + f*x])^m*Tan[e + f*x]^3,x]
[Out]
((b*Sec[e + f*x])^m*(-m^(-1) + Sec[e + f*x]^2/(2 + m)))/f
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Maple [C] time = 0.175, size = 2707, normalized size = 63. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*sec(f*x+e))^m*tan(f*x+e)^3,x)
[Out]
-1/(2+m)/f/(exp(2*I*(f*x+e))+1)^2/m*(2^m*b^m*exp(I*(Re(f*x)+Re(e)))^m/((exp(2*I*(f*x+e))+1)^m)*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))^3*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*csgn(I*exp(I*(f*x+e))/(exp(2*I*(f*x+e))+1))^2*cs
gn(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*exp(I*(f*x+e)))
*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*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(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(4*I*f*x)*exp(4*I*e)+2*2^m*b^m*exp(I*(Re
(f*x)+Re(e)))^m/((exp(2*I*(f*x+e))+1)^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))^3*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*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*exp(I*(f*x+e)))*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*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(1/2*I*Pi*csgn(I*b/(exp(2*I*(f*x+e))+1)*exp(I*(f*x+e)))^2*cs
gn(I*b)*m)*exp(4*I*f*x)*exp(4*I*e)-2*2^m*b^m*exp(I*(Re(f*x)+Re(e)))^m/((exp(2*I*(f*x+e))+1)^m)*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))^3*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*csgn(I*exp(I*(f*x+e))/(exp(2*I*(f*x+e))+1))^2*cs
gn(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*exp(I*(f*x+e)))
*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*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(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(2*I*f*x)*exp(2*I*e)+4*2^m*b^m*exp(I*(Re
(f*x)+Re(e)))^m/((exp(2*I*(f*x+e))+1)^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))^3*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*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*exp(I*(f*x+e)))*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*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(1/2*I*Pi*csgn(I*b/(exp(2*I*(f*x+e))+1)*exp(I*(f*x+e)))^2*cs
gn(I*b)*m)*exp(2*I*f*x)*exp(2*I*e)+2^m*b^m*exp(I*(Re(f*x)+Re(e)))^m/((exp(2*I*(f*x+e))+1)^m)*m*exp(-1/2*m*(I*P
i*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)-I*Pi*csg
n(I*b/(exp(2*I*(f*x+e))+1)*exp(I*(f*x+e)))^2*csgn(I*b)+I*Pi*csgn(I*exp(I*(f*x+e))/(exp(2*I*(f*x+e))+1))^3-I*Pi
*csgn(I*exp(I*(f*x+e))/(exp(2*I*(f*x+e))+1))^2*csgn(I*exp(I*(f*x+e)))-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))+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))-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+I*Pi*csgn(I*b/(exp(2*I*(f*x+e))+1)*exp(I*(f*x+e)))^3+2*Im(f*x)+2*Im(e)))+2*2^m*b^m*exp(I*(Re
(f*x)+Re(e)))^m/((exp(2*I*(f*x+e))+1)^m)*exp(-1/2*m*(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)-I*Pi*csgn(I*b/(exp(2*I*(f*x+e))+1)*exp(I*(f*x+e)))^2*csgn(I*b)
+I*Pi*csgn(I*exp(I*(f*x+e))/(exp(2*I*(f*x+e))+1))^3-I*Pi*csgn(I*exp(I*(f*x+e))/(exp(2*I*(f*x+e))+1))^2*csgn(I*
exp(I*(f*x+e)))-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))+I*Pi*csgn(I*ex
p(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))-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+I*Pi*csgn(I*b/(exp(2*I*(f*x+e))+1)*e
xp(I*(f*x+e)))^3+2*Im(f*x)+2*Im(e))))
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Maxima [A] time = 1.00428, size = 69, normalized size = 1.6 \begin{align*} -\frac{\frac{b^{m} \cos \left (f x + e\right )^{-m}}{m} - \frac{b^{m} \cos \left (f x + e\right )^{-m}}{{\left (m + 2\right )} \cos \left (f x + e\right )^{2}}}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*sec(f*x+e))^m*tan(f*x+e)^3,x, algorithm="maxima")
[Out]
-(b^m*cos(f*x + e)^(-m)/m - b^m*cos(f*x + e)^(-m)/((m + 2)*cos(f*x + e)^2))/f
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Fricas [A] time = 1.64096, size = 112, normalized size = 2.6 \begin{align*} -\frac{{\left ({\left (m + 2\right )} \cos \left (f x + e\right )^{2} - m\right )} \left (\frac{b}{\cos \left (f x + e\right )}\right )^{m}}{{\left (f m^{2} + 2 \, f m\right )} \cos \left (f x + e\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*sec(f*x+e))^m*tan(f*x+e)^3,x, algorithm="fricas")
[Out]
-((m + 2)*cos(f*x + e)^2 - m)*(b/cos(f*x + e))^m/((f*m^2 + 2*f*m)*cos(f*x + e)^2)
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*sec(f*x+e))**m*tan(f*x+e)**3,x)
[Out]
Exception raised: TypeError
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \sec \left (f x + e\right )\right )^{m} \tan \left (f x + e\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*sec(f*x+e))^m*tan(f*x+e)^3,x, algorithm="giac")
[Out]
integrate((b*sec(f*x + e))^m*tan(f*x + e)^3, x)