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(f*x+e))^m/f/m+(b*sec(f*x+e))^(2+m)/b^2/f/(2+m)
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Rubi [A] time = 0.05, 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 =
{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 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 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])
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*}
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Mathematica [A] time = 0.12, 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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fricas [A] time = 0.62, size = 50, normalized size = 1.16 \[ -\frac {{\left ({\left (m + 2\right )} \cos \left (f x + e\right )^{2} - m\right )} \left (\frac {b}{\cos \left (f x + e\right )}\right )^{m}}{{\left (f m^{2} + 2 \, f m\right )} \cos \left (f x + e\right )^{2}} \]
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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \sec \left (f x + e\right )\right )^{m} \tan \left (f x + e\right )^{3}\,{d x} \]
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)
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maple [C] time = 0.54, size = 2707, normalized size = 62.95 \[ \text {Expression too large to display} \]
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*(m/((exp(2*I*(f*x+e))+1)^m)*exp(I*(Re(f*x)+Re(e)))^m*2^m*b^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/((exp(2*I*(f*x+e)
)+1)^m)*exp(I*(Re(f*x)+Re(e)))^m*2^m*b^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*m/((exp(2*I*(f*x+e))+1)^m)*exp(I*(Re(f*x)+Re(e)))^m*2^m*b^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/((exp(2*I*(f*x+e)
)+1)^m)*exp(I*(Re(f*x)+Re(e)))^m*2^m*b^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)+m/((exp(2*I*(f*x+e))+1)^m)*exp(I*(Re(f*x)+Re(e)))^m*2^m*b^m*exp(-1/2*m*(I*P
i*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*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)))^3-I*Pi*csgn(I*b/(exp(2*I*(f*x+e))+1)*exp(I*(f*x+e)))^2*csgn(I*b)+2*Im(e)+2*Im(f*x)))+2/((exp(2*I*(f*x+e)
)+1)^m)*exp(I*(Re(f*x)+Re(e)))^m*2^m*b^m*exp(-1/2*m*(I*Pi*csgn(I*exp(I*(f*x+e))/(exp(2*I*(f*x+e))+1))^3-I*Pi*c
sgn(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)*ex
p(I*(f*x+e)))^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)+I*Pi*csgn(I*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)))^2*csgn(I*b)+2*Im(e)+2*Im(f*x))))
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maxima [A] time = 0.89, size = 51, normalized size = 1.19 \[ -\frac {\frac {b^{m} \cos \left (f x + e\right )^{-m}}{m} - \frac {b^{m} \cos \left (f x + e\right )^{-m}}{{\left (m + 2\right )} \cos \left (f x + e\right )^{2}}}{f} \]
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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mupad [B] time = 3.39, size = 87, normalized size = 2.02 \[ -\frac {{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^m\,\left (8\,\cos \left (2\,e+2\,f\,x\right )-m+2\,\cos \left (4\,e+4\,f\,x\right )+m\,\cos \left (4\,e+4\,f\,x\right )+6\right )}{f\,m\,\left (m+2\right )\,\left (4\,\cos \left (2\,e+2\,f\,x\right )+\cos \left (4\,e+4\,f\,x\right )+3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(tan(e + f*x)^3*(b/cos(e + f*x))^m,x)
[Out]
-((b/cos(e + f*x))^m*(8*cos(2*e + 2*f*x) - m + 2*cos(4*e + 4*f*x) + m*cos(4*e + 4*f*x) + 6))/(f*m*(m + 2)*(4*c
os(2*e + 2*f*x) + cos(4*e + 4*f*x) + 3))
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \begin {cases} x \left (b \sec {\relax (e )}\right )^{m} \tan ^{3}{\relax (e )} & \text {for}\: f = 0 \\\frac {\int \frac {\tan ^{3}{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )}}\, dx}{b^{2}} & \text {for}\: m = -2 \\- \frac {\log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac {\tan ^{2}{\left (e + f x \right )}}{2 f} & \text {for}\: m = 0 \\\frac {b^{m} m \tan ^{2}{\left (e + f x \right )} \sec ^{m}{\left (e + f x \right )}}{f m^{2} + 2 f m} - \frac {2 b^{m} \sec ^{m}{\left (e + f x \right )}}{f m^{2} + 2 f m} & \text {otherwise} \end {cases} \]
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]
Piecewise((x*(b*sec(e))**m*tan(e)**3, Eq(f, 0)), (Integral(tan(e + f*x)**3/sec(e + f*x)**2, x)/b**2, Eq(m, -2)
), (-log(tan(e + f*x)**2 + 1)/(2*f) + tan(e + f*x)**2/(2*f), Eq(m, 0)), (b**m*m*tan(e + f*x)**2*sec(e + f*x)**
m/(f*m**2 + 2*f*m) - 2*b**m*sec(e + f*x)**m/(f*m**2 + 2*f*m), True))
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