3.6.51 \(\int e^{m x} \sec ^3(x) \, dx\) [551]
Optimal. Leaf size=51 \[ \frac {8 e^{(3 i+m) x} \, _2F_1\left (3,\frac {1}{2} (3-i m);\frac {1}{2} (5-i m);-e^{2 i x}\right )}{3 i+m} \]
[Out]
8*exp((3*I+m)*x)*hypergeom([3, 3/2-1/2*I*m],[5/2-1/2*I*m],-exp(2*I*x))/(3*I+m)
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Rubi [A]
time = 0.03, antiderivative size = 77, normalized size of antiderivative = 1.51, number of steps
used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4533, 4536}
\begin {gather*} (-m+i) \left (-e^{(m+i) x}\right ) \text {Hypergeometric2F1}\left (1,\frac {1}{2} (1-i m),\frac {1}{2} (3-i m),-e^{2 i x}\right )-\frac {1}{2} m e^{m x} \sec (x)+\frac {1}{2} e^{m x} \tan (x) \sec (x) \end {gather*}
Antiderivative was successfully verified.
[In]
Int[E^(m*x)*Sec[x]^3,x]
[Out]
-(E^((I + m)*x)*(I - m)*Hypergeometric2F1[1, (1 - I*m)/2, (3 - I*m)/2, -E^((2*I)*x)]) - (E^(m*x)*m*Sec[x])/2 +
(E^(m*x)*Sec[x]*Tan[x])/2
Rule 4533
Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sec[(d_.) + (e_.)*(x_)]^(n_), x_Symbol] :> Simp[(-b)*c*Log[F]*F^(c*(a +
b*x))*(Sec[d + e*x]^(n - 2)/(e^2*(n - 1)*(n - 2))), x] + (Dist[(e^2*(n - 2)^2 + b^2*c^2*Log[F]^2)/(e^2*(n - 1)
*(n - 2)), Int[F^(c*(a + b*x))*Sec[d + e*x]^(n - 2), x], x] + Simp[F^(c*(a + b*x))*Sec[d + e*x]^(n - 1)*(Sin[d
+ e*x]/(e*(n - 1))), x]) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[b^2*c^2*Log[F]^2 + e^2*(n - 2)^2, 0] && GtQ[n
, 1] && NeQ[n, 2]
Rule 4536
Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sec[(d_.) + (e_.)*(x_)]^(n_.), x_Symbol] :> Simp[2^n*E^(I*n*(d + e*x))*(
F^(c*(a + b*x))/(I*e*n + b*c*Log[F]))*Hypergeometric2F1[n, n/2 - I*b*c*(Log[F]/(2*e)), 1 + n/2 - I*b*c*(Log[F]
/(2*e)), -E^(2*I*(d + e*x))], x] /; FreeQ[{F, a, b, c, d, e}, x] && IntegerQ[n]
Rubi steps
\begin {align*} \int e^{m x} \sec ^3(x) \, dx &=-\frac {1}{2} e^{m x} m \sec (x)+\frac {1}{2} e^{m x} \sec (x) \tan (x)+\frac {1}{2} \left (1+m^2\right ) \int e^{m x} \sec (x) \, dx\\ &=-e^{(i+m) x} (i-m) \, _2F_1\left (1,\frac {1}{2} (1-i m);\frac {1}{2} (3-i m);-e^{2 i x}\right )-\frac {1}{2} e^{m x} m \sec (x)+\frac {1}{2} e^{m x} \sec (x) \tan (x)\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 66, normalized size = 1.29 \begin {gather*} \frac {1}{2} e^{m x} \left (2 e^{i x} (-i+m) \, _2F_1\left (1,\frac {1}{2}-\frac {i m}{2};\frac {3}{2}-\frac {i m}{2};-e^{2 i x}\right )+\sec (x) (-m+\tan (x))\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[E^(m*x)*Sec[x]^3,x]
[Out]
(E^(m*x)*(2*E^(I*x)*(-I + m)*Hypergeometric2F1[1, 1/2 - (I/2)*m, 3/2 - (I/2)*m, -E^((2*I)*x)] + Sec[x]*(-m + T
an[x])))/2
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{m x}}{\cos \left (x \right )^{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(exp(m*x)/cos(x)^3,x)
[Out]
int(exp(m*x)/cos(x)^3,x)
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(exp(m*x)/cos(x)^3,x, algorithm="maxima")
[Out]
8*(48*m*cos(x)*e^(m*x) + 6*(m^2 - 15)*e^(m*x)*sin(x) + ((m^3 + 25*m)*cos(3*x)*e^(m*x) + 48*m*cos(x)*e^(m*x) -
3*(m^2 + 25)*e^(m*x)*sin(3*x) + 6*(m^2 - 15)*e^(m*x)*sin(x))*cos(6*x) + 3*((m^3 + 25*m)*cos(3*x)*e^(m*x) + 48*
m*cos(x)*e^(m*x) - 3*(m^2 + 25)*e^(m*x)*sin(3*x) + 6*(m^2 - 15)*e^(m*x)*sin(x))*cos(4*x) + (3*(m^3 + 25*m)*cos
(2*x)*e^(m*x) + 9*(m^2 + 25)*e^(m*x)*sin(2*x) + (m^3 + 25*m)*e^(m*x))*cos(3*x) + 18*(8*m*cos(x)*e^(m*x) + (m^2
- 15)*e^(m*x)*sin(x))*cos(2*x) - 6*(m^4 + (m^4 + 34*m^2 + 225)*cos(6*x)^2 + 9*(m^4 + 34*m^2 + 225)*cos(4*x)^2
+ 9*(m^4 + 34*m^2 + 225)*cos(2*x)^2 + (m^4 + 34*m^2 + 225)*sin(6*x)^2 + 9*(m^4 + 34*m^2 + 225)*sin(4*x)^2 + 1
8*(m^4 + 34*m^2 + 225)*sin(4*x)*sin(2*x) + 9*(m^4 + 34*m^2 + 225)*sin(2*x)^2 + 34*m^2 + 2*(m^4 + 34*m^2 + 3*(m
^4 + 34*m^2 + 225)*cos(4*x) + 3*(m^4 + 34*m^2 + 225)*cos(2*x) + 225)*cos(6*x) + 6*(m^4 + 34*m^2 + 3*(m^4 + 34*
m^2 + 225)*cos(2*x) + 225)*cos(4*x) + 6*(m^4 + 34*m^2 + 225)*cos(2*x) + 6*((m^4 + 34*m^2 + 225)*sin(4*x) + (m^
4 + 34*m^2 + 225)*sin(2*x))*sin(6*x) + 225)*integrate(((m^2 - 15)*cos(x)*e^(m*x) - 8*m*e^(m*x)*sin(x) + ((m^2
- 15)*cos(x)*e^(m*x) - 8*m*e^(m*x)*sin(x))*cos(8*x) + 4*((m^2 - 15)*cos(x)*e^(m*x) - 8*m*e^(m*x)*sin(x))*cos(6
*x) + 6*((m^2 - 15)*cos(x)*e^(m*x) - 8*m*e^(m*x)*sin(x))*cos(4*x) + 4*((m^2 - 15)*cos(x)*e^(m*x) - 8*m*e^(m*x)
*sin(x))*cos(2*x) + (8*m*cos(x)*e^(m*x) + (m^2 - 15)*e^(m*x)*sin(x))*sin(8*x) + 4*(8*m*cos(x)*e^(m*x) + (m^2 -
15)*e^(m*x)*sin(x))*sin(6*x) + 6*(8*m*cos(x)*e^(m*x) + (m^2 - 15)*e^(m*x)*sin(x))*sin(4*x) + 4*(8*m*cos(x)*e^
(m*x) + (m^2 - 15)*e^(m*x)*sin(x))*sin(2*x))/(m^4 + (m^4 + 34*m^2 + 225)*cos(8*x)^2 + 16*(m^4 + 34*m^2 + 225)*
cos(6*x)^2 + 36*(m^4 + 34*m^2 + 225)*cos(4*x)^2 + 16*(m^4 + 34*m^2 + 225)*cos(2*x)^2 + (m^4 + 34*m^2 + 225)*si
n(8*x)^2 + 16*(m^4 + 34*m^2 + 225)*sin(6*x)^2 + 36*(m^4 + 34*m^2 + 225)*sin(4*x)^2 + 48*(m^4 + 34*m^2 + 225)*s
in(4*x)*sin(2*x) + 16*(m^4 + 34*m^2 + 225)*sin(2*x)^2 + 34*m^2 + 2*(m^4 + 34*m^2 + 4*(m^4 + 34*m^2 + 225)*cos(
6*x) + 6*(m^4 + 34*m^2 + 225)*cos(4*x) + 4*(m^4 + 34*m^2 + 225)*cos(2*x) + 225)*cos(8*x) + 8*(m^4 + 34*m^2 + 6
*(m^4 + 34*m^2 + 225)*cos(4*x) + 4*(m^4 + 34*m^2 + 225)*cos(2*x) + 225)*cos(6*x) + 12*(m^4 + 34*m^2 + 4*(m^4 +
34*m^2 + 225)*cos(2*x) + 225)*cos(4*x) + 8*(m^4 + 34*m^2 + 225)*cos(2*x) + 4*(2*(m^4 + 34*m^2 + 225)*sin(6*x)
+ 3*(m^4 + 34*m^2 + 225)*sin(4*x) + 2*(m^4 + 34*m^2 + 225)*sin(2*x))*sin(8*x) + 16*(3*(m^4 + 34*m^2 + 225)*si
n(4*x) + 2*(m^4 + 34*m^2 + 225)*sin(2*x))*sin(6*x) + 225), x) - 6*(m^5 + 34*m^3 + (m^5 + 34*m^3 + 225*m)*cos(6
*x)^2 + 9*(m^5 + 34*m^3 + 225*m)*cos(4*x)^2 + 9*(m^5 + 34*m^3 + 225*m)*cos(2*x)^2 + (m^5 + 34*m^3 + 225*m)*sin
(6*x)^2 + 9*(m^5 + 34*m^3 + 225*m)*sin(4*x)^2 + 18*(m^5 + 34*m^3 + 225*m)*sin(4*x)*sin(2*x) + 9*(m^5 + 34*m^3
+ 225*m)*sin(2*x)^2 + 2*(m^5 + 34*m^3 + 3*(m^5 + 34*m^3 + 225*m)*cos(4*x) + 3*(m^5 + 34*m^3 + 225*m)*cos(2*x)
+ 225*m)*cos(6*x) + 6*(m^5 + 34*m^3 + 3*(m^5 + 34*m^3 + 225*m)*cos(2*x) + 225*m)*cos(4*x) + 6*(m^5 + 34*m^3 +
225*m)*cos(2*x) + 6*((m^5 + 34*m^3 + 225*m)*sin(4*x) + (m^5 + 34*m^3 + 225*m)*sin(2*x))*sin(6*x) + 225*m)*inte
grate((8*m*cos(x)*e^(m*x) + (m^2 - 15)*e^(m*x)*sin(x) + (8*m*cos(x)*e^(m*x) + (m^2 - 15)*e^(m*x)*sin(x))*cos(8
*x) + 4*(8*m*cos(x)*e^(m*x) + (m^2 - 15)*e^(m*x)*sin(x))*cos(6*x) + 6*(8*m*cos(x)*e^(m*x) + (m^2 - 15)*e^(m*x)
*sin(x))*cos(4*x) + 4*(8*m*cos(x)*e^(m*x) + (m^2 - 15)*e^(m*x)*sin(x))*cos(2*x) - ((m^2 - 15)*cos(x)*e^(m*x) -
8*m*e^(m*x)*sin(x))*sin(8*x) - 4*((m^2 - 15)*cos(x)*e^(m*x) - 8*m*e^(m*x)*sin(x))*sin(6*x) - 6*((m^2 - 15)*co
s(x)*e^(m*x) - 8*m*e^(m*x)*sin(x))*sin(4*x) - 4*((m^2 - 15)*cos(x)*e^(m*x) - 8*m*e^(m*x)*sin(x))*sin(2*x))/(m^
4 + (m^4 + 34*m^2 + 225)*cos(8*x)^2 + 16*(m^4 + 34*m^2 + 225)*cos(6*x)^2 + 36*(m^4 + 34*m^2 + 225)*cos(4*x)^2
+ 16*(m^4 + 34*m^2 + 225)*cos(2*x)^2 + (m^4 + 34*m^2 + 225)*sin(8*x)^2 + 16*(m^4 + 34*m^2 + 225)*sin(6*x)^2 +
36*(m^4 + 34*m^2 + 225)*sin(4*x)^2 + 48*(m^4 + 34*m^2 + 225)*sin(4*x)*sin(2*x) + 16*(m^4 + 34*m^2 + 225)*sin(2
*x)^2 + 34*m^2 + 2*(m^4 + 34*m^2 + 4*(m^4 + 34*m^2 + 225)*cos(6*x) + 6*(m^4 + 34*m^2 + 225)*cos(4*x) + 4*(m^4
+ 34*m^2 + 225)*cos(2*x) + 225)*cos(8*x) + 8*(m^4 + 34*m^2 + 6*(m^4 + 34*m^2 + 225)*cos(4*x) + 4*(m^4 + 34*m^2
+ 225)*cos(2*x) + 225)*cos(6*x) + 12*(m^4 + 34*m^2 + 4*(m^4 + 34*m^2 + 225)*cos(2*x) + 225)*cos(4*x) + 8*(m^4
+ 34*m^2 + 225)*cos(2*x) + 4*(2*(m^4 + 34*m^2 + 225)*sin(6*x) + 3*(m^4 + 34*m^2 + 225)*sin(4*x) + 2*(m^4 + 34
*m^2 + 225)*sin(2*x))*sin(8*x) + 16*(3*(m^4 + 34*m^2 + 225)*sin(4*x) + 2*(m^4 + 34*m^2 + 225)*sin(2*x))*sin(6*
x) + 225), x) + (3*(m^2 + 25)*cos(3*x)*e^(m*x) - 6*(m^2 - 15)*cos(x)*e^(m*x) + (m^3 + 25*m)*e^(m*x)*sin(3*x) +
48*m*e^(m*x)*sin(x))*sin(6*x) + 3*(3*(m^2 + 25)*cos(3*x)*e^(m*x) - 6*(m^2 - 15)*cos(x)*e^(m*x) + (m^3 + 25*m)
*e^(m*x)*sin(3*x) + 48*m*e^(m*x)*sin(x))*sin(4*x) - 3*(3*(m^2 + 25)*cos(2*x)*e^(m*x) - (m^3 + 25*m)*e^(m*x)*si
n(2*x) + (m^2 + 25)*e^(m*x))*sin(3*x) - 18*((m^2 - 15)*cos(x)*e^(m*x) - 8*m*e^(m*x)*sin(x))*sin(2*x))/(m^4 + (
m^4 + 34*m^2 + 225)*cos(6*x)^2 + 9*(m^4 + 34*m^...
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Fricas [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(exp(m*x)/cos(x)^3,x, algorithm="fricas")
[Out]
integral(e^(m*x)/cos(x)^3, x)
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{m x}}{\cos ^{3}{\left (x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(exp(m*x)/cos(x)**3,x)
[Out]
Integral(exp(m*x)/cos(x)**3, x)
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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(exp(m*x)/cos(x)^3,x, algorithm="giac")
[Out]
integrate(e^(m*x)/cos(x)^3, x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\mathrm {e}}^{m\,x}}{{\cos \left (x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(exp(m*x)/cos(x)^3,x)
[Out]
int(exp(m*x)/cos(x)^3, x)
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