3.1.17 \(\int F^{c (a+b x)} \sec ^4(d+e x) \, dx\) [17]
Optimal. Leaf size=143 \[ -\frac {2 e^{2 i (d+e x)} F^{c (a+b x)} \, _2F_1\left (2,1-\frac {i b c \log (F)}{2 e};2-\frac {i b c \log (F)}{2 e};-e^{2 i (d+e x)}\right ) (2 i e-b c \log (F))}{3 e^2}-\frac {b c F^{c (a+b x)} \log (F) \sec ^2(d+e x)}{6 e^2}+\frac {F^{c (a+b x)} \sec ^2(d+e x) \tan (d+e x)}{3 e} \]
[Out]
-2/3*exp(2*I*(e*x+d))*F^(c*(b*x+a))*hypergeom([2, 1-1/2*I*b*c*ln(F)/e],[2-1/2*I*b*c*ln(F)/e],-exp(2*I*(e*x+d))
)*(2*I*e-b*c*ln(F))/e^2-1/6*b*c*F^(c*(b*x+a))*ln(F)*sec(e*x+d)^2/e^2+1/3*F^(c*(b*x+a))*sec(e*x+d)^2*tan(e*x+d)
/e
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Rubi [A]
time = 0.04, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {4533, 4536}
\begin {gather*} -\frac {2 e^{2 i (d+e x)} F^{c (a+b x)} (-b c \log (F)+2 i e) \, _2F_1\left (2,1-\frac {i b c \log (F)}{2 e};2-\frac {i b c \log (F)}{2 e};-e^{2 i (d+e x)}\right )}{3 e^2}-\frac {b c \log (F) \sec ^2(d+e x) F^{c (a+b x)}}{6 e^2}+\frac {\tan (d+e x) \sec ^2(d+e x) F^{c (a+b x)}}{3 e} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[F^(c*(a + b*x))*Sec[d + e*x]^4,x]
[Out]
(-2*E^((2*I)*(d + e*x))*F^(c*(a + b*x))*Hypergeometric2F1[2, 1 - ((I/2)*b*c*Log[F])/e, 2 - ((I/2)*b*c*Log[F])/
e, -E^((2*I)*(d + e*x))]*((2*I)*e - b*c*Log[F]))/(3*e^2) - (b*c*F^(c*(a + b*x))*Log[F]*Sec[d + e*x]^2)/(6*e^2)
+ (F^(c*(a + b*x))*Sec[d + e*x]^2*Tan[d + e*x])/(3*e)
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 F^{c (a+b x)} \sec ^4(d+e x) \, dx &=-\frac {b c F^{c (a+b x)} \log (F) \sec ^2(d+e x)}{6 e^2}+\frac {F^{c (a+b x)} \sec ^2(d+e x) \tan (d+e x)}{3 e}+\frac {1}{6} \left (4+\frac {b^2 c^2 \log ^2(F)}{e^2}\right ) \int F^{c (a+b x)} \sec ^2(d+e x) \, dx\\ &=-\frac {2 e^{2 i (d+e x)} F^{c (a+b x)} \, _2F_1\left (2,1-\frac {i b c \log (F)}{2 e};2-\frac {i b c \log (F)}{2 e};-e^{2 i (d+e x)}\right ) (2 i e-b c \log (F))}{3 e^2}-\frac {b c F^{c (a+b x)} \log (F) \sec ^2(d+e x)}{6 e^2}+\frac {F^{c (a+b x)} \sec ^2(d+e x) \tan (d+e x)}{3 e}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 111, normalized size = 0.78 \begin {gather*} \frac {F^{c (a+b x)} \left (4 e^{2 i (d+e x)} \, _2F_1\left (2,1-\frac {i b c \log (F)}{2 e};2-\frac {i b c \log (F)}{2 e};-e^{2 i (d+e x)}\right ) (-2 i e+b c \log (F))+\sec ^2(d+e x) (-b c \log (F)+2 e \tan (d+e x))\right )}{6 e^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[F^(c*(a + b*x))*Sec[d + e*x]^4,x]
[Out]
(F^(c*(a + b*x))*(4*E^((2*I)*(d + e*x))*Hypergeometric2F1[2, 1 - ((I/2)*b*c*Log[F])/e, 2 - ((I/2)*b*c*Log[F])/
e, -E^((2*I)*(d + e*x))]*((-2*I)*e + b*c*Log[F]) + Sec[d + e*x]^2*(-(b*c*Log[F]) + 2*e*Tan[d + e*x])))/(6*e^2)
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Maple [F]
time = 0.21, size = 0, normalized size = 0.00 \[\int F^{c \left (b x +a \right )} \left (\sec ^{4}\left (e x +d \right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(F^(c*(b*x+a))*sec(e*x+d)^4,x)
[Out]
int(F^(c*(b*x+a))*sec(e*x+d)^4,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(F^(c*(b*x+a))*sec(e*x+d)^4,x, algorithm="maxima")
[Out]
16*(6*(F^(a*c)*b^5*c^5*log(F)^5 + 100*F^(a*c)*b^3*c^3*e^2*log(F)^3 + 2304*F^(a*c)*b*c*e^4*log(F))*F^(b*c*x)*co
s(4*x*e + 4*d)^2 + 320*(F^(a*c)*b^3*c^3*e^2*log(F)^3 + 64*F^(a*c)*b*c*e^4*log(F))*F^(b*c*x)*cos(2*x*e + 2*d)^2
+ 6*(F^(a*c)*b^5*c^5*log(F)^5 + 100*F^(a*c)*b^3*c^3*e^2*log(F)^3 + 2304*F^(a*c)*b*c*e^4*log(F))*F^(b*c*x)*sin
(4*x*e + 4*d)^2 + 320*(F^(a*c)*b^3*c^3*e^2*log(F)^3 + 64*F^(a*c)*b*c*e^4*log(F))*F^(b*c*x)*sin(2*x*e + 2*d)^2
- 560*(F^(a*c)*b^3*c^3*e^2*log(F)^3 - 32*F^(a*c)*b*c*e^4*log(F))*F^(b*c*x)*cos(2*x*e + 2*d) + 40*(F^(a*c)*b^4*
c^4*e*log(F)^4 - 104*F^(a*c)*b^2*c^2*e^3*log(F)^2)*F^(b*c*x)*sin(2*x*e + 2*d) - 160*(F^(a*c)*b^3*c^3*e^2*log(F
)^3 - 20*F^(a*c)*b*c*e^4*log(F))*F^(b*c*x) + ((F^(a*c)*b^5*c^5*log(F)^5 + 100*F^(a*c)*b^3*c^3*e^2*log(F)^3 + 2
304*F^(a*c)*b*c*e^4*log(F))*F^(b*c*x)*cos(4*x*e + 4*d) + 80*(F^(a*c)*b^3*c^3*e^2*log(F)^3 + 64*F^(a*c)*b*c*e^4
*log(F))*F^(b*c*x)*cos(2*x*e + 2*d) - 4*(F^(a*c)*b^4*c^4*e*log(F)^4 + 100*F^(a*c)*b^2*c^2*e^3*log(F)^2 + 2304*
F^(a*c)*e^5)*F^(b*c*x)*sin(4*x*e + 4*d) + 8*(F^(a*c)*b^4*c^4*e*log(F)^4 + 40*F^(a*c)*b^2*c^2*e^3*log(F)^2 - 15
36*F^(a*c)*e^5)*F^(b*c*x)*sin(2*x*e + 2*d) - 160*(F^(a*c)*b^3*c^3*e^2*log(F)^3 - 20*F^(a*c)*b*c*e^4*log(F))*F^
(b*c*x))*cos(8*x*e + 8*d) + 4*((F^(a*c)*b^5*c^5*log(F)^5 + 100*F^(a*c)*b^3*c^3*e^2*log(F)^3 + 2304*F^(a*c)*b*c
*e^4*log(F))*F^(b*c*x)*cos(4*x*e + 4*d) + 80*(F^(a*c)*b^3*c^3*e^2*log(F)^3 + 64*F^(a*c)*b*c*e^4*log(F))*F^(b*c
*x)*cos(2*x*e + 2*d) - 4*(F^(a*c)*b^4*c^4*e*log(F)^4 + 100*F^(a*c)*b^2*c^2*e^3*log(F)^2 + 2304*F^(a*c)*e^5)*F^
(b*c*x)*sin(4*x*e + 4*d) + 8*(F^(a*c)*b^4*c^4*e*log(F)^4 + 40*F^(a*c)*b^2*c^2*e^3*log(F)^2 - 1536*F^(a*c)*e^5)
*F^(b*c*x)*sin(2*x*e + 2*d) - 160*(F^(a*c)*b^3*c^3*e^2*log(F)^3 - 20*F^(a*c)*b*c*e^4*log(F))*F^(b*c*x))*cos(6*
x*e + 6*d) + (4*(F^(a*c)*b^5*c^5*log(F)^5 + 220*F^(a*c)*b^3*c^3*e^2*log(F)^3 + 9984*F^(a*c)*b*c*e^4*log(F))*F^
(b*c*x)*cos(2*x*e + 2*d) + 64*(F^(a*c)*b^4*c^4*e*log(F)^4 + 55*F^(a*c)*b^2*c^2*e^3*log(F)^2 - 576*F^(a*c)*e^5)
*F^(b*c*x)*sin(2*x*e + 2*d) + (F^(a*c)*b^5*c^5*log(F)^5 - 860*F^(a*c)*b^3*c^3*e^2*log(F)^3 + 21504*F^(a*c)*b*c
*e^4*log(F))*F^(b*c*x))*cos(4*x*e + 4*d) + 16*(F^(a*c)*b^7*c^7*e^2*log(F)^7 + 116*F^(a*c)*b^5*c^5*e^4*log(F)^5
+ 3904*F^(a*c)*b^3*c^3*e^6*log(F)^3 + 36864*F^(a*c)*b*c*e^8*log(F) + (F^(a*c)*b^7*c^7*e^2*log(F)^7 + 116*F^(a
*c)*b^5*c^5*e^4*log(F)^5 + 3904*F^(a*c)*b^3*c^3*e^6*log(F)^3 + 36864*F^(a*c)*b*c*e^8*log(F))*cos(8*x*e + 8*d)^
2 + 16*(F^(a*c)*b^7*c^7*e^2*log(F)^7 + 116*F^(a*c)*b^5*c^5*e^4*log(F)^5 + 3904*F^(a*c)*b^3*c^3*e^6*log(F)^3 +
36864*F^(a*c)*b*c*e^8*log(F))*cos(6*x*e + 6*d)^2 + 36*(F^(a*c)*b^7*c^7*e^2*log(F)^7 + 116*F^(a*c)*b^5*c^5*e^4*
log(F)^5 + 3904*F^(a*c)*b^3*c^3*e^6*log(F)^3 + 36864*F^(a*c)*b*c*e^8*log(F))*cos(4*x*e + 4*d)^2 + 16*(F^(a*c)*
b^7*c^7*e^2*log(F)^7 + 116*F^(a*c)*b^5*c^5*e^4*log(F)^5 + 3904*F^(a*c)*b^3*c^3*e^6*log(F)^3 + 36864*F^(a*c)*b*
c*e^8*log(F))*cos(2*x*e + 2*d)^2 + (F^(a*c)*b^7*c^7*e^2*log(F)^7 + 116*F^(a*c)*b^5*c^5*e^4*log(F)^5 + 3904*F^(
a*c)*b^3*c^3*e^6*log(F)^3 + 36864*F^(a*c)*b*c*e^8*log(F))*sin(8*x*e + 8*d)^2 + 16*(F^(a*c)*b^7*c^7*e^2*log(F)^
7 + 116*F^(a*c)*b^5*c^5*e^4*log(F)^5 + 3904*F^(a*c)*b^3*c^3*e^6*log(F)^3 + 36864*F^(a*c)*b*c*e^8*log(F))*sin(6
*x*e + 6*d)^2 + 36*(F^(a*c)*b^7*c^7*e^2*log(F)^7 + 116*F^(a*c)*b^5*c^5*e^4*log(F)^5 + 3904*F^(a*c)*b^3*c^3*e^6
*log(F)^3 + 36864*F^(a*c)*b*c*e^8*log(F))*sin(4*x*e + 4*d)^2 + 48*(F^(a*c)*b^7*c^7*e^2*log(F)^7 + 116*F^(a*c)*
b^5*c^5*e^4*log(F)^5 + 3904*F^(a*c)*b^3*c^3*e^6*log(F)^3 + 36864*F^(a*c)*b*c*e^8*log(F))*sin(4*x*e + 4*d)*sin(
2*x*e + 2*d) + 16*(F^(a*c)*b^7*c^7*e^2*log(F)^7 + 116*F^(a*c)*b^5*c^5*e^4*log(F)^5 + 3904*F^(a*c)*b^3*c^3*e^6*
log(F)^3 + 36864*F^(a*c)*b*c*e^8*log(F))*sin(2*x*e + 2*d)^2 + 2*(F^(a*c)*b^7*c^7*e^2*log(F)^7 + 116*F^(a*c)*b^
5*c^5*e^4*log(F)^5 + 3904*F^(a*c)*b^3*c^3*e^6*log(F)^3 + 36864*F^(a*c)*b*c*e^8*log(F) + 4*(F^(a*c)*b^7*c^7*e^2
*log(F)^7 + 116*F^(a*c)*b^5*c^5*e^4*log(F)^5 + 3904*F^(a*c)*b^3*c^3*e^6*log(F)^3 + 36864*F^(a*c)*b*c*e^8*log(F
))*cos(6*x*e + 6*d) + 6*(F^(a*c)*b^7*c^7*e^2*log(F)^7 + 116*F^(a*c)*b^5*c^5*e^4*log(F)^5 + 3904*F^(a*c)*b^3*c^
3*e^6*log(F)^3 + 36864*F^(a*c)*b*c*e^8*log(F))*cos(4*x*e + 4*d) + 4*(F^(a*c)*b^7*c^7*e^2*log(F)^7 + 116*F^(a*c
)*b^5*c^5*e^4*log(F)^5 + 3904*F^(a*c)*b^3*c^3*e^6*log(F)^3 + 36864*F^(a*c)*b*c*e^8*log(F))*cos(2*x*e + 2*d))*c
os(8*x*e + 8*d) + 8*(F^(a*c)*b^7*c^7*e^2*log(F)^7 + 116*F^(a*c)*b^5*c^5*e^4*log(F)^5 + 3904*F^(a*c)*b^3*c^3*e^
6*log(F)^3 + 36864*F^(a*c)*b*c*e^8*log(F) + 6*(F^(a*c)*b^7*c^7*e^2*log(F)^7 + 116*F^(a*c)*b^5*c^5*e^4*log(F)^5
+ 3904*F^(a*c)*b^3*c^3*e^6*log(F)^3 + 36864*F^(a*c)*b*c*e^8*log(F))*cos(4*x*e + 4*d) + 4*(F^(a*c)*b^7*c^7*e^2
*log(F)^7 + 116*F^(a*c)*b^5*c^5*e^4*log(F)^5 + 3904*F^(a*c)*b^3*c^3*e^6*log(F)^3 + 36864*F^(a*c)*b*c*e^8*log(F
))*cos(2*x*e + 2*d))*cos(6*x*e + 6*d) + 12*(F^(a*c)*b^7*c^7*e^2*log(F)^7 + 116*F^(a*c)*b^5*c^5*e^4*log(F)^5 +
3904*F^(a*c)*b^3*c^3*e^6*log(F)^3 + 36864*F^(a*c)*b*c*e^8*log(F) + 4*(F^(a*c)*b^7*c^7*e^2*log(F)^7 + 116*F^(a*
c)*b^5*c^5*e^4*log(F)^5 + 3904*F^(a*c)*b^3*c^3*...
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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(F^(c*(b*x+a))*sec(e*x+d)^4,x, algorithm="fricas")
[Out]
integral(F^(b*c*x + a*c)*sec(x*e + d)^4, x)
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(F**(c*(b*x+a))*sec(e*x+d)**4,x)
[Out]
Timed out
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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(F^(c*(b*x+a))*sec(e*x+d)^4,x, algorithm="giac")
[Out]
integrate(F^((b*x + a)*c)*sec(e*x + d)^4, x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {F^{c\,\left (a+b\,x\right )}}{{\cos \left (d+e\,x\right )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(F^(c*(a + b*x))/cos(d + e*x)^4,x)
[Out]
int(F^(c*(a + b*x))/cos(d + e*x)^4, x)
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