3.202 \(\int (a+b \sec (e+f x))^{5/2} (c+d \sec (e+f x)) \, dx\)
Optimal. Leaf size=442 \[ \frac{2 \sqrt{a+b} \left (a^2 b (45 c-23 d)+15 a^3 d-a b^2 (35 c-17 d)+b^3 (5 c-9 d)\right ) \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right ),\frac{a+b}{a-b}\right )}{15 b f}-\frac{2 (a-b) \sqrt{a+b} \left (23 a^2 d+35 a b c+9 b^2 d\right ) \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{15 b f}-\frac{2 a^2 c \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{f}+\frac{2 b (8 a d+5 b c) \tan (e+f x) \sqrt{a+b \sec (e+f x)}}{15 f}+\frac{2 b d \tan (e+f x) (a+b \sec (e+f x))^{3/2}}{5 f} \]
[Out]
(-2*(a - b)*Sqrt[a + b]*(35*a*b*c + 23*a^2*d + 9*b^2*d)*Cot[e + f*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[e + f*x]]
/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/
(15*b*f) + (2*Sqrt[a + b]*(a^2*b*(45*c - 23*d) - a*b^2*(35*c - 17*d) + b^3*(5*c - 9*d) + 15*a^3*d)*Cot[e + f*x
]*EllipticF[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)
]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(15*b*f) - (2*a^2*Sqrt[a + b]*c*Cot[e + f*x]*EllipticPi[(a + b)/a,
ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*
(1 + Sec[e + f*x]))/(a - b))])/f + (2*b*(5*b*c + 8*a*d)*Sqrt[a + b*Sec[e + f*x]]*Tan[e + f*x])/(15*f) + (2*b*d
*(a + b*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(5*f)
________________________________________________________________________________________
Rubi [A] time = 0.629278, antiderivative size = 442, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used =
{3918, 4056, 4058, 3921, 3784, 3832, 4004} \[ \frac{2 \sqrt{a+b} \left (a^2 b (45 c-23 d)+15 a^3 d-a b^2 (35 c-17 d)+b^3 (5 c-9 d)\right ) \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{15 b f}-\frac{2 (a-b) \sqrt{a+b} \left (23 a^2 d+35 a b c+9 b^2 d\right ) \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{15 b f}-\frac{2 a^2 c \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{f}+\frac{2 b (8 a d+5 b c) \tan (e+f x) \sqrt{a+b \sec (e+f x)}}{15 f}+\frac{2 b d \tan (e+f x) (a+b \sec (e+f x))^{3/2}}{5 f} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*Sec[e + f*x])^(5/2)*(c + d*Sec[e + f*x]),x]
[Out]
(-2*(a - b)*Sqrt[a + b]*(35*a*b*c + 23*a^2*d + 9*b^2*d)*Cot[e + f*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[e + f*x]]
/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/
(15*b*f) + (2*Sqrt[a + b]*(a^2*b*(45*c - 23*d) - a*b^2*(35*c - 17*d) + b^3*(5*c - 9*d) + 15*a^3*d)*Cot[e + f*x
]*EllipticF[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)
]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(15*b*f) - (2*a^2*Sqrt[a + b]*c*Cot[e + f*x]*EllipticPi[(a + b)/a,
ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*
(1 + Sec[e + f*x]))/(a - b))])/f + (2*b*(5*b*c + 8*a*d)*Sqrt[a + b*Sec[e + f*x]]*Tan[e + f*x])/(15*f) + (2*b*d
*(a + b*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(5*f)
Rule 3918
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)), x_Symbol] :> -Simp[(b*
d*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1))/(f*m), x] + Dist[1/m, Int[(a + b*Csc[e + f*x])^(m - 2)*Simp[a^2*c
*m + (b^2*d*(m - 1) + 2*a*b*c*m + a^2*d*m)*Csc[e + f*x] + b*(b*c*m + a*d*(2*m - 1))*Csc[e + f*x]^2, x], x], x]
/; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && GtQ[m, 1] && NeQ[a^2 - b^2, 0] && IntegerQ[2*m]
Rule 4056
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_.) +
(a_))^(m_.), x_Symbol] :> -Simp[(C*Cot[e + f*x]*(a + b*Csc[e + f*x])^m)/(f*(m + 1)), x] + Dist[1/(m + 1), Int
[(a + b*Csc[e + f*x])^(m - 1)*Simp[a*A*(m + 1) + ((A*b + a*B)*(m + 1) + b*C*m)*Csc[e + f*x] + (b*B*(m + 1) + a
*C*m)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && IGtQ[2*m, 0]
Rule 4058
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_
.) + (a_)], x_Symbol] :> Int[(A + (B - C)*Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]], x] + Dist[C, Int[(Csc[e + f*
x]*(1 + Csc[e + f*x]))/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0
]
Rule 3921
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[c, In
t[1/Sqrt[a + b*Csc[e + f*x]], x], x] + Dist[d, Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ[{a,
b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
Rule 3784
Int[1/Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[(2*Rt[a + b, 2]*Sqrt[(b*(1 - Csc[c + d*x])
)/(a + b)]*Sqrt[-((b*(1 + Csc[c + d*x]))/(a - b))]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Csc[c + d*x]]/Rt[a
+ b, 2]], (a + b)/(a - b)])/(a*d*Cot[c + d*x]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]
Rule 3832
Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[(-2*Rt[a + b, 2]*Sqr
t[(b*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Csc[e + f*x]))/(a - b))]*EllipticF[ArcSin[Sqrt[a + b*Csc[e +
f*x]]/Rt[a + b, 2]], (a + b)/(a - b)])/(b*f*Cot[e + f*x]), x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]
Rule 4004
Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)
], x_Symbol] :> Simp[(-2*(A*b - a*B)*Rt[a + (b*B)/A, 2]*Sqrt[(b*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Cs
c[e + f*x]))/(a - b))]*EllipticE[ArcSin[Sqrt[a + b*Csc[e + f*x]]/Rt[a + (b*B)/A, 2]], (a*A + b*B)/(a*A - b*B)]
)/(b^2*f*Cot[e + f*x]), x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[a^2 - b^2, 0] && EqQ[A^2 - B^2, 0]
Rubi steps
\begin{align*} \int (a+b \sec (e+f x))^{5/2} (c+d \sec (e+f x)) \, dx &=\frac{2 b d (a+b \sec (e+f x))^{3/2} \tan (e+f x)}{5 f}+\frac{2}{5} \int \sqrt{a+b \sec (e+f x)} \left (\frac{5 a^2 c}{2}+\frac{1}{2} \left (10 a b c+5 a^2 d+3 b^2 d\right ) \sec (e+f x)+\frac{1}{2} b (5 b c+8 a d) \sec ^2(e+f x)\right ) \, dx\\ &=\frac{2 b (5 b c+8 a d) \sqrt{a+b \sec (e+f x)} \tan (e+f x)}{15 f}+\frac{2 b d (a+b \sec (e+f x))^{3/2} \tan (e+f x)}{5 f}+\frac{4}{15} \int \frac{\frac{15 a^3 c}{4}+\frac{1}{4} \left (45 a^2 b c+5 b^3 c+15 a^3 d+17 a b^2 d\right ) \sec (e+f x)+\frac{1}{4} b \left (35 a b c+23 a^2 d+9 b^2 d\right ) \sec ^2(e+f x)}{\sqrt{a+b \sec (e+f x)}} \, dx\\ &=\frac{2 b (5 b c+8 a d) \sqrt{a+b \sec (e+f x)} \tan (e+f x)}{15 f}+\frac{2 b d (a+b \sec (e+f x))^{3/2} \tan (e+f x)}{5 f}+\frac{4}{15} \int \frac{\frac{15 a^3 c}{4}+\left (-\frac{1}{4} b \left (35 a b c+23 a^2 d+9 b^2 d\right )+\frac{1}{4} \left (45 a^2 b c+5 b^3 c+15 a^3 d+17 a b^2 d\right )\right ) \sec (e+f x)}{\sqrt{a+b \sec (e+f x)}} \, dx+\frac{1}{15} \left (b \left (35 a b c+23 a^2 d+9 b^2 d\right )\right ) \int \frac{\sec (e+f x) (1+\sec (e+f x))}{\sqrt{a+b \sec (e+f x)}} \, dx\\ &=-\frac{2 (a-b) \sqrt{a+b} \left (35 a b c+23 a^2 d+9 b^2 d\right ) \cot (e+f x) E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (1+\sec (e+f x))}{a-b}}}{15 b f}+\frac{2 b (5 b c+8 a d) \sqrt{a+b \sec (e+f x)} \tan (e+f x)}{15 f}+\frac{2 b d (a+b \sec (e+f x))^{3/2} \tan (e+f x)}{5 f}+\left (a^3 c\right ) \int \frac{1}{\sqrt{a+b \sec (e+f x)}} \, dx+\frac{1}{15} \left (a^2 b (45 c-23 d)-a b^2 (35 c-17 d)+b^3 (5 c-9 d)+15 a^3 d\right ) \int \frac{\sec (e+f x)}{\sqrt{a+b \sec (e+f x)}} \, dx\\ &=-\frac{2 (a-b) \sqrt{a+b} \left (35 a b c+23 a^2 d+9 b^2 d\right ) \cot (e+f x) E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (1+\sec (e+f x))}{a-b}}}{15 b f}+\frac{2 \sqrt{a+b} \left (a^2 b (45 c-23 d)-a b^2 (35 c-17 d)+b^3 (5 c-9 d)+15 a^3 d\right ) \cot (e+f x) F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (1+\sec (e+f x))}{a-b}}}{15 b f}-\frac{2 a^2 \sqrt{a+b} c \cot (e+f x) \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (1+\sec (e+f x))}{a-b}}}{f}+\frac{2 b (5 b c+8 a d) \sqrt{a+b \sec (e+f x)} \tan (e+f x)}{15 f}+\frac{2 b d (a+b \sec (e+f x))^{3/2} \tan (e+f x)}{5 f}\\ \end{align*}
Mathematica [B] time = 25.2858, size = 7168, normalized size = 16.22 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[(a + b*Sec[e + f*x])^(5/2)*(c + d*Sec[e + f*x]),x]
[Out]
Result too large to show
________________________________________________________________________________________
Maple [B] time = 0.687, size = 3285, normalized size = 7.4 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*sec(f*x+e))^(5/2)*(c+d*sec(f*x+e)),x)
[Out]
2/15/f*(1+cos(f*x+e))^2*(1/cos(f*x+e)*(a*cos(f*x+e)+b))^(1/2)*(-1+cos(f*x+e))^2*(-23*cos(f*x+e)^4*a^3*d+23*sin
(f*x+e)*cos(f*x+e)^2*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*Ellipti
cE((-1+cos(f*x+e))/sin(f*x+e),((a-b)/(a+b))^(1/2))*a^2*b*d+35*sin(f*x+e)*cos(f*x+e)^2*(cos(f*x+e)/(1+cos(f*x+e
)))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*EllipticE((-1+cos(f*x+e))/sin(f*x+e),((a-b)/(a+b))^(
1/2))*a*b^2*c+9*sin(f*x+e)*cos(f*x+e)^2*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x
+e)))^(1/2)*EllipticE((-1+cos(f*x+e))/sin(f*x+e),((a-b)/(a+b))^(1/2))*a*b^2*d-17*sin(f*x+e)*cos(f*x+e)^3*(cos(
f*x+e)/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*EllipticF((-1+cos(f*x+e))/sin(f*x
+e),((a-b)/(a+b))^(1/2))*a*b^2*d+35*sin(f*x+e)*cos(f*x+e)^3*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(
f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*EllipticE((-1+cos(f*x+e))/sin(f*x+e),((a-b)/(a+b))^(1/2))*a^2*b*c-35*cos(f*x+e
)^4*a^2*b*c-11*cos(f*x+e)^4*a^2*b*d-5*cos(f*x+e)^4*a*b^2*c-9*cos(f*x+e)^4*a*b^2*d+35*cos(f*x+e)^3*a^2*b*c-23*c
os(f*x+e)^3*a^2*b*d-35*cos(f*x+e)^3*a*b^2*c-5*cos(f*x+e)^3*a*b^2*d+34*cos(f*x+e)^2*a^2*b*d-9*sin(f*x+e)*cos(f*
x+e)^3*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*EllipticF((-1+cos(f*x
+e))/sin(f*x+e),((a-b)/(a+b))^(1/2))*b^3*d+23*sin(f*x+e)*cos(f*x+e)^3*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(1/(a+
b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*EllipticE((-1+cos(f*x+e))/sin(f*x+e),((a-b)/(a+b))^(1/2))*a^3*d+9*si
n(f*x+e)*cos(f*x+e)^3*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*Ellipt
icE((-1+cos(f*x+e))/sin(f*x+e),((a-b)/(a+b))^(1/2))*b^3*d-30*sin(f*x+e)*cos(f*x+e)^3*(cos(f*x+e)/(1+cos(f*x+e)
))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*EllipticPi((-1+cos(f*x+e))/sin(f*x+e),-1,((a-b)/(a+b)
)^(1/2))*a^3*c+15*sin(f*x+e)*cos(f*x+e)^2*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f
*x+e)))^(1/2)*EllipticF((-1+cos(f*x+e))/sin(f*x+e),((a-b)/(a+b))^(1/2))*a^3*c-15*sin(f*x+e)*cos(f*x+e)^2*(cos(
f*x+e)/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*EllipticF((-1+cos(f*x+e))/sin(f*x
+e),((a-b)/(a+b))^(1/2))*a^3*d-5*sin(f*x+e)*cos(f*x+e)^2*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x
+e)+b)/(1+cos(f*x+e)))^(1/2)*EllipticF((-1+cos(f*x+e))/sin(f*x+e),((a-b)/(a+b))^(1/2))*b^3*c-9*sin(f*x+e)*cos(
f*x+e)^2*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*EllipticF((-1+cos(f
*x+e))/sin(f*x+e),((a-b)/(a+b))^(1/2))*b^3*d+23*sin(f*x+e)*cos(f*x+e)^2*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(1/(
a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*EllipticE((-1+cos(f*x+e))/sin(f*x+e),((a-b)/(a+b))^(1/2))*a^3*d+9*
sin(f*x+e)*cos(f*x+e)^2*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*Elli
pticE((-1+cos(f*x+e))/sin(f*x+e),((a-b)/(a+b))^(1/2))*b^3*d-30*sin(f*x+e)*cos(f*x+e)^2*(cos(f*x+e)/(1+cos(f*x+
e)))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*EllipticPi((-1+cos(f*x+e))/sin(f*x+e),-1,((a-b)/(a+
b))^(1/2))*a^3*c+15*sin(f*x+e)*cos(f*x+e)^3*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos
(f*x+e)))^(1/2)*EllipticF((-1+cos(f*x+e))/sin(f*x+e),((a-b)/(a+b))^(1/2))*a^3*c-15*sin(f*x+e)*cos(f*x+e)^3*(co
s(f*x+e)/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*EllipticF((-1+cos(f*x+e))/sin(f
*x+e),((a-b)/(a+b))^(1/2))*a^3*d-5*sin(f*x+e)*cos(f*x+e)^3*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f
*x+e)+b)/(1+cos(f*x+e)))^(1/2)*EllipticF((-1+cos(f*x+e))/sin(f*x+e),((a-b)/(a+b))^(1/2))*b^3*c-23*sin(f*x+e)*c
os(f*x+e)^3*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*EllipticF((-1+co
s(f*x+e))/sin(f*x+e),((a-b)/(a+b))^(1/2))*a^2*b*d-35*sin(f*x+e)*cos(f*x+e)^3*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)
*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*EllipticF((-1+cos(f*x+e))/sin(f*x+e),((a-b)/(a+b))^(1/2))*a*b
^2*c+23*cos(f*x+e)^3*a^3*d-9*cos(f*x+e)^3*b^3*d+6*cos(f*x+e)^2*b^3*d-5*cos(f*x+e)^3*b^3*c+5*cos(f*x+e)*b^3*c+4
0*cos(f*x+e)^2*a*b^2*c+14*cos(f*x+e)*a*b^2*d-45*sin(f*x+e)*cos(f*x+e)^2*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(1/(
a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*EllipticF((-1+cos(f*x+e))/sin(f*x+e),((a-b)/(a+b))^(1/2))*a^2*b*c-
23*sin(f*x+e)*cos(f*x+e)^2*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*E
llipticF((-1+cos(f*x+e))/sin(f*x+e),((a-b)/(a+b))^(1/2))*a^2*b*d-35*sin(f*x+e)*cos(f*x+e)^2*(cos(f*x+e)/(1+cos
(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*EllipticF((-1+cos(f*x+e))/sin(f*x+e),((a-b)/(a
+b))^(1/2))*a*b^2*c-17*sin(f*x+e)*cos(f*x+e)^2*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(1+
cos(f*x+e)))^(1/2)*EllipticF((-1+cos(f*x+e))/sin(f*x+e),((a-b)/(a+b))^(1/2))*a*b^2*d+35*sin(f*x+e)*cos(f*x+e)^
2*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*EllipticE((-1+cos(f*x+e))/
sin(f*x+e),((a-b)/(a+b))^(1/2))*a^2*b*c-45*sin(f*x+e)*cos(f*x+e)^3*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*
(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*EllipticF((-1+cos(f*x+e))/sin(f*x+e),((a-b)/(a+b))^(1/2))*a^2*b*c+3*b^3
*d+23*sin(f*x+e)*cos(f*x+e)^3*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2
)*EllipticE((-1+cos(f*x+e))/sin(f*x+e),((a-b)/(a+b))^(1/2))*a^2*b*d+35*sin(f*x+e)*cos(f*x+e)^3*(cos(f*x+e)/(1+
cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*EllipticE((-1+cos(f*x+e))/sin(f*x+e),((a-b)
/(a+b))^(1/2))*a*b^2*c+9*sin(f*x+e)*cos(f*x+e)^3*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(
1+cos(f*x+e)))^(1/2)*EllipticE((-1+cos(f*x+e))/sin(f*x+e),((a-b)/(a+b))^(1/2))*a*b^2*d)/(a*cos(f*x+e)+b)/cos(f
*x+e)^2/sin(f*x+e)^5
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac{5}{2}}{\left (d \sec \left (f x + e\right ) + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*sec(f*x+e))^(5/2)*(c+d*sec(f*x+e)),x, algorithm="maxima")
[Out]
integrate((b*sec(f*x + e) + a)^(5/2)*(d*sec(f*x + e) + c), x)
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b^{2} d \sec \left (f x + e\right )^{3} + a^{2} c +{\left (b^{2} c + 2 \, a b d\right )} \sec \left (f x + e\right )^{2} +{\left (2 \, a b c + a^{2} d\right )} \sec \left (f x + e\right )\right )} \sqrt{b \sec \left (f x + e\right ) + a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*sec(f*x+e))^(5/2)*(c+d*sec(f*x+e)),x, algorithm="fricas")
[Out]
integral((b^2*d*sec(f*x + e)^3 + a^2*c + (b^2*c + 2*a*b*d)*sec(f*x + e)^2 + (2*a*b*c + a^2*d)*sec(f*x + e))*sq
rt(b*sec(f*x + e) + a), x)
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*sec(f*x+e))**(5/2)*(c+d*sec(f*x+e)),x)
[Out]
Timed out
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac{5}{2}}{\left (d \sec \left (f x + e\right ) + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*sec(f*x+e))^(5/2)*(c+d*sec(f*x+e)),x, algorithm="giac")
[Out]
integrate((b*sec(f*x + e) + a)^(5/2)*(d*sec(f*x + e) + c), x)