3.980 \(\int \frac{a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{7/2}} \, dx\)

Optimal. Leaf size=519 \[ \frac{2 \left (-2 a^2 b (3 B+C)+9 a^3 C+a b^2 (B-3 C)+3 b^3 B\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right ),\frac{a+b}{a-b}\right )}{3 a^2 d \sqrt{a+b} \left (a^2-b^2\right )}+\frac{2 b^2 \left (7 a^2 b B-11 a^3 C+3 a b^2 C-3 b^3 B\right ) \tan (c+d x)}{3 a^2 d \left (a^2-b^2\right )^2 \sqrt{a+b \sec (c+d x)}}+\frac{2 b^2 (b B-2 a C) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}+\frac{2 \left (7 a^2 b B-11 a^3 C+3 a b^2 C-3 b^3 B\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{3 a^2 d (a-b) (a+b)^{3/2}}-\frac{2 \sqrt{a+b} (b B-a C) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{a^3 d} \]

[Out]

(2*(7*a^2*b*B - 3*b^3*B - 11*a^3*C + 3*a*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a
+ b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*a^2*(
a - b)*(a + b)^(3/2)*d) + (2*(3*b^3*B + a*b^2*(B - 3*C) + 9*a^3*C - 2*a^2*b*(3*B + C))*Cot[c + d*x]*EllipticF[
ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*
(1 + Sec[c + d*x]))/(a - b))])/(3*a^2*Sqrt[a + b]*(a^2 - b^2)*d) - (2*Sqrt[a + b]*(b*B - a*C)*Cot[c + d*x]*Ell
ipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/
(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^3*d) + (2*b^2*(b*B - 2*a*C)*Tan[c + d*x])/(3*a*(a^2 - b^2
)*d*(a + b*Sec[c + d*x])^(3/2)) + (2*b^2*(7*a^2*b*B - 3*b^3*B - 11*a^3*C + 3*a*b^2*C)*Tan[c + d*x])/(3*a^2*(a^
2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])

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Rubi [A]  time = 0.999887, antiderivative size = 519, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 50, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {24, 3923, 4060, 4058, 3921, 3784, 3832, 4004} \[ \frac{2 b^2 \left (7 a^2 b B-11 a^3 C+3 a b^2 C-3 b^3 B\right ) \tan (c+d x)}{3 a^2 d \left (a^2-b^2\right )^2 \sqrt{a+b \sec (c+d x)}}+\frac{2 b^2 (b B-2 a C) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}+\frac{2 \left (-2 a^2 b (3 B+C)+9 a^3 C+a b^2 (B-3 C)+3 b^3 B\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{3 a^2 d \sqrt{a+b} \left (a^2-b^2\right )}+\frac{2 \left (7 a^2 b B-11 a^3 C+3 a b^2 C-3 b^3 B\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{3 a^2 d (a-b) (a+b)^{3/2}}-\frac{2 \sqrt{a+b} (b B-a C) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{a^3 d} \]

Antiderivative was successfully verified.

[In]

Int[(a*b*B - a^2*C + b^2*B*Sec[c + d*x] + b^2*C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(7/2),x]

[Out]

(2*(7*a^2*b*B - 3*b^3*B - 11*a^3*C + 3*a*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a
+ b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*a^2*(
a - b)*(a + b)^(3/2)*d) + (2*(3*b^3*B + a*b^2*(B - 3*C) + 9*a^3*C - 2*a^2*b*(3*B + C))*Cot[c + d*x]*EllipticF[
ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*
(1 + Sec[c + d*x]))/(a - b))])/(3*a^2*Sqrt[a + b]*(a^2 - b^2)*d) - (2*Sqrt[a + b]*(b*B - a*C)*Cot[c + d*x]*Ell
ipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/
(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^3*d) + (2*b^2*(b*B - 2*a*C)*Tan[c + d*x])/(3*a*(a^2 - b^2
)*d*(a + b*Sec[c + d*x])^(3/2)) + (2*b^2*(7*a^2*b*B - 3*b^3*B - 11*a^3*C + 3*a*b^2*C)*Tan[c + d*x])/(3*a^2*(a^
2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])

Rule 24

Int[(u_.)*((a_) + (b_.)*(v_))^(m_)*((A_.) + (B_.)*(v_) + (C_.)*(v_)^2), x_Symbol] :> Dist[1/b^2, Int[u*(a + b*
v)^(m + 1)*Simp[b*B - a*C + b*C*v, x], x], x] /; FreeQ[{a, b, A, B, C}, x] && EqQ[A*b^2 - a*b*B + a^2*C, 0] &&
LeQ[m, -1]

Rule 3923

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)), x_Symbol] :> Simp[(b*(
b*c - a*d)*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1))/(a*f*(m + 1)*(a^2 - b^2)), x] + Dist[1/(a*(m + 1)*(a^2 -
b^2)), Int[(a + b*Csc[e + f*x])^(m + 1)*Simp[c*(a^2 - b^2)*(m + 1) - (a*(b*c - a*d)*(m + 1))*Csc[e + f*x] + b
*(b*c - a*d)*(m + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && LtQ[m,
-1] && NeQ[a^2 - b^2, 0] && IntegerQ[2*m]

Rule 4060

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_.) +
(a_))^(m_), x_Symbol] :> Simp[((A*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1))/(a*f*(m + 1
)*(a^2 - b^2)), x] + Dist[1/(a*(m + 1)*(a^2 - b^2)), Int[(a + b*Csc[e + f*x])^(m + 1)*Simp[A*(a^2 - b^2)*(m +
1) - a*(A*b - a*B + b*C)*(m + 1)*Csc[e + f*x] + (A*b^2 - a*b*B + a^2*C)*(m + 2)*Csc[e + f*x]^2, x], x], x] /;
FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]

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 \frac{a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{7/2}} \, dx &=\frac{\int \frac{b^2 (b B-a C)+b^3 C \sec (c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx}{b^2}\\ &=\frac{2 b^2 (b B-2 a C) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}-\frac{2 \int \frac{-\frac{3}{2} b^2 \left (a^2-b^2\right ) (b B-a C)+\frac{3}{2} a b^3 (b B-2 a C) \sec (c+d x)-\frac{1}{2} b^4 (b B-2 a C) \sec ^2(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx}{3 a b^2 \left (a^2-b^2\right )}\\ &=\frac{2 b^2 (b B-2 a C) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}+\frac{2 b^2 \left (7 a^2 b B-3 b^3 B-11 a^3 C+3 a b^2 C\right ) \tan (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sqrt{a+b \sec (c+d x)}}+\frac{4 \int \frac{\frac{3}{4} b^2 \left (a^2-b^2\right )^2 (b B-a C)-\frac{1}{4} a b^3 \left (6 a^2 b B-2 b^3 B-9 a^3 C+a b^2 C\right ) \sec (c+d x)-\frac{1}{4} b^4 \left (7 a^2 b B-3 b^3 B-11 a^3 C+3 a b^2 C\right ) \sec ^2(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx}{3 a^2 b^2 \left (a^2-b^2\right )^2}\\ &=\frac{2 b^2 (b B-2 a C) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}+\frac{2 b^2 \left (7 a^2 b B-3 b^3 B-11 a^3 C+3 a b^2 C\right ) \tan (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sqrt{a+b \sec (c+d x)}}+\frac{4 \int \frac{\frac{3}{4} b^2 \left (a^2-b^2\right )^2 (b B-a C)+\left (-\frac{1}{4} a b^3 \left (6 a^2 b B-2 b^3 B-9 a^3 C+a b^2 C\right )+\frac{1}{4} b^4 \left (7 a^2 b B-3 b^3 B-11 a^3 C+3 a b^2 C\right )\right ) \sec (c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx}{3 a^2 b^2 \left (a^2-b^2\right )^2}-\frac{\left (b^2 \left (7 a^2 b B-3 b^3 B-11 a^3 C+3 a b^2 C\right )\right ) \int \frac{\sec (c+d x) (1+\sec (c+d x))}{\sqrt{a+b \sec (c+d x)}} \, dx}{3 a^2 \left (a^2-b^2\right )^2}\\ &=\frac{2 \left (7 a^2 b B-3 b^3 B-11 a^3 C+3 a b^2 C\right ) \cot (c+d x) E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (1+\sec (c+d x))}{a-b}}}{3 a^2 (a-b) (a+b)^{3/2} d}+\frac{2 b^2 (b B-2 a C) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}+\frac{2 b^2 \left (7 a^2 b B-3 b^3 B-11 a^3 C+3 a b^2 C\right ) \tan (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sqrt{a+b \sec (c+d x)}}+\frac{(b B-a C) \int \frac{1}{\sqrt{a+b \sec (c+d x)}} \, dx}{a^2}+\frac{\left (b \left (3 b^3 B+a b^2 (B-3 C)+9 a^3 C-2 a^2 b (3 B+C)\right )\right ) \int \frac{\sec (c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx}{3 a^2 (a-b) (a+b)^2}\\ &=\frac{2 \left (7 a^2 b B-3 b^3 B-11 a^3 C+3 a b^2 C\right ) \cot (c+d x) E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (1+\sec (c+d x))}{a-b}}}{3 a^2 (a-b) (a+b)^{3/2} d}+\frac{2 \left (3 b^3 B+a b^2 (B-3 C)+9 a^3 C-2 a^2 b (3 B+C)\right ) \cot (c+d x) F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (1+\sec (c+d x))}{a-b}}}{3 a^2 (a-b) (a+b)^{3/2} d}-\frac{2 \sqrt{a+b} (b B-a C) \cot (c+d x) \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (1+\sec (c+d x))}{a-b}}}{a^3 d}+\frac{2 b^2 (b B-2 a C) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}+\frac{2 b^2 \left (7 a^2 b B-3 b^3 B-11 a^3 C+3 a b^2 C\right ) \tan (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sqrt{a+b \sec (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 14.9712, size = 814, normalized size = 1.57 \[ \frac{\sec ^2(c+d x) (b B-a C+b C \sec (c+d x)) \left (\frac{2 b \left (11 C a^3-7 b B a^2-3 b^2 C a+3 b^3 B\right ) \sin (c+d x)}{3 a^2 \left (b^2-a^2\right )^2}-\frac{2 \left (b^4 B \sin (c+d x)-2 a b^3 C \sin (c+d x)\right )}{3 a^2 \left (a^2-b^2\right ) (b+a \cos (c+d x))^2}-\frac{2 \left (4 B \sin (c+d x) b^5-5 a C \sin (c+d x) b^4-8 a^2 B \sin (c+d x) b^3+13 a^3 C \sin (c+d x) b^2\right )}{3 a^2 \left (a^2-b^2\right )^2 (b+a \cos (c+d x))}\right ) (b+a \cos (c+d x))^3}{d (b C-a \cos (c+d x) C+b B \cos (c+d x)) (a+b \sec (c+d x))^{5/2}}+\frac{2 (b B-a C+b C \sec (c+d x)) \left (-a b (a+b) \left (11 C a^3-7 b B a^2-3 b^2 C a+3 b^3 B\right ) E\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right )|\frac{a-b}{a+b}\right ) \sqrt{\frac{(b+a \cos (c+d x)) \sec ^2\left (\frac{1}{2} (c+d x)\right )}{a+b}} \sec ^2\left (\frac{1}{2} (c+d x)\right )-b (a+b) \left (-12 C a^4+b (9 B+C) a^3-2 b^2 (B-3 C) a^2-3 b^3 (2 B+C) a+3 b^4 B\right ) \text{EllipticF}\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right ),\frac{a-b}{a+b}\right ) \sqrt{\frac{(b+a \cos (c+d x)) \sec ^2\left (\frac{1}{2} (c+d x)\right )}{a+b}} \sec ^2\left (\frac{1}{2} (c+d x)\right )-3 (a-b)^2 (a+b)^2 (b B-a C) \left ((a-b) \text{EllipticF}\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right ),\frac{a-b}{a+b}\right )+2 a \Pi \left (-1;-\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right )|\frac{a-b}{a+b}\right )\right ) \sqrt{\frac{(b+a \cos (c+d x)) \sec ^2\left (\frac{1}{2} (c+d x)\right )}{a+b}} \sec ^2\left (\frac{1}{2} (c+d x)\right )-a b \left (11 C a^3-7 b B a^2-3 b^2 C a+3 b^3 B\right ) (b+a \cos (c+d x)) \left (\cos (c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right )\right )^{3/2} \sec (c+d x) \tan \left (\frac{1}{2} (c+d x)\right )\right ) (b+a \cos (c+d x))^2}{3 a^3 \left (a^2-b^2\right )^2 d (b C-a \cos (c+d x) C+b B \cos (c+d x)) \left (\cos (c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right )\right )^{3/2} (a+b \sec (c+d x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*b*B - a^2*C + b^2*B*Sec[c + d*x] + b^2*C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(7/2),x]

[Out]

((b + a*Cos[c + d*x])^3*Sec[c + d*x]^2*(b*B - a*C + b*C*Sec[c + d*x])*((2*b*(-7*a^2*b*B + 3*b^3*B + 11*a^3*C -
3*a*b^2*C)*Sin[c + d*x])/(3*a^2*(-a^2 + b^2)^2) - (2*(b^4*B*Sin[c + d*x] - 2*a*b^3*C*Sin[c + d*x]))/(3*a^2*(a
^2 - b^2)*(b + a*Cos[c + d*x])^2) - (2*(-8*a^2*b^3*B*Sin[c + d*x] + 4*b^5*B*Sin[c + d*x] + 13*a^3*b^2*C*Sin[c
+ d*x] - 5*a*b^4*C*Sin[c + d*x]))/(3*a^2*(a^2 - b^2)^2*(b + a*Cos[c + d*x]))))/(d*(b*C + b*B*Cos[c + d*x] - a*
C*Cos[c + d*x])*(a + b*Sec[c + d*x])^(5/2)) + (2*(b + a*Cos[c + d*x])^2*(b*B - a*C + b*C*Sec[c + d*x])*(-(a*b*
(a + b)*(-7*a^2*b*B + 3*b^3*B + 11*a^3*C - 3*a*b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec
[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]) - b*(a + b)*(3*b^4*B - 2*a^2*b^2*(B -
3*C) - 12*a^4*C - 3*a*b^3*(2*B + C) + a^3*b*(9*B + C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*S
ec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - 3*(a - b)^2*(a + b)^2*(b*B - a*C)*
((a - b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*a*EllipticPi[-1, -ArcSin[Tan[(c + d*x)/2]],
(a - b)/(a + b)])*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - a*b*(-7*a^2*b*B
+ 3*b^3*B + 11*a^3*C - 3*a*b^2*C)*(b + a*Cos[c + d*x])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*T
an[(c + d*x)/2]))/(3*a^3*(a^2 - b^2)^2*d*(b*C + b*B*Cos[c + d*x] - a*C*Cos[c + d*x])*(Cos[c + d*x]*Sec[(c + d*
x)/2]^2)^(3/2)*(a + b*Sec[c + d*x])^(5/2))

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Maple [B]  time = 0.466, size = 7862, normalized size = 15.2 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*a*b-a^2*C+b^2*B*sec(d*x+c)+b^2*C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^(7/2),x)

[Out]

result too large to display

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Maxima [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*B-a^2*C+b^2*B*sec(d*x+c)+b^2*C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^(7/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (C b \sec \left (d x + c\right ) - C a + B b\right )} \sqrt{b \sec \left (d x + c\right ) + a}}{b^{3} \sec \left (d x + c\right )^{3} + 3 \, a b^{2} \sec \left (d x + c\right )^{2} + 3 \, a^{2} b \sec \left (d x + c\right ) + a^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*b*B-a^2*C+b^2*B*sec(d*x+c)+b^2*C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^(7/2),x, algorithm="fricas")

[Out]

integral((C*b*sec(d*x + c) - C*a + B*b)*sqrt(b*sec(d*x + c) + a)/(b^3*sec(d*x + c)^3 + 3*a*b^2*sec(d*x + c)^2
+ 3*a^2*b*sec(d*x + c) + a^3), 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*B-a**2*C+b**2*B*sec(d*x+c)+b**2*C*sec(d*x+c)**2)/(a+b*sec(d*x+c))**(7/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{C b^{2} \sec \left (d x + c\right )^{2} + B b^{2} \sec \left (d x + c\right ) - C a^{2} + B a b}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*b*B-a^2*C+b^2*B*sec(d*x+c)+b^2*C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^(7/2),x, algorithm="giac")

[Out]

integrate((C*b^2*sec(d*x + c)^2 + B*b^2*sec(d*x + c) - C*a^2 + B*a*b)/(b*sec(d*x + c) + a)^(7/2), x)