3.416 \(\int \frac{1}{(a+b \cos (d+e x)+c \sin (d+e x))^{7/2}} \, dx\)
Optimal. Leaf size=490 \[ -\frac{16 a \sqrt{\frac{a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt{b^2+c^2}}} \text{EllipticF}\left (\frac{1}{2} \left (-\tan ^{-1}(b,c)+d+e x\right ),\frac{2 \sqrt{b^2+c^2}}{a+\sqrt{b^2+c^2}}\right )}{15 e \left (a^2-b^2-c^2\right )^2 \sqrt{a+b \cos (d+e x)+c \sin (d+e x)}}+\frac{2 \left (23 a^2+9 \left (b^2+c^2\right )\right ) \sqrt{a+b \cos (d+e x)+c \sin (d+e x)} E\left (\frac{1}{2} \left (d+e x-\tan ^{-1}(b,c)\right )|\frac{2 \sqrt{b^2+c^2}}{a+\sqrt{b^2+c^2}}\right )}{15 e \left (a^2-b^2-c^2\right )^3 \sqrt{\frac{a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt{b^2+c^2}}}}+\frac{2 \left (c \left (23 a^2+9 \left (b^2+c^2\right )\right ) \cos (d+e x)-b \left (23 a^2+9 \left (b^2+c^2\right )\right ) \sin (d+e x)\right )}{15 e \left (a^2-b^2-c^2\right )^3 \sqrt{a+b \cos (d+e x)+c \sin (d+e x)}}+\frac{16 (a c \cos (d+e x)-a b \sin (d+e x))}{15 e \left (a^2-b^2-c^2\right )^2 (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}+\frac{2 (c \cos (d+e x)-b \sin (d+e x))}{5 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^{5/2}} \]
[Out]
(2*(c*Cos[d + e*x] - b*Sin[d + e*x]))/(5*(a^2 - b^2 - c^2)*e*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(5/2)) + (1
6*(a*c*Cos[d + e*x] - a*b*Sin[d + e*x]))/(15*(a^2 - b^2 - c^2)^2*e*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(3/2)
) + (2*(23*a^2 + 9*(b^2 + c^2))*EllipticE[(d + e*x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2]
)]*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]])/(15*(a^2 - b^2 - c^2)^3*e*Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e
*x])/(a + Sqrt[b^2 + c^2])]) - (16*a*EllipticF[(d + e*x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 +
c^2])]*Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])])/(15*(a^2 - b^2 - c^2)^2*e*Sqrt[a +
b*Cos[d + e*x] + c*Sin[d + e*x]]) + (2*(c*(23*a^2 + 9*(b^2 + c^2))*Cos[d + e*x] - b*(23*a^2 + 9*(b^2 + c^2))*S
in[d + e*x]))/(15*(a^2 - b^2 - c^2)^3*e*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]])
________________________________________________________________________________________
Rubi [A] time = 0.619259, antiderivative size = 490, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used =
{3129, 3156, 3149, 3119, 2653, 3127, 2661} \[ -\frac{16 a \sqrt{\frac{a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt{b^2+c^2}}} F\left (\frac{1}{2} \left (d+e x-\tan ^{-1}(b,c)\right )|\frac{2 \sqrt{b^2+c^2}}{a+\sqrt{b^2+c^2}}\right )}{15 e \left (a^2-b^2-c^2\right )^2 \sqrt{a+b \cos (d+e x)+c \sin (d+e x)}}+\frac{2 \left (23 a^2+9 \left (b^2+c^2\right )\right ) \sqrt{a+b \cos (d+e x)+c \sin (d+e x)} E\left (\frac{1}{2} \left (d+e x-\tan ^{-1}(b,c)\right )|\frac{2 \sqrt{b^2+c^2}}{a+\sqrt{b^2+c^2}}\right )}{15 e \left (a^2-b^2-c^2\right )^3 \sqrt{\frac{a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt{b^2+c^2}}}}+\frac{2 \left (c \left (23 a^2+9 \left (b^2+c^2\right )\right ) \cos (d+e x)-b \left (23 a^2+9 \left (b^2+c^2\right )\right ) \sin (d+e x)\right )}{15 e \left (a^2-b^2-c^2\right )^3 \sqrt{a+b \cos (d+e x)+c \sin (d+e x)}}+\frac{16 (a c \cos (d+e x)-a b \sin (d+e x))}{15 e \left (a^2-b^2-c^2\right )^2 (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}+\frac{2 (c \cos (d+e x)-b \sin (d+e x))}{5 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^{5/2}} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(-7/2),x]
[Out]
(2*(c*Cos[d + e*x] - b*Sin[d + e*x]))/(5*(a^2 - b^2 - c^2)*e*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(5/2)) + (1
6*(a*c*Cos[d + e*x] - a*b*Sin[d + e*x]))/(15*(a^2 - b^2 - c^2)^2*e*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(3/2)
) + (2*(23*a^2 + 9*(b^2 + c^2))*EllipticE[(d + e*x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2]
)]*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]])/(15*(a^2 - b^2 - c^2)^3*e*Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e
*x])/(a + Sqrt[b^2 + c^2])]) - (16*a*EllipticF[(d + e*x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 +
c^2])]*Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])])/(15*(a^2 - b^2 - c^2)^2*e*Sqrt[a +
b*Cos[d + e*x] + c*Sin[d + e*x]]) + (2*(c*(23*a^2 + 9*(b^2 + c^2))*Cos[d + e*x] - b*(23*a^2 + 9*(b^2 + c^2))*S
in[d + e*x]))/(15*(a^2 - b^2 - c^2)^3*e*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]])
Rule 3129
Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^(n_), x_Symbol] :> Simp[((-(c*Cos[d
+ e*x]) + b*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1))/(e*(n + 1)*(a^2 - b^2 - c^2)), x] +
Dist[1/((n + 1)*(a^2 - b^2 - c^2)), Int[(a*(n + 1) - b*(n + 2)*Cos[d + e*x] - c*(n + 2)*Sin[d + e*x])*(a + b*C
os[d + e*x] + c*Sin[d + e*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && LtQ[n
, -1] && NeQ[n, -3/2]
Rule 3156
Int[((a_.) + cos[(d_.) + (e_.)*(x_)]*(b_.) + (c_.)*sin[(d_.) + (e_.)*(x_)])^(n_)*((A_.) + cos[(d_.) + (e_.)*(x
_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_)]), x_Symbol] :> -Simp[((c*B - b*C - (a*C - c*A)*Cos[d + e*x] + (a*B -
b*A)*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1))/(e*(n + 1)*(a^2 - b^2 - c^2)), x] + Dist[1/
((n + 1)*(a^2 - b^2 - c^2)), Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1)*Simp[(n + 1)*(a*A - b*B - c*C)
+ (n + 2)*(a*B - b*A)*Cos[d + e*x] + (n + 2)*(a*C - c*A)*Sin[d + e*x], x], x], x] /; FreeQ[{a, b, c, d, e, A,
B, C}, x] && LtQ[n, -1] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[n, -2]
Rule 3149
Int[((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_)])/Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.)
+ (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]], x_Symbol] :> Dist[B/b, Int[Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]
, x], x] + Dist[(A*b - a*B)/b, Int[1/Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]], x], x] /; FreeQ[{a, b, c, d, e
, A, B, C}, x] && EqQ[B*c - b*C, 0] && NeQ[A*b - a*B, 0]
Rule 3119
Int[Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*C
os[d + e*x] + c*Sin[d + e*x]]/Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])], Int[Sqrt[a/(a
+ Sqrt[b^2 + c^2]) + (Sqrt[b^2 + c^2]*Cos[d + e*x - ArcTan[b, c]])/(a + Sqrt[b^2 + c^2])], x], x] /; FreeQ[{a
, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[b^2 + c^2, 0] && !GtQ[a + Sqrt[b^2 + c^2], 0]
Rule 2653
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*Sqrt[a + b]*EllipticE[(1*(c - Pi/2 + d*x)
)/2, (2*b)/(a + b)])/d, x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Rule 3127
Int[1/Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a +
b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])]/Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]], Int[1/Sqrt[
a/(a + Sqrt[b^2 + c^2]) + (Sqrt[b^2 + c^2]*Cos[d + e*x - ArcTan[b, c]])/(a + Sqrt[b^2 + c^2])], x], x] /; Free
Q[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[b^2 + c^2, 0] && !GtQ[a + Sqrt[b^2 + c^2], 0]
Rule 2661
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, (2*b)
/(a + b)])/(d*Sqrt[a + b]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Rubi steps
\begin{align*} \int \frac{1}{(a+b \cos (d+e x)+c \sin (d+e x))^{7/2}} \, dx &=\frac{2 (c \cos (d+e x)-b \sin (d+e x))}{5 \left (a^2-b^2-c^2\right ) e (a+b \cos (d+e x)+c \sin (d+e x))^{5/2}}-\frac{2 \int \frac{-\frac{5 a}{2}+\frac{3}{2} b \cos (d+e x)+\frac{3}{2} c \sin (d+e x)}{(a+b \cos (d+e x)+c \sin (d+e x))^{5/2}} \, dx}{5 \left (a^2-b^2-c^2\right )}\\ &=\frac{2 (c \cos (d+e x)-b \sin (d+e x))}{5 \left (a^2-b^2-c^2\right ) e (a+b \cos (d+e x)+c \sin (d+e x))^{5/2}}+\frac{16 (a c \cos (d+e x)-a b \sin (d+e x))}{15 \left (a^2-b^2-c^2\right )^2 e (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}+\frac{4 \int \frac{\frac{3}{4} \left (5 a^2+3 \left (b^2+c^2\right )\right )-2 a b \cos (d+e x)-2 a c \sin (d+e x)}{(a+b \cos (d+e x)+c \sin (d+e x))^{3/2}} \, dx}{15 \left (a^2-b^2-c^2\right )^2}\\ &=\frac{2 (c \cos (d+e x)-b \sin (d+e x))}{5 \left (a^2-b^2-c^2\right ) e (a+b \cos (d+e x)+c \sin (d+e x))^{5/2}}+\frac{16 (a c \cos (d+e x)-a b \sin (d+e x))}{15 \left (a^2-b^2-c^2\right )^2 e (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}+\frac{2 \left (c \left (23 a^2+9 \left (b^2+c^2\right )\right ) \cos (d+e x)-b \left (23 a^2+9 \left (b^2+c^2\right )\right ) \sin (d+e x)\right )}{15 \left (a^2-b^2-c^2\right )^3 e \sqrt{a+b \cos (d+e x)+c \sin (d+e x)}}-\frac{8 \int \frac{-\frac{1}{8} a \left (15 a^2+17 \left (b^2+c^2\right )\right )-\frac{1}{8} b \left (23 a^2+9 \left (b^2+c^2\right )\right ) \cos (d+e x)-\frac{1}{8} c \left (23 a^2+9 \left (b^2+c^2\right )\right ) \sin (d+e x)}{\sqrt{a+b \cos (d+e x)+c \sin (d+e x)}} \, dx}{15 \left (a^2-b^2-c^2\right )^3}\\ &=\frac{2 (c \cos (d+e x)-b \sin (d+e x))}{5 \left (a^2-b^2-c^2\right ) e (a+b \cos (d+e x)+c \sin (d+e x))^{5/2}}+\frac{16 (a c \cos (d+e x)-a b \sin (d+e x))}{15 \left (a^2-b^2-c^2\right )^2 e (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}+\frac{2 \left (c \left (23 a^2+9 \left (b^2+c^2\right )\right ) \cos (d+e x)-b \left (23 a^2+9 \left (b^2+c^2\right )\right ) \sin (d+e x)\right )}{15 \left (a^2-b^2-c^2\right )^3 e \sqrt{a+b \cos (d+e x)+c \sin (d+e x)}}-\frac{(8 a) \int \frac{1}{\sqrt{a+b \cos (d+e x)+c \sin (d+e x)}} \, dx}{15 \left (a^2-b^2-c^2\right )^2}+\frac{\left (23 a^2+9 \left (b^2+c^2\right )\right ) \int \sqrt{a+b \cos (d+e x)+c \sin (d+e x)} \, dx}{15 \left (a^2-b^2-c^2\right )^3}\\ &=\frac{2 (c \cos (d+e x)-b \sin (d+e x))}{5 \left (a^2-b^2-c^2\right ) e (a+b \cos (d+e x)+c \sin (d+e x))^{5/2}}+\frac{16 (a c \cos (d+e x)-a b \sin (d+e x))}{15 \left (a^2-b^2-c^2\right )^2 e (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}+\frac{2 \left (c \left (23 a^2+9 \left (b^2+c^2\right )\right ) \cos (d+e x)-b \left (23 a^2+9 \left (b^2+c^2\right )\right ) \sin (d+e x)\right )}{15 \left (a^2-b^2-c^2\right )^3 e \sqrt{a+b \cos (d+e x)+c \sin (d+e x)}}+\frac{\left (\left (23 a^2+9 \left (b^2+c^2\right )\right ) \sqrt{a+b \cos (d+e x)+c \sin (d+e x)}\right ) \int \sqrt{\frac{a}{a+\sqrt{b^2+c^2}}+\frac{\sqrt{b^2+c^2} \cos \left (d+e x-\tan ^{-1}(b,c)\right )}{a+\sqrt{b^2+c^2}}} \, dx}{15 \left (a^2-b^2-c^2\right )^3 \sqrt{\frac{a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt{b^2+c^2}}}}-\frac{\left (8 a \sqrt{\frac{a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt{b^2+c^2}}}\right ) \int \frac{1}{\sqrt{\frac{a}{a+\sqrt{b^2+c^2}}+\frac{\sqrt{b^2+c^2} \cos \left (d+e x-\tan ^{-1}(b,c)\right )}{a+\sqrt{b^2+c^2}}}} \, dx}{15 \left (a^2-b^2-c^2\right )^2 \sqrt{a+b \cos (d+e x)+c \sin (d+e x)}}\\ &=\frac{2 (c \cos (d+e x)-b \sin (d+e x))}{5 \left (a^2-b^2-c^2\right ) e (a+b \cos (d+e x)+c \sin (d+e x))^{5/2}}+\frac{16 (a c \cos (d+e x)-a b \sin (d+e x))}{15 \left (a^2-b^2-c^2\right )^2 e (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}+\frac{2 \left (23 a^2+9 \left (b^2+c^2\right )\right ) E\left (\frac{1}{2} \left (d+e x-\tan ^{-1}(b,c)\right )|\frac{2 \sqrt{b^2+c^2}}{a+\sqrt{b^2+c^2}}\right ) \sqrt{a+b \cos (d+e x)+c \sin (d+e x)}}{15 \left (a^2-b^2-c^2\right )^3 e \sqrt{\frac{a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt{b^2+c^2}}}}-\frac{16 a F\left (\frac{1}{2} \left (d+e x-\tan ^{-1}(b,c)\right )|\frac{2 \sqrt{b^2+c^2}}{a+\sqrt{b^2+c^2}}\right ) \sqrt{\frac{a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt{b^2+c^2}}}}{15 \left (a^2-b^2-c^2\right )^2 e \sqrt{a+b \cos (d+e x)+c \sin (d+e x)}}+\frac{2 \left (c \left (23 a^2+9 \left (b^2+c^2\right )\right ) \cos (d+e x)-b \left (23 a^2+9 \left (b^2+c^2\right )\right ) \sin (d+e x)\right )}{15 \left (a^2-b^2-c^2\right )^3 e \sqrt{a+b \cos (d+e x)+c \sin (d+e x)}}\\ \end{align*}
Mathematica [C] time = 6.62076, size = 4116, normalized size = 8.4 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(-7/2),x]
[Out]
(Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]*((-2*(b^2 + c^2)*(23*a^2 + 9*b^2 + 9*c^2))/(15*b*c*(-a^2 + b^2 + c^
2)^3) + (2*(a*c + b^2*Sin[d + e*x] + c^2*Sin[d + e*x]))/(5*b*(-a^2 + b^2 + c^2)*(a + b*Cos[d + e*x] + c*Sin[d
+ e*x])^3) - (2*(5*a^2*c + 3*b^2*c + 3*c^3 + 8*a*b^2*Sin[d + e*x] + 8*a*c^2*Sin[d + e*x]))/(15*b*(-a^2 + b^2 +
c^2)^2*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^2) + (2*(15*a^3*c + 17*a*b^2*c + 17*a*c^3 + 23*a^2*b^2*Sin[d + e
*x] + 9*b^4*Sin[d + e*x] + 23*a^2*c^2*Sin[d + e*x] + 18*b^2*c^2*Sin[d + e*x] + 9*c^4*Sin[d + e*x]))/(15*b*(-a^
2 + b^2 + c^2)^3*(a + b*Cos[d + e*x] + c*Sin[d + e*x]))))/e - (2*a^3*AppellF1[1/2, 1/2, 1/2, 3/2, -((a + Sqrt[
1 + b^2/c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(1 - a/(Sqrt[1 + b^2/c^2]*c))*c)), -((a + Sqrt[1
+ b^2/c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(-1 - a/(Sqrt[1 + b^2/c^2]*c))*c))]*Sec[d + e*x +
ArcTan[b/c]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] - c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]])/(a + c*Sqrt[
(b^2 + c^2)/c^2])]*Sqrt[a + c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2]
+ c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]])/(-a + c*Sqrt[(b^2 + c^2)/c^2])])/(Sqrt[1 + b^2/c^2]*c*(-
a^2 + b^2 + c^2)^3*e) - (34*a*b^2*AppellF1[1/2, 1/2, 1/2, 3/2, -((a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + ArcTan
[b/c]])/(Sqrt[1 + b^2/c^2]*(1 - a/(Sqrt[1 + b^2/c^2]*c))*c)), -((a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + ArcTan[
b/c]])/(Sqrt[1 + b^2/c^2]*(-1 - a/(Sqrt[1 + b^2/c^2]*c))*c))]*Sec[d + e*x + ArcTan[b/c]]*Sqrt[(c*Sqrt[(b^2 + c
^2)/c^2] - c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]])/(a + c*Sqrt[(b^2 + c^2)/c^2])]*Sqrt[a + c*Sqrt[
(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] + c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e
*x + ArcTan[b/c]])/(-a + c*Sqrt[(b^2 + c^2)/c^2])])/(15*Sqrt[1 + b^2/c^2]*c*(-a^2 + b^2 + c^2)^3*e) - (34*a*c*
AppellF1[1/2, 1/2, 1/2, 3/2, -((a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(1 - a/
(Sqrt[1 + b^2/c^2]*c))*c)), -((a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(-1 - a/
(Sqrt[1 + b^2/c^2]*c))*c))]*Sec[d + e*x + ArcTan[b/c]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] - c*Sqrt[(b^2 + c^2)/c^2]
*Sin[d + e*x + ArcTan[b/c]])/(a + c*Sqrt[(b^2 + c^2)/c^2])]*Sqrt[a + c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + Arc
Tan[b/c]]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] + c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]])/(-a + c*Sqrt[(b
^2 + c^2)/c^2])])/(15*Sqrt[1 + b^2/c^2]*(-a^2 + b^2 + c^2)^3*e) - (23*a^2*b^2*(-((c*AppellF1[-1/2, -1/2, -1/2,
1/2, -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(1 - a/(b*Sqrt[1 + c^2/b^2])
))), -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(-1 - a/(b*Sqrt[1 + c^2/b^2])
)))]*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] - b*Sqrt[(b^2 + c^2)/b^2]*
Cos[d + e*x - ArcTan[c/b]])/(a + b*Sqrt[(b^2 + c^2)/b^2])]*Sqrt[a + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcT
an[c/b]]]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(-a + b*Sqrt[(b^
2 + c^2)/b^2])])) - ((2*b*(a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]))/(b^2 + c^2) - (c*Sin[d + e*x -
ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]))/Sqrt[a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]]))/(15*c*(-a^2 +
b^2 + c^2)^3*e) - (3*b^4*(-((c*AppellF1[-1/2, -1/2, -1/2, 1/2, -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTa
n[c/b]])/(b*Sqrt[1 + c^2/b^2]*(1 - a/(b*Sqrt[1 + c^2/b^2])))), -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan
[c/b]])/(b*Sqrt[1 + c^2/b^2]*(-1 - a/(b*Sqrt[1 + c^2/b^2]))))]*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2
]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] - b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(a + b*Sqrt[(b^2 + c^2)/
b^2])]*Sqrt[a + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]]]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] + b*Sqrt[(b^
2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(-a + b*Sqrt[(b^2 + c^2)/b^2])])) - ((2*b*(a + b*Sqrt[1 + c^2/b^2]*C
os[d + e*x - ArcTan[c/b]]))/(b^2 + c^2) - (c*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]))/Sqrt[a + b*Sqr
t[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]]))/(5*c*(-a^2 + b^2 + c^2)^3*e) - (23*a^2*c*(-((c*AppellF1[-1/2, -1/
2, -1/2, 1/2, -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(1 - a/(b*Sqrt[1 + c
^2/b^2])))), -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(-1 - a/(b*Sqrt[1 + c
^2/b^2]))))]*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] - b*Sqrt[(b^2 + c^
2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(a + b*Sqrt[(b^2 + c^2)/b^2])]*Sqrt[a + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*
x - ArcTan[c/b]]]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(-a + b*
Sqrt[(b^2 + c^2)/b^2])])) - ((2*b*(a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]))/(b^2 + c^2) - (c*Sin[d
+ e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]))/Sqrt[a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]]))/(15*(
-a^2 + b^2 + c^2)^3*e) - (6*b^2*c*(-((c*AppellF1[-1/2, -1/2, -1/2, 1/2, -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x
- ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(1 - a/(b*Sqrt[1 + c^2/b^2])))), -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x
- ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(-1 - a/(b*Sqrt[1 + c^2/b^2]))))]*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 +
c^2/b^2]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] - b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(a + b*Sqrt[(b^2
+ c^2)/b^2])]*Sqrt[a + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]]]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] + b*
Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(-a + b*Sqrt[(b^2 + c^2)/b^2])])) - ((2*b*(a + b*Sqrt[1 + c^
2/b^2]*Cos[d + e*x - ArcTan[c/b]]))/(b^2 + c^2) - (c*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]))/Sqrt[a
+ b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]]))/(5*(-a^2 + b^2 + c^2)^3*e) - (3*c^3*(-((c*AppellF1[-1/2,
-1/2, -1/2, 1/2, -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(1 - a/(b*Sqrt[1
+ c^2/b^2])))), -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(-1 - a/(b*Sqrt[1
+ c^2/b^2]))))]*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] - b*Sqrt[(b^2 +
c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(a + b*Sqrt[(b^2 + c^2)/b^2])]*Sqrt[a + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d +
e*x - ArcTan[c/b]]]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(-a +
b*Sqrt[(b^2 + c^2)/b^2])])) - ((2*b*(a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]))/(b^2 + c^2) - (c*Si
n[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]))/Sqrt[a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]]))/(5
*(-a^2 + b^2 + c^2)^3*e)
________________________________________________________________________________________
Maple [B] time = 94., size = 3876, normalized size = 7.9 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(a+b*cos(e*x+d)+c*sin(e*x+d))^(7/2),x)
[Out]
(-(-b^2*sin(e*x+d-arctan(-b,c))-c^2*sin(e*x+d-arctan(-b,c))-a*(b^2+c^2)^(1/2))*cos(e*x+d-arctan(-b,c))^2/(b^2+
c^2)^(1/2))^(1/2)/(b^2+c^2)^(1/2)*(1/8/a*(b^4+2*b^2*c^2+c^4)/(a^2-b^2-c^2)*(cos(e*x+d-arctan(-b,c))^2*((b^2+c^
2)^(1/2)*sin(e*x+d-arctan(-b,c))+a))^(1/2)/(b^2*sin(e*x+d-arctan(-b,c))+c^2*sin(e*x+d-arctan(-b,c))-a*(b^2+c^2
)^(1/2))^2+1/5/(a^2-b^2-c^2)/(b^2+c^2)^(1/2)*(cos(e*x+d-arctan(-b,c))^2*((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c
))+a))^(1/2)/(sin(e*x+d-arctan(-b,c))+1/(b^2+c^2)^(1/2)*a)^3-3/32*(b^2+c^2)^(1/2)*(5*a^2*b^2+5*a^2*c^2-b^4-2*b
^2*c^2-c^4)/(a^2-b^2-c^2)^2/a^2*(cos(e*x+d-arctan(-b,c))^2*((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a))^(1/2)/
(b^2*sin(e*x+d-arctan(-b,c))+c^2*sin(e*x+d-arctan(-b,c))-a*(b^2+c^2)^(1/2))+8/15/(a^2-b^2-c^2)^2*a*(cos(e*x+d-
arctan(-b,c))^2*((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a))^(1/2)/(sin(e*x+d-arctan(-b,c))+1/(b^2+c^2)^(1/2)*
a)^2+1/15*(b^2+c^2)*cos(e*x+d-arctan(-b,c))^2/(a^2-b^2-c^2)^3*(23*a^2+9*b^2+9*c^2)/(cos(e*x+d-arctan(-b,c))^2*
((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a))^(1/2)+2*(1/64*(11*a^2+b^2+c^2)*(b^2+c^2)^(1/2)/a/(a^2-b^2-c^2)^2-
4/15*a*(b^2+c^2)^(1/2)/(a^2-b^2-c^2)^2+1/30*a*(b^2+c^2)^(1/2)*(23*a^2+9*b^2+9*c^2)/(a^2-b^2-c^2)^3)*(1/(b^2+c^
2)^(1/2)*a-1)*((-(b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))-a)/(-a+(b^2+c^2)^(1/2)))^(1/2)*((-sin(e*x+d-arctan(-b
,c))+1)*(b^2+c^2)^(1/2)/(a+(b^2+c^2)^(1/2)))^(1/2)*((1+sin(e*x+d-arctan(-b,c)))*(b^2+c^2)^(1/2)/(-a+(b^2+c^2)^
(1/2)))^(1/2)/(cos(e*x+d-arctan(-b,c))^2*((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a))^(1/2)*EllipticF(((-(b^2+
c^2)^(1/2)*sin(e*x+d-arctan(-b,c))-a)/(-a+(b^2+c^2)^(1/2)))^(1/2),((a-(b^2+c^2)^(1/2))/(a+(b^2+c^2)^(1/2)))^(1
/2))+2*(-3/64*(5*a^2*b^2+5*a^2*c^2-b^4-2*b^2*c^2-c^4)/(a^2-b^2-c^2)^2/a^2+1/30*(b^2+c^2)*(23*a^2+9*b^2+9*c^2)/
(a^2-b^2-c^2)^3)*(1/(b^2+c^2)^(1/2)*a-1)*((-(b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))-a)/(-a+(b^2+c^2)^(1/2)))^(
1/2)*((-sin(e*x+d-arctan(-b,c))+1)*(b^2+c^2)^(1/2)/(a+(b^2+c^2)^(1/2)))^(1/2)*((1+sin(e*x+d-arctan(-b,c)))*(b^
2+c^2)^(1/2)/(-a+(b^2+c^2)^(1/2)))^(1/2)/(cos(e*x+d-arctan(-b,c))^2*((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a
))^(1/2)*((-1/(b^2+c^2)^(1/2)*a-1)*EllipticE(((-(b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))-a)/(-a+(b^2+c^2)^(1/2)
))^(1/2),((a-(b^2+c^2)^(1/2))/(a+(b^2+c^2)^(1/2)))^(1/2))+EllipticF(((-(b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))
-a)/(-a+(b^2+c^2)^(1/2)))^(1/2),((a-(b^2+c^2)^(1/2))/(a+(b^2+c^2)^(1/2)))^(1/2)))-1/64*(43*a^4*b^2+43*a^4*c^2+
2*a^2*b^4+4*a^2*b^2*c^2+2*a^2*c^4+3*b^6+9*b^4*c^2+9*b^2*c^4+3*c^6)/(a^2-b^2-c^2)^2/a^3/(b^2+c^2)^(1/2)*(1/(b^2
+c^2)^(1/2)*a-1)*((-(b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))-a)/(-a+(b^2+c^2)^(1/2)))^(1/2)*((-sin(e*x+d-arctan
(-b,c))+1)*(b^2+c^2)^(1/2)/(a+(b^2+c^2)^(1/2)))^(1/2)*((1+sin(e*x+d-arctan(-b,c)))*(b^2+c^2)^(1/2)/(-a+(b^2+c^
2)^(1/2)))^(1/2)/(cos(e*x+d-arctan(-b,c))^2*((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a))^(1/2)*EllipticPi(((-(
b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))-a)/(-a+(b^2+c^2)^(1/2)))^(1/2),-1/2*(-1/(b^2+c^2)^(1/2)*a+1)*(b^2+c^2)^
(1/2)/a,((a-(b^2+c^2)^(1/2))/(a+(b^2+c^2)^(1/2)))^(1/2))-1/8/a/(a^2-b^2-c^2)*(b^2+c^2)^(3/2)*(cos(e*x+d-arctan
(-b,c))^2*(b^2+c^2)*((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a))^(1/2)/(b^2*sin(e*x+d-arctan(-b,c))+c^2*sin(e*
x+d-arctan(-b,c))-a*(b^2+c^2)^(1/2))^2+1/5/(a^2-b^2-c^2)/(b^2+c^2)*(cos(e*x+d-arctan(-b,c))^2*(b^2+c^2)*((b^2+
c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a))^(1/2)/(sin(e*x+d-arctan(-b,c))+1/(b^2+c^2)^(1/2)*a)^3+3/32*(5*a^2*b^2+5
*a^2*c^2-b^4-2*b^2*c^2-c^4)/(a^2-b^2-c^2)^2/a^2*(cos(e*x+d-arctan(-b,c))^2*(b^2+c^2)*((b^2+c^2)^(1/2)*sin(e*x+
d-arctan(-b,c))+a))^(1/2)/(b^2*sin(e*x+d-arctan(-b,c))+c^2*sin(e*x+d-arctan(-b,c))-a*(b^2+c^2)^(1/2))+8/15/(a^
2-b^2-c^2)^2*a/(b^2+c^2)^(1/2)*(cos(e*x+d-arctan(-b,c))^2*(b^2+c^2)*((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a
))^(1/2)/(sin(e*x+d-arctan(-b,c))+1/(b^2+c^2)^(1/2)*a)^2-1/15*(b^2+c^2)^(1/2)*(-b^2-c^2)*cos(e*x+d-arctan(-b,c
))^2/(a^2-b^2-c^2)^3*(23*a^2+9*b^2+9*c^2)/(cos(e*x+d-arctan(-b,c))^2*(b^2+c^2)*((b^2+c^2)^(1/2)*sin(e*x+d-arct
an(-b,c))+a))^(1/2)+2*(-1/64*(11*a^2*b^2+11*a^2*c^2+b^4+2*b^2*c^2+c^4)/a/(a^2-b^2-c^2)^2-4/15*a*(b^2+c^2)/(a^2
-b^2-c^2)^2+1/30*a*(b^2+c^2)*(23*a^2+9*b^2+9*c^2)/(a^2-b^2-c^2)^3)*(1/(b^2+c^2)^(1/2)*a-1)*((-(b^2+c^2)^(1/2)*
sin(e*x+d-arctan(-b,c))-a)/(-a+(b^2+c^2)^(1/2)))^(1/2)*((-sin(e*x+d-arctan(-b,c))+1)*(b^2+c^2)^(1/2)/(a+(b^2+c
^2)^(1/2)))^(1/2)*((1+sin(e*x+d-arctan(-b,c)))*(b^2+c^2)^(1/2)/(-a+(b^2+c^2)^(1/2)))^(1/2)/(cos(e*x+d-arctan(-
b,c))^2*(b^2+c^2)*((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a))^(1/2)*EllipticF(((-(b^2+c^2)^(1/2)*sin(e*x+d-ar
ctan(-b,c))-a)/(-a+(b^2+c^2)^(1/2)))^(1/2),((a-(b^2+c^2)^(1/2))/(a+(b^2+c^2)^(1/2)))^(1/2))+2*(3/64*(b^2+c^2)^
(1/2)*(5*a^2*b^2+5*a^2*c^2-b^4-2*b^2*c^2-c^4)/(a^2-b^2-c^2)^2/a^2-1/30*(b^2+c^2)^(3/2)*(23*a^2+9*b^2+9*c^2)/(a
^2-b^2-c^2)^3+1/30*(b^2+c^2)^(1/2)*(2*b^2+2*c^2)/(a^2-b^2-c^2)^3*(23*a^2+9*b^2+9*c^2))*(1/(b^2+c^2)^(1/2)*a-1)
*((-(b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))-a)/(-a+(b^2+c^2)^(1/2)))^(1/2)*((-sin(e*x+d-arctan(-b,c))+1)*(b^2+
c^2)^(1/2)/(a+(b^2+c^2)^(1/2)))^(1/2)*((1+sin(e*x+d-arctan(-b,c)))*(b^2+c^2)^(1/2)/(-a+(b^2+c^2)^(1/2)))^(1/2)
/(cos(e*x+d-arctan(-b,c))^2*(b^2+c^2)*((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a))^(1/2)*((-1/(b^2+c^2)^(1/2)*
a-1)*EllipticE(((-(b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))-a)/(-a+(b^2+c^2)^(1/2)))^(1/2),((a-(b^2+c^2)^(1/2))/
(a+(b^2+c^2)^(1/2)))^(1/2))+EllipticF(((-(b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))-a)/(-a+(b^2+c^2)^(1/2)))^(1/2
),((a-(b^2+c^2)^(1/2))/(a+(b^2+c^2)^(1/2)))^(1/2)))+1/64*(43*a^4*b^2+43*a^4*c^2+2*a^2*b^4+4*a^2*b^2*c^2+2*a^2*
c^4+3*b^6+9*b^4*c^2+9*b^2*c^4+3*c^6)/(a^2-b^2-c^2)^2/a^3*(1/(b^2+c^2)^(1/2)*a-1)*((-(b^2+c^2)^(1/2)*sin(e*x+d-
arctan(-b,c))-a)/(-a+(b^2+c^2)^(1/2)))^(1/2)*((-sin(e*x+d-arctan(-b,c))+1)*(b^2+c^2)^(1/2)/(a+(b^2+c^2)^(1/2))
)^(1/2)*((1+sin(e*x+d-arctan(-b,c)))*(b^2+c^2)^(1/2)/(-a+(b^2+c^2)^(1/2)))^(1/2)/(cos(e*x+d-arctan(-b,c))^2*(b
^2+c^2)*((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a))^(1/2)*EllipticPi(((-(b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c
))-a)/(-a+(b^2+c^2)^(1/2)))^(1/2),-1/2*(-1/(b^2+c^2)^(1/2)*a+1)*(b^2+c^2)^(1/2)/a,((a-(b^2+c^2)^(1/2))/(a+(b^2
+c^2)^(1/2)))^(1/2)))/cos(e*x+d-arctan(-b,c))/((b^2*sin(e*x+d-arctan(-b,c))+c^2*sin(e*x+d-arctan(-b,c))+a*(b^2
+c^2)^(1/2))/(b^2+c^2)^(1/2))^(1/2)/e
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \cos \left (e x + d\right ) + c \sin \left (e x + d\right ) + a\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*cos(e*x+d)+c*sin(e*x+d))^(7/2),x, algorithm="maxima")
[Out]
integrate((b*cos(e*x + d) + c*sin(e*x + d) + a)^(-7/2), x)
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b \cos \left (e x + d\right ) + c \sin \left (e x + d\right ) + a}}{{\left (b^{4} - 6 \, b^{2} c^{2} + c^{4}\right )} \cos \left (e x + d\right )^{4} + a^{4} + 6 \, a^{2} c^{2} + c^{4} + 4 \,{\left (a b^{3} - 3 \, a b c^{2}\right )} \cos \left (e x + d\right )^{3} + 2 \,{\left (3 \, a^{2} b^{2} - c^{4} - 3 \,{\left (a^{2} - b^{2}\right )} c^{2}\right )} \cos \left (e x + d\right )^{2} + 4 \,{\left (a^{3} b + 3 \, a b c^{2}\right )} \cos \left (e x + d\right ) + 4 \,{\left (a^{3} c + a c^{3} +{\left (b^{3} c - b c^{3}\right )} \cos \left (e x + d\right )^{3} +{\left (3 \, a b^{2} c - a c^{3}\right )} \cos \left (e x + d\right )^{2} +{\left (3 \, a^{2} b c + b c^{3}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*cos(e*x+d)+c*sin(e*x+d))^(7/2),x, algorithm="fricas")
[Out]
integral(sqrt(b*cos(e*x + d) + c*sin(e*x + d) + a)/((b^4 - 6*b^2*c^2 + c^4)*cos(e*x + d)^4 + a^4 + 6*a^2*c^2 +
c^4 + 4*(a*b^3 - 3*a*b*c^2)*cos(e*x + d)^3 + 2*(3*a^2*b^2 - c^4 - 3*(a^2 - b^2)*c^2)*cos(e*x + d)^2 + 4*(a^3*
b + 3*a*b*c^2)*cos(e*x + d) + 4*(a^3*c + a*c^3 + (b^3*c - b*c^3)*cos(e*x + d)^3 + (3*a*b^2*c - a*c^3)*cos(e*x
+ d)^2 + (3*a^2*b*c + b*c^3)*cos(e*x + d))*sin(e*x + d)), 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(1/(a+b*cos(e*x+d)+c*sin(e*x+d))**(7/2),x)
[Out]
Timed out
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \cos \left (e x + d\right ) + c \sin \left (e x + d\right ) + a\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*cos(e*x+d)+c*sin(e*x+d))^(7/2),x, algorithm="giac")
[Out]
integrate((b*cos(e*x + d) + c*sin(e*x + d) + a)^(-7/2), x)