3.87 \(\int \frac {1}{(a+b \cos (c+d x))^3 (e \sin (c+d x))^{5/2}} \, dx\)
Optimal. Leaf size=629 \[ \frac {a \left (8 a^2+69 b^2\right ) \sqrt {\sin (c+d x)} F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{12 d e^2 \left (a^2-b^2\right )^3 \sqrt {e \sin (c+d x)}}-\frac {7 a b^2 \left (9 a^2+2 b^2\right ) \sqrt {\sin (c+d x)} \Pi \left (\frac {2 b}{b-\sqrt {b^2-a^2}};\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{8 d e^2 \left (a^2-b^2\right )^3 \left (a^2-b \left (b-\sqrt {b^2-a^2}\right )\right ) \sqrt {e \sin (c+d x)}}-\frac {7 a b^2 \left (9 a^2+2 b^2\right ) \sqrt {\sin (c+d x)} \Pi \left (\frac {2 b}{b+\sqrt {b^2-a^2}};\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{8 d e^2 \left (a^2-b^2\right )^3 \left (a^2-b \left (\sqrt {b^2-a^2}+b\right )\right ) \sqrt {e \sin (c+d x)}}-\frac {11 a b}{4 d e \left (a^2-b^2\right )^2 (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))}-\frac {b}{2 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))^2}+\frac {7 b \left (9 a^2+2 b^2\right )-a \left (8 a^2+69 b^2\right ) \cos (c+d x)}{12 d e \left (a^2-b^2\right )^3 (e \sin (c+d x))^{3/2}}+\frac {7 b^{5/2} \left (9 a^2+2 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{8 d e^{5/2} \left (b^2-a^2\right )^{15/4}}+\frac {7 b^{5/2} \left (9 a^2+2 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{8 d e^{5/2} \left (b^2-a^2\right )^{15/4}} \]
[Out]
7/8*b^(5/2)*(9*a^2+2*b^2)*arctan(b^(1/2)*(e*sin(d*x+c))^(1/2)/(-a^2+b^2)^(1/4)/e^(1/2))/(-a^2+b^2)^(15/4)/d/e^
(5/2)+7/8*b^(5/2)*(9*a^2+2*b^2)*arctanh(b^(1/2)*(e*sin(d*x+c))^(1/2)/(-a^2+b^2)^(1/4)/e^(1/2))/(-a^2+b^2)^(15/
4)/d/e^(5/2)-1/2*b/(a^2-b^2)/d/e/(a+b*cos(d*x+c))^2/(e*sin(d*x+c))^(3/2)-11/4*a*b/(a^2-b^2)^2/d/e/(a+b*cos(d*x
+c))/(e*sin(d*x+c))^(3/2)+1/12*(7*b*(9*a^2+2*b^2)-a*(8*a^2+69*b^2)*cos(d*x+c))/(a^2-b^2)^3/d/e/(e*sin(d*x+c))^
(3/2)-1/12*a*(8*a^2+69*b^2)*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/4*Pi+1/2*d*x)*EllipticF(cos(1/2*c+
1/4*Pi+1/2*d*x),2^(1/2))*sin(d*x+c)^(1/2)/(a^2-b^2)^3/d/e^2/(e*sin(d*x+c))^(1/2)+7/8*a*b^2*(9*a^2+2*b^2)*(sin(
1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/4*Pi+1/2*d*x)*EllipticPi(cos(1/2*c+1/4*Pi+1/2*d*x),2*b/(b-(-a^2+b^2
)^(1/2)),2^(1/2))*sin(d*x+c)^(1/2)/(a^2-b^2)^3/d/e^2/(a^2-b*(b-(-a^2+b^2)^(1/2)))/(e*sin(d*x+c))^(1/2)+7/8*a*b
^2*(9*a^2+2*b^2)*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/4*Pi+1/2*d*x)*EllipticPi(cos(1/2*c+1/4*Pi+1/2
*d*x),2*b/(b+(-a^2+b^2)^(1/2)),2^(1/2))*sin(d*x+c)^(1/2)/(a^2-b^2)^3/d/e^2/(a^2-b*(b+(-a^2+b^2)^(1/2)))/(e*sin
(d*x+c))^(1/2)
________________________________________________________________________________________
Rubi [A] time = 1.77, antiderivative size = 629, normalized size of antiderivative = 1.00,
number of steps used = 15, number of rules used = 13, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules
used = {2694, 2864, 2866, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205}
\[ \frac {7 b^{5/2} \left (9 a^2+2 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{8 d e^{5/2} \left (b^2-a^2\right )^{15/4}}+\frac {7 b^{5/2} \left (9 a^2+2 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{8 d e^{5/2} \left (b^2-a^2\right )^{15/4}}+\frac {a \left (8 a^2+69 b^2\right ) \sqrt {\sin (c+d x)} F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{12 d e^2 \left (a^2-b^2\right )^3 \sqrt {e \sin (c+d x)}}-\frac {7 a b^2 \left (9 a^2+2 b^2\right ) \sqrt {\sin (c+d x)} \Pi \left (\frac {2 b}{b-\sqrt {b^2-a^2}};\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{8 d e^2 \left (a^2-b^2\right )^3 \left (a^2-b \left (b-\sqrt {b^2-a^2}\right )\right ) \sqrt {e \sin (c+d x)}}-\frac {7 a b^2 \left (9 a^2+2 b^2\right ) \sqrt {\sin (c+d x)} \Pi \left (\frac {2 b}{b+\sqrt {b^2-a^2}};\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{8 d e^2 \left (a^2-b^2\right )^3 \left (a^2-b \left (\sqrt {b^2-a^2}+b\right )\right ) \sqrt {e \sin (c+d x)}}-\frac {11 a b}{4 d e \left (a^2-b^2\right )^2 (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))}-\frac {b}{2 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))^2}+\frac {7 b \left (9 a^2+2 b^2\right )-a \left (8 a^2+69 b^2\right ) \cos (c+d x)}{12 d e \left (a^2-b^2\right )^3 (e \sin (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
Int[1/((a + b*Cos[c + d*x])^3*(e*Sin[c + d*x])^(5/2)),x]
[Out]
(7*b^(5/2)*(9*a^2 + 2*b^2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*(-a^2 + b^2
)^(15/4)*d*e^(5/2)) + (7*b^(5/2)*(9*a^2 + 2*b^2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sq
rt[e])])/(8*(-a^2 + b^2)^(15/4)*d*e^(5/2)) - b/(2*(a^2 - b^2)*d*e*(a + b*Cos[c + d*x])^2*(e*Sin[c + d*x])^(3/2
)) - (11*a*b)/(4*(a^2 - b^2)^2*d*e*(a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(3/2)) + (7*b*(9*a^2 + 2*b^2) - a*(8*
a^2 + 69*b^2)*Cos[c + d*x])/(12*(a^2 - b^2)^3*d*e*(e*Sin[c + d*x])^(3/2)) + (a*(8*a^2 + 69*b^2)*EllipticF[(c -
Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(12*(a^2 - b^2)^3*d*e^2*Sqrt[e*Sin[c + d*x]]) - (7*a*b^2*(9*a^2 + 2*b^2
)*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*(a^2 - b^2)^3*(a^2 -
b*(b - Sqrt[-a^2 + b^2]))*d*e^2*Sqrt[e*Sin[c + d*x]]) - (7*a*b^2*(9*a^2 + 2*b^2)*EllipticPi[(2*b)/(b + Sqrt[-a
^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*(a^2 - b^2)^3*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*e^2
*Sqrt[e*Sin[c + d*x]])
Rule 205
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]
Rule 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rule 212
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]
Rule 329
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]
Rule 2641
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{c, d}, x]
Rule 2642
Int[1/Sqrt[(b_)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[Sin[c + d*x]]/Sqrt[b*Sin[c + d*x]], Int[1/Sqr
t[Sin[c + d*x]], x], x] /; FreeQ[{b, c, d}, x]
Rule 2694
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Simp[(b*(g
*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1))/(f*g*(a^2 - b^2)*(m + 1)), x] + Dist[1/((a^2 - b^2)*(m +
1)), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1)*(a*(m + 1) - b*(m + p + 2)*Sin[e + f*x]), x], x] /; F
reeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*p]
Rule 2702
Int[1/(Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> With[{q = Rt[
-a^2 + b^2, 2]}, -Dist[a/(2*q), Int[1/(Sqrt[g*Cos[e + f*x]]*(q + b*Cos[e + f*x])), x], x] + (Dist[(b*g)/f, Sub
st[Int[1/(Sqrt[x]*(g^2*(a^2 - b^2) + b^2*x^2)), x], x, g*Cos[e + f*x]], x] - Dist[a/(2*q), Int[1/(Sqrt[g*Cos[e
+ f*x]]*(q - b*Cos[e + f*x])), x], x])] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0]
Rule 2805
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp
[(2*EllipticPi[(2*b)/(a + b), (1*(e - Pi/2 + f*x))/2, (2*d)/(c + d)])/(f*(a + b)*Sqrt[c + d]), x] /; FreeQ[{a,
b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 0]
Rule 2807
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist
[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt[c + d*Sin[e + f*x]], Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d*
Sin[e + f*x])/(c + d)]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && N
eQ[c^2 - d^2, 0] && !GtQ[c + d, 0]
Rule 2864
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> -Simp[((b*c - a*d)*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1))/(f*g*(a
^2 - b^2)*(m + 1)), x] + Dist[1/((a^2 - b^2)*(m + 1)), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1)*Sim
p[(a*c - b*d)*(m + 1) - (b*c - a*d)*(m + p + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x]
&& NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]
Rule 2866
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> Simp[((g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1)*(b*c - a*d - (a*c -
b*d)*Sin[e + f*x]))/(f*g*(a^2 - b^2)*(p + 1)), x] + Dist[1/(g^2*(a^2 - b^2)*(p + 1)), Int[(g*Cos[e + f*x])^(p
+ 2)*(a + b*Sin[e + f*x])^m*Simp[c*(a^2*(p + 2) - b^2*(m + p + 2)) + a*b*d*m + b*(a*c - b*d)*(m + p + 3)*Sin[e
+ f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && LtQ[p, -1] && IntegerQ[2*m]
Rule 2867
Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]))/((a_) + (b_.)*sin[(e_.) + (
f_.)*(x_)]), x_Symbol] :> Dist[d/b, Int[(g*Cos[e + f*x])^p, x], x] + Dist[(b*c - a*d)/b, Int[(g*Cos[e + f*x])^
p/(a + b*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[a^2 - b^2, 0]
Rubi steps
\begin {align*} \int \frac {1}{(a+b \cos (c+d x))^3 (e \sin (c+d x))^{5/2}} \, dx &=-\frac {b}{2 \left (a^2-b^2\right ) d e (a+b \cos (c+d x))^2 (e \sin (c+d x))^{3/2}}-\frac {\int \frac {-2 a+\frac {7}{2} b \cos (c+d x)}{(a+b \cos (c+d x))^2 (e \sin (c+d x))^{5/2}} \, dx}{2 \left (a^2-b^2\right )}\\ &=-\frac {b}{2 \left (a^2-b^2\right ) d e (a+b \cos (c+d x))^2 (e \sin (c+d x))^{3/2}}-\frac {11 a b}{4 \left (a^2-b^2\right )^2 d e (a+b \cos (c+d x)) (e \sin (c+d x))^{3/2}}+\frac {\int \frac {\frac {1}{2} \left (4 a^2+7 b^2\right )-\frac {55}{4} a b \cos (c+d x)}{(a+b \cos (c+d x)) (e \sin (c+d x))^{5/2}} \, dx}{2 \left (a^2-b^2\right )^2}\\ &=-\frac {b}{2 \left (a^2-b^2\right ) d e (a+b \cos (c+d x))^2 (e \sin (c+d x))^{3/2}}-\frac {11 a b}{4 \left (a^2-b^2\right )^2 d e (a+b \cos (c+d x)) (e \sin (c+d x))^{3/2}}+\frac {7 b \left (9 a^2+2 b^2\right )-a \left (8 a^2+69 b^2\right ) \cos (c+d x)}{12 \left (a^2-b^2\right )^3 d e (e \sin (c+d x))^{3/2}}-\frac {\int \frac {\frac {1}{4} \left (-4 a^4+60 a^2 b^2+21 b^4\right )-\frac {1}{8} a b \left (8 a^2+69 b^2\right ) \cos (c+d x)}{(a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}} \, dx}{3 \left (a^2-b^2\right )^3 e^2}\\ &=-\frac {b}{2 \left (a^2-b^2\right ) d e (a+b \cos (c+d x))^2 (e \sin (c+d x))^{3/2}}-\frac {11 a b}{4 \left (a^2-b^2\right )^2 d e (a+b \cos (c+d x)) (e \sin (c+d x))^{3/2}}+\frac {7 b \left (9 a^2+2 b^2\right )-a \left (8 a^2+69 b^2\right ) \cos (c+d x)}{12 \left (a^2-b^2\right )^3 d e (e \sin (c+d x))^{3/2}}-\frac {\left (7 b^2 \left (9 a^2+2 b^2\right )\right ) \int \frac {1}{(a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}} \, dx}{8 \left (a^2-b^2\right )^3 e^2}+\frac {\left (a \left (8 a^2+69 b^2\right )\right ) \int \frac {1}{\sqrt {e \sin (c+d x)}} \, dx}{24 \left (a^2-b^2\right )^3 e^2}\\ &=-\frac {b}{2 \left (a^2-b^2\right ) d e (a+b \cos (c+d x))^2 (e \sin (c+d x))^{3/2}}-\frac {11 a b}{4 \left (a^2-b^2\right )^2 d e (a+b \cos (c+d x)) (e \sin (c+d x))^{3/2}}+\frac {7 b \left (9 a^2+2 b^2\right )-a \left (8 a^2+69 b^2\right ) \cos (c+d x)}{12 \left (a^2-b^2\right )^3 d e (e \sin (c+d x))^{3/2}}-\frac {\left (7 a b^2 \left (9 a^2+2 b^2\right )\right ) \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {-a^2+b^2}-b \sin (c+d x)\right )} \, dx}{16 \left (-a^2+b^2\right )^{7/2} e^2}-\frac {\left (7 a b^2 \left (9 a^2+2 b^2\right )\right ) \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {-a^2+b^2}+b \sin (c+d x)\right )} \, dx}{16 \left (-a^2+b^2\right )^{7/2} e^2}+\frac {\left (7 b^3 \left (9 a^2+2 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (\left (a^2-b^2\right ) e^2+b^2 x^2\right )} \, dx,x,e \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^3 d e}+\frac {\left (a \left (8 a^2+69 b^2\right ) \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{24 \left (a^2-b^2\right )^3 e^2 \sqrt {e \sin (c+d x)}}\\ &=-\frac {b}{2 \left (a^2-b^2\right ) d e (a+b \cos (c+d x))^2 (e \sin (c+d x))^{3/2}}-\frac {11 a b}{4 \left (a^2-b^2\right )^2 d e (a+b \cos (c+d x)) (e \sin (c+d x))^{3/2}}+\frac {7 b \left (9 a^2+2 b^2\right )-a \left (8 a^2+69 b^2\right ) \cos (c+d x)}{12 \left (a^2-b^2\right )^3 d e (e \sin (c+d x))^{3/2}}+\frac {a \left (8 a^2+69 b^2\right ) F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{12 \left (a^2-b^2\right )^3 d e^2 \sqrt {e \sin (c+d x)}}+\frac {\left (7 b^3 \left (9 a^2+2 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\left (a^2-b^2\right ) e^2+b^2 x^4} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{4 \left (a^2-b^2\right )^3 d e}-\frac {\left (7 a b^2 \left (9 a^2+2 b^2\right ) \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)} \left (\sqrt {-a^2+b^2}-b \sin (c+d x)\right )} \, dx}{16 \left (-a^2+b^2\right )^{7/2} e^2 \sqrt {e \sin (c+d x)}}-\frac {\left (7 a b^2 \left (9 a^2+2 b^2\right ) \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)} \left (\sqrt {-a^2+b^2}+b \sin (c+d x)\right )} \, dx}{16 \left (-a^2+b^2\right )^{7/2} e^2 \sqrt {e \sin (c+d x)}}\\ &=-\frac {b}{2 \left (a^2-b^2\right ) d e (a+b \cos (c+d x))^2 (e \sin (c+d x))^{3/2}}-\frac {11 a b}{4 \left (a^2-b^2\right )^2 d e (a+b \cos (c+d x)) (e \sin (c+d x))^{3/2}}+\frac {7 b \left (9 a^2+2 b^2\right )-a \left (8 a^2+69 b^2\right ) \cos (c+d x)}{12 \left (a^2-b^2\right )^3 d e (e \sin (c+d x))^{3/2}}+\frac {a \left (8 a^2+69 b^2\right ) F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{12 \left (a^2-b^2\right )^3 d e^2 \sqrt {e \sin (c+d x)}}+\frac {7 a b^2 \left (9 a^2+2 b^2\right ) \Pi \left (\frac {2 b}{b-\sqrt {-a^2+b^2}};\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{8 \left (-a^2+b^2\right )^{7/2} \left (b-\sqrt {-a^2+b^2}\right ) d e^2 \sqrt {e \sin (c+d x)}}-\frac {7 a b^2 \left (9 a^2+2 b^2\right ) \Pi \left (\frac {2 b}{b+\sqrt {-a^2+b^2}};\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{8 \left (-a^2+b^2\right )^{7/2} \left (b+\sqrt {-a^2+b^2}\right ) d e^2 \sqrt {e \sin (c+d x)}}+\frac {\left (7 b^3 \left (9 a^2+2 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} e-b x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{8 \left (-a^2+b^2\right )^{7/2} d e^2}+\frac {\left (7 b^3 \left (9 a^2+2 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} e+b x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{8 \left (-a^2+b^2\right )^{7/2} d e^2}\\ &=\frac {7 b^{5/2} \left (9 a^2+2 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 \left (-a^2+b^2\right )^{15/4} d e^{5/2}}+\frac {7 b^{5/2} \left (9 a^2+2 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 \left (-a^2+b^2\right )^{15/4} d e^{5/2}}-\frac {b}{2 \left (a^2-b^2\right ) d e (a+b \cos (c+d x))^2 (e \sin (c+d x))^{3/2}}-\frac {11 a b}{4 \left (a^2-b^2\right )^2 d e (a+b \cos (c+d x)) (e \sin (c+d x))^{3/2}}+\frac {7 b \left (9 a^2+2 b^2\right )-a \left (8 a^2+69 b^2\right ) \cos (c+d x)}{12 \left (a^2-b^2\right )^3 d e (e \sin (c+d x))^{3/2}}+\frac {a \left (8 a^2+69 b^2\right ) F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{12 \left (a^2-b^2\right )^3 d e^2 \sqrt {e \sin (c+d x)}}+\frac {7 a b^2 \left (9 a^2+2 b^2\right ) \Pi \left (\frac {2 b}{b-\sqrt {-a^2+b^2}};\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{8 \left (-a^2+b^2\right )^{7/2} \left (b-\sqrt {-a^2+b^2}\right ) d e^2 \sqrt {e \sin (c+d x)}}-\frac {7 a b^2 \left (9 a^2+2 b^2\right ) \Pi \left (\frac {2 b}{b+\sqrt {-a^2+b^2}};\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{8 \left (-a^2+b^2\right )^{7/2} \left (b+\sqrt {-a^2+b^2}\right ) d e^2 \sqrt {e \sin (c+d x)}}\\ \end {align*}
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Mathematica [C] time = 14.49, size = 1308, normalized size = 2.08 \[ \frac {\left (\frac {15 a b^3}{4 \left (a^2-b^2\right )^3 (a+b \cos (c+d x))}+\frac {b^3}{2 \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}-\frac {2 \left (\cos (c+d x) a^3-3 b a^2+3 b^2 \cos (c+d x) a-b^3\right ) \csc ^2(c+d x)}{3 \left (a^2-b^2\right )^3}\right ) \sin ^3(c+d x)}{d (e \sin (c+d x))^{5/2}}+\frac {\left (\frac {2 \left (8 b a^3+69 b^3 a\right ) \left (a+b \sqrt {1-\sin ^2(c+d x)}\right ) \left (\frac {5 b \left (a^2-b^2\right ) \sqrt {\sin (c+d x)} \sqrt {1-\sin ^2(c+d x)} F_1\left (\frac {1}{4};-\frac {1}{2},1;\frac {5}{4};\sin ^2(c+d x),\frac {b^2 \sin ^2(c+d x)}{b^2-a^2}\right )}{\left (2 \left (2 F_1\left (\frac {5}{4};-\frac {1}{2},2;\frac {9}{4};\sin ^2(c+d x),\frac {b^2 \sin ^2(c+d x)}{b^2-a^2}\right ) b^2+\left (a^2-b^2\right ) F_1\left (\frac {5}{4};\frac {1}{2},1;\frac {9}{4};\sin ^2(c+d x),\frac {b^2 \sin ^2(c+d x)}{b^2-a^2}\right )\right ) \sin ^2(c+d x)-5 \left (a^2-b^2\right ) F_1\left (\frac {1}{4};-\frac {1}{2},1;\frac {5}{4};\sin ^2(c+d x),\frac {b^2 \sin ^2(c+d x)}{b^2-a^2}\right )\right ) \left (a^2+b^2 \left (\sin ^2(c+d x)-1\right )\right )}+\frac {a \left (-2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {\sin (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )+2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {\sin (c+d x)}}{\sqrt [4]{a^2-b^2}}+1\right )-\log \left (b \sin (c+d x)-\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\sin (c+d x)}+\sqrt {a^2-b^2}\right )+\log \left (b \sin (c+d x)+\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\sin (c+d x)}+\sqrt {a^2-b^2}\right )\right )}{4 \sqrt {2} \sqrt {b} \left (a^2-b^2\right )^{3/4}}\right ) \cos ^2(c+d x)}{(a+b \cos (c+d x)) \left (1-\sin ^2(c+d x)\right )}+\frac {2 \left (8 a^4-120 b^2 a^2-42 b^4\right ) \left (a+b \sqrt {1-\sin ^2(c+d x)}\right ) \left (\frac {5 a \left (a^2-b^2\right ) F_1\left (\frac {1}{4};\frac {1}{2},1;\frac {5}{4};\sin ^2(c+d x),\frac {b^2 \sin ^2(c+d x)}{b^2-a^2}\right ) \sqrt {\sin (c+d x)}}{\sqrt {1-\sin ^2(c+d x)} \left (5 \left (a^2-b^2\right ) F_1\left (\frac {1}{4};\frac {1}{2},1;\frac {5}{4};\sin ^2(c+d x),\frac {b^2 \sin ^2(c+d x)}{b^2-a^2}\right )-2 \left (2 F_1\left (\frac {5}{4};\frac {1}{2},2;\frac {9}{4};\sin ^2(c+d x),\frac {b^2 \sin ^2(c+d x)}{b^2-a^2}\right ) b^2+\left (b^2-a^2\right ) F_1\left (\frac {5}{4};\frac {3}{2},1;\frac {9}{4};\sin ^2(c+d x),\frac {b^2 \sin ^2(c+d x)}{b^2-a^2}\right )\right ) \sin ^2(c+d x)\right ) \left (a^2+b^2 \left (\sin ^2(c+d x)-1\right )\right )}-\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) \sqrt {b} \left (2 \tan ^{-1}\left (1-\frac {(1+i) \sqrt {b} \sqrt {\sin (c+d x)}}{\sqrt [4]{b^2-a^2}}\right )-2 \tan ^{-1}\left (\frac {(1+i) \sqrt {b} \sqrt {\sin (c+d x)}}{\sqrt [4]{b^2-a^2}}+1\right )+\log \left (i b \sin (c+d x)-(1+i) \sqrt {b} \sqrt [4]{b^2-a^2} \sqrt {\sin (c+d x)}+\sqrt {b^2-a^2}\right )-\log \left (i b \sin (c+d x)+(1+i) \sqrt {b} \sqrt [4]{b^2-a^2} \sqrt {\sin (c+d x)}+\sqrt {b^2-a^2}\right )\right )}{\left (b^2-a^2\right )^{3/4}}\right ) \cos (c+d x)}{(a+b \cos (c+d x)) \sqrt {1-\sin ^2(c+d x)}}\right ) \sin ^{\frac {5}{2}}(c+d x)}{24 (a-b)^3 (a+b)^3 d (e \sin (c+d x))^{5/2}} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[1/((a + b*Cos[c + d*x])^3*(e*Sin[c + d*x])^(5/2)),x]
[Out]
((b^3/(2*(a^2 - b^2)^2*(a + b*Cos[c + d*x])^2) + (15*a*b^3)/(4*(a^2 - b^2)^3*(a + b*Cos[c + d*x])) - (2*(-3*a^
2*b - b^3 + a^3*Cos[c + d*x] + 3*a*b^2*Cos[c + d*x])*Csc[c + d*x]^2)/(3*(a^2 - b^2)^3))*Sin[c + d*x]^3)/(d*(e*
Sin[c + d*x])^(5/2)) + (Sin[c + d*x]^(5/2)*((2*(8*a^3*b + 69*a*b^3)*Cos[c + d*x]^2*(a + b*Sqrt[1 - Sin[c + d*x
]^2])*((a*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[
b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c
+ d*x]] + b*Sin[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Si
n[c + d*x]]))/(4*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)) + (5*b*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x
]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Sin[c + d*x]]*Sqrt[1 - Sin[c + d*x]^2])/((-5*(a^2 - b^2)*AppellF1
[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 9/4,
Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + (a^2 - b^2)*AppellF1[5/4, 1/2, 1, 9/4, Sin[c + d*x]^2, (b
^2*Sin[c + d*x]^2)/(-a^2 + b^2)])*Sin[c + d*x]^2)*(a^2 + b^2*(-1 + Sin[c + d*x]^2)))))/((a + b*Cos[c + d*x])*(
1 - Sin[c + d*x]^2)) + (2*(8*a^4 - 120*a^2*b^2 - 42*b^4)*Cos[c + d*x]*(a + b*Sqrt[1 - Sin[c + d*x]^2])*(((-1/8
+ I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)
*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*S
qrt[Sin[c + d*x]] + I*b*Sin[c + d*x]] - Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c +
d*x]] + I*b*Sin[c + d*x]]))/(-a^2 + b^2)^(3/4) + (5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2,
(b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Sin[c + d*x]])/(Sqrt[1 - Sin[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4,
1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Sin[c +
d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Sin[c + d*x]^2, (b^2*Sin[
c + d*x]^2)/(-a^2 + b^2)])*Sin[c + d*x]^2)*(a^2 + b^2*(-1 + Sin[c + d*x]^2)))))/((a + b*Cos[c + d*x])*Sqrt[1 -
Sin[c + d*x]^2])))/(24*(a - b)^3*(a + b)^3*d*(e*Sin[c + d*x])^(5/2))
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*cos(d*x+c))^3/(e*sin(d*x+c))^(5/2),x, algorithm="fricas")
[Out]
Timed out
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*cos(d*x+c))^3/(e*sin(d*x+c))^(5/2),x, algorithm="giac")
[Out]
Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:Evaluation time:
13.57Unable to divide, perhaps due to rounding error%%%{%%{[-268435456,0]:[1,0,-2]%%},[4,9,0,2,1]%%%}+%%%{%%{[
805306368,0]:[1,0,-2]%%},[4,8,1,2,1]%%%}+%%%{%%{[-2147483648,0]:[1,0,-2]%%},[4,6,3,2,1]%%%}+%%%{%%{[1610612736
,0]:[1,0,-2]%%},[4,5,4,2,1]%%%}+%%%{%%{[1610612736,0]:[1,0,-2]%%},[4,4,5,2,1]%%%}+%%%{%%{[-2147483648,0]:[1,0,
-2]%%},[4,3,6,2,1]%%%}+%%%{%%{[805306368,0]:[1,0,-2]%%},[4,1,8,2,1]%%%}+%%%{%%{[-268435456,0]:[1,0,-2]%%},[4,0
,9,2,1]%%%}+%%%{%%{[-536870912,0]:[1,0,-2]%%},[2,9,0,2,1]%%%}+%%%{%%{[536870912,0]:[1,0,-2]%%},[2,8,1,2,1]%%%}
+%%%{%%{[2147483648,0]:[1,0,-2]%%},[2,7,2,2,1]%%%}+%%%{%%{[-2147483648,0]:[1,0,-2]%%},[2,6,3,2,1]%%%}+%%%{%%{[
-3221225472,0]:[1,0,-2]%%},[2,5,4,2,1]%%%}+%%%{%%{[3221225472,0]:[1,0,-2]%%},[2,4,5,2,1]%%%}+%%%{%%{[214748364
8,0]:[1,0,-2]%%},[2,3,6,2,1]%%%}+%%%{%%{[-2147483648,0]:[1,0,-2]%%},[2,2,7,2,1]%%%}+%%%{%%{[-536870912,0]:[1,0
,-2]%%},[2,1,8,2,1]%%%}+%%%{%%{[536870912,0]:[1,0,-2]%%},[2,0,9,2,1]%%%}+%%%{%%{[-268435456,0]:[1,0,-2]%%},[0,
9,0,2,1]%%%}+%%%{%%{[-268435456,0]:[1,0,-2]%%},[0,8,1,2,1]%%%}+%%%{%%{[1073741824,0]:[1,0,-2]%%},[0,7,2,2,1]%%
%}+%%%{%%{[1073741824,0]:[1,0,-2]%%},[0,6,3,2,1]%%%}+%%%{%%{[-1610612736,0]:[1,0,-2]%%},[0,5,4,2,1]%%%}+%%%{%%
{[-1610612736,0]:[1,0,-2]%%},[0,4,5,2,1]%%%}+%%%{%%{[1073741824,0]:[1,0,-2]%%},[0,3,6,2,1]%%%}+%%%{%%{[1073741
824,0]:[1,0,-2]%%},[0,2,7,2,1]%%%}+%%%{%%{[-268435456,0]:[1,0,-2]%%},[0,1,8,2,1]%%%}+%%%{%%{[-268435456,0]:[1,
0,-2]%%},[0,0,9,2,1]%%%} / %%%{1,[4,2,0,0,0]%%%}+%%%{-2,[4,1,1,0,0]%%%}+%%%{1,[4,0,2,0,0]%%%}+%%%{2,[2,2,0,0,0
]%%%}+%%%{-2,[2,0,2,0,0]%%%}+%%%{1,[0,2,0,0,0]%%%}+%%%{2,[0,1,1,0,0]%%%}+%%%{1,[0,0,2,0,0]%%%} Error: Bad Argu
ment Value
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maple [B] time = 3.14, size = 4661, normalized size = 7.41 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(a+b*cos(d*x+c))^3/(e*sin(d*x+c))^(5/2),x)
[Out]
63/16/d/e*b^3/(a-b)^3/(a+b)^3*(e^2*(a^2-b^2)/b^2)^(1/4)/(a^2*e^2-b^2*e^2)*2^(1/2)*arctan(2^(1/2)/(e^2*(a^2-b^2
)/b^2)^(1/4)*(e*sin(d*x+c))^(1/2)+1)*a^2+63/16/d/e*b^3/(a-b)^3/(a+b)^3*(e^2*(a^2-b^2)/b^2)^(1/4)/(a^2*e^2-b^2*
e^2)*2^(1/2)*arctan(2^(1/2)/(e^2*(a^2-b^2)/b^2)^(1/4)*(e*sin(d*x+c))^(1/2)-1)*a^2+63/32/d/e*b^3/(a-b)^3/(a+b)^
3*(e^2*(a^2-b^2)/b^2)^(1/4)/(a^2*e^2-b^2*e^2)*2^(1/2)*ln((e*sin(d*x+c)+(e^2*(a^2-b^2)/b^2)^(1/4)*(e*sin(d*x+c)
)^(1/2)*2^(1/2)+(e^2*(a^2-b^2)/b^2)^(1/2))/(e*sin(d*x+c)-(e^2*(a^2-b^2)/b^2)^(1/4)*(e*sin(d*x+c))^(1/2)*2^(1/2
)+(e^2*(a^2-b^2)/b^2)^(1/2)))*a^2+1/d/e^2*a/cos(d*x+c)/(e*sin(d*x+c))^(1/2)/(a^2-b^2)^3/(cos(d*x+c)^2-1)*(-sin
(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(5/2)*EllipticF((-sin(d*x+c)+1)^(1/2),1/2*2^(1/2))*b^2-1/2/
d/e^2*sin(d*x+c)*cos(d*x+c)*a/(e*sin(d*x+c))^(1/2)*b^4/(a+b)^2/(a-b)^2/(a^2-b^2)/(-cos(d*x+c)^2*b^2+a^2)-1/d/e
^2*sin(d*x+c)*cos(d*x+c)*a/(e*sin(d*x+c))^(1/2)*b^4/(a+b)/(a-b)/(a^2-b^2)/(-cos(d*x+c)^2*b^2+a^2)^2-13/4/d/e^2
*sin(d*x+c)*cos(d*x+c)*a/(e*sin(d*x+c))^(1/2)*b^4/(a+b)/(a-b)/(a^2-b^2)^2/(-cos(d*x+c)^2*b^2+a^2)+3/2/d/e^2*si
n(d*x+c)*cos(d*x+c)/a/(e*sin(d*x+c))^(1/2)*b^6/(a+b)/(a-b)/(a^2-b^2)^2/(-cos(d*x+c)^2*b^2+a^2)-3/2/d/e^2*sin(d
*x+c)*cos(d*x+c)/a/(e*sin(d*x+c))^(1/2)*b^6/(a+b)^2/(a-b)^2/(a^2-b^2)/(-cos(d*x+c)^2*b^2+a^2)+2/3/d/e*b^3/(a^2
-b^2)^3/(e*sin(d*x+c))^(3/2)+1/2/d/e^2*a^3/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b/(a-b)^3/(a+b)^3/(-a^2+b^2)^(1/2)*
(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)/(1-(-a^2+b^2)^(1/2)/b)*EllipticPi((-sin(d*x+c)+1
)^(1/2),1/(1-(-a^2+b^2)^(1/2)/b),1/2*2^(1/2))-1/2/d/e^2*a^3/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b/(a-b)^3/(a+b)^3/
(-a^2+b^2)^(1/2)*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)/(1+(-a^2+b^2)^(1/2)/b)*Elliptic
Pi((-sin(d*x+c)+1)^(1/2),1/(1+(-a^2+b^2)^(1/2)/b),1/2*2^(1/2))+3/2/d/e^2*a/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^3
/(a-b)^3/(a+b)^3/(-a^2+b^2)^(1/2)*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)/(1-(-a^2+b^2)^
(1/2)/b)*EllipticPi((-sin(d*x+c)+1)^(1/2),1/(1-(-a^2+b^2)^(1/2)/b),1/2*2^(1/2))-3/2/d/e^2*a/cos(d*x+c)/(e*sin(
d*x+c))^(1/2)*b^3/(a-b)^3/(a+b)^3/(-a^2+b^2)^(1/2)*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/
2)/(1+(-a^2+b^2)^(1/2)/b)*EllipticPi((-sin(d*x+c)+1)^(1/2),1/(1+(-a^2+b^2)^(1/2)/b),1/2*2^(1/2))+13/8/d/e^2*a/
cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^3/(a+b)^2/(a-b)^2/(a^2-b^2)/(-a^2+b^2)^(1/2)*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*
x+c)+2)^(1/2)*sin(d*x+c)^(1/2)/(1-(-a^2+b^2)^(1/2)/b)*EllipticPi((-sin(d*x+c)+1)^(1/2),1/(1-(-a^2+b^2)^(1/2)/b
),1/2*2^(1/2))+9/4/d/e^2*a/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^3/(a+b)/(a-b)/(a^2-b^2)^2/(-a^2+b^2)^(1/2)*(-sin(
d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)/(1+(-a^2+b^2)^(1/2)/b)*EllipticPi((-sin(d*x+c)+1)^(1/2
),1/(1+(-a^2+b^2)^(1/2)/b),1/2*2^(1/2))-3/4/d/e^2/a/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^5/(a+b)/(a-b)/(a^2-b^2)^
2/(-a^2+b^2)^(1/2)*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)/(1+(-a^2+b^2)^(1/2)/b)*Ellipt
icPi((-sin(d*x+c)+1)^(1/2),1/(1+(-a^2+b^2)^(1/2)/b),1/2*2^(1/2))+5/8/d/e^2*a^3/cos(d*x+c)/(e*sin(d*x+c))^(1/2)
*b/(a+b)^2/(a-b)^2/(a^2-b^2)/(-a^2+b^2)^(1/2)*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)/(1
-(-a^2+b^2)^(1/2)/b)*EllipticPi((-sin(d*x+c)+1)^(1/2),1/(1-(-a^2+b^2)^(1/2)/b),1/2*2^(1/2))-3/4/d/e^2/a/cos(d*
x+c)/(e*sin(d*x+c))^(1/2)*b^5/(a+b)^2/(a-b)^2/(a^2-b^2)/(-a^2+b^2)^(1/2)*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2
)^(1/2)*sin(d*x+c)^(1/2)/(1-(-a^2+b^2)^(1/2)/b)*EllipticPi((-sin(d*x+c)+1)^(1/2),1/(1-(-a^2+b^2)^(1/2)/b),1/2*
2^(1/2))-5/8/d/e^2*a^3/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b/(a+b)^2/(a-b)^2/(a^2-b^2)/(-a^2+b^2)^(1/2)*(-sin(d*x+
c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)/(1+(-a^2+b^2)^(1/2)/b)*EllipticPi((-sin(d*x+c)+1)^(1/2),1/
(1+(-a^2+b^2)^(1/2)/b),1/2*2^(1/2))+3/4/d/e^2/a/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^5/(a+b)^2/(a-b)^2/(a^2-b^2)/
(-a^2+b^2)^(1/2)*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)/(1+(-a^2+b^2)^(1/2)/b)*Elliptic
Pi((-sin(d*x+c)+1)^(1/2),1/(1+(-a^2+b^2)^(1/2)/b),1/2*2^(1/2))+45/16/d/e^2*a^3/cos(d*x+c)/(e*sin(d*x+c))^(1/2)
*b/(a+b)/(a-b)/(a^2-b^2)^2/(-a^2+b^2)^(1/2)*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)/(1-(
-a^2+b^2)^(1/2)/b)*EllipticPi((-sin(d*x+c)+1)^(1/2),1/(1-(-a^2+b^2)^(1/2)/b),1/2*2^(1/2))-9/4/d/e^2*a/cos(d*x+
c)/(e*sin(d*x+c))^(1/2)*b^3/(a+b)/(a-b)/(a^2-b^2)^2/(-a^2+b^2)^(1/2)*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1
/2)*sin(d*x+c)^(1/2)/(1-(-a^2+b^2)^(1/2)/b)*EllipticPi((-sin(d*x+c)+1)^(1/2),1/(1-(-a^2+b^2)^(1/2)/b),1/2*2^(1
/2))+3/4/d/e^2/a/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^5/(a+b)/(a-b)/(a^2-b^2)^2/(-a^2+b^2)^(1/2)*(-sin(d*x+c)+1)^
(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)/(1-(-a^2+b^2)^(1/2)/b)*EllipticPi((-sin(d*x+c)+1)^(1/2),1/(1-(-a
^2+b^2)^(1/2)/b),1/2*2^(1/2))-45/16/d/e^2*a^3/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b/(a+b)/(a-b)/(a^2-b^2)^2/(-a^2+
b^2)^(1/2)*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)/(1+(-a^2+b^2)^(1/2)/b)*EllipticPi((-s
in(d*x+c)+1)^(1/2),1/(1+(-a^2+b^2)^(1/2)/b),1/2*2^(1/2))-13/8/d/e^2*a/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^3/(a+b
)^2/(a-b)^2/(a^2-b^2)/(-a^2+b^2)^(1/2)*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)/(1+(-a^2+
b^2)^(1/2)/b)*EllipticPi((-sin(d*x+c)+1)^(1/2),1/(1+(-a^2+b^2)^(1/2)/b),1/2*2^(1/2))+1/3/d/e^2*a^3/cos(d*x+c)/
(e*sin(d*x+c))^(1/2)/(a^2-b^2)^3/(cos(d*x+c)^2-1)*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(5/2
)*EllipticF((-sin(d*x+c)+1)^(1/2),1/2*2^(1/2))+2/d/e^2*a*cos(d*x+c)/(e*sin(d*x+c))^(1/2)/(a^2-b^2)^3/(cos(d*x+
c)^2-1)*sin(d*x+c)*b^2+7/8/d/e*b^5/(a-b)^3/(a+b)^3*(e^2*(a^2-b^2)/b^2)^(1/4)/(a^2*e^2-b^2*e^2)*2^(1/2)*arctan(
2^(1/2)/(e^2*(a^2-b^2)/b^2)^(1/4)*(e*sin(d*x+c))^(1/2)+1)+7/8/d/e*b^5/(a-b)^3/(a+b)^3*(e^2*(a^2-b^2)/b^2)^(1/4
)/(a^2*e^2-b^2*e^2)*2^(1/2)*arctan(2^(1/2)/(e^2*(a^2-b^2)/b^2)^(1/4)*(e*sin(d*x+c))^(1/2)-1)+7/16/d/e*b^5/(a-b
)^3/(a+b)^3*(e^2*(a^2-b^2)/b^2)^(1/4)/(a^2*e^2-b^2*e^2)*2^(1/2)*ln((e*sin(d*x+c)+(e^2*(a^2-b^2)/b^2)^(1/4)*(e*
sin(d*x+c))^(1/2)*2^(1/2)+(e^2*(a^2-b^2)/b^2)^(1/2))/(e*sin(d*x+c)-(e^2*(a^2-b^2)/b^2)^(1/4)*(e*sin(d*x+c))^(1
/2)*2^(1/2)+(e^2*(a^2-b^2)/b^2)^(1/2)))+2/d/e*b/(a^2-b^2)^3/(e*sin(d*x+c))^(3/2)*a^2+1/2/d/e*b^7/(a-b)^3/(a+b)
^3/(-b^2*cos(d*x+c)^2*e^2+a^2*e^2)^2*(e*sin(d*x+c))^(5/2)-1/2/d*e*b^7/(a-b)^3/(a+b)^3/(-b^2*cos(d*x+c)^2*e^2+a
^2*e^2)^2*(e*sin(d*x+c))^(1/2)-1/4/d/e^2*a/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^2/(a+b)^2/(a-b)^2/(a^2-b^2)*(-sin
(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)*EllipticF((-sin(d*x+c)+1)^(1/2),1/2*2^(1/2))-13/8/d/e
^2*a/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^2/(a+b)/(a-b)/(a^2-b^2)^2*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*
sin(d*x+c)^(1/2)*EllipticF((-sin(d*x+c)+1)^(1/2),1/2*2^(1/2))+3/4/d/e^2/a/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^4/
(a+b)/(a-b)/(a^2-b^2)^2*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)*EllipticF((-sin(d*x+c)+1
)^(1/2),1/2*2^(1/2))-3/4/d/e^2/a/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^4/(a+b)^2/(a-b)^2/(a^2-b^2)*(-sin(d*x+c)+1)
^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)*EllipticF((-sin(d*x+c)+1)^(1/2),1/2*2^(1/2))-15/4/d*e*b^5/(a-b)
^3/(a+b)^3/(-b^2*cos(d*x+c)^2*e^2+a^2*e^2)^2*(e*sin(d*x+c))^(1/2)*a^2+13/4/d/e*b^5/(a-b)^3/(a+b)^3/(-b^2*cos(d
*x+c)^2*e^2+a^2*e^2)^2*(e*sin(d*x+c))^(5/2)*a^2+17/4/d*e*b^3/(a-b)^3/(a+b)^3/(-b^2*cos(d*x+c)^2*e^2+a^2*e^2)^2
*(e*sin(d*x+c))^(1/2)*a^4+2/3/d/e^2*a^3*cos(d*x+c)/(e*sin(d*x+c))^(1/2)/(a^2-b^2)^3/(cos(d*x+c)^2-1)*sin(d*x+c
)
________________________________________________________________________________________
maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*cos(d*x+c))^3/(e*sin(d*x+c))^(5/2),x, algorithm="maxima")
[Out]
Timed out
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (e\,\sin \left (c+d\,x\right )\right )}^{5/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/((e*sin(c + d*x))^(5/2)*(a + b*cos(c + d*x))^3),x)
[Out]
int(1/((e*sin(c + d*x))^(5/2)*(a + b*cos(c + d*x))^3), x)
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*cos(d*x+c))**3/(e*sin(d*x+c))**(5/2),x)
[Out]
Timed out
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