3.8.36 \(\int \frac {(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^{7/2}} \, dx\) [736]

Optimal. Leaf size=460 \[ \frac {2 (b c-a d)^2 \cos (e+f x)}{5 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{5/2}}-\frac {4 (b c-a d) \left (4 a c d+b \left (c^2-5 d^2\right )\right ) \cos (e+f x)}{15 d \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^{3/2}}+\frac {2 \left (a^2 d^2 \left (23 c^2+9 d^2\right )-a b \left (6 c^3 d+58 c d^3\right )-b^2 \left (2 c^4-19 c^2 d^2-15 d^4\right )\right ) \cos (e+f x)}{15 d \left (c^2-d^2\right )^3 f \sqrt {c+d \sin (e+f x)}}+\frac {2 \left (a^2 d^2 \left (23 c^2+9 d^2\right )-a b \left (6 c^3 d+58 c d^3\right )-b^2 \left (2 c^4-19 c^2 d^2-15 d^4\right )\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{15 d^2 \left (c^2-d^2\right )^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {4 (b c-a d) \left (4 a c d+b \left (c^2-5 d^2\right )\right ) F\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{15 d^2 \left (c^2-d^2\right )^2 f \sqrt {c+d \sin (e+f x)}} \]

[Out]

2/5*(-a*d+b*c)^2*cos(f*x+e)/d/(c^2-d^2)/f/(c+d*sin(f*x+e))^(5/2)-4/15*(-a*d+b*c)*(4*a*c*d+b*(c^2-5*d^2))*cos(f
*x+e)/d/(c^2-d^2)^2/f/(c+d*sin(f*x+e))^(3/2)+2/15*(a^2*d^2*(23*c^2+9*d^2)-a*b*(6*c^3*d+58*c*d^3)-b^2*(2*c^4-19
*c^2*d^2-15*d^4))*cos(f*x+e)/d/(c^2-d^2)^3/f/(c+d*sin(f*x+e))^(1/2)-2/15*(a^2*d^2*(23*c^2+9*d^2)-a*b*(6*c^3*d+
58*c*d^3)-b^2*(2*c^4-19*c^2*d^2-15*d^4))*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*Ellipti
cE(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2)*(d/(c+d))^(1/2))*(c+d*sin(f*x+e))^(1/2)/d^2/(c^2-d^2)^3/f/((c+d*sin(f*x+e
))/(c+d))^(1/2)-4/15*(-a*d+b*c)*(4*a*c*d+b*(c^2-5*d^2))*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1
/2*f*x)*EllipticF(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2)*(d/(c+d))^(1/2))*((c+d*sin(f*x+e))/(c+d))^(1/2)/d^2/(c^2-d
^2)^2/f/(c+d*sin(f*x+e))^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.56, antiderivative size = 460, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2869, 2833, 2831, 2742, 2740, 2734, 2732} \begin {gather*} \frac {2 \left (a^2 d^2 \left (23 c^2+9 d^2\right )-a b \left (6 c^3 d+58 c d^3\right )-\left (b^2 \left (2 c^4-19 c^2 d^2-15 d^4\right )\right )\right ) \cos (e+f x)}{15 d f \left (c^2-d^2\right )^3 \sqrt {c+d \sin (e+f x)}}+\frac {2 \left (a^2 d^2 \left (23 c^2+9 d^2\right )-a b \left (6 c^3 d+58 c d^3\right )-\left (b^2 \left (2 c^4-19 c^2 d^2-15 d^4\right )\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{15 d^2 f \left (c^2-d^2\right )^3 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {2 (b c-a d)^2 \cos (e+f x)}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}-\frac {4 \left (4 a c d+b \left (c^2-5 d^2\right )\right ) (b c-a d) \cos (e+f x)}{15 d f \left (c^2-d^2\right )^2 (c+d \sin (e+f x))^{3/2}}+\frac {4 \left (4 a c d+b \left (c^2-5 d^2\right )\right ) (b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} F\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{15 d^2 f \left (c^2-d^2\right )^2 \sqrt {c+d \sin (e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^(7/2),x]

[Out]

(2*(b*c - a*d)^2*Cos[e + f*x])/(5*d*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(5/2)) - (4*(b*c - a*d)*(4*a*c*d + b*(c
^2 - 5*d^2))*Cos[e + f*x])/(15*d*(c^2 - d^2)^2*f*(c + d*Sin[e + f*x])^(3/2)) + (2*(a^2*d^2*(23*c^2 + 9*d^2) -
a*b*(6*c^3*d + 58*c*d^3) - b^2*(2*c^4 - 19*c^2*d^2 - 15*d^4))*Cos[e + f*x])/(15*d*(c^2 - d^2)^3*f*Sqrt[c + d*S
in[e + f*x]]) + (2*(a^2*d^2*(23*c^2 + 9*d^2) - a*b*(6*c^3*d + 58*c*d^3) - b^2*(2*c^4 - 19*c^2*d^2 - 15*d^4))*E
llipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(15*d^2*(c^2 - d^2)^3*f*Sqrt[(c + d*Sin[
e + f*x])/(c + d)]) + (4*(b*c - a*d)*(4*a*c*d + b*(c^2 - 5*d^2))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*
Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(15*d^2*(c^2 - d^2)^2*f*Sqrt[c + d*Sin[e + f*x]])

Rule 2732

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2
+ d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2734

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c +
d*x])/(a + b)], Int[Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2740

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*Sqrt[a + b]))*EllipticF[(1/2)*(c - P
i/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2742

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a
+ b*Sin[c + d*x]], Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2831

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[(b*c
- a*d)/b, Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] + Dist[d/b, Int[Sqrt[a + b*Sin[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 2833

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

Rule 2869

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

Rubi steps

\begin {align*} \int \frac {(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^{7/2}} \, dx &=\frac {2 (b c-a d)^2 \cos (e+f x)}{5 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{5/2}}+\frac {2 \int \frac {\frac {5}{2} d \left (\left (a^2+b^2\right ) c-2 a b d\right )+\frac {1}{2} \left (6 a b c d-3 a^2 d^2+b^2 \left (2 c^2-5 d^2\right )\right ) \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}} \, dx}{5 d \left (c^2-d^2\right )}\\ &=\frac {2 (b c-a d)^2 \cos (e+f x)}{5 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{5/2}}-\frac {4 (b c-a d) \left (4 a c d+b \left (c^2-5 d^2\right )\right ) \cos (e+f x)}{15 d \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^{3/2}}-\frac {4 \int \frac {\frac {3}{4} d \left (16 a b c d-a^2 \left (5 c^2+3 d^2\right )-b^2 \left (3 c^2+5 d^2\right )\right )-\frac {1}{2} (b c-a d) \left (b c^2+4 a c d-5 b d^2\right ) \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx}{15 d \left (c^2-d^2\right )^2}\\ &=\frac {2 (b c-a d)^2 \cos (e+f x)}{5 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{5/2}}-\frac {4 (b c-a d) \left (4 a c d+b \left (c^2-5 d^2\right )\right ) \cos (e+f x)}{15 d \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^{3/2}}+\frac {2 \left (a^2 d^2 \left (23 c^2+9 d^2\right )-a b \left (6 c^3 d+58 c d^3\right )-b^2 \left (2 c^4-19 c^2 d^2-15 d^4\right )\right ) \cos (e+f x)}{15 d \left (c^2-d^2\right )^3 f \sqrt {c+d \sin (e+f x)}}+\frac {8 \int \frac {-\frac {1}{8} d \left (2 a b d \left (27 c^2+5 d^2\right )-b^2 c \left (7 c^2+25 d^2\right )-a^2 \left (15 c^3+17 c d^2\right )\right )+\frac {1}{8} \left (a^2 d^2 \left (23 c^2+9 d^2\right )-a b \left (6 c^3 d+58 c d^3\right )-b^2 \left (2 c^4-19 c^2 d^2-15 d^4\right )\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx}{15 d \left (c^2-d^2\right )^3}\\ &=\frac {2 (b c-a d)^2 \cos (e+f x)}{5 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{5/2}}-\frac {4 (b c-a d) \left (4 a c d+b \left (c^2-5 d^2\right )\right ) \cos (e+f x)}{15 d \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^{3/2}}+\frac {2 \left (a^2 d^2 \left (23 c^2+9 d^2\right )-a b \left (6 c^3 d+58 c d^3\right )-b^2 \left (2 c^4-19 c^2 d^2-15 d^4\right )\right ) \cos (e+f x)}{15 d \left (c^2-d^2\right )^3 f \sqrt {c+d \sin (e+f x)}}+\frac {\left (a^2 d^2 \left (23 c^2+9 d^2\right )-a b \left (6 c^3 d+58 c d^3\right )-b^2 \left (2 c^4-19 c^2 d^2-15 d^4\right )\right ) \int \sqrt {c+d \sin (e+f x)} \, dx}{15 d^2 \left (c^2-d^2\right )^3}--\frac {\left (8 \left (-\frac {1}{8} d^2 \left (2 a b d \left (27 c^2+5 d^2\right )-b^2 c \left (7 c^2+25 d^2\right )-a^2 \left (15 c^3+17 c d^2\right )\right )-\frac {1}{8} c \left (a^2 d^2 \left (23 c^2+9 d^2\right )-a b \left (6 c^3 d+58 c d^3\right )-b^2 \left (2 c^4-19 c^2 d^2-15 d^4\right )\right )\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{15 d^2 \left (c^2-d^2\right )^3}\\ &=\frac {2 (b c-a d)^2 \cos (e+f x)}{5 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{5/2}}-\frac {4 (b c-a d) \left (4 a c d+b \left (c^2-5 d^2\right )\right ) \cos (e+f x)}{15 d \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^{3/2}}+\frac {2 \left (a^2 d^2 \left (23 c^2+9 d^2\right )-a b \left (6 c^3 d+58 c d^3\right )-b^2 \left (2 c^4-19 c^2 d^2-15 d^4\right )\right ) \cos (e+f x)}{15 d \left (c^2-d^2\right )^3 f \sqrt {c+d \sin (e+f x)}}+\frac {\left (\left (a^2 d^2 \left (23 c^2+9 d^2\right )-a b \left (6 c^3 d+58 c d^3\right )-b^2 \left (2 c^4-19 c^2 d^2-15 d^4\right )\right ) \sqrt {c+d \sin (e+f x)}\right ) \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}} \, dx}{15 d^2 \left (c^2-d^2\right )^3 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}--\frac {\left (8 \left (-\frac {1}{8} d^2 \left (2 a b d \left (27 c^2+5 d^2\right )-b^2 c \left (7 c^2+25 d^2\right )-a^2 \left (15 c^3+17 c d^2\right )\right )-\frac {1}{8} c \left (a^2 d^2 \left (23 c^2+9 d^2\right )-a b \left (6 c^3 d+58 c d^3\right )-b^2 \left (2 c^4-19 c^2 d^2-15 d^4\right )\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right ) \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}} \, dx}{15 d^2 \left (c^2-d^2\right )^3 \sqrt {c+d \sin (e+f x)}}\\ &=\frac {2 (b c-a d)^2 \cos (e+f x)}{5 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{5/2}}-\frac {4 (b c-a d) \left (4 a c d+b \left (c^2-5 d^2\right )\right ) \cos (e+f x)}{15 d \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^{3/2}}+\frac {2 \left (a^2 d^2 \left (23 c^2+9 d^2\right )-a b \left (6 c^3 d+58 c d^3\right )-b^2 \left (2 c^4-19 c^2 d^2-15 d^4\right )\right ) \cos (e+f x)}{15 d \left (c^2-d^2\right )^3 f \sqrt {c+d \sin (e+f x)}}+\frac {2 \left (a^2 d^2 \left (23 c^2+9 d^2\right )-a b \left (6 c^3 d+58 c d^3\right )-b^2 \left (2 c^4-19 c^2 d^2-15 d^4\right )\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{15 d^2 \left (c^2-d^2\right )^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {4 (b c-a d) \left (b c^2+4 a c d-5 b d^2\right ) F\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{15 d^2 \left (c^2-d^2\right )^2 f \sqrt {c+d \sin (e+f x)}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 5.36, size = 424, normalized size = 0.92 \begin {gather*} \frac {2 \left (-\frac {\left (d^2 \left (-2 a b d \left (27 c^2+5 d^2\right )+b^2 c \left (7 c^2+25 d^2\right )+a^2 \left (15 c^3+17 c d^2\right )\right ) F\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-\left (-a^2 d^2 \left (23 c^2+9 d^2\right )+a b \left (6 c^3 d+58 c d^3\right )+b^2 \left (2 c^4-19 c^2 d^2-15 d^4\right )\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c F\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )\right )\right ) \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{5/2}}{(c-d)^3 (c+d)}+\frac {d \cos (e+f x) \left (3 (b c-a d)^2 \left (c^2-d^2\right )^2-2 \left (c^2-d^2\right ) \left (-4 a^2 c d^2+a b d \left (3 c^2+5 d^2\right )+b^2 \left (c^3-5 c d^2\right )\right ) (c+d \sin (e+f x))-\left (-a^2 d^2 \left (23 c^2+9 d^2\right )+a b \left (6 c^3 d+58 c d^3\right )+b^2 \left (2 c^4-19 c^2 d^2-15 d^4\right )\right ) (c+d \sin (e+f x))^2\right )}{\left (c^2-d^2\right )^3}\right )}{15 d^2 f (c+d \sin (e+f x))^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^(7/2),x]

[Out]

(2*(-(((d^2*(-2*a*b*d*(27*c^2 + 5*d^2) + b^2*c*(7*c^2 + 25*d^2) + a^2*(15*c^3 + 17*c*d^2))*EllipticF[(-2*e + P
i - 2*f*x)/4, (2*d)/(c + d)] - (-(a^2*d^2*(23*c^2 + 9*d^2)) + a*b*(6*c^3*d + 58*c*d^3) + b^2*(2*c^4 - 19*c^2*d
^2 - 15*d^4))*((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*EllipticF[(-2*e + Pi - 2*f*x)/4, (2
*d)/(c + d)]))*((c + d*Sin[e + f*x])/(c + d))^(5/2))/((c - d)^3*(c + d))) + (d*Cos[e + f*x]*(3*(b*c - a*d)^2*(
c^2 - d^2)^2 - 2*(c^2 - d^2)*(-4*a^2*c*d^2 + a*b*d*(3*c^2 + 5*d^2) + b^2*(c^3 - 5*c*d^2))*(c + d*Sin[e + f*x])
- (-(a^2*d^2*(23*c^2 + 9*d^2)) + a*b*(6*c^3*d + 58*c*d^3) + b^2*(2*c^4 - 19*c^2*d^2 - 15*d^4))*(c + d*Sin[e +
f*x])^2))/(c^2 - d^2)^3))/(15*d^2*f*(c + d*Sin[e + f*x])^(5/2))

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1449\) vs. \(2(502)=1004\).
time = 42.31, size = 1450, normalized size = 3.15

method	result	size
default	\(\text {Expression too large to display}\)	\(1450\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^(7/2),x,method=_RETURNVERBOSE)

[Out]

(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*(b^2/d^2*(2*d*cos(f*x+e)^2/(c^2-d^2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)
^(1/2)+2*c/(c^2-d^2)*(1/d*c-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin(f*x+e)-1)*
d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^
(1/2))+2/(c^2-d^2)*d*(1/d*c-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin(f*x+e)-1)*
d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*((-1/d*c-1)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((
c-d)/(c+d))^(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))))+(a^2*d^2-2*a*b*c*d+b^2*c^2)
/d^2*(2/5/(c^2-d^2)/d^2*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(sin(f*x+e)+1/d*c)^3+16/15*c/(c^2-d^2)^2/d*(-(
-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(sin(f*x+e)+1/d*c)^2+2/15*d*cos(f*x+e)^2/(c^2-d^2)^3*(23*c^2+9*d^2)/(-(-d
*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)+2*(15*c^3+17*c*d^2)/(15*c^6-45*c^4*d^2+45*c^2*d^4-15*d^6)*(1/d*c-1)*((c+d*s
in(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin(f*x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos
(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+2/15*d*(23*c^2+9*d^2)/(c^2-d^2)
^3*(1/d*c-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin(f*x+e)-1)*d/(c-d))^(1/2)/(-(
-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*((-1/d*c-1)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))
+EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))))+2*b*(a*d-b*c)/d^2*(2/3/(c^2-d^2)/d*(-(-d*sin(
f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(sin(f*x+e)+1/d*c)^2+8/3*d*cos(f*x+e)^2/(c^2-d^2)^2*c/(-(-d*sin(f*x+e)-c)*cos(f*
x+e)^2)^(1/2)+2*(3*c^2+d^2)/(3*c^4-6*c^2*d^2+3*d^4)*(1/d*c-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))
/(c+d))^(1/2)*((-sin(f*x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+
e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+8/3*d*c/(c^2-d^2)^2*(1/d*c-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f
*x+e))/(c+d))^(1/2)*((-sin(f*x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*((-1/d*c-1)*Ellipt
icE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))
^(1/2)))))/cos(f*x+e)/(c+d*sin(f*x+e))^(1/2)/f

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^(7/2),x, algorithm="maxima")

[Out]

integrate((b*sin(f*x + e) + a)^2/(d*sin(f*x + e) + c)^(7/2), x)

________________________________________________________________________________________

Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.35, size = 2321, normalized size = 5.05 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^(7/2),x, algorithm="fricas")

[Out]

1/45*((3*sqrt(2)*(4*b^2*c^6*d^2 + 12*a*b*c^5*d^3 - 46*a*b*c^3*d^5 - 30*a*b*c*d^7 - (a^2 + 17*b^2)*c^4*d^4 + 3*
(11*a^2 + 15*b^2)*c^2*d^6)*cos(f*x + e)^2 + (sqrt(2)*(4*b^2*c^5*d^3 + 12*a*b*c^4*d^4 - 46*a*b*c^2*d^6 - 30*a*b
*d^8 - (a^2 + 17*b^2)*c^3*d^5 + 3*(11*a^2 + 15*b^2)*c*d^7)*cos(f*x + e)^2 - sqrt(2)*(12*b^2*c^7*d + 36*a*b*c^6
*d^2 - 126*a*b*c^4*d^4 - 136*a*b*c^2*d^6 - 30*a*b*d^8 - (3*a^2 + 47*b^2)*c^5*d^3 + 2*(49*a^2 + 59*b^2)*c^3*d^5
+ 3*(11*a^2 + 15*b^2)*c*d^7))*sin(f*x + e) - sqrt(2)*(4*b^2*c^8 + 12*a*b*c^7*d - 10*a*b*c^5*d^3 - 168*a*b*c^3
*d^5 - 90*a*b*c*d^7 - (a^2 + 5*b^2)*c^6*d^2 + 6*(5*a^2 - b^2)*c^4*d^4 + 9*(11*a^2 + 15*b^2)*c^2*d^6))*sqrt(I*d
)*weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) - 3*I*d
*sin(f*x + e) - 2*I*c)/d) + (3*sqrt(2)*(4*b^2*c^6*d^2 + 12*a*b*c^5*d^3 - 46*a*b*c^3*d^5 - 30*a*b*c*d^7 - (a^2
+ 17*b^2)*c^4*d^4 + 3*(11*a^2 + 15*b^2)*c^2*d^6)*cos(f*x + e)^2 + (sqrt(2)*(4*b^2*c^5*d^3 + 12*a*b*c^4*d^4 - 4
6*a*b*c^2*d^6 - 30*a*b*d^8 - (a^2 + 17*b^2)*c^3*d^5 + 3*(11*a^2 + 15*b^2)*c*d^7)*cos(f*x + e)^2 - sqrt(2)*(12*
b^2*c^7*d + 36*a*b*c^6*d^2 - 126*a*b*c^4*d^4 - 136*a*b*c^2*d^6 - 30*a*b*d^8 - (3*a^2 + 47*b^2)*c^5*d^3 + 2*(49
*a^2 + 59*b^2)*c^3*d^5 + 3*(11*a^2 + 15*b^2)*c*d^7))*sin(f*x + e) - sqrt(2)*(4*b^2*c^8 + 12*a*b*c^7*d - 10*a*b
*c^5*d^3 - 168*a*b*c^3*d^5 - 90*a*b*c*d^7 - (a^2 + 5*b^2)*c^6*d^2 + 6*(5*a^2 - b^2)*c^4*d^4 + 9*(11*a^2 + 15*b
^2)*c^2*d^6))*sqrt(-I*d)*weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, 1/3*(
3*d*cos(f*x + e) + 3*I*d*sin(f*x + e) + 2*I*c)/d) + 3*(3*sqrt(2)*(2*I*b^2*c^5*d^3 + 6*I*a*b*c^4*d^4 + 58*I*a*b
*c^2*d^6 - I*(23*a^2 + 19*b^2)*c^3*d^5 - 3*I*(3*a^2 + 5*b^2)*c*d^7)*cos(f*x + e)^2 + (sqrt(2)*(2*I*b^2*c^4*d^4
+ 6*I*a*b*c^3*d^5 + 58*I*a*b*c*d^7 - I*(23*a^2 + 19*b^2)*c^2*d^6 - 3*I*(3*a^2 + 5*b^2)*d^8)*cos(f*x + e)^2 +
sqrt(2)*(-6*I*b^2*c^6*d^2 - 18*I*a*b*c^5*d^3 - 180*I*a*b*c^3*d^5 - 58*I*a*b*c*d^7 + I*(69*a^2 + 55*b^2)*c^4*d^
4 + 2*I*(25*a^2 + 32*b^2)*c^2*d^6 + 3*I*(3*a^2 + 5*b^2)*d^8))*sin(f*x + e) + sqrt(2)*(-2*I*b^2*c^7*d - 6*I*a*b
*c^6*d^2 - 76*I*a*b*c^4*d^4 - 174*I*a*b*c^2*d^6 + I*(23*a^2 + 13*b^2)*c^5*d^3 + 6*I*(13*a^2 + 12*b^2)*c^3*d^5
+ 9*I*(3*a^2 + 5*b^2)*c*d^7))*sqrt(I*d)*weierstrassZeta(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/
d^3, weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) - 3*
I*d*sin(f*x + e) - 2*I*c)/d)) + 3*(3*sqrt(2)*(-2*I*b^2*c^5*d^3 - 6*I*a*b*c^4*d^4 - 58*I*a*b*c^2*d^6 + I*(23*a^
2 + 19*b^2)*c^3*d^5 + 3*I*(3*a^2 + 5*b^2)*c*d^7)*cos(f*x + e)^2 + (sqrt(2)*(-2*I*b^2*c^4*d^4 - 6*I*a*b*c^3*d^5
- 58*I*a*b*c*d^7 + I*(23*a^2 + 19*b^2)*c^2*d^6 + 3*I*(3*a^2 + 5*b^2)*d^8)*cos(f*x + e)^2 + sqrt(2)*(6*I*b^2*c
^6*d^2 + 18*I*a*b*c^5*d^3 + 180*I*a*b*c^3*d^5 + 58*I*a*b*c*d^7 - I*(69*a^2 + 55*b^2)*c^4*d^4 - 2*I*(25*a^2 + 3
2*b^2)*c^2*d^6 - 3*I*(3*a^2 + 5*b^2)*d^8))*sin(f*x + e) + sqrt(2)*(2*I*b^2*c^7*d + 6*I*a*b*c^6*d^2 + 76*I*a*b*
c^4*d^4 + 174*I*a*b*c^2*d^6 - I*(23*a^2 + 13*b^2)*c^5*d^3 - 6*I*(13*a^2 + 12*b^2)*c^3*d^5 - 9*I*(3*a^2 + 5*b^2
)*c*d^7))*sqrt(-I*d)*weierstrassZeta(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, weierstrassPI
nverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) + 3*I*d*sin(f*x + e)
+ 2*I*c)/d)) - 6*((2*b^2*c^4*d^4 + 6*a*b*c^3*d^5 + 58*a*b*c*d^7 - (23*a^2 + 19*b^2)*c^2*d^6 - 3*(3*a^2 + 5*b^2
)*d^8)*cos(f*x + e)^3 - 2*(3*b^2*c^5*d^3 + 9*a*b*c^4*d^4 + 60*a*b*c^2*d^6 - 5*a*b*d^8 - (27*a^2 + 25*b^2)*c^3*
d^5 - 5*(a^2 + 2*b^2)*c*d^7)*cos(f*x + e)*sin(f*x + e) - (b^2*c^6*d^2 + 18*a*b*c^5*d^3 + 56*a*b*c^3*d^5 + 54*a
*b*c*d^7 - (34*a^2 + 23*b^2)*c^4*d^4 - 9*(2*a^2 + 3*b^2)*c^2*d^6 - 3*(4*a^2 + 5*b^2)*d^8)*cos(f*x + e))*sqrt(d
*sin(f*x + e) + c))/(3*(c^7*d^5 - 3*c^5*d^7 + 3*c^3*d^9 - c*d^11)*f*cos(f*x + e)^2 - (c^9*d^3 - 6*c^5*d^7 + 8*
c^3*d^9 - 3*c*d^11)*f + ((c^6*d^6 - 3*c^4*d^8 + 3*c^2*d^10 - d^12)*f*cos(f*x + e)^2 - (3*c^8*d^4 - 8*c^6*d^6 +
6*c^4*d^8 - d^12)*f)*sin(f*x + e))

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(f*x+e))**2/(c+d*sin(f*x+e))**(7/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^(7/2),x, algorithm="giac")

[Out]

integrate((b*sin(f*x + e) + a)^2/(d*sin(f*x + e) + c)^(7/2), x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\sin \left (e+f\,x\right )\right )}^2}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{7/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*sin(e + f*x))^2/(c + d*sin(e + f*x))^(7/2),x)

[Out]

int((a + b*sin(e + f*x))^2/(c + d*sin(e + f*x))^(7/2), x)

________________________________________________________________________________________