\(\int \frac {(3+3 \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{7/2}} \, dx\) [585]

Optimal result

Integrand size = 29, antiderivative size = 180 \[ \int \frac {(3+3 \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{7/2}} \, dx=\frac {18 (c-d) \cos (e+f x) \sqrt {3+3 \sin (e+f x)}}{5 d (c+d) f (c+d \sin (e+f x))^{5/2}}+\frac {18 (c-d) (3 c+11 d) \cos (e+f x)}{5 d^2 (c+d)^2 f \sqrt {3+3 \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac {18 \left (3 c^2+14 c d+43 d^2\right ) \cos (e+f x)}{5 d^2 (c+d)^3 f \sqrt {3+3 \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \]

[Out]

2/15*a^3*(c-d)*(3*c+11*d)*cos(f*x+e)/d^2/(c+d)^2/f/(c+d*sin(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(1/2)+2/5*a^2*(c-d)
*cos(f*x+e)*(a+a*sin(f*x+e))^(1/2)/d/(c+d)/f/(c+d*sin(f*x+e))^(5/2)-2/15*a^3*(3*c^2+14*c*d+43*d^2)*cos(f*x+e)/
d^2/(c+d)^3/f/(a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.05, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2841, 3059, 2850} \[ \int \frac {(3+3 \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{7/2}} \, dx=-\frac {2 a^3 \left (3 c^2+14 c d+43 d^2\right ) \cos (e+f x)}{15 d^2 f (c+d)^3 \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}+\frac {2 a^3 (c-d) (3 c+11 d) \cos (e+f x)}{15 d^2 f (c+d)^2 \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}+\frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{5 d f (c+d) (c+d \sin (e+f x))^{5/2}} \]

[In]

Int[(a + a*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^(7/2),x]

[Out]

(2*a^2*(c - d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(5*d*(c + d)*f*(c + d*Sin[e + f*x])^(5/2)) + (2*a^3*(c -
d)*(3*c + 11*d)*Cos[e + f*x])/(15*d^2*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2)) - (2*a
^3*(3*c^2 + 14*c*d + 43*d^2)*Cos[e + f*x])/(15*d^2*(c + d)^3*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x
]])

Rule 2841

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[(-b^2)*(b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(b*c
+ a*d))), x] + Dist[b^2/(d*(n + 1)*(b*c + a*d)), Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1
)*Simp[a*c*(m - 2) - b*d*(m - 2*n - 4) - (b*c*(m - 1) - a*d*(m + 2*n + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{
a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && LtQ[n, -1
] && (IntegersQ[2*m, 2*n] || IntegerQ[m + 1/2] || (IntegerQ[m] && EqQ[c, 0]))

Rule 2850

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(3/2), x_Symbol] :> Sim
p[-2*b^2*(Cos[e + f*x]/(f*(b*c + a*d)*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])), x] /; FreeQ[{a, b,
c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 3059

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.
) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*(B*c - A*d)*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n
+ 1)*(b*c + a*d)*Sqrt[a + b*Sin[e + f*x]])), x] + Dist[(A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)))/(2*d*(n +
1)*(b*c + a*d)), Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{5 d (c+d) f (c+d \sin (e+f x))^{5/2}}-\frac {(2 a) \int \frac {\sqrt {a+a \sin (e+f x)} \left (\frac {1}{2} a (c-11 d)-\frac {1}{2} a (3 c+7 d) \sin (e+f x)\right )}{(c+d \sin (e+f x))^{5/2}} \, dx}{5 d (c+d)} \\ & = \frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{5 d (c+d) f (c+d \sin (e+f x))^{5/2}}+\frac {2 a^3 (c-d) (3 c+11 d) \cos (e+f x)}{15 d^2 (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}+\frac {\left (a^2 \left (3 c^2+14 c d+43 d^2\right )\right ) \int \frac {\sqrt {a+a \sin (e+f x)}}{(c+d \sin (e+f x))^{3/2}} \, dx}{15 d^2 (c+d)^2} \\ & = \frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{5 d (c+d) f (c+d \sin (e+f x))^{5/2}}+\frac {2 a^3 (c-d) (3 c+11 d) \cos (e+f x)}{15 d^2 (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac {2 a^3 \left (3 c^2+14 c d+43 d^2\right ) \cos (e+f x)}{15 d^2 (c+d)^3 f \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.76 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.84 \[ \int \frac {(3+3 \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{7/2}} \, dx=-\frac {3 \sqrt {3} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))^{5/2} \left (89 c^2+42 c d+49 d^2-\left (3 c^2+14 c d+43 d^2\right ) \cos (2 (e+f x))+4 \left (7 c^2+46 c d+7 d^2\right ) \sin (e+f x)\right )}{5 (c+d)^3 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 (c+d \sin (e+f x))^{5/2}} \]

[In]

Integrate[(3 + 3*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^(7/2),x]

[Out]

(-3*Sqrt[3]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^(5/2)*(89*c^2 + 42*c*d + 49*d^2 - (3*c^2
+ 14*c*d + 43*d^2)*Cos[2*(e + f*x)] + 4*(7*c^2 + 46*c*d + 7*d^2)*Sin[e + f*x]))/(5*(c + d)^3*f*(Cos[(e + f*x)/
2] + Sin[(e + f*x)/2])^5*(c + d*Sin[e + f*x])^(5/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(551\) vs. \(2(171)=342\).

Time = 4.24 (sec) , antiderivative size = 552, normalized size of antiderivative = 3.07

method	result	size
default	\(-\frac {2 \sec \left (f x +e \right ) \sqrt {a \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {c +d \sin \left (f x +e \right )}\, \left (-9 \sin \left (f x +e \right ) c^{2} d^{3}-32 \sin \left (f x +e \right ) c^{4} d +58 c^{3} \left (\cos ^{2}\left (f x +e \right )\right ) d^{2}-64 c^{2} d^{3} \left (\sin ^{2}\left (f x +e \right )\right )-27 \left (\cos ^{2}\left (f x +e \right )\right ) c^{4} d -32 c \,d^{4} \left (\sin ^{3}\left (f x +e \right )\right )+73 \left (\sin ^{3}\left (f x +e \right )\right ) c^{2} d^{3}+32 c^{5}+64 \sin \left (f x +e \right ) c^{3} d^{2}+14 \left (\sin ^{2}\left (f x +e \right )\right ) \left (\cos ^{4}\left (f x +e \right )\right ) c \,d^{4}+31 \left (\sin ^{3}\left (f x +e \right )\right ) \left (\cos ^{2}\left (f x +e \right )\right ) c^{2} d^{3}-19 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right ) c^{4} d -3 d^{5} \left (\sin ^{3}\left (f x +e \right )\right )-43 \left (\sin ^{4}\left (f x +e \right )\right ) \left (\cos ^{2}\left (f x +e \right )\right ) d^{5}-37 \left (\sin ^{2}\left (f x +e \right )\right ) \left (\cos ^{2}\left (f x +e \right )\right ) c \,d^{4}-86 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right ) c^{3} d^{2}-64 c^{3} d^{2}+11 \left (\cos ^{2}\left (f x +e \right )\right ) c^{5}+9 \left (\cos ^{4}\left (f x +e \right )\right ) c^{4} d +3 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) c^{5}+32 c \,d^{4} \left (\sin ^{2}\left (f x +e \right )\right )+32 c^{4} d +51 \left (\sin ^{3}\left (f x +e \right )\right ) \left (\cos ^{2}\left (f x +e \right )\right ) c \,d^{4}+70 \left (\sin ^{2}\left (f x +e \right )\right ) \left (\cos ^{2}\left (f x +e \right )\right ) c^{2} d^{3}+3 \left (\cos ^{4}\left (f x +e \right )\right ) \left (\sin ^{2}\left (f x +e \right )\right ) c^{2} d^{3}-9 \sin \left (f x +e \right ) \left (\cos ^{4}\left (f x +e \right )\right ) c^{3} d^{2}+9 \left (\cos ^{4}\left (f x +e \right )\right ) c^{3} d^{2}-29 \left (\sin ^{5}\left (f x +e \right )\right ) d^{5}+32 d^{5} \left (\sin ^{4}\left (f x +e \right )\right )-32 c^{5} \sin \left (f x +e \right )\right ) a^{2}}{15 f {\left (\left (\cos ^{2}\left (f x +e \right )\right ) d^{2}+c^{2}-d^{2}\right )}^{3} \left (c +d \right )^{3}}\)	\(552\)

[In]

int((a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(7/2),x,method=_RETURNVERBOSE)

[Out]

-2/15/f*sec(f*x+e)*(a*(sin(f*x+e)+1))^(1/2)*(c+d*sin(f*x+e))^(1/2)*(-43*sin(f*x+e)^4*cos(f*x+e)^2*d^5-9*sin(f*
x+e)*c^2*d^3+58*c^3*cos(f*x+e)^2*d^2+73*sin(f*x+e)^3*c^2*d^3-32*sin(f*x+e)*c^4*d+70*sin(f*x+e)^2*cos(f*x+e)^2*
c^2*d^3+32*c^5+64*sin(f*x+e)*c^3*d^2-29*sin(f*x+e)^5*d^5+51*sin(f*x+e)^3*cos(f*x+e)^2*c*d^4+9*cos(f*x+e)^4*c^4
*d-64*c^3*d^2+9*cos(f*x+e)^4*c^3*d^2-27*cos(f*x+e)^2*c^4*d+32*c^4*d+3*cos(f*x+e)^2*sin(f*x+e)*c^5+3*cos(f*x+e)
^4*sin(f*x+e)^2*c^2*d^3+14*sin(f*x+e)^2*cos(f*x+e)^4*c*d^4+31*sin(f*x+e)^3*cos(f*x+e)^2*c^2*d^3-9*sin(f*x+e)*c
os(f*x+e)^4*c^3*d^2-37*sin(f*x+e)^2*cos(f*x+e)^2*c*d^4-19*sin(f*x+e)*cos(f*x+e)^2*c^4*d-86*sin(f*x+e)*cos(f*x+
e)^2*c^3*d^2+11*cos(f*x+e)^2*c^5+32*d^5*sin(f*x+e)^4-3*d^5*sin(f*x+e)^3-32*c*d^4*sin(f*x+e)^3-64*c^2*d^3*sin(f
*x+e)^2+32*c*d^4*sin(f*x+e)^2-32*c^5*sin(f*x+e))*a^2/(cos(f*x+e)^2*d^2+c^2-d^2)^3/(c+d)^3

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 654 vs. \(2 (171) = 342\).

Time = 0.33 (sec) , antiderivative size = 654, normalized size of antiderivative = 3.63 \[ \int \frac {(3+3 \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{7/2}} \, dx=-\frac {2 \, {\left (32 \, a^{2} c^{2} - 64 \, a^{2} c d + 32 \, a^{2} d^{2} - {\left (3 \, a^{2} c^{2} + 14 \, a^{2} c d + 43 \, a^{2} d^{2}\right )} \cos \left (f x + e\right )^{3} + {\left (11 \, a^{2} c^{2} + 78 \, a^{2} c d - 29 \, a^{2} d^{2}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (23 \, a^{2} c^{2} + 14 \, a^{2} c d + 23 \, a^{2} d^{2}\right )} \cos \left (f x + e\right ) - {\left (32 \, a^{2} c^{2} - 64 \, a^{2} c d + 32 \, a^{2} d^{2} - {\left (3 \, a^{2} c^{2} + 14 \, a^{2} c d + 43 \, a^{2} d^{2}\right )} \cos \left (f x + e\right )^{2} - 2 \, {\left (7 \, a^{2} c^{2} + 46 \, a^{2} c d + 7 \, a^{2} d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c}}{15 \, {\left ({\left (c^{3} d^{3} + 3 \, c^{2} d^{4} + 3 \, c d^{5} + d^{6}\right )} f \cos \left (f x + e\right )^{4} - 3 \, {\left (c^{4} d^{2} + 3 \, c^{3} d^{3} + 3 \, c^{2} d^{4} + c d^{5}\right )} f \cos \left (f x + e\right )^{3} - {\left (3 \, c^{5} d + 12 \, c^{4} d^{2} + 20 \, c^{3} d^{3} + 18 \, c^{2} d^{4} + 9 \, c d^{5} + 2 \, d^{6}\right )} f \cos \left (f x + e\right )^{2} + {\left (c^{6} + 3 \, c^{5} d + 6 \, c^{4} d^{2} + 10 \, c^{3} d^{3} + 9 \, c^{2} d^{4} + 3 \, c d^{5}\right )} f \cos \left (f x + e\right ) + {\left (c^{6} + 6 \, c^{5} d + 15 \, c^{4} d^{2} + 20 \, c^{3} d^{3} + 15 \, c^{2} d^{4} + 6 \, c d^{5} + d^{6}\right )} f - {\left ({\left (c^{3} d^{3} + 3 \, c^{2} d^{4} + 3 \, c d^{5} + d^{6}\right )} f \cos \left (f x + e\right )^{3} + {\left (3 \, c^{4} d^{2} + 10 \, c^{3} d^{3} + 12 \, c^{2} d^{4} + 6 \, c d^{5} + d^{6}\right )} f \cos \left (f x + e\right )^{2} - {\left (3 \, c^{5} d + 9 \, c^{4} d^{2} + 10 \, c^{3} d^{3} + 6 \, c^{2} d^{4} + 3 \, c d^{5} + d^{6}\right )} f \cos \left (f x + e\right ) - {\left (c^{6} + 6 \, c^{5} d + 15 \, c^{4} d^{2} + 20 \, c^{3} d^{3} + 15 \, c^{2} d^{4} + 6 \, c d^{5} + d^{6}\right )} f\right )} \sin \left (f x + e\right )\right )}} \]

[In]

integrate((a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(7/2),x, algorithm="fricas")

[Out]

-2/15*(32*a^2*c^2 - 64*a^2*c*d + 32*a^2*d^2 - (3*a^2*c^2 + 14*a^2*c*d + 43*a^2*d^2)*cos(f*x + e)^3 + (11*a^2*c
^2 + 78*a^2*c*d - 29*a^2*d^2)*cos(f*x + e)^2 + 2*(23*a^2*c^2 + 14*a^2*c*d + 23*a^2*d^2)*cos(f*x + e) - (32*a^2
*c^2 - 64*a^2*c*d + 32*a^2*d^2 - (3*a^2*c^2 + 14*a^2*c*d + 43*a^2*d^2)*cos(f*x + e)^2 - 2*(7*a^2*c^2 + 46*a^2*
c*d + 7*a^2*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/((c^3*d^3 + 3*c
^2*d^4 + 3*c*d^5 + d^6)*f*cos(f*x + e)^4 - 3*(c^4*d^2 + 3*c^3*d^3 + 3*c^2*d^4 + c*d^5)*f*cos(f*x + e)^3 - (3*c
^5*d + 12*c^4*d^2 + 20*c^3*d^3 + 18*c^2*d^4 + 9*c*d^5 + 2*d^6)*f*cos(f*x + e)^2 + (c^6 + 3*c^5*d + 6*c^4*d^2 +
10*c^3*d^3 + 9*c^2*d^4 + 3*c*d^5)*f*cos(f*x + e) + (c^6 + 6*c^5*d + 15*c^4*d^2 + 20*c^3*d^3 + 15*c^2*d^4 + 6*
c*d^5 + d^6)*f - ((c^3*d^3 + 3*c^2*d^4 + 3*c*d^5 + d^6)*f*cos(f*x + e)^3 + (3*c^4*d^2 + 10*c^3*d^3 + 12*c^2*d^
4 + 6*c*d^5 + d^6)*f*cos(f*x + e)^2 - (3*c^5*d + 9*c^4*d^2 + 10*c^3*d^3 + 6*c^2*d^4 + 3*c*d^5 + d^6)*f*cos(f*x
+ e) - (c^6 + 6*c^5*d + 15*c^4*d^2 + 20*c^3*d^3 + 15*c^2*d^4 + 6*c*d^5 + d^6)*f)*sin(f*x + e))

Sympy [F(-1)]

Timed out. \[ \int \frac {(3+3 \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{7/2}} \, dx=\text {Timed out} \]

[In]

integrate((a+a*sin(f*x+e))**(5/2)/(c+d*sin(f*x+e))**(7/2),x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 464 vs. \(2 (171) = 342\).

Time = 0.36 (sec) , antiderivative size = 464, normalized size of antiderivative = 2.58 \[ \int \frac {(3+3 \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{7/2}} \, dx=-\frac {2 \, {\left ({\left (43 \, c^{3} + 14 \, c^{2} d + 3 \, c d^{2}\right )} a^{\frac {5}{2}} - \frac {{\left (15 \, c^{3} - 256 \, c^{2} d - 53 \, c d^{2} - 6 \, d^{3}\right )} a^{\frac {5}{2}} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {{\left (113 \, c^{3} - 116 \, c^{2} d + 493 \, c d^{2} + 50 \, d^{3}\right )} a^{\frac {5}{2}} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {5 \, {\left (17 \, c^{3} - 82 \, c^{2} d + 65 \, c d^{2} - 60 \, d^{3}\right )} a^{\frac {5}{2}} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {5 \, {\left (17 \, c^{3} - 82 \, c^{2} d + 65 \, c d^{2} - 60 \, d^{3}\right )} a^{\frac {5}{2}} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {{\left (113 \, c^{3} - 116 \, c^{2} d + 493 \, c d^{2} + 50 \, d^{3}\right )} a^{\frac {5}{2}} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {{\left (15 \, c^{3} - 256 \, c^{2} d - 53 \, c d^{2} - 6 \, d^{3}\right )} a^{\frac {5}{2}} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {{\left (43 \, c^{3} + 14 \, c^{2} d + 3 \, c d^{2}\right )} a^{\frac {5}{2}} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}}\right )} {\left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}}{15 \, {\left (c^{3} + 3 \, c^{2} d + 3 \, c d^{2} + d^{3} + \frac {{\left (c^{3} + 3 \, c^{2} d + 3 \, c d^{2} + d^{3}\right )} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )} {\left (c + \frac {2 \, d \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {c \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )}^{\frac {7}{2}} f} \]

[In]

integrate((a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(7/2),x, algorithm="maxima")

[Out]

-2/15*((43*c^3 + 14*c^2*d + 3*c*d^2)*a^(5/2) - (15*c^3 - 256*c^2*d - 53*c*d^2 - 6*d^3)*a^(5/2)*sin(f*x + e)/(c
os(f*x + e) + 1) + (113*c^3 - 116*c^2*d + 493*c*d^2 + 50*d^3)*a^(5/2)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 5*
(17*c^3 - 82*c^2*d + 65*c*d^2 - 60*d^3)*a^(5/2)*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 5*(17*c^3 - 82*c^2*d + 6
5*c*d^2 - 60*d^3)*a^(5/2)*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - (113*c^3 - 116*c^2*d + 493*c*d^2 + 50*d^3)*a^(
5/2)*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + (15*c^3 - 256*c^2*d - 53*c*d^2 - 6*d^3)*a^(5/2)*sin(f*x + e)^6/(cos
(f*x + e) + 1)^6 - (43*c^3 + 14*c^2*d + 3*c*d^2)*a^(5/2)*sin(f*x + e)^7/(cos(f*x + e) + 1)^7)*(sin(f*x + e)^2/
(cos(f*x + e) + 1)^2 + 1)/((c^3 + 3*c^2*d + 3*c*d^2 + d^3 + (c^3 + 3*c^2*d + 3*c*d^2 + d^3)*sin(f*x + e)^2/(co
s(f*x + e) + 1)^2)*(c + 2*d*sin(f*x + e)/(cos(f*x + e) + 1) + c*sin(f*x + e)^2/(cos(f*x + e) + 1)^2)^(7/2)*f)

Giac [F(-1)]

Timed out. \[ \int \frac {(3+3 \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{7/2}} \, dx=\text {Timed out} \]

[In]

integrate((a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(7/2),x, algorithm="giac")

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 17.94 (sec) , antiderivative size = 590, normalized size of antiderivative = 3.28 \[ \int \frac {(3+3 \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{7/2}} \, dx=-\frac {\sqrt {c+d\,\sin \left (e+f\,x\right )}\,\left (\frac {8\,a^2\,{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (15\,c^2-10\,c\,d+7\,d^2\right )}{3\,d^3\,f\,{\left (c+d\right )}^3}+\frac {4\,a^2\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (5\,c^2+34\,c\,d-3\,d^2\right )}{3\,d^3\,f\,{\left (c+d\right )}^3}-\frac {8\,a^2\,{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (c^2\,15{}\mathrm {i}-c\,d\,10{}\mathrm {i}+d^2\,7{}\mathrm {i}\right )}{3\,d^3\,f\,{\left (c+d\right )}^3}-\frac {4\,a^2\,{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (c^2\,5{}\mathrm {i}+c\,d\,34{}\mathrm {i}-d^2\,3{}\mathrm {i}\right )}{3\,d^3\,f\,{\left (c+d\right )}^3}-\frac {4\,a^2\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (3\,c^2+14\,c\,d+43\,d^2\right )}{15\,d^3\,f\,{\left (c+d\right )}^3}+\frac {4\,a^2\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (c^2\,3{}\mathrm {i}+c\,d\,14{}\mathrm {i}+d^2\,43{}\mathrm {i}\right )}{15\,d^3\,f\,{\left (c+d\right )}^3}\right )}{{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}+\frac {{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^3}{{\left (c+d\right )}^3}-\frac {3\,{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\left (4\,c^2+2\,c\,d+d^2\right )}{d^2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,\left (6\,c+d\right )}{d}+\frac {{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\left (8\,c^3+12\,c^2\,d+12\,c\,d^2+3\,d^3\right )}{d^3}+\frac {{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\left (c\,6{}\mathrm {i}+d\,1{}\mathrm {i}\right )}{d}-\frac {3\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^3\,\left (4\,c^2+2\,c\,d+d^2\right )}{d^2\,{\left (c+d\right )}^3}+\frac {{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^3\,\left (8\,c^3+12\,c^2\,d+12\,c\,d^2+3\,d^3\right )}{d^3\,{\left (c+d\right )}^3}} \]

[In]

int((a + a*sin(e + f*x))^(5/2)/(c + d*sin(e + f*x))^(7/2),x)

[Out]

-((c + d*sin(e + f*x))^(1/2)*((8*a^2*exp(e*4i + f*x*4i)*(a + a*sin(e + f*x))^(1/2)*(15*c^2 - 10*c*d + 7*d^2))/
(3*d^3*f*(c + d)^3) + (4*a^2*exp(e*2i + f*x*2i)*(a + a*sin(e + f*x))^(1/2)*(34*c*d + 5*c^2 - 3*d^2))/(3*d^3*f*
(c + d)^3) - (8*a^2*exp(e*3i + f*x*3i)*(a + a*sin(e + f*x))^(1/2)*(c^2*15i - c*d*10i + d^2*7i))/(3*d^3*f*(c +
d)^3) - (4*a^2*exp(e*5i + f*x*5i)*(a + a*sin(e + f*x))^(1/2)*(c*d*34i + c^2*5i - d^2*3i))/(3*d^3*f*(c + d)^3)
- (4*a^2*exp(e*6i + f*x*6i)*(a + a*sin(e + f*x))^(1/2)*(14*c*d + 3*c^2 + 43*d^2))/(15*d^3*f*(c + d)^3) + (4*a^
2*exp(e*1i + f*x*1i)*(a + a*sin(e + f*x))^(1/2)*(c*d*14i + c^2*3i + d^2*43i))/(15*d^3*f*(c + d)^3)))/(exp(e*7i
+ f*x*7i) + (c*1i + d*1i)^3/(c + d)^3 - (3*exp(e*5i + f*x*5i)*(2*c*d + 4*c^2 + d^2))/d^2 - (exp(e*1i + f*x*1i
)*(6*c + d))/d + (exp(e*3i + f*x*3i)*(12*c*d^2 + 12*c^2*d + 8*c^3 + 3*d^3))/d^3 + (exp(e*6i + f*x*6i)*(c*6i +
d*1i))/d - (3*exp(e*2i + f*x*2i)*(c*1i + d*1i)^3*(2*c*d + 4*c^2 + d^2))/(d^2*(c + d)^3) + (exp(e*4i + f*x*4i)*
(c*1i + d*1i)^3*(12*c*d^2 + 12*c^2*d + 8*c^3 + 3*d^3))/(d^3*(c + d)^3))