\(\int \frac {(3+3 \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^{7/2}} \, dx\) [577]

Optimal result

Integrand size = 29, antiderivative size = 163 \[ \int \frac {(3+3 \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^{7/2}} \, dx=\frac {18 (c-d) \cos (e+f x)}{5 d (c+d) f \sqrt {3+3 \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}-\frac {6 (c+9 d) \cos (e+f x)}{5 d (c+d)^2 f \sqrt {3+3 \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac {12 (c+9 d) \cos (e+f x)}{5 d (c+d)^3 f \sqrt {3+3 \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \]

[Out]

2/5*a^2*(c-d)*cos(f*x+e)/d/(c+d)/f/(c+d*sin(f*x+e))^(5/2)/(a+a*sin(f*x+e))^(1/2)-2/15*a^2*(c+9*d)*cos(f*x+e)/d
/(c+d)^2/f/(c+d*sin(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(1/2)-4/15*a^2*(c+9*d)*cos(f*x+e)/d/(c+d)^3/f/(a+a*sin(f*x+
e))^(1/2)/(c+d*sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.06, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2841, 21, 2851, 2850} \[ \int \frac {(3+3 \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^{7/2}} \, dx=-\frac {4 a^2 (c+9 d) \cos (e+f x)}{15 d f (c+d)^3 \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}-\frac {2 a^2 (c+9 d) \cos (e+f x)}{15 d f (c+d)^2 \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}+\frac {2 a^2 (c-d) \cos (e+f x)}{5 d f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}} \]

[In]

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

[Out]

(2*a^2*(c - d)*Cos[e + f*x])/(5*d*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(5/2)) - (2*a^2*(c +
9*d)*Cos[e + f*x])/(15*d*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2)) - (4*a^2*(c + 9*d)*
Cos[e + f*x])/(15*d*(c + d)^3*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*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 2851

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

Rubi steps \begin{align*} \text {integral}& = \frac {2 a^2 (c-d) \cos (e+f x)}{5 d (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}-\frac {(2 a) \int \frac {-\frac {1}{2} a (c+9 d)-\frac {1}{2} a (c+9 d) \sin (e+f x)}{\sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}} \, dx}{5 d (c+d)} \\ & = \frac {2 a^2 (c-d) \cos (e+f x)}{5 d (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}+\frac {(a (c+9 d)) \int \frac {\sqrt {a+a \sin (e+f x)}}{(c+d \sin (e+f x))^{5/2}} \, dx}{5 d (c+d)} \\ & = \frac {2 a^2 (c-d) \cos (e+f x)}{5 d (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}-\frac {2 a^2 (c+9 d) \cos (e+f x)}{15 d (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}+\frac {(2 a (c+9 d)) \int \frac {\sqrt {a+a \sin (e+f x)}}{(c+d \sin (e+f x))^{3/2}} \, dx}{15 d (c+d)^2} \\ & = \frac {2 a^2 (c-d) \cos (e+f x)}{5 d (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}-\frac {2 a^2 (c+9 d) \cos (e+f x)}{15 d (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac {4 a^2 (c+9 d) \cos (e+f x)}{15 d (c+d)^3 f \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.97 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.87 \[ \int \frac {(3+3 \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^{7/2}} \, dx=-\frac {2 \sqrt {3} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))^{3/2} \left (25 c^2+13 c d+12 d^2-d (c+9 d) \cos (2 (e+f x))+\left (5 c^2+46 c d+9 d^2\right ) \sin (e+f x)\right )}{5 (c+d)^3 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 (c+d \sin (e+f x))^{5/2}} \]

[In]

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

[Out]

(-2*Sqrt[3]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^(3/2)*(25*c^2 + 13*c*d + 12*d^2 - d*(c +
9*d)*Cos[2*(e + f*x)] + (5*c^2 + 46*c*d + 9*d^2)*Sin[e + f*x]))/(5*(c + d)^3*f*(Cos[(e + f*x)/2] + Sin[(e + f*
x)/2])^3*(c + d*Sin[e + f*x])^(5/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(470\) vs. \(2(154)=308\).

Time = 3.78 (sec) , antiderivative size = 471, normalized size of antiderivative = 2.89

method	result	size
default	\(\frac {2 \sec \left (f x +e \right ) \sqrt {a \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {c +d \sin \left (f x +e \right )}\, \left (9 \sin \left (f x +e \right ) c^{2} d^{3}+28 \sin \left (f x +e \right ) c^{4} d -36 c^{3} \left (\cos ^{2}\left (f x +e \right )\right ) d^{2}+40 c^{2} d^{3} \left (\sin ^{2}\left (f x +e \right )\right )+16 \left (\cos ^{2}\left (f x +e \right )\right ) c^{4} d +4 c \,d^{4} \left (\sin ^{3}\left (f x +e \right )\right )-49 \left (\sin ^{3}\left (f x +e \right )\right ) c^{2} d^{3}-4 c \,d^{4} \left (\sin ^{2}\left (f x +e \right )\right )-20 c^{5}-24 \sin \left (f x +e \right ) c^{3} d^{2}-2 \left (\sin ^{2}\left (f x +e \right )\right ) \left (\cos ^{4}\left (f x +e \right )\right ) c \,d^{4}-\left (\sin ^{3}\left (f x +e \right )\right ) \left (\cos ^{2}\left (f x +e \right )\right ) c^{2} d^{3}+13 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right ) c^{4} d +18 \left (\sin ^{4}\left (f x +e \right )\right ) \left (\cos ^{2}\left (f x +e \right )\right ) d^{5}-3 \left (\sin ^{2}\left (f x +e \right )\right ) \left (\cos ^{2}\left (f x +e \right )\right ) c \,d^{4}+54 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right ) c^{3} d^{2}+24 c^{3} d^{2}-5 \left (\cos ^{2}\left (f x +e \right )\right ) c^{5}-28 c^{4} d -10 \left (\sin ^{3}\left (f x +e \right )\right ) \left (\cos ^{2}\left (f x +e \right )\right ) c \,d^{4}-58 \left (\sin ^{2}\left (f x +e \right )\right ) \left (\cos ^{2}\left (f x +e \right )\right ) c^{2} d^{3}+9 \left (\cos ^{4}\left (f x +e \right )\right ) c^{3} d^{2}+9 \left (\sin ^{5}\left (f x +e \right )\right ) d^{5}-12 d^{5} \left (\sin ^{4}\left (f x +e \right )\right )+3 d^{5} \left (\sin ^{3}\left (f x +e \right )\right )+20 c^{5} \sin \left (f x +e \right )\right ) a}{15 f {\left (\left (\cos ^{2}\left (f x +e \right )\right ) d^{2}+c^{2}-d^{2}\right )}^{3} \left (c +d \right )^{3}}\)	\(471\)

[In]

int((a+a*sin(f*x+e))^(3/2)/(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)*(18*sin(f*x+e)^4*cos(f*x+e)^2*d^5+9*sin(f*x+
e)*c^2*d^3-36*c^3*cos(f*x+e)^2*d^2-49*sin(f*x+e)^3*c^2*d^3+28*sin(f*x+e)*c^4*d-58*sin(f*x+e)^2*cos(f*x+e)^2*c^
2*d^3-20*c^5-24*sin(f*x+e)*c^3*d^2+9*sin(f*x+e)^5*d^5-10*sin(f*x+e)^3*cos(f*x+e)^2*c*d^4+24*c^3*d^2+9*cos(f*x+
e)^4*c^3*d^2+16*cos(f*x+e)^2*c^4*d-28*c^4*d-2*sin(f*x+e)^2*cos(f*x+e)^4*c*d^4-sin(f*x+e)^3*cos(f*x+e)^2*c^2*d^
3-3*sin(f*x+e)^2*cos(f*x+e)^2*c*d^4+13*sin(f*x+e)*cos(f*x+e)^2*c^4*d+54*sin(f*x+e)*cos(f*x+e)^2*c^3*d^2-5*cos(
f*x+e)^2*c^5-12*d^5*sin(f*x+e)^4+3*d^5*sin(f*x+e)^3+4*c*d^4*sin(f*x+e)^3+40*c^2*d^3*sin(f*x+e)^2-4*c*d^4*sin(f
*x+e)^2+20*c^5*sin(f*x+e))*a/(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. 598 vs. \(2 (154) = 308\).

Time = 0.33 (sec) , antiderivative size = 598, normalized size of antiderivative = 3.67 \[ \int \frac {(3+3 \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^{7/2}} \, dx=\frac {2 \, {\left (2 \, {\left (a c d + 9 \, a d^{2}\right )} \cos \left (f x + e\right )^{3} - 20 \, a c^{2} + 32 \, a c d - 12 \, a d^{2} - {\left (5 \, a c^{2} + 44 \, a c d - 9 \, a d^{2}\right )} \cos \left (f x + e\right )^{2} - {\left (25 \, a c^{2} + 14 \, a c d + 21 \, a d^{2}\right )} \cos \left (f x + e\right ) + {\left (20 \, a c^{2} - 32 \, a c d + 12 \, a d^{2} - 2 \, {\left (a c d + 9 \, a d^{2}\right )} \cos \left (f x + e\right )^{2} - {\left (5 \, a c^{2} + 46 \, a c d + 9 \, a d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c}}{15 \, {\left ({\left (c^{3} d^{3} + 3 \, c^{2} d^{4} + 3 \, c d^{5} + d^{6}\right )} f \cos \left (f x + e\right )^{4} - 3 \, {\left (c^{4} d^{2} + 3 \, c^{3} d^{3} + 3 \, c^{2} d^{4} + c d^{5}\right )} f \cos \left (f x + e\right )^{3} - {\left (3 \, c^{5} d + 12 \, c^{4} d^{2} + 20 \, c^{3} d^{3} + 18 \, c^{2} d^{4} + 9 \, c d^{5} + 2 \, d^{6}\right )} f \cos \left (f x + e\right )^{2} + {\left (c^{6} + 3 \, c^{5} d + 6 \, c^{4} d^{2} + 10 \, c^{3} d^{3} + 9 \, c^{2} d^{4} + 3 \, c d^{5}\right )} f \cos \left (f x + e\right ) + {\left (c^{6} + 6 \, c^{5} d + 15 \, c^{4} d^{2} + 20 \, c^{3} d^{3} + 15 \, c^{2} d^{4} + 6 \, c d^{5} + d^{6}\right )} f - {\left ({\left (c^{3} d^{3} + 3 \, c^{2} d^{4} + 3 \, c d^{5} + d^{6}\right )} f \cos \left (f x + e\right )^{3} + {\left (3 \, c^{4} d^{2} + 10 \, c^{3} d^{3} + 12 \, c^{2} d^{4} + 6 \, c d^{5} + d^{6}\right )} f \cos \left (f x + e\right )^{2} - {\left (3 \, c^{5} d + 9 \, c^{4} d^{2} + 10 \, c^{3} d^{3} + 6 \, c^{2} d^{4} + 3 \, c d^{5} + d^{6}\right )} f \cos \left (f x + e\right ) - {\left (c^{6} + 6 \, c^{5} d + 15 \, c^{4} d^{2} + 20 \, c^{3} d^{3} + 15 \, c^{2} d^{4} + 6 \, c d^{5} + d^{6}\right )} f\right )} \sin \left (f x + e\right )\right )}} \]

[In]

integrate((a+a*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^(7/2),x, algorithm="fricas")

[Out]

2/15*(2*(a*c*d + 9*a*d^2)*cos(f*x + e)^3 - 20*a*c^2 + 32*a*c*d - 12*a*d^2 - (5*a*c^2 + 44*a*c*d - 9*a*d^2)*cos
(f*x + e)^2 - (25*a*c^2 + 14*a*c*d + 21*a*d^2)*cos(f*x + e) + (20*a*c^2 - 32*a*c*d + 12*a*d^2 - 2*(a*c*d + 9*a
*d^2)*cos(f*x + e)^2 - (5*a*c^2 + 46*a*c*d + 9*a*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqr
t(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. \[ \int \frac {(3+3 \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^{7/2}} \, dx=\text {Timed out} \]

[In]

integrate((a+a*sin(f*x+e))**(3/2)/(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. 505 vs. \(2 (154) = 308\).

Time = 0.35 (sec) , antiderivative size = 505, normalized size of antiderivative = 3.10 \[ \int \frac {(3+3 \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^{7/2}} \, dx=-\frac {2 \, {\left ({\left (25 \, c^{3} + 12 \, c^{2} d + 3 \, c d^{2}\right )} a^{\frac {3}{2}} - \frac {{\left (15 \, c^{3} - 130 \, c^{2} d - 39 \, c d^{2} - 6 \, d^{3}\right )} a^{\frac {3}{2}} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {{\left (65 \, c^{3} - 78 \, c^{2} d + 223 \, c d^{2} + 30 \, d^{3}\right )} a^{\frac {3}{2}} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {5 \, {\left (11 \, c^{3} - 44 \, c^{2} d + 33 \, c d^{2} - 24 \, d^{3}\right )} a^{\frac {3}{2}} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {5 \, {\left (11 \, c^{3} - 44 \, c^{2} d + 33 \, c d^{2} - 24 \, d^{3}\right )} a^{\frac {3}{2}} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {{\left (65 \, c^{3} - 78 \, c^{2} d + 223 \, c d^{2} + 30 \, d^{3}\right )} a^{\frac {3}{2}} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {{\left (15 \, c^{3} - 130 \, c^{2} d - 39 \, c d^{2} - 6 \, d^{3}\right )} a^{\frac {3}{2}} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {{\left (25 \, c^{3} + 12 \, c^{2} d + 3 \, c d^{2}\right )} a^{\frac {3}{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 )}^{2}}{15 \, {\left (c^{3} + 3 \, c^{2} d + 3 \, c d^{2} + d^{3} + \frac {2 \, {\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}} + \frac {{\left (c^{3} + 3 \, c^{2} d + 3 \, c d^{2} + d^{3}\right )} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}\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))^(3/2)/(c+d*sin(f*x+e))^(7/2),x, algorithm="maxima")

[Out]

-2/15*((25*c^3 + 12*c^2*d + 3*c*d^2)*a^(3/2) - (15*c^3 - 130*c^2*d - 39*c*d^2 - 6*d^3)*a^(3/2)*sin(f*x + e)/(c
os(f*x + e) + 1) + (65*c^3 - 78*c^2*d + 223*c*d^2 + 30*d^3)*a^(3/2)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 5*(1
1*c^3 - 44*c^2*d + 33*c*d^2 - 24*d^3)*a^(3/2)*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 5*(11*c^3 - 44*c^2*d + 33*
c*d^2 - 24*d^3)*a^(3/2)*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - (65*c^3 - 78*c^2*d + 223*c*d^2 + 30*d^3)*a^(3/2)
*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + (15*c^3 - 130*c^2*d - 39*c*d^2 - 6*d^3)*a^(3/2)*sin(f*x + e)^6/(cos(f*x
+ e) + 1)^6 - (25*c^3 + 12*c^2*d + 3*c*d^2)*a^(3/2)*sin(f*x + e)^7/(cos(f*x + e) + 1)^7)*(sin(f*x + e)^2/(cos
(f*x + e) + 1)^2 + 1)^2/((c^3 + 3*c^2*d + 3*c*d^2 + d^3 + 2*(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^3 + 3*c^2*d + 3*c*d^2 + d^3)*sin(f*x + e)^4/(cos(f*x + e) + 1)^4)*(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. \[ \int \frac {(3+3 \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^{7/2}} \, dx=\text {Timed out} \]

[In]

integrate((a+a*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^(7/2),x, algorithm="giac")

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 18.53 (sec) , antiderivative size = 541, normalized size of antiderivative = 3.32 \[ \int \frac {(3+3 \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^{7/2}} \, dx=\frac {\sqrt {c+d\,\sin \left (e+f\,x\right )}\,\left (\frac {8\,a\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\left (c+9\,d\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{15\,d^2\,f\,{\left (c+d\right )}^3}-\frac {8\,a\,{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (9\,c^2-4\,c\,d+3\,d^2\right )}{3\,d^3\,f\,{\left (c+d\right )}^3}+\frac {8\,a\,{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (c^2\,9{}\mathrm {i}-c\,d\,4{}\mathrm {i}+d^2\,3{}\mathrm {i}\right )}{3\,d^3\,f\,{\left (c+d\right )}^3}-\frac {8\,a\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,\left (c\,1{}\mathrm {i}+d\,9{}\mathrm {i}\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{15\,d^2\,f\,{\left (c+d\right )}^3}+\frac {8\,a\,c\,{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\left (c\,1{}\mathrm {i}+d\,9{}\mathrm {i}\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{3\,d^3\,f\,{\left (c+d\right )}^3}-\frac {8\,a\,c\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,\left (c+9\,d\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{3\,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))^(3/2)/(c + d*sin(e + f*x))^(7/2),x)

[Out]

((c + d*sin(e + f*x))^(1/2)*((8*a*exp(e*6i + f*x*6i)*(c + 9*d)*(a + a*sin(e + f*x))^(1/2))/(15*d^2*f*(c + d)^3
) - (8*a*exp(e*4i + f*x*4i)*(a + a*sin(e + f*x))^(1/2)*(9*c^2 - 4*c*d + 3*d^2))/(3*d^3*f*(c + d)^3) + (8*a*exp
(e*3i + f*x*3i)*(a + a*sin(e + f*x))^(1/2)*(c^2*9i - c*d*4i + d^2*3i))/(3*d^3*f*(c + d)^3) - (8*a*exp(e*1i + f
*x*1i)*(c*1i + d*9i)*(a + a*sin(e + f*x))^(1/2))/(15*d^2*f*(c + d)^3) + (8*a*c*exp(e*5i + f*x*5i)*(c*1i + d*9i
)*(a + a*sin(e + f*x))^(1/2))/(3*d^3*f*(c + d)^3) - (8*a*c*exp(e*2i + f*x*2i)*(c + 9*d)*(a + a*sin(e + f*x))^(
1/2))/(3*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))