\(\int \frac {(c+d \sin (e+f x))^3}{(3+3 \sin (e+f x))^{3/2}} \, dx\) [550]

Optimal result

Integrand size = 27, antiderivative size = 178 \[ \int \frac {(c+d \sin (e+f x))^3}{(3+3 \sin (e+f x))^{3/2}} \, dx=-\frac {(c-d)^2 (c+11 d) \text {arctanh}\left (\frac {\sqrt {\frac {3}{2}} \cos (e+f x)}{\sqrt {3+3 \sin (e+f x)}}\right )}{6 \sqrt {6} f}+\frac {d \left (3 c^2-24 c d+13 d^2\right ) \cos (e+f x)}{9 f \sqrt {3+3 \sin (e+f x)}}+\frac {(3 c-7 d) d^2 \cos (e+f x) \sqrt {3+3 \sin (e+f x)}}{54 f}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (3+3 \sin (e+f x))^{3/2}} \]

[Out]

-1/2*(c-d)*cos(f*x+e)*(c+d*sin(f*x+e))^2/f/(a+a*sin(f*x+e))^(3/2)-1/4*(c-d)^2*(c+11*d)*arctanh(1/2*cos(f*x+e)*
a^(1/2)*2^(1/2)/(a+a*sin(f*x+e))^(1/2))/a^(3/2)/f*2^(1/2)+1/3*d*(3*c^2-24*c*d+13*d^2)*cos(f*x+e)/a/f/(a+a*sin(
f*x+e))^(1/2)+1/6*(3*c-7*d)*d^2*cos(f*x+e)*(a+a*sin(f*x+e))^(1/2)/a^2/f

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2844, 3047, 3102, 2830, 2728, 212} \[ \int \frac {(c+d \sin (e+f x))^3}{(3+3 \sin (e+f x))^{3/2}} \, dx=-\frac {(c+11 d) (c-d)^2 \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{2 \sqrt {2} a^{3/2} f}+\frac {d^2 (3 c-7 d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{6 a^2 f}+\frac {d \left (3 c^2-24 c d+13 d^2\right ) \cos (e+f x)}{3 a f \sqrt {a \sin (e+f x)+a}}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a \sin (e+f x)+a)^{3/2}} \]

[In]

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

[Out]

-1/2*((c - d)^2*(c + 11*d)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[2]*a^(3/2
)*f) + (d*(3*c^2 - 24*c*d + 13*d^2)*Cos[e + f*x])/(3*a*f*Sqrt[a + a*Sin[e + f*x]]) + ((3*c - 7*d)*d^2*Cos[e +
f*x]*Sqrt[a + a*Sin[e + f*x]])/(6*a^2*f) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(2*f*(a + a*Sin[e + f
*x])^(3/2))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2728

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

Rule 2830

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

Rule 2844

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

Rule 3047

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

Rule 3102

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Dist[1/(
b*(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x],
x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a+a \sin (e+f x))^{3/2}}-\frac {\int \frac {(c+d \sin (e+f x)) \left (-\frac {1}{2} a \left (c^2+7 c d-4 d^2\right )+\frac {1}{2} a (3 c-7 d) d \sin (e+f x)\right )}{\sqrt {a+a \sin (e+f x)}} \, dx}{2 a^2} \\ & = -\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a+a \sin (e+f x))^{3/2}}-\frac {\int \frac {-\frac {1}{2} a c \left (c^2+7 c d-4 d^2\right )+\left (\frac {1}{2} a c (3 c-7 d) d-\frac {1}{2} a d \left (c^2+7 c d-4 d^2\right )\right ) \sin (e+f x)+\frac {1}{2} a (3 c-7 d) d^2 \sin ^2(e+f x)}{\sqrt {a+a \sin (e+f x)}} \, dx}{2 a^2} \\ & = \frac {(3 c-7 d) d^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{6 a^2 f}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a+a \sin (e+f x))^{3/2}}-\frac {\int \frac {-\frac {1}{4} a^2 \left (3 c^3+21 c^2 d-15 c d^2+7 d^3\right )+\frac {1}{2} a^2 d \left (3 c^2-24 c d+13 d^2\right ) \sin (e+f x)}{\sqrt {a+a \sin (e+f x)}} \, dx}{3 a^3} \\ & = \frac {d \left (3 c^2-24 c d+13 d^2\right ) \cos (e+f x)}{3 a f \sqrt {a+a \sin (e+f x)}}+\frac {(3 c-7 d) d^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{6 a^2 f}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a+a \sin (e+f x))^{3/2}}+\frac {\left ((c-d)^2 (c+11 d)\right ) \int \frac {1}{\sqrt {a+a \sin (e+f x)}} \, dx}{4 a} \\ & = \frac {d \left (3 c^2-24 c d+13 d^2\right ) \cos (e+f x)}{3 a f \sqrt {a+a \sin (e+f x)}}+\frac {(3 c-7 d) d^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{6 a^2 f}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a+a \sin (e+f x))^{3/2}}-\frac {\left ((c-d)^2 (c+11 d)\right ) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{2 a f} \\ & = -\frac {(c-d)^2 (c+11 d) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{2 \sqrt {2} a^{3/2} f}+\frac {d \left (3 c^2-24 c d+13 d^2\right ) \cos (e+f x)}{3 a f \sqrt {a+a \sin (e+f x)}}+\frac {(3 c-7 d) d^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{6 a^2 f}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a+a \sin (e+f x))^{3/2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.66 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.86 \[ \int \frac {(c+d \sin (e+f x))^3}{(3+3 \sin (e+f x))^{3/2}} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (6 (c-d)^3 \sin \left (\frac {1}{2} (e+f x)\right )-3 (c-d)^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )+(3+3 i) (-1)^{3/4} (c-d)^2 (c+11 d) \text {arctanh}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \left (-1+\tan \left (\frac {1}{4} (e+f x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2-18 (2 c-d) d^2 \cos \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2-2 d^3 \cos \left (\frac {3}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2+18 (2 c-d) d^2 \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2-2 d^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2 \sin \left (\frac {3}{2} (e+f x)\right )\right )}{18 \sqrt {3} f (1+\sin (e+f x))^{3/2}} \]

[In]

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

[Out]

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(6*(c - d)^3*Sin[(e + f*x)/2] - 3*(c - d)^3*(Cos[(e + f*x)/2] + Sin[(e
+ f*x)/2]) + (3 + 3*I)*(-1)^(3/4)*(c - d)^2*(c + 11*d)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(e + f*x)/4])]
*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 - 18*(2*c - d)*d^2*Cos[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)
/2])^2 - 2*d^3*Cos[(3*(e + f*x))/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + 18*(2*c - d)*d^2*Sin[(e + f*x)/2
]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 - 2*d^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2*Sin[(3*(e + f*x))/2]
))/(18*Sqrt[3]*f*(1 + Sin[e + f*x])^(3/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(489\) vs. \(2(169)=338\).

Time = 3.18 (sec) , antiderivative size = 490, normalized size of antiderivative = 2.75

method	result	size
default	\(\frac {\left (\sin \left (f x +e \right ) \left (8 \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} d^{3} \sqrt {a}-72 a^{\frac {3}{2}} c \,d^{2} \sqrt {a -a \sin \left (f x +e \right )}+24 a^{\frac {3}{2}} d^{3} \sqrt {a -a \sin \left (f x +e \right )}-3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} c^{3}-27 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} c^{2} d +63 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} c \,d^{2}-33 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} d^{3}\right )+8 \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} d^{3} \sqrt {a}-6 \sqrt {a -a \sin \left (f x +e \right )}\, a^{\frac {3}{2}} c^{3}+18 \sqrt {a -a \sin \left (f x +e \right )}\, a^{\frac {3}{2}} c^{2} d -90 a^{\frac {3}{2}} c \,d^{2} \sqrt {a -a \sin \left (f x +e \right )}+30 a^{\frac {3}{2}} d^{3} \sqrt {a -a \sin \left (f x +e \right )}-3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} c^{3}-27 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} c^{2} d +63 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} c \,d^{2}-33 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} d^{3}\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}}{12 a^{\frac {7}{2}} \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\)	\(490\)
parts	\(-\frac {c^{3} \left (\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} \sin \left (f x +e \right )+2 \sqrt {a -a \sin \left (f x +e \right )}\, a^{\frac {3}{2}}+\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2}\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}}{4 a^{\frac {7}{2}} \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}+\frac {d^{3} \left (8 \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \sin \left (f x +e \right ) \sqrt {a}-33 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} \sin \left (f x +e \right )+8 \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {a}+24 \sqrt {a -a \sin \left (f x +e \right )}\, \sin \left (f x +e \right ) a^{\frac {3}{2}}-33 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2}+30 \sqrt {a -a \sin \left (f x +e \right )}\, a^{\frac {3}{2}}\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}}{12 a^{\frac {7}{2}} \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}-\frac {3 c \,d^{2} \left (-7 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a \sin \left (f x +e \right )+8 \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {a}\, \sin \left (f x +e \right )-7 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a +10 \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {a}\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}}{4 a^{\frac {5}{2}} \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}+\frac {3 c^{2} d \left (-3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a \sin \left (f x +e \right )-3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a +2 \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {a}\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}}{4 a^{\frac {5}{2}} \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\)	\(600\)

[In]

int((c+d*sin(f*x+e))^3/(a+a*sin(f*x+e))^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/12/a^(7/2)*(sin(f*x+e)*(8*(a-a*sin(f*x+e))^(3/2)*d^3*a^(1/2)-72*a^(3/2)*c*d^2*(a-a*sin(f*x+e))^(1/2)+24*a^(3
/2)*d^3*(a-a*sin(f*x+e))^(1/2)-3*2^(1/2)*arctanh(1/2*(a-a*sin(f*x+e))^(1/2)*2^(1/2)/a^(1/2))*a^2*c^3-27*2^(1/2
)*arctanh(1/2*(a-a*sin(f*x+e))^(1/2)*2^(1/2)/a^(1/2))*a^2*c^2*d+63*2^(1/2)*arctanh(1/2*(a-a*sin(f*x+e))^(1/2)*
2^(1/2)/a^(1/2))*a^2*c*d^2-33*2^(1/2)*arctanh(1/2*(a-a*sin(f*x+e))^(1/2)*2^(1/2)/a^(1/2))*a^2*d^3)+8*(a-a*sin(
f*x+e))^(3/2)*d^3*a^(1/2)-6*(a-a*sin(f*x+e))^(1/2)*a^(3/2)*c^3+18*(a-a*sin(f*x+e))^(1/2)*a^(3/2)*c^2*d-90*a^(3
/2)*c*d^2*(a-a*sin(f*x+e))^(1/2)+30*a^(3/2)*d^3*(a-a*sin(f*x+e))^(1/2)-3*2^(1/2)*arctanh(1/2*(a-a*sin(f*x+e))^
(1/2)*2^(1/2)/a^(1/2))*a^2*c^3-27*2^(1/2)*arctanh(1/2*(a-a*sin(f*x+e))^(1/2)*2^(1/2)/a^(1/2))*a^2*c^2*d+63*2^(
1/2)*arctanh(1/2*(a-a*sin(f*x+e))^(1/2)*2^(1/2)/a^(1/2))*a^2*c*d^2-33*2^(1/2)*arctanh(1/2*(a-a*sin(f*x+e))^(1/
2)*2^(1/2)/a^(1/2))*a^2*d^3)*(-a*(sin(f*x+e)-1))^(1/2)/cos(f*x+e)/(a+a*sin(f*x+e))^(1/2)/f

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 494 vs. \(2 (169) = 338\).

Time = 0.30 (sec) , antiderivative size = 494, normalized size of antiderivative = 2.78 \[ \int \frac {(c+d \sin (e+f x))^3}{(3+3 \sin (e+f x))^{3/2}} \, dx=-\frac {3 \, \sqrt {2} {\left (2 \, c^{3} + 18 \, c^{2} d - 42 \, c d^{2} + 22 \, d^{3} - {\left (c^{3} + 9 \, c^{2} d - 21 \, c d^{2} + 11 \, d^{3}\right )} \cos \left (f x + e\right )^{2} + {\left (c^{3} + 9 \, c^{2} d - 21 \, c d^{2} + 11 \, d^{3}\right )} \cos \left (f x + e\right ) + {\left (2 \, c^{3} + 18 \, c^{2} d - 42 \, c d^{2} + 22 \, d^{3} + {\left (c^{3} + 9 \, c^{2} d - 21 \, c d^{2} + 11 \, d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a} \log \left (-\frac {a \cos \left (f x + e\right )^{2} - 2 \, \sqrt {2} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {a} {\left (\cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right )} + 3 \, a \cos \left (f x + e\right ) - {\left (a \cos \left (f x + e\right ) - 2 \, a\right )} \sin \left (f x + e\right ) + 2 \, a}{\cos \left (f x + e\right )^{2} - {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right ) + 4 \, {\left (4 \, d^{3} \cos \left (f x + e\right )^{3} - 3 \, c^{3} + 9 \, c^{2} d - 9 \, c d^{2} + 3 \, d^{3} - 4 \, {\left (9 \, c d^{2} - 4 \, d^{3}\right )} \cos \left (f x + e\right )^{2} - 3 \, {\left (c^{3} - 3 \, c^{2} d + 15 \, c d^{2} - 5 \, d^{3}\right )} \cos \left (f x + e\right ) - {\left (4 \, d^{3} \cos \left (f x + e\right )^{2} - 3 \, c^{3} + 9 \, c^{2} d - 9 \, c d^{2} + 3 \, d^{3} + 12 \, {\left (3 \, c d^{2} - d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{24 \, {\left (a^{2} f \cos \left (f x + e\right )^{2} - a^{2} f \cos \left (f x + e\right ) - 2 \, a^{2} f - {\left (a^{2} f \cos \left (f x + e\right ) + 2 \, a^{2} f\right )} \sin \left (f x + e\right )\right )}} \]

[In]

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

[Out]

-1/24*(3*sqrt(2)*(2*c^3 + 18*c^2*d - 42*c*d^2 + 22*d^3 - (c^3 + 9*c^2*d - 21*c*d^2 + 11*d^3)*cos(f*x + e)^2 +
(c^3 + 9*c^2*d - 21*c*d^2 + 11*d^3)*cos(f*x + e) + (2*c^3 + 18*c^2*d - 42*c*d^2 + 22*d^3 + (c^3 + 9*c^2*d - 21
*c*d^2 + 11*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a)*log(-(a*cos(f*x + e)^2 - 2*sqrt(2)*sqrt(a*sin(f*x + e) +
a)*sqrt(a)*(cos(f*x + e) - sin(f*x + e) + 1) + 3*a*cos(f*x + e) - (a*cos(f*x + e) - 2*a)*sin(f*x + e) + 2*a)/(
cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) + 4*(4*d^3*cos(f*x + e)^3 - 3*c^3 + 9*c^
2*d - 9*c*d^2 + 3*d^3 - 4*(9*c*d^2 - 4*d^3)*cos(f*x + e)^2 - 3*(c^3 - 3*c^2*d + 15*c*d^2 - 5*d^3)*cos(f*x + e)
- (4*d^3*cos(f*x + e)^2 - 3*c^3 + 9*c^2*d - 9*c*d^2 + 3*d^3 + 12*(3*c*d^2 - d^3)*cos(f*x + e))*sin(f*x + e))*
sqrt(a*sin(f*x + e) + a))/(a^2*f*cos(f*x + e)^2 - a^2*f*cos(f*x + e) - 2*a^2*f - (a^2*f*cos(f*x + e) + 2*a^2*f
)*sin(f*x + e))

Sympy [F]

\[ \int \frac {(c+d \sin (e+f x))^3}{(3+3 \sin (e+f x))^{3/2}} \, dx=\int \frac {\left (c + d \sin {\left (e + f x \right )}\right )^{3}}{\left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate((c+d*sin(f*x+e))**3/(a+a*sin(f*x+e))**(3/2),x)

[Out]

Integral((c + d*sin(e + f*x))**3/(a*(sin(e + f*x) + 1))**(3/2), x)

Maxima [F]

\[ \int \frac {(c+d \sin (e+f x))^3}{(3+3 \sin (e+f x))^{3/2}} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{3}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

integrate((d*sin(f*x + e) + c)^3/(a*sin(f*x + e) + a)^(3/2), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 362 vs. \(2 (169) = 338\).

Time = 0.39 (sec) , antiderivative size = 362, normalized size of antiderivative = 2.03 \[ \int \frac {(c+d \sin (e+f x))^3}{(3+3 \sin (e+f x))^{3/2}} \, dx=\frac {\frac {3 \, \sqrt {2} {\left (\sqrt {a} c^{3} + 9 \, \sqrt {a} c^{2} d - 21 \, \sqrt {a} c d^{2} + 11 \, \sqrt {a} d^{3}\right )} \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {3 \, \sqrt {2} {\left (\sqrt {a} c^{3} + 9 \, \sqrt {a} c^{2} d - 21 \, \sqrt {a} c d^{2} + 11 \, \sqrt {a} d^{3}\right )} \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {6 \, \sqrt {2} {\left (\sqrt {a} c^{3} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 3 \, \sqrt {a} c^{2} d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 3 \, \sqrt {a} c d^{2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {a} d^{3} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )} a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {16 \, \sqrt {2} {\left (2 \, a^{\frac {9}{2}} d^{3} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 9 \, a^{\frac {9}{2}} c d^{2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 3 \, a^{\frac {9}{2}} d^{3} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{6} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{24 \, f} \]

[In]

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

[Out]

1/24*(3*sqrt(2)*(sqrt(a)*c^3 + 9*sqrt(a)*c^2*d - 21*sqrt(a)*c*d^2 + 11*sqrt(a)*d^3)*log(sin(-1/4*pi + 1/2*f*x
+ 1/2*e) + 1)/(a^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))) - 3*sqrt(2)*(sqrt(a)*c^3 + 9*sqrt(a)*c^2*d - 21*sqrt(a
)*c*d^2 + 11*sqrt(a)*d^3)*log(-sin(-1/4*pi + 1/2*f*x + 1/2*e) + 1)/(a^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))) -
6*sqrt(2)*(sqrt(a)*c^3*sin(-1/4*pi + 1/2*f*x + 1/2*e) - 3*sqrt(a)*c^2*d*sin(-1/4*pi + 1/2*f*x + 1/2*e) + 3*sq
rt(a)*c*d^2*sin(-1/4*pi + 1/2*f*x + 1/2*e) - sqrt(a)*d^3*sin(-1/4*pi + 1/2*f*x + 1/2*e))/((sin(-1/4*pi + 1/2*f
*x + 1/2*e)^2 - 1)*a^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))) - 16*sqrt(2)*(2*a^(9/2)*d^3*sin(-1/4*pi + 1/2*f*x
+ 1/2*e)^3 - 9*a^(9/2)*c*d^2*sin(-1/4*pi + 1/2*f*x + 1/2*e) + 3*a^(9/2)*d^3*sin(-1/4*pi + 1/2*f*x + 1/2*e))/(a
^6*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))))/f

Mupad [F(-1)]

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

[In]

int((c + d*sin(e + f*x))^3/(a + a*sin(e + f*x))^(3/2),x)

[Out]

int((c + d*sin(e + f*x))^3/(a + a*sin(e + f*x))^(3/2), x)