\(\int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx\) [343]

Optimal result

Integrand size = 31, antiderivative size = 184 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\frac {2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{a^{3/2} d}-\frac {344 \cos (c+d x)}{105 a d \sqrt {a+a \sin (c+d x)}}-\frac {16 \cos (c+d x) \sin ^2(c+d x)}{35 a d \sqrt {a+a \sin (c+d x)}}+\frac {2 \cos (c+d x) \sin ^3(c+d x)}{7 a d \sqrt {a+a \sin (c+d x)}}+\frac {76 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{105 a^2 d} \]

[Out]

2*arctanh(1/2*cos(d*x+c)*a^(1/2)*2^(1/2)/(a+a*sin(d*x+c))^(1/2))/a^(3/2)/d*2^(1/2)-344/105*cos(d*x+c)/a/d/(a+a
*sin(d*x+c))^(1/2)-16/35*cos(d*x+c)*sin(d*x+c)^2/a/d/(a+a*sin(d*x+c))^(1/2)+2/7*cos(d*x+c)*sin(d*x+c)^3/a/d/(a
+a*sin(d*x+c))^(1/2)+76/105*cos(d*x+c)*(a+a*sin(d*x+c))^(1/2)/a^2/d

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {2958, 3062, 3047, 3102, 2830, 2728, 212} \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\frac {2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{a^{3/2} d}+\frac {76 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{105 a^2 d}+\frac {2 \sin ^3(c+d x) \cos (c+d x)}{7 a d \sqrt {a \sin (c+d x)+a}}-\frac {16 \sin ^2(c+d x) \cos (c+d x)}{35 a d \sqrt {a \sin (c+d x)+a}}-\frac {344 \cos (c+d x)}{105 a d \sqrt {a \sin (c+d x)+a}} \]

[In]

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

[Out]

(2*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(3/2)*d) - (344*Cos[c + d*x]
)/(105*a*d*Sqrt[a + a*Sin[c + d*x]]) - (16*Cos[c + d*x]*Sin[c + d*x]^2)/(35*a*d*Sqrt[a + a*Sin[c + d*x]]) + (2
*Cos[c + d*x]*Sin[c + d*x]^3)/(7*a*d*Sqrt[a + a*Sin[c + d*x]]) + (76*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(1
05*a^2*d)

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 2958

Int[cos[(e_.) + (f_.)*(x_)]^2*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)
, x_Symbol] :> Dist[1/b^2, Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x])^(m + 1)*(a - b*Sin[e + f*x]), x], x] /;
FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[2*m, 2*n]

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 3062

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^n/(f*
(m + n + 1))), x] + Dist[1/(b*(m + n + 1)), Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n - 1)*Simp[A*b*c
*(m + n + 1) + B*(a*c*m + b*d*n) + (A*b*d*(m + n + 1) + B*(a*d*m + b*c*n))*Sin[e + f*x], x], x], x] /; FreeQ[{
a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[n, 0] &&
(IntegerQ[n] || EqQ[m + 1/2, 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 {\int \frac {\sin ^3(c+d x) (a-a \sin (c+d x))}{\sqrt {a+a \sin (c+d x)}} \, dx}{a^2} \\ & = \frac {2 \cos (c+d x) \sin ^3(c+d x)}{7 a d \sqrt {a+a \sin (c+d x)}}+\frac {2 \int \frac {\sin ^2(c+d x) \left (-3 a^2+4 a^2 \sin (c+d x)\right )}{\sqrt {a+a \sin (c+d x)}} \, dx}{7 a^3} \\ & = -\frac {16 \cos (c+d x) \sin ^2(c+d x)}{35 a d \sqrt {a+a \sin (c+d x)}}+\frac {2 \cos (c+d x) \sin ^3(c+d x)}{7 a d \sqrt {a+a \sin (c+d x)}}+\frac {4 \int \frac {\sin (c+d x) \left (8 a^3-\frac {19}{2} a^3 \sin (c+d x)\right )}{\sqrt {a+a \sin (c+d x)}} \, dx}{35 a^4} \\ & = -\frac {16 \cos (c+d x) \sin ^2(c+d x)}{35 a d \sqrt {a+a \sin (c+d x)}}+\frac {2 \cos (c+d x) \sin ^3(c+d x)}{7 a d \sqrt {a+a \sin (c+d x)}}+\frac {4 \int \frac {8 a^3 \sin (c+d x)-\frac {19}{2} a^3 \sin ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx}{35 a^4} \\ & = -\frac {16 \cos (c+d x) \sin ^2(c+d x)}{35 a d \sqrt {a+a \sin (c+d x)}}+\frac {2 \cos (c+d x) \sin ^3(c+d x)}{7 a d \sqrt {a+a \sin (c+d x)}}+\frac {76 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{105 a^2 d}+\frac {8 \int \frac {-\frac {19 a^4}{4}+\frac {43}{2} a^4 \sin (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx}{105 a^5} \\ & = -\frac {344 \cos (c+d x)}{105 a d \sqrt {a+a \sin (c+d x)}}-\frac {16 \cos (c+d x) \sin ^2(c+d x)}{35 a d \sqrt {a+a \sin (c+d x)}}+\frac {2 \cos (c+d x) \sin ^3(c+d x)}{7 a d \sqrt {a+a \sin (c+d x)}}+\frac {76 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{105 a^2 d}-\frac {2 \int \frac {1}{\sqrt {a+a \sin (c+d x)}} \, dx}{a} \\ & = -\frac {344 \cos (c+d x)}{105 a d \sqrt {a+a \sin (c+d x)}}-\frac {16 \cos (c+d x) \sin ^2(c+d x)}{35 a d \sqrt {a+a \sin (c+d x)}}+\frac {2 \cos (c+d x) \sin ^3(c+d x)}{7 a d \sqrt {a+a \sin (c+d x)}}+\frac {76 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{105 a^2 d}+\frac {4 \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\frac {a \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{a d} \\ & = \frac {2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{a^{3/2} d}-\frac {344 \cos (c+d x)}{105 a d \sqrt {a+a \sin (c+d x)}}-\frac {16 \cos (c+d x) \sin ^2(c+d x)}{35 a d \sqrt {a+a \sin (c+d x)}}+\frac {2 \cos (c+d x) \sin ^3(c+d x)}{7 a d \sqrt {a+a \sin (c+d x)}}+\frac {76 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{105 a^2 d} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 3.06 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.09 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\frac {\sqrt {a (1+\sin (c+d x))} \left ((1680+1680 i) (-1)^{3/4} \text {arctanh}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \sec \left (\frac {d x}{4}\right ) \left (\cos \left (\frac {1}{4} (2 c+d x)\right )-\sin \left (\frac {1}{4} (2 c+d x)\right )\right )\right )-1365 \cos \left (\frac {1}{2} (c+d x)\right )+245 \cos \left (\frac {3}{2} (c+d x)\right )+63 \cos \left (\frac {5}{2} (c+d x)\right )-15 \cos \left (\frac {7}{2} (c+d x)\right )+1365 \sin \left (\frac {1}{2} (c+d x)\right )+245 \sin \left (\frac {3}{2} (c+d x)\right )-63 \sin \left (\frac {5}{2} (c+d x)\right )-15 \sin \left (\frac {7}{2} (c+d x)\right )\right )}{420 a^2 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )} \]

[In]

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

[Out]

(Sqrt[a*(1 + Sin[c + d*x])]*((1680 + 1680*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*Sec[(d*x)/4]*(Cos[(2*c
+ d*x)/4] - Sin[(2*c + d*x)/4])] - 1365*Cos[(c + d*x)/2] + 245*Cos[(3*(c + d*x))/2] + 63*Cos[(5*(c + d*x))/2]
- 15*Cos[(7*(c + d*x))/2] + 1365*Sin[(c + d*x)/2] + 245*Sin[(3*(c + d*x))/2] - 63*Sin[(5*(c + d*x))/2] - 15*Si
n[(7*(c + d*x))/2]))/(420*a^2*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.80

method	result	size
default	\(\frac {2 \left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (105 a^{\frac {7}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )-15 \left (a -a \sin \left (d x +c \right )\right )^{\frac {7}{2}}+21 \left (a -a \sin \left (d x +c \right )\right )^{\frac {5}{2}} a -35 a^{2} \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}}-105 a^{3} \sqrt {a -a \sin \left (d x +c \right )}\right )}{105 a^{5} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\)	\(148\)

[In]

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

[Out]

2/105*(1+sin(d*x+c))*(-a*(sin(d*x+c)-1))^(1/2)*(105*a^(7/2)*2^(1/2)*arctanh(1/2*(a-a*sin(d*x+c))^(1/2)*2^(1/2)
/a^(1/2))-15*(a-a*sin(d*x+c))^(7/2)+21*(a-a*sin(d*x+c))^(5/2)*a-35*a^2*(a-a*sin(d*x+c))^(3/2)-105*a^3*(a-a*sin
(d*x+c))^(1/2))/a^5/cos(d*x+c)/(a+a*sin(d*x+c))^(1/2)/d

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.41 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\frac {\frac {105 \, \sqrt {2} {\left (a \cos \left (d x + c\right ) + a \sin \left (d x + c\right ) + a\right )} \log \left (-\frac {\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right ) + \frac {2 \, \sqrt {2} \sqrt {a \sin \left (d x + c\right ) + a} {\left (\cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 1\right )}}{\sqrt {a}} + 3 \, \cos \left (d x + c\right ) + 2}{\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}\right )}{\sqrt {a}} - 2 \, {\left (15 \, \cos \left (d x + c\right )^{4} - 24 \, \cos \left (d x + c\right )^{3} - 92 \, \cos \left (d x + c\right )^{2} + {\left (15 \, \cos \left (d x + c\right )^{3} + 39 \, \cos \left (d x + c\right )^{2} - 53 \, \cos \left (d x + c\right ) - 211\right )} \sin \left (d x + c\right ) + 158 \, \cos \left (d x + c\right ) + 211\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{105 \, {\left (a^{2} d \cos \left (d x + c\right ) + a^{2} d \sin \left (d x + c\right ) + a^{2} d\right )}} \]

[In]

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

[Out]

1/105*(105*sqrt(2)*(a*cos(d*x + c) + a*sin(d*x + c) + a)*log(-(cos(d*x + c)^2 - (cos(d*x + c) - 2)*sin(d*x + c
) + 2*sqrt(2)*sqrt(a*sin(d*x + c) + a)*(cos(d*x + c) - sin(d*x + c) + 1)/sqrt(a) + 3*cos(d*x + c) + 2)/(cos(d*
x + c)^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2))/sqrt(a) - 2*(15*cos(d*x + c)^4 - 24*cos(d*x +
c)^3 - 92*cos(d*x + c)^2 + (15*cos(d*x + c)^3 + 39*cos(d*x + c)^2 - 53*cos(d*x + c) - 211)*sin(d*x + c) + 158*
cos(d*x + c) + 211)*sqrt(a*sin(d*x + c) + a))/(a^2*d*cos(d*x + c) + a^2*d*sin(d*x + c) + a^2*d)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.99 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=-\frac {\frac {105 \, \sqrt {2} \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{\frac {3}{2}} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {105 \, \sqrt {2} \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{\frac {3}{2}} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {2 \, \sqrt {2} {\left (120 \, a^{\frac {25}{2}} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 84 \, a^{\frac {25}{2}} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 70 \, a^{\frac {25}{2}} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 105 \, a^{\frac {25}{2}} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a^{14} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{105 \, d} \]

[In]

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

[Out]

-1/105*(105*sqrt(2)*log(sin(-1/4*pi + 1/2*d*x + 1/2*c) + 1)/(a^(3/2)*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))) - 10
5*sqrt(2)*log(-sin(-1/4*pi + 1/2*d*x + 1/2*c) + 1)/(a^(3/2)*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))) - 2*sqrt(2)*(
120*a^(25/2)*sin(-1/4*pi + 1/2*d*x + 1/2*c)^7 - 84*a^(25/2)*sin(-1/4*pi + 1/2*d*x + 1/2*c)^5 + 70*a^(25/2)*sin
(-1/4*pi + 1/2*d*x + 1/2*c)^3 + 105*a^(25/2)*sin(-1/4*pi + 1/2*d*x + 1/2*c))/(a^14*sgn(cos(-1/4*pi + 1/2*d*x +
1/2*c))))/d

Mupad [F(-1)]

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

[In]

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

[Out]

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