\(\int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx\) [483]

Optimal result

Integrand size = 31, antiderivative size = 169 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=-\frac {4 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{a^{5/2} d}+\frac {4 \cos ^5(c+d x)}{7 d (a+a \sin (c+d x))^{5/2}}+\frac {2 \cos ^3(c+d x)}{3 a d (a+a \sin (c+d x))^{3/2}}-\frac {2 \cos ^5(c+d x)}{7 a d (a+a \sin (c+d x))^{3/2}}+\frac {4 \cos (c+d x)}{a^2 d \sqrt {a+a \sin (c+d x)}} \]

[Out]

4/7*cos(d*x+c)^5/d/(a+a*sin(d*x+c))^(5/2)+2/3*cos(d*x+c)^3/a/d/(a+a*sin(d*x+c))^(3/2)-2/7*cos(d*x+c)^5/a/d/(a+
a*sin(d*x+c))^(3/2)-4*arctanh(1/2*cos(d*x+c)*a^(1/2)*2^(1/2)/(a+a*sin(d*x+c))^(1/2))/a^(5/2)/d*2^(1/2)+4*cos(d
*x+c)/a^2/d/(a+a*sin(d*x+c))^(1/2)

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {2957, 2939, 2758, 2728, 212} \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=-\frac {4 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{a^{5/2} d}+\frac {4 \cos (c+d x)}{a^2 d \sqrt {a \sin (c+d x)+a}}-\frac {2 \cos ^5(c+d x)}{7 a d (a \sin (c+d x)+a)^{3/2}}+\frac {4 \cos ^5(c+d x)}{7 d (a \sin (c+d x)+a)^{5/2}}+\frac {2 \cos ^3(c+d x)}{3 a d (a \sin (c+d x)+a)^{3/2}} \]

[In]

Int[(Cos[c + d*x]^4*Sin[c + d*x]^2)/(a + a*Sin[c + d*x])^(5/2),x]

[Out]

(-4*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(5/2)*d) + (4*Cos[c + d*x]^
5)/(7*d*(a + a*Sin[c + d*x])^(5/2)) + (2*Cos[c + d*x]^3)/(3*a*d*(a + a*Sin[c + d*x])^(3/2)) - (2*Cos[c + d*x]^
5)/(7*a*d*(a + a*Sin[c + d*x])^(3/2)) + (4*Cos[c + d*x])/(a^2*d*Sqrt[a + a*Sin[c + d*x]])

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 2758

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[g*(g*C
os[e + f*x])^(p - 1)*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + p))), x] + Dist[g^2*((p - 1)/(a*(m + p))), Int[(g
*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0]
&& LtQ[m, -1] && GtQ[p, 1] && (GtQ[m, -2] || EqQ[2*m + p + 1, 0] || (EqQ[m, -2] && IntegerQ[p])) && NeQ[m + p,
0] && IntegersQ[2*m, 2*p]

Rule 2939

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

Rule 2957

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*sin[(e_.) + (f_.)*(x_)]^2*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)
, x_Symbol] :> Simp[(-(g*Cos[e + f*x])^(p + 1))*((a + b*Sin[e + f*x])^(m + 1)/(b*f*g*(m + p + 2))), x] + Dist[
1/(b*(m + p + 2)), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^m*(b*(m + 1) - a*(p + 1)*Sin[e + f*x]), x], x]
/; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && NeQ[m + p + 2, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \cos ^5(c+d x)}{7 a d (a+a \sin (c+d x))^{3/2}}+\frac {2 \int \frac {\cos ^4(c+d x) \left (-\frac {3 a}{2}-5 a \sin (c+d x)\right )}{(a+a \sin (c+d x))^{5/2}} \, dx}{7 a} \\ & = \frac {4 \cos ^5(c+d x)}{7 d (a+a \sin (c+d x))^{5/2}}-\frac {2 \cos ^5(c+d x)}{7 a d (a+a \sin (c+d x))^{3/2}}+\int \frac {\cos ^4(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx \\ & = \frac {4 \cos ^5(c+d x)}{7 d (a+a \sin (c+d x))^{5/2}}+\frac {2 \cos ^3(c+d x)}{3 a d (a+a \sin (c+d x))^{3/2}}-\frac {2 \cos ^5(c+d x)}{7 a d (a+a \sin (c+d x))^{3/2}}+\frac {2 \int \frac {\cos ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx}{a} \\ & = \frac {4 \cos ^5(c+d x)}{7 d (a+a \sin (c+d x))^{5/2}}+\frac {2 \cos ^3(c+d x)}{3 a d (a+a \sin (c+d x))^{3/2}}-\frac {2 \cos ^5(c+d x)}{7 a d (a+a \sin (c+d x))^{3/2}}+\frac {4 \cos (c+d x)}{a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {4 \int \frac {1}{\sqrt {a+a \sin (c+d x)}} \, dx}{a^2} \\ & = \frac {4 \cos ^5(c+d x)}{7 d (a+a \sin (c+d x))^{5/2}}+\frac {2 \cos ^3(c+d x)}{3 a d (a+a \sin (c+d x))^{3/2}}-\frac {2 \cos ^5(c+d x)}{7 a d (a+a \sin (c+d x))^{3/2}}+\frac {4 \cos (c+d x)}{a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {8 \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^2 d} \\ & = -\frac {4 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{a^{5/2} d}+\frac {4 \cos ^5(c+d x)}{7 d (a+a \sin (c+d x))^{5/2}}+\frac {2 \cos ^3(c+d x)}{3 a d (a+a \sin (c+d x))^{3/2}}-\frac {2 \cos ^5(c+d x)}{7 a d (a+a \sin (c+d x))^{3/2}}+\frac {4 \cos (c+d x)}{a^2 d \sqrt {a+a \sin (c+d x)}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 4.02 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.19 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=-\frac {\sqrt {a (1+\sin (c+d x))} \left ((672+672 i) (-1)^{3/4} \text {arctanh}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \sec \left (\frac {d x}{4}\right ) \left (\cos \left (\frac {1}{4} (2 c+d x)\right )-\sin \left (\frac {1}{4} (2 c+d x)\right )\right )\right )-525 \cos \left (\frac {1}{2} (c+d x)\right )+91 \cos \left (\frac {3}{2} (c+d x)\right )+21 \cos \left (\frac {5}{2} (c+d x)\right )-3 \cos \left (\frac {7}{2} (c+d x)\right )+525 \sin \left (\frac {1}{2} (c+d x)\right )+91 \sin \left (\frac {3}{2} (c+d x)\right )-21 \sin \left (\frac {5}{2} (c+d x)\right )-3 \sin \left (\frac {7}{2} (c+d x)\right )\right )}{84 a^3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )} \]

[In]

Integrate[(Cos[c + d*x]^4*Sin[c + d*x]^2)/(a + a*Sin[c + d*x])^(5/2),x]

[Out]

-1/84*(Sqrt[a*(1 + Sin[c + d*x])]*((672 + 672*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])] - 525*Cos[(c + d*x)/2] + 91*Cos[(3*(c + d*x))/2] + 21*Cos[(5*(c + d*x))/2
] - 3*Cos[(7*(c + d*x))/2] + 525*Sin[(c + d*x)/2] + 91*Sin[(3*(c + d*x))/2] - 21*Sin[(5*(c + d*x))/2] - 3*Sin[
(7*(c + d*x))/2]))/(a^3*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))

Maple [A] (verified)

Time = 0.13 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.78

method	result	size
default	\(-\frac {2 \left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (42 a^{\frac {7}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )-3 \left (a -a \sin \left (d x +c \right )\right )^{\frac {7}{2}}-7 a^{2} \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}}-42 a^{3} \sqrt {a -a \sin \left (d x +c \right )}\right )}{21 a^{6} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\)	\(132\)

[In]

int(cos(d*x+c)^4*sin(d*x+c)^2/(a+a*sin(d*x+c))^(5/2),x,method=_RETURNVERBOSE)

[Out]

-2/21*(1+sin(d*x+c))*(-a*(sin(d*x+c)-1))^(1/2)*(42*a^(7/2)*2^(1/2)*arctanh(1/2*(a-a*sin(d*x+c))^(1/2)*2^(1/2)/
a^(1/2))-3*(a-a*sin(d*x+c))^(7/2)-7*a^2*(a-a*sin(d*x+c))^(3/2)-42*a^3*(a-a*sin(d*x+c))^(1/2))/a^6/cos(d*x+c)/(
a+a*sin(d*x+c))^(1/2)/d

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.53 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\frac {2 \, {\left (\frac {21 \, \sqrt {2} {\left (a \cos \left (d x + c\right ) + a \sin \left (d x + c\right ) + a\right )} \log \left (-\frac {\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right ) - \frac {2 \, \sqrt {2} \sqrt {a \sin \left (d x + c\right ) + a} {\left (\cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 1\right )}}{\sqrt {a}} + 3 \, \cos \left (d x + c\right ) + 2}{\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}\right )}{\sqrt {a}} + {\left (3 \, \cos \left (d x + c\right )^{4} - 9 \, \cos \left (d x + c\right )^{3} - 31 \, \cos \left (d x + c\right )^{2} + {\left (3 \, \cos \left (d x + c\right )^{3} + 12 \, \cos \left (d x + c\right )^{2} - 19 \, \cos \left (d x + c\right ) - 80\right )} \sin \left (d x + c\right ) + 61 \, \cos \left (d x + c\right ) + 80\right )} \sqrt {a \sin \left (d x + c\right ) + a}\right )}}{21 \, {\left (a^{3} d \cos \left (d x + c\right ) + a^{3} d \sin \left (d x + c\right ) + a^{3} d\right )}} \]

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)^2/(a+a*sin(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

2/21*(21*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) + (3*cos(d*x + c)^4 - 9*cos(d*x + c)^3 -
31*cos(d*x + c)^2 + (3*cos(d*x + c)^3 + 12*cos(d*x + c)^2 - 19*cos(d*x + c) - 80)*sin(d*x + c) + 61*cos(d*x +
c) + 80)*sqrt(a*sin(d*x + c) + a))/(a^3*d*cos(d*x + c) + a^3*d*sin(d*x + c) + a^3*d)

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**4*sin(d*x+c)**2/(a+a*sin(d*x+c))**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\int { \frac {\cos \left (d x + c\right )^{4} \sin \left (d x + c\right )^{2}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)^2/(a+a*sin(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

integrate(cos(d*x + c)^4*sin(d*x + c)^2/(a*sin(d*x + c) + a)^(5/2), x)

Giac [A] (verification not implemented)

none

Time = 0.47 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.96 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\frac {2 \, {\left (\frac {21 \, \sqrt {2} \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{\frac {5}{2}} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {21 \, \sqrt {2} \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{\frac {5}{2}} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {2 \, \sqrt {2} {\left (12 \, a^{\frac {37}{2}} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 7 \, a^{\frac {37}{2}} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 21 \, a^{\frac {37}{2}} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a^{21} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}\right )}}{21 \, d} \]

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)^2/(a+a*sin(d*x+c))^(5/2),x, algorithm="giac")

[Out]

2/21*(21*sqrt(2)*log(sin(-1/4*pi + 1/2*d*x + 1/2*c) + 1)/(a^(5/2)*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))) - 21*sq
rt(2)*log(-sin(-1/4*pi + 1/2*d*x + 1/2*c) + 1)/(a^(5/2)*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))) - 2*sqrt(2)*(12*a
^(37/2)*sin(-1/4*pi + 1/2*d*x + 1/2*c)^7 + 7*a^(37/2)*sin(-1/4*pi + 1/2*d*x + 1/2*c)^3 + 21*a^(37/2)*sin(-1/4*
pi + 1/2*d*x + 1/2*c))/(a^21*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))))/d

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^4\,{\sin \left (c+d\,x\right )}^2}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{5/2}} \,d x \]

[In]

int((cos(c + d*x)^4*sin(c + d*x)^2)/(a + a*sin(c + d*x))^(5/2),x)

[Out]

int((cos(c + d*x)^4*sin(c + d*x)^2)/(a + a*sin(c + d*x))^(5/2), x)