\(\int \frac {(3+3 \sin (e+f x))^3}{\sqrt {c-c \sin (e+f x)}} \, dx\) [311]

Optimal result

Integrand size = 28, antiderivative size = 137 \[ \int \frac {(3+3 \sin (e+f x))^3}{\sqrt {c-c \sin (e+f x)}} \, dx=\frac {216 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{\sqrt {c} f}-\frac {54 c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^{5/2}}-\frac {36 c \cos ^3(e+f x)}{f (c-c \sin (e+f x))^{3/2}}-\frac {216 \cos (e+f x)}{f \sqrt {c-c \sin (e+f x)}} \]

[Out]

-2/5*a^3*c^2*cos(f*x+e)^5/f/(c-c*sin(f*x+e))^(5/2)-4/3*a^3*c*cos(f*x+e)^3/f/(c-c*sin(f*x+e))^(3/2)+8*a^3*arcta
nh(1/2*cos(f*x+e)*c^(1/2)*2^(1/2)/(c-c*sin(f*x+e))^(1/2))*2^(1/2)/f/c^(1/2)-8*a^3*cos(f*x+e)/f/(c-c*sin(f*x+e)
)^(1/2)

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.10, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2815, 2758, 2728, 212} \[ \int \frac {(3+3 \sin (e+f x))^3}{\sqrt {c-c \sin (e+f x)}} \, dx=\frac {8 \sqrt {2} a^3 \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{\sqrt {c} f}-\frac {2 a^3 c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^{5/2}}-\frac {4 a^3 c \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}-\frac {8 a^3 \cos (e+f x)}{f \sqrt {c-c \sin (e+f x)}} \]

[In]

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

[Out]

(8*Sqrt[2]*a^3*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(Sqrt[c]*f) - (2*a^3*c^2*Co
s[e + f*x]^5)/(5*f*(c - c*Sin[e + f*x])^(5/2)) - (4*a^3*c*Cos[e + f*x]^3)/(3*f*(c - c*Sin[e + f*x])^(3/2)) - (
8*a^3*Cos[e + f*x])/(f*Sqrt[c - c*Sin[e + f*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 2815

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

Rubi steps \begin{align*} \text {integral}& = \left (a^3 c^3\right ) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^{7/2}} \, dx \\ & = -\frac {2 a^3 c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^{5/2}}+\left (2 a^3 c^2\right ) \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^{5/2}} \, dx \\ & = -\frac {2 a^3 c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^{5/2}}-\frac {4 a^3 c \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}+\left (4 a^3 c\right ) \int \frac {\cos ^2(e+f x)}{(c-c \sin (e+f x))^{3/2}} \, dx \\ & = -\frac {2 a^3 c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^{5/2}}-\frac {4 a^3 c \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}-\frac {8 a^3 \cos (e+f x)}{f \sqrt {c-c \sin (e+f x)}}+\left (8 a^3\right ) \int \frac {1}{\sqrt {c-c \sin (e+f x)}} \, dx \\ & = -\frac {2 a^3 c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^{5/2}}-\frac {4 a^3 c \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}-\frac {8 a^3 \cos (e+f x)}{f \sqrt {c-c \sin (e+f x)}}-\frac {\left (16 a^3\right ) \text {Subst}\left (\int \frac {1}{2 c-x^2} \, dx,x,-\frac {c \cos (e+f x)}{\sqrt {c-c \sin (e+f x)}}\right )}{f} \\ & = \frac {8 \sqrt {2} a^3 \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{\sqrt {c} f}-\frac {2 a^3 c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^{5/2}}-\frac {4 a^3 c \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}-\frac {8 a^3 \cos (e+f x)}{f \sqrt {c-c \sin (e+f x)}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.74 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.12 \[ \int \frac {(3+3 \sin (e+f x))^3}{\sqrt {c-c \sin (e+f x)}} \, dx=-\frac {9 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left ((480+480 i) \sqrt [4]{-1} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [4]{-1} \left (1+\tan \left (\frac {1}{4} (e+f x)\right )\right )\right )+330 \cos \left (\frac {1}{2} (e+f x)\right )-35 \cos \left (\frac {3}{2} (e+f x)\right )-3 \cos \left (\frac {5}{2} (e+f x)\right )+330 \sin \left (\frac {1}{2} (e+f x)\right )+35 \sin \left (\frac {3}{2} (e+f x)\right )-3 \sin \left (\frac {5}{2} (e+f x)\right )\right )}{10 f \sqrt {c-c \sin (e+f x)}} \]

[In]

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

[Out]

(-9*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*((480 + 480*I)*(-1)^(1/4)*ArcTan[(1/2 + I/2)*(-1)^(1/4)*(1 + Tan[(e
+ f*x)/4])] + 330*Cos[(e + f*x)/2] - 35*Cos[(3*(e + f*x))/2] - 3*Cos[(5*(e + f*x))/2] + 330*Sin[(e + f*x)/2] +
35*Sin[(3*(e + f*x))/2] - 3*Sin[(5*(e + f*x))/2]))/(10*f*Sqrt[c - c*Sin[e + f*x]])

Maple [A] (verified)

Time = 2.41 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.94

method	result	size
default	\(\frac {2 \left (\sin \left (f x +e \right )-1\right ) \sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, a^{3} \left (-60 c^{\frac {5}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )+3 \left (c \left (\sin \left (f x +e \right )+1\right )\right )^{\frac {5}{2}}+10 c \left (c \left (\sin \left (f x +e \right )+1\right )\right )^{\frac {3}{2}}+60 \sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, c^{2}\right )}{15 c^{3} \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\)	\(129\)
parts	\(-\frac {a^{3} \left (\sin \left (f x +e \right )-1\right ) \sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{\sqrt {c}\, \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}+\frac {a^{3} \left (\sin \left (f x +e \right )-1\right ) \sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, \left (-15 c^{\frac {5}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )+6 \left (c \left (\sin \left (f x +e \right )+1\right )\right )^{\frac {5}{2}}-10 c \left (c \left (\sin \left (f x +e \right )+1\right )\right )^{\frac {3}{2}}+30 \sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, c^{2}\right )}{15 c^{3} \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}-\frac {3 a^{3} \left (\sin \left (f x +e \right )-1\right ) \sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, \left (\sqrt {c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )-2 \sqrt {c \left (\sin \left (f x +e \right )+1\right )}\right )}{c \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}-\frac {a^{3} \left (\sin \left (f x +e \right )-1\right ) \sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, \left (3 c^{\frac {3}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )-2 \left (c \left (\sin \left (f x +e \right )+1\right )\right )^{\frac {3}{2}}\right )}{c^{2} \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\)	\(397\)

[In]

int((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/15*(sin(f*x+e)-1)*(c*(sin(f*x+e)+1))^(1/2)*a^3*(-60*c^(5/2)*2^(1/2)*arctanh(1/2*(c*(sin(f*x+e)+1))^(1/2)*2^(
1/2)/c^(1/2))+3*(c*(sin(f*x+e)+1))^(5/2)+10*c*(c*(sin(f*x+e)+1))^(3/2)+60*(c*(sin(f*x+e)+1))^(1/2)*c^2)/c^3/co
s(f*x+e)/(c-c*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. 265 vs. \(2 (132) = 264\).

Time = 0.29 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.93 \[ \int \frac {(3+3 \sin (e+f x))^3}{\sqrt {c-c \sin (e+f x)}} \, dx=\frac {2 \, {\left (\frac {30 \, \sqrt {2} {\left (a^{3} c \cos \left (f x + e\right ) - a^{3} c \sin \left (f x + e\right ) + a^{3} c\right )} \log \left (-\frac {\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) - 2\right )} \sin \left (f x + e\right ) + \frac {2 \, \sqrt {2} \sqrt {-c \sin \left (f x + e\right ) + c} {\left (\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right )}}{\sqrt {c}} + 3 \, \cos \left (f x + e\right ) + 2}{\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right )}{\sqrt {c}} + {\left (3 \, a^{3} \cos \left (f x + e\right )^{3} + 19 \, a^{3} \cos \left (f x + e\right )^{2} - 76 \, a^{3} \cos \left (f x + e\right ) - 92 \, a^{3} + {\left (3 \, a^{3} \cos \left (f x + e\right )^{2} - 16 \, a^{3} \cos \left (f x + e\right ) - 92 \, a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}\right )}}{15 \, {\left (c f \cos \left (f x + e\right ) - c f \sin \left (f x + e\right ) + c f\right )}} \]

[In]

integrate((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

2/15*(30*sqrt(2)*(a^3*c*cos(f*x + e) - a^3*c*sin(f*x + e) + a^3*c)*log(-(cos(f*x + e)^2 + (cos(f*x + e) - 2)*s
in(f*x + e) + 2*sqrt(2)*sqrt(-c*sin(f*x + e) + c)*(cos(f*x + e) + sin(f*x + e) + 1)/sqrt(c) + 3*cos(f*x + e) +
2)/(cos(f*x + e)^2 + (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2))/sqrt(c) + (3*a^3*cos(f*x + e)^3 + 1
9*a^3*cos(f*x + e)^2 - 76*a^3*cos(f*x + e) - 92*a^3 + (3*a^3*cos(f*x + e)^2 - 16*a^3*cos(f*x + e) - 92*a^3)*si
n(f*x + e))*sqrt(-c*sin(f*x + e) + c))/(c*f*cos(f*x + e) - c*f*sin(f*x + e) + c*f)

Sympy [F]

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

[In]

integrate((a+a*sin(f*x+e))**3/(c-c*sin(f*x+e))**(1/2),x)

[Out]

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

Maxima [F]

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

[In]

integrate((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate((a*sin(f*x + e) + a)^3/sqrt(-c*sin(f*x + e) + c), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 290 vs. \(2 (132) = 264\).

Time = 0.33 (sec) , antiderivative size = 290, normalized size of antiderivative = 2.12 \[ \int \frac {(3+3 \sin (e+f x))^3}{\sqrt {c-c \sin (e+f x)}} \, dx=\frac {4 \, {\left (\frac {15 \, \sqrt {2} a^{3} \log \left (-\frac {\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1}\right )}{\sqrt {c} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {4 \, \sqrt {2} {\left (23 \, a^{3} \sqrt {c} - \frac {70 \, a^{3} \sqrt {c} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} + \frac {140 \, a^{3} \sqrt {c} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{2}}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{2}} - \frac {90 \, a^{3} \sqrt {c} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{3}}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{3}} + \frac {45 \, a^{3} \sqrt {c} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{4}}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{4}}\right )}}{c {\left (\frac {\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} - 1\right )}^{5} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}\right )}}{15 \, f} \]

[In]

integrate((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^(1/2),x, algorithm="giac")

[Out]

4/15*(15*sqrt(2)*a^3*log(-(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1))/(sqrt(c)*
sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))) - 4*sqrt(2)*(23*a^3*sqrt(c) - 70*a^3*sqrt(c)*(cos(-1/4*pi + 1/2*f*x + 1/2
*e) - 1)/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1) + 140*a^3*sqrt(c)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^2/(cos(-1
/4*pi + 1/2*f*x + 1/2*e) + 1)^2 - 90*a^3*sqrt(c)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^3/(cos(-1/4*pi + 1/2*f*x
+ 1/2*e) + 1)^3 + 45*a^3*sqrt(c)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^4/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^
4)/(c*((cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1) - 1)^5*sgn(sin(-1/4*pi + 1/2*
f*x + 1/2*e))))/f

Mupad [F(-1)]

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

[In]

int((a + a*sin(e + f*x))^3/(c - c*sin(e + f*x))^(1/2),x)

[Out]

int((a + a*sin(e + f*x))^3/(c - c*sin(e + f*x))^(1/2), x)