Integrand size = 28, antiderivative size = 151 \[ \int \frac {(a+a \sin (e+f x))^3}{\sqrt {c-c \sin (e+f x)}} \, dx=\frac {8 \sqrt {2} a^3 \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{\sqrt {c} f}-\frac {2 a^3 c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^{5/2}}-\frac {4 a^3 c \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}-\frac {8 a^3 \cos (e+f x)}{f \sqrt {c-c \sin (e+f x)}} \] Output:
8*2^(1/2)*a^3*arctanh(1/2*c^(1/2)*cos(f*x+e)*2^(1/2)/(c-c*sin(f*x+e))^(1/2 ))/c^(1/2)/f-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*c os(f*x+e)^3/f/(c-c*sin(f*x+e))^(3/2)-8*a^3*cos(f*x+e)/f/(c-c*sin(f*x+e))^( 1/2)
Result contains complex when optimal does not.
Time = 7.98 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.03 \[ \int \frac {(a+a \sin (e+f x))^3}{\sqrt {c-c \sin (e+f x)}} \, dx=-\frac {a^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left ((480+480 i) \sqrt [4]{-1} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [4]{-1} \left (1+\tan \left (\frac {1}{4} (e+f x)\right )\right )\right )+330 \cos \left (\frac {1}{2} (e+f x)\right )-35 \cos \left (\frac {3}{2} (e+f x)\right )-3 \cos \left (\frac {5}{2} (e+f x)\right )+330 \sin \left (\frac {1}{2} (e+f x)\right )+35 \sin \left (\frac {3}{2} (e+f x)\right )-3 \sin \left (\frac {5}{2} (e+f x)\right )\right )}{30 f \sqrt {c-c \sin (e+f x)}} \] Input:
Integrate[(a + a*Sin[e + f*x])^3/Sqrt[c - c*Sin[e + f*x]],x]
Output:
-1/30*(a^3*(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]))/(f*Sqrt[c - c*Sin [e + f*x]])
Time = 0.84 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.08, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {3042, 3215, 3042, 3158, 3042, 3158, 3042, 3158, 3042, 3128, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^3}{\sqrt {c-c \sin (e+f x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^3}{\sqrt {c-c \sin (e+f x)}}dx\) |
| \(\Big \downarrow \) 3215 |
| \(\displaystyle a^3 c^3 \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^{7/2}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \int \frac {\cos (e+f x)^6}{(c-c \sin (e+f x))^{7/2}}dx\) |
| \(\Big \downarrow \) 3158 |
| \(\displaystyle a^3 c^3 \left (\frac {2 \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^{5/2}}dx}{c}-\frac {2 \cos ^5(e+f x)}{5 c f (c-c \sin (e+f x))^{5/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \left (\frac {2 \int \frac {\cos (e+f x)^4}{(c-c \sin (e+f x))^{5/2}}dx}{c}-\frac {2 \cos ^5(e+f x)}{5 c f (c-c \sin (e+f x))^{5/2}}\right )\) |
| \(\Big \downarrow \) 3158 |
| \(\displaystyle a^3 c^3 \left (\frac {2 \left (\frac {2 \int \frac {\cos ^2(e+f x)}{(c-c \sin (e+f x))^{3/2}}dx}{c}-\frac {2 \cos ^3(e+f x)}{3 c f (c-c \sin (e+f x))^{3/2}}\right )}{c}-\frac {2 \cos ^5(e+f x)}{5 c f (c-c \sin (e+f x))^{5/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \left (\frac {2 \left (\frac {2 \int \frac {\cos (e+f x)^2}{(c-c \sin (e+f x))^{3/2}}dx}{c}-\frac {2 \cos ^3(e+f x)}{3 c f (c-c \sin (e+f x))^{3/2}}\right )}{c}-\frac {2 \cos ^5(e+f x)}{5 c f (c-c \sin (e+f x))^{5/2}}\right )\) |
| \(\Big \downarrow \) 3158 |
| \(\displaystyle a^3 c^3 \left (\frac {2 \left (\frac {2 \left (\frac {2 \int \frac {1}{\sqrt {c-c \sin (e+f x)}}dx}{c}-\frac {2 \cos (e+f x)}{c f \sqrt {c-c \sin (e+f x)}}\right )}{c}-\frac {2 \cos ^3(e+f x)}{3 c f (c-c \sin (e+f x))^{3/2}}\right )}{c}-\frac {2 \cos ^5(e+f x)}{5 c f (c-c \sin (e+f x))^{5/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \left (\frac {2 \left (\frac {2 \left (\frac {2 \int \frac {1}{\sqrt {c-c \sin (e+f x)}}dx}{c}-\frac {2 \cos (e+f x)}{c f \sqrt {c-c \sin (e+f x)}}\right )}{c}-\frac {2 \cos ^3(e+f x)}{3 c f (c-c \sin (e+f x))^{3/2}}\right )}{c}-\frac {2 \cos ^5(e+f x)}{5 c f (c-c \sin (e+f x))^{5/2}}\right )\) |
| \(\Big \downarrow \) 3128 |
| \(\displaystyle a^3 c^3 \left (\frac {2 \left (\frac {2 \left (-\frac {4 \int \frac {1}{2 c-\frac {c^2 \cos ^2(e+f x)}{c-c \sin (e+f x)}}d\left (-\frac {c \cos (e+f x)}{\sqrt {c-c \sin (e+f x)}}\right )}{c f}-\frac {2 \cos (e+f x)}{c f \sqrt {c-c \sin (e+f x)}}\right )}{c}-\frac {2 \cos ^3(e+f x)}{3 c f (c-c \sin (e+f x))^{3/2}}\right )}{c}-\frac {2 \cos ^5(e+f x)}{5 c f (c-c \sin (e+f x))^{5/2}}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle a^3 c^3 \left (\frac {2 \left (\frac {2 \left (\frac {2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{c^{3/2} f}-\frac {2 \cos (e+f x)}{c f \sqrt {c-c \sin (e+f x)}}\right )}{c}-\frac {2 \cos ^3(e+f x)}{3 c f (c-c \sin (e+f x))^{3/2}}\right )}{c}-\frac {2 \cos ^5(e+f x)}{5 c f (c-c \sin (e+f x))^{5/2}}\right )\) |
Input:
Int[(a + a*Sin[e + f*x])^3/Sqrt[c - c*Sin[e + f*x]],x]
Output:
a^3*c^3*((-2*Cos[e + f*x]^5)/(5*c*f*(c - c*Sin[e + f*x])^(5/2)) + (2*((-2* Cos[e + f*x]^3)/(3*c*f*(c - c*Sin[e + f*x])^(3/2)) + (2*((2*Sqrt[2]*ArcTan h[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(c^(3/2)*f) - (2*Cos[e + f*x])/(c*f*Sqrt[c - c*Sin[e + f*x]])))/c))/c)
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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2/d Subst[Int[1/(2*a - x^2), x], x, b*(Cos[c + d*x]/Sqrt[a + b*Sin[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[g*(g*Cos[e + f*x])^(p - 1)*((a + b*Sin[e + f*x ])^(m + 1)/(b*f*(m + p))), x] + Simp[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] && In tegersQ[2*m, 2*p]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[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] && ((Lt Q[m, 0] && GtQ[n, 0]) || LtQ[0, n, m] || LtQ[m, n, 0]))
Time = 2.11 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.85
| method | result | size |
| default | \(-\frac {2 \left (-1+\sin \left (f x +e \right )\right ) \sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, a^{3} \left (60 c^{\frac {5}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )-3 \left (c \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {5}{2}}-10 \left (c \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {3}{2}} c -60 c^{2} \sqrt {c \left (1+\sin \left (f x +e \right )\right )}\right )}{15 c^{3} \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(129\) |
| parts | \(-\frac {a^{3} \left (-1+\sin \left (f x +e \right )\right ) \sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{\sqrt {c}\, \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}-\frac {a^{3} \left (-1+\sin \left (f x +e \right )\right ) \sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \left (15 c^{\frac {5}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )-6 \left (c \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {5}{2}}+10 \left (c \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {3}{2}} c -30 c^{2} \sqrt {c \left (1+\sin \left (f x +e \right )\right )}\right )}{15 c^{3} \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}+\frac {3 a^{3} \left (-1+\sin \left (f x +e \right )\right ) \sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \left (-\sqrt {c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )+2 \sqrt {c \left (1+\sin \left (f x +e \right )\right )}\right )}{c \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}-\frac {a^{3} \left (-1+\sin \left (f x +e \right )\right ) \sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \left (3 c^{\frac {3}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )-2 \left (c \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {3}{2}}\right )}{c^{2} \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(398\) |
Input:
int((a+sin(f*x+e)*a)^3/(c-c*sin(f*x+e))^(1/2),x,method=_RETURNVERBOSE)
Output:
-2/15*(-1+sin(f*x+e))*(c*(1+sin(f*x+e)))^(1/2)*a^3*(60*c^(5/2)*2^(1/2)*arc tanh(1/2*(c*(1+sin(f*x+e)))^(1/2)*2^(1/2)/c^(1/2))-3*(c*(1+sin(f*x+e)))^(5 /2)-10*(c*(1+sin(f*x+e)))^(3/2)*c-60*c^2*(c*(1+sin(f*x+e)))^(1/2))/c^3/cos (f*x+e)/(c-c*sin(f*x+e))^(1/2)/f
Leaf count of result is larger than twice the leaf count of optimal. 265 vs. \(2 (132) = 264\).
Time = 0.09 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.75 \[ \int \frac {(a+a \sin (e+f x))^3}{\sqrt {c-c \sin (e+f x)}} \, dx=\frac {2 \, {\left (\frac {30 \, \sqrt {2} {\left (a^{3} c \cos \left (f x + e\right ) - a^{3} c \sin \left (f x + e\right ) + a^{3} c\right )} \log \left (-\frac {\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) - 2\right )} \sin \left (f x + e\right ) + \frac {2 \, \sqrt {2} \sqrt {-c \sin \left (f x + e\right ) + c} {\left (\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right )}}{\sqrt {c}} + 3 \, \cos \left (f x + e\right ) + 2}{\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right )}{\sqrt {c}} + {\left (3 \, a^{3} \cos \left (f x + e\right )^{3} + 19 \, a^{3} \cos \left (f x + e\right )^{2} - 76 \, a^{3} \cos \left (f x + e\right ) - 92 \, a^{3} + {\left (3 \, a^{3} \cos \left (f x + e\right )^{2} - 16 \, a^{3} \cos \left (f x + e\right ) - 92 \, a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}\right )}}{15 \, {\left (c f \cos \left (f x + e\right ) - c f \sin \left (f x + e\right ) + c f\right )}} \] Input:
integrate((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^(1/2),x, algorithm="fricas")
Output:
2/15*(30*sqrt(2)*(a^3*c*cos(f*x + e) - a^3*c*sin(f*x + e) + a^3*c)*log(-(c os(f*x + e)^2 + (cos(f*x + e) - 2)*sin(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))/s qrt(c) + (3*a^3*cos(f*x + e)^3 + 19*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)*sin(f* x + e))*sqrt(-c*sin(f*x + e) + c))/(c*f*cos(f*x + e) - c*f*sin(f*x + e) + c*f)
\[ \int \frac {(a+a \sin (e+f x))^3}{\sqrt {c-c \sin (e+f x)}} \, dx=a^{3} \left (\int \frac {3 \sin {\left (e + f x \right )}}{\sqrt {- c \sin {\left (e + f x \right )} + c}}\, dx + \int \frac {3 \sin ^{2}{\left (e + f x \right )}}{\sqrt {- c \sin {\left (e + f x \right )} + c}}\, dx + \int \frac {\sin ^{3}{\left (e + f x \right )}}{\sqrt {- c \sin {\left (e + f x \right )} + c}}\, dx + \int \frac {1}{\sqrt {- c \sin {\left (e + f x \right )} + c}}\, dx\right ) \] Input:
integrate((a+a*sin(f*x+e))**3/(c-c*sin(f*x+e))**(1/2),x)
Output:
a**3*(Integral(3*sin(e + f*x)/sqrt(-c*sin(e + f*x) + c), x) + Integral(3*s in(e + f*x)**2/sqrt(-c*sin(e + f*x) + c), x) + Integral(sin(e + f*x)**3/sq rt(-c*sin(e + f*x) + c), x) + Integral(1/sqrt(-c*sin(e + f*x) + c), x))
\[ \int \frac {(a+a \sin (e+f x))^3}{\sqrt {c-c \sin (e+f x)}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{3}}{\sqrt {-c \sin \left (f x + e\right ) + c}} \,d x } \] Input:
integrate((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^(1/2),x, algorithm="maxima")
Output:
integrate((a*sin(f*x + e) + a)^3/sqrt(-c*sin(f*x + e) + c), x)
Exception generated. \[ \int \frac {(a+a \sin (e+f x))^3}{\sqrt {c-c \sin (e+f x)}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^(1/2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {(a+a \sin (e+f x))^3}{\sqrt {c-c \sin (e+f x)}} \, dx=\int \frac {{\left (a+a\,\sin \left (e+f\,x\right )\right )}^3}{\sqrt {c-c\,\sin \left (e+f\,x\right )}} \,d x \] Input:
int((a + a*sin(e + f*x))^3/(c - c*sin(e + f*x))^(1/2),x)
Output:
int((a + a*sin(e + f*x))^3/(c - c*sin(e + f*x))^(1/2), x)
\[ \int \frac {(a+a \sin (e+f x))^3}{\sqrt {c-c \sin (e+f x)}} \, dx=\frac {\sqrt {c}\, a^{3} \left (-\left (\int \frac {\sqrt {-\sin \left (f x +e \right )+1}}{\sin \left (f x +e \right )-1}d x \right )-\left (\int \frac {\sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{3}}{\sin \left (f x +e \right )-1}d x \right )-3 \left (\int \frac {\sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{2}}{\sin \left (f x +e \right )-1}d x \right )-3 \left (\int \frac {\sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )}{\sin \left (f x +e \right )-1}d x \right )\right )}{c} \] Input:
int((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^(1/2),x)
Output:
(sqrt(c)*a**3*( - int(sqrt( - sin(e + f*x) + 1)/(sin(e + f*x) - 1),x) - in t((sqrt( - sin(e + f*x) + 1)*sin(e + f*x)**3)/(sin(e + f*x) - 1),x) - 3*in t((sqrt( - sin(e + f*x) + 1)*sin(e + f*x)**2)/(sin(e + f*x) - 1),x) - 3*in t((sqrt( - sin(e + f*x) + 1)*sin(e + f*x))/(sin(e + f*x) - 1),x)))/c