Integrand size = 27, antiderivative size = 192 \[ \int \frac {(c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^{3/2}} \, dx=-\frac {(c-d)^2 (c+11 d) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{2 \sqrt {2} a^{3/2} f}+\frac {d \left (3 c^2-24 c d+13 d^2\right ) \cos (e+f x)}{3 a f \sqrt {a+a \sin (e+f x)}}+\frac {(3 c-7 d) d^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{6 a^2 f}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a+a \sin (e+f x))^{3/2}} \] Output:
-1/4*(c-d)^2*(c+11*d)*arctanh(1/2*a^(1/2)*cos(f*x+e)*2^(1/2)/(a+a*sin(f*x+ e))^(1/2))*2^(1/2)/a^(3/2)/f+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-1/2*(c-d)*cos(f*x+e)*(c+d*sin(f*x+e))^2/f/(a+a*sin(f*x+e))^(3/2)
Result contains complex when optimal does not.
Time = 1.72 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.71 \[ \int \frac {(c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^{3/2}} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (6 (c-d)^3 \sin \left (\frac {1}{2} (e+f x)\right )-3 (c-d)^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )+(3+3 i) (-1)^{3/4} (c-d)^2 (c+11 d) \text {arctanh}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \left (-1+\tan \left (\frac {1}{4} (e+f x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2-18 (2 c-d) d^2 \cos \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2-2 d^3 \cos \left (\frac {3}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2+18 (2 c-d) d^2 \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2-2 d^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2 \sin \left (\frac {3}{2} (e+f x)\right )\right )}{6 f (a (1+\sin (e+f x)))^{3/2}} \] Input:
Integrate[(c + d*Sin[e + f*x])^3/(a + a*Sin[e + f*x])^(3/2),x]
Output:
((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]*(C os[(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]))/(6*f*(a*(1 + Sin[e + f*x]))^(3/2))
Time = 1.08 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.05, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {3042, 3244, 27, 3042, 3447, 3042, 3502, 27, 3042, 3230, 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.
| \(\displaystyle \int \frac {(c+d \sin (e+f x))^3}{(a \sin (e+f x)+a)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(c+d \sin (e+f x))^3}{(a \sin (e+f x)+a)^{3/2}}dx\) |
| \(\Big \downarrow \) 3244 |
| \(\displaystyle -\frac {\int -\frac {(c+d \sin (e+f x)) \left (a \left (c^2+7 d c-4 d^2\right )-a (3 c-7 d) d \sin (e+f x)\right )}{2 \sqrt {\sin (e+f x) a+a}}dx}{2 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a \sin (e+f x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(c+d \sin (e+f x)) \left (a \left (c^2+7 d c-4 d^2\right )-a (3 c-7 d) d \sin (e+f x)\right )}{\sqrt {\sin (e+f x) a+a}}dx}{4 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a \sin (e+f x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(c+d \sin (e+f x)) \left (a \left (c^2+7 d c-4 d^2\right )-a (3 c-7 d) d \sin (e+f x)\right )}{\sqrt {\sin (e+f x) a+a}}dx}{4 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a \sin (e+f x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {\int \frac {-a (3 c-7 d) d^2 \sin ^2(e+f x)+\left (a d \left (c^2+7 d c-4 d^2\right )-a c (3 c-7 d) d\right ) \sin (e+f x)+a c \left (c^2+7 d c-4 d^2\right )}{\sqrt {\sin (e+f x) a+a}}dx}{4 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a \sin (e+f x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {-a (3 c-7 d) d^2 \sin (e+f x)^2+\left (a d \left (c^2+7 d c-4 d^2\right )-a c (3 c-7 d) d\right ) \sin (e+f x)+a c \left (c^2+7 d c-4 d^2\right )}{\sqrt {\sin (e+f x) a+a}}dx}{4 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a \sin (e+f x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {2 \int \frac {a^2 \left (3 c^3+21 d c^2-15 d^2 c+7 d^3\right )-2 a^2 d \left (3 c^2-24 d c+13 d^2\right ) \sin (e+f x)}{2 \sqrt {\sin (e+f x) a+a}}dx}{3 a}+\frac {2 d^2 (3 c-7 d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}}{4 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a \sin (e+f x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {a^2 \left (3 c^3+21 d c^2-15 d^2 c+7 d^3\right )-2 a^2 d \left (3 c^2-24 d c+13 d^2\right ) \sin (e+f x)}{\sqrt {\sin (e+f x) a+a}}dx}{3 a}+\frac {2 d^2 (3 c-7 d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}}{4 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a \sin (e+f x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {a^2 \left (3 c^3+21 d c^2-15 d^2 c+7 d^3\right )-2 a^2 d \left (3 c^2-24 d c+13 d^2\right ) \sin (e+f x)}{\sqrt {\sin (e+f x) a+a}}dx}{3 a}+\frac {2 d^2 (3 c-7 d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}}{4 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a \sin (e+f x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 3230 |
| \(\displaystyle \frac {\frac {3 a^2 (c-d)^2 (c+11 d) \int \frac {1}{\sqrt {\sin (e+f x) a+a}}dx+\frac {4 a^2 d \left (3 c^2-24 c d+13 d^2\right ) \cos (e+f x)}{f \sqrt {a \sin (e+f x)+a}}}{3 a}+\frac {2 d^2 (3 c-7 d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}}{4 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a \sin (e+f x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3 a^2 (c-d)^2 (c+11 d) \int \frac {1}{\sqrt {\sin (e+f x) a+a}}dx+\frac {4 a^2 d \left (3 c^2-24 c d+13 d^2\right ) \cos (e+f x)}{f \sqrt {a \sin (e+f x)+a}}}{3 a}+\frac {2 d^2 (3 c-7 d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}}{4 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a \sin (e+f x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 3128 |
| \(\displaystyle \frac {\frac {\frac {4 a^2 d \left (3 c^2-24 c d+13 d^2\right ) \cos (e+f x)}{f \sqrt {a \sin (e+f x)+a}}-\frac {6 a^2 (c-d)^2 (c+11 d) \int \frac {1}{2 a-\frac {a^2 \cos ^2(e+f x)}{\sin (e+f x) a+a}}d\frac {a \cos (e+f x)}{\sqrt {\sin (e+f x) a+a}}}{f}}{3 a}+\frac {2 d^2 (3 c-7 d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}}{4 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a \sin (e+f x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {4 a^2 d \left (3 c^2-24 c d+13 d^2\right ) \cos (e+f x)}{f \sqrt {a \sin (e+f x)+a}}-\frac {3 \sqrt {2} a^{3/2} (c-d)^2 (c+11 d) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{f}}{3 a}+\frac {2 d^2 (3 c-7 d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}}{4 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a \sin (e+f x)+a)^{3/2}}\) |
Input:
Int[(c + d*Sin[e + f*x])^3/(a + a*Sin[e + f*x])^(3/2),x]
Output:
-1/2*((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(f*(a + a*Sin[e + f*x]) ^(3/2)) + ((2*(3*c - 7*d)*d^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(3*f) + ((-3*Sqrt[2]*a^(3/2)*(c - d)^2*(c + 11*d)*ArcTanh[(Sqrt[a]*Cos[e + f*x] )/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/f + (4*a^2*d*(3*c^2 - 24*c*d + 13*d ^2)*Cos[e + f*x])/(f*Sqrt[a + a*Sin[e + f*x]]))/(3*a))/(4*a^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[((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] + Simp[(a*d*m + b*c*(m + 1))/(b*(m + 1)) Int[(a + b*Sin[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)]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(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] + Simp[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]))
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]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[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]
Leaf count of result is larger than twice the leaf count of optimal. \(489\) vs. \(2(169)=338\).
Time = 4.61 (sec) , antiderivative size = 490, normalized size of antiderivative = 2.55
| method | result | size |
| default | \(-\frac {\left (\sin \left (f x +e \right ) \left (3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -\sin \left (f x +e \right ) a}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} c^{3}+27 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -\sin \left (f x +e \right ) a}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} c^{2} d -63 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -\sin \left (f x +e \right ) a}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} c \,d^{2}+33 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -\sin \left (f x +e \right ) a}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} d^{3}-8 \left (a -\sin \left (f x +e \right ) a \right )^{\frac {3}{2}} d^{3} \sqrt {a}+72 a^{\frac {3}{2}} d^{2} c \sqrt {a -\sin \left (f x +e \right ) a}-24 a^{\frac {3}{2}} d^{3} \sqrt {a -\sin \left (f x +e \right ) a}\right )+3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -\sin \left (f x +e \right ) a}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} c^{3}+27 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -\sin \left (f x +e \right ) a}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} c^{2} d -63 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -\sin \left (f x +e \right ) a}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} c \,d^{2}+33 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -\sin \left (f x +e \right ) a}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} d^{3}-8 \left (a -\sin \left (f x +e \right ) a \right )^{\frac {3}{2}} d^{3} \sqrt {a}+6 \sqrt {a -\sin \left (f x +e \right ) a}\, a^{\frac {3}{2}} c^{3}-18 \sqrt {a -\sin \left (f x +e \right ) a}\, a^{\frac {3}{2}} c^{2} d +90 a^{\frac {3}{2}} d^{2} c \sqrt {a -\sin \left (f x +e \right ) a}-30 a^{\frac {3}{2}} d^{3} \sqrt {a -\sin \left (f x +e \right ) a}\right ) \sqrt {-a \left (-1+\sin \left (f x +e \right )\right )}}{12 a^{\frac {7}{2}} \cos \left (f x +e \right ) \sqrt {a +\sin \left (f x +e \right ) a}\, f}\) | \(490\) |
| parts | \(-\frac {c^{3} \left (\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -\sin \left (f x +e \right ) a}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} \sin \left (f x +e \right )+\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -\sin \left (f x +e \right ) a}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2}+2 \sqrt {a -\sin \left (f x +e \right ) a}\, a^{\frac {3}{2}}\right ) \sqrt {-a \left (-1+\sin \left (f x +e \right )\right )}}{4 a^{\frac {7}{2}} \cos \left (f x +e \right ) \sqrt {a +\sin \left (f x +e \right ) a}\, f}-\frac {d^{3} \left (33 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -\sin \left (f x +e \right ) a}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} \sin \left (f x +e \right )-8 \left (a -\sin \left (f x +e \right ) a \right )^{\frac {3}{2}} \sin \left (f x +e \right ) \sqrt {a}+33 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -\sin \left (f x +e \right ) a}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2}-8 \left (a -\sin \left (f x +e \right ) a \right )^{\frac {3}{2}} \sqrt {a}-24 \sqrt {a -\sin \left (f x +e \right ) a}\, \sin \left (f x +e \right ) a^{\frac {3}{2}}-30 \sqrt {a -\sin \left (f x +e \right ) a}\, a^{\frac {3}{2}}\right ) \sqrt {-a \left (-1+\sin \left (f x +e \right )\right )}}{12 a^{\frac {7}{2}} \cos \left (f x +e \right ) \sqrt {a +\sin \left (f x +e \right ) a}\, f}+\frac {3 c \,d^{2} \left (7 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -\sin \left (f x +e \right ) a}\, \sqrt {2}}{2 \sqrt {a}}\right ) a \sin \left (f x +e \right )+7 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -\sin \left (f x +e \right ) a}\, \sqrt {2}}{2 \sqrt {a}}\right ) a -8 \sqrt {a -\sin \left (f x +e \right ) a}\, \sqrt {a}\, \sin \left (f x +e \right )-10 \sqrt {a -\sin \left (f x +e \right ) a}\, \sqrt {a}\right ) \sqrt {-a \left (-1+\sin \left (f x +e \right )\right )}}{4 a^{\frac {5}{2}} \cos \left (f x +e \right ) \sqrt {a +\sin \left (f x +e \right ) a}\, f}-\frac {3 c^{2} d \left (3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -\sin \left (f x +e \right ) a}\, \sqrt {2}}{2 \sqrt {a}}\right ) a \sin \left (f x +e \right )+3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -\sin \left (f x +e \right ) a}\, \sqrt {2}}{2 \sqrt {a}}\right ) a -2 \sqrt {a -\sin \left (f x +e \right ) a}\, \sqrt {a}\right ) \sqrt {-a \left (-1+\sin \left (f x +e \right )\right )}}{4 a^{\frac {5}{2}} \cos \left (f x +e \right ) \sqrt {a +\sin \left (f x +e \right ) a}\, f}\) | \(600\) |
Input:
int((c+d*sin(f*x+e))^3/(a+sin(f*x+e)*a)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/12/a^(7/2)*(sin(f*x+e)*(3*2^(1/2)*arctanh(1/2*(a-sin(f*x+e)*a)^(1/2)*2^ (1/2)/a^(1/2))*a^2*c^3+27*2^(1/2)*arctanh(1/2*(a-sin(f*x+e)*a)^(1/2)*2^(1/ 2)/a^(1/2))*a^2*c^2*d-63*2^(1/2)*arctanh(1/2*(a-sin(f*x+e)*a)^(1/2)*2^(1/2 )/a^(1/2))*a^2*c*d^2+33*2^(1/2)*arctanh(1/2*(a-sin(f*x+e)*a)^(1/2)*2^(1/2) /a^(1/2))*a^2*d^3-8*(a-sin(f*x+e)*a)^(3/2)*d^3*a^(1/2)+72*a^(3/2)*d^2*c*(a -sin(f*x+e)*a)^(1/2)-24*a^(3/2)*d^3*(a-sin(f*x+e)*a)^(1/2))+3*2^(1/2)*arct anh(1/2*(a-sin(f*x+e)*a)^(1/2)*2^(1/2)/a^(1/2))*a^2*c^3+27*2^(1/2)*arctanh (1/2*(a-sin(f*x+e)*a)^(1/2)*2^(1/2)/a^(1/2))*a^2*c^2*d-63*2^(1/2)*arctanh( 1/2*(a-sin(f*x+e)*a)^(1/2)*2^(1/2)/a^(1/2))*a^2*c*d^2+33*2^(1/2)*arctanh(1 /2*(a-sin(f*x+e)*a)^(1/2)*2^(1/2)/a^(1/2))*a^2*d^3-8*(a-sin(f*x+e)*a)^(3/2 )*d^3*a^(1/2)+6*(a-sin(f*x+e)*a)^(1/2)*a^(3/2)*c^3-18*(a-sin(f*x+e)*a)^(1/ 2)*a^(3/2)*c^2*d+90*a^(3/2)*d^2*c*(a-sin(f*x+e)*a)^(1/2)-30*a^(3/2)*d^3*(a -sin(f*x+e)*a)^(1/2))*(-a*(-1+sin(f*x+e)))^(1/2)/cos(f*x+e)/(a+sin(f*x+e)* a)^(1/2)/f
Leaf count of result is larger than twice the leaf count of optimal. 494 vs. \(2 (169) = 338\).
Time = 0.10 (sec) , antiderivative size = 494, normalized size of antiderivative = 2.57 \[ \int \frac {(c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^{3/2}} \, dx=-\frac {3 \, \sqrt {2} {\left (2 \, c^{3} + 18 \, c^{2} d - 42 \, c d^{2} + 22 \, d^{3} - {\left (c^{3} + 9 \, c^{2} d - 21 \, c d^{2} + 11 \, d^{3}\right )} \cos \left (f x + e\right )^{2} + {\left (c^{3} + 9 \, c^{2} d - 21 \, c d^{2} + 11 \, d^{3}\right )} \cos \left (f x + e\right ) + {\left (2 \, c^{3} + 18 \, c^{2} d - 42 \, c d^{2} + 22 \, d^{3} + {\left (c^{3} + 9 \, c^{2} d - 21 \, c d^{2} + 11 \, d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a} \log \left (-\frac {a \cos \left (f x + e\right )^{2} - 2 \, \sqrt {2} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {a} {\left (\cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right )} + 3 \, a \cos \left (f x + e\right ) - {\left (a \cos \left (f x + e\right ) - 2 \, a\right )} \sin \left (f x + e\right ) + 2 \, a}{\cos \left (f x + e\right )^{2} - {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right ) + 4 \, {\left (4 \, d^{3} \cos \left (f x + e\right )^{3} - 3 \, c^{3} + 9 \, c^{2} d - 9 \, c d^{2} + 3 \, d^{3} - 4 \, {\left (9 \, c d^{2} - 4 \, d^{3}\right )} \cos \left (f x + e\right )^{2} - 3 \, {\left (c^{3} - 3 \, c^{2} d + 15 \, c d^{2} - 5 \, d^{3}\right )} \cos \left (f x + e\right ) - {\left (4 \, d^{3} \cos \left (f x + e\right )^{2} - 3 \, c^{3} + 9 \, c^{2} d - 9 \, c d^{2} + 3 \, d^{3} + 12 \, {\left (3 \, c d^{2} - d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{24 \, {\left (a^{2} f \cos \left (f x + e\right )^{2} - a^{2} f \cos \left (f x + e\right ) - 2 \, a^{2} f - {\left (a^{2} f \cos \left (f x + e\right ) + 2 \, a^{2} f\right )} \sin \left (f x + e\right )\right )}} \] Input:
integrate((c+d*sin(f*x+e))^3/(a+a*sin(f*x+e))^(3/2),x, algorithm="fricas")
Output:
-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)*co s(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))
\[ \int \frac {(c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^{3/2}} \, dx=\int \frac {\left (c + d \sin {\left (e + f x \right )}\right )^{3}}{\left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((c+d*sin(f*x+e))**3/(a+a*sin(f*x+e))**(3/2),x)
Output:
Integral((c + d*sin(e + f*x))**3/(a*(sin(e + f*x) + 1))**(3/2), x)
\[ \int \frac {(c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^{3/2}} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{3}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((c+d*sin(f*x+e))^3/(a+a*sin(f*x+e))^(3/2),x, algorithm="maxima")
Output:
integrate((d*sin(f*x + e) + c)^3/(a*sin(f*x + e) + a)^(3/2), x)
Exception generated. \[ \int \frac {(c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((c+d*sin(f*x+e))^3/(a+a*sin(f*x+e))^(3/2),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. \[ \int \frac {(c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^{3/2}} \, dx=\int \frac {{\left (c+d\,\sin \left (e+f\,x\right )\right )}^3}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \] Input:
int((c + d*sin(e + f*x))^3/(a + a*sin(e + f*x))^(3/2),x)
Output:
int((c + d*sin(e + f*x))^3/(a + a*sin(e + f*x))^(3/2), x)
\[ \int \frac {(c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^{3/2}} \, dx=\frac {\sqrt {a}\, \left (\left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}}{\sin \left (f x +e \right )^{2}+2 \sin \left (f x +e \right )+1}d x \right ) c^{3}+\left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{3}}{\sin \left (f x +e \right )^{2}+2 \sin \left (f x +e \right )+1}d x \right ) d^{3}+3 \left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{2}}{\sin \left (f x +e \right )^{2}+2 \sin \left (f x +e \right )+1}d x \right ) c \,d^{2}+3 \left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )}{\sin \left (f x +e \right )^{2}+2 \sin \left (f x +e \right )+1}d x \right ) c^{2} d \right )}{a^{2}} \] Input:
int((c+d*sin(f*x+e))^3/(a+a*sin(f*x+e))^(3/2),x)
Output:
(sqrt(a)*(int(sqrt(sin(e + f*x) + 1)/(sin(e + f*x)**2 + 2*sin(e + f*x) + 1 ),x)*c**3 + int((sqrt(sin(e + f*x) + 1)*sin(e + f*x)**3)/(sin(e + f*x)**2 + 2*sin(e + f*x) + 1),x)*d**3 + 3*int((sqrt(sin(e + f*x) + 1)*sin(e + f*x) **2)/(sin(e + f*x)**2 + 2*sin(e + f*x) + 1),x)*c*d**2 + 3*int((sqrt(sin(e + f*x) + 1)*sin(e + f*x))/(sin(e + f*x)**2 + 2*sin(e + f*x) + 1),x)*c**2*d ))/a**2