Integrand size = 27, antiderivative size = 178 \[ \int \frac {(c+d \sin (e+f x))^3}{\sqrt {a+a \sin (e+f x)}} \, dx=-\frac {\sqrt {2} (c-d)^3 \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{\sqrt {a} f}-\frac {4 d \left (21 c^2-12 c d+7 d^2\right ) \cos (e+f x)}{15 f \sqrt {a+a \sin (e+f x)}}-\frac {2 (9 c-d) d^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{15 a f}-\frac {2 d \cos (e+f x) (c+d \sin (e+f x))^2}{5 f \sqrt {a+a \sin (e+f x)}} \] Output:
-2^(1/2)*(c-d)^3*arctanh(1/2*a^(1/2)*cos(f*x+e)*2^(1/2)/(a+a*sin(f*x+e))^( 1/2))/a^(1/2)/f-4/15*d*(21*c^2-12*c*d+7*d^2)*cos(f*x+e)/f/(a+a*sin(f*x+e)) ^(1/2)-2/15*(9*c-d)*d^2*cos(f*x+e)*(a+a*sin(f*x+e))^(1/2)/a/f-2/5*d*cos(f* x+e)*(c+d*sin(f*x+e))^2/f/(a+a*sin(f*x+e))^(1/2)
Result contains complex when optimal does not.
Time = 1.25 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.87 \[ \int \frac {(c+d \sin (e+f x))^3}{\sqrt {a+a \sin (e+f x)}} \, dx=-\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left ((-60-60 i) (-1)^{3/4} (c-d)^3 \text {arctanh}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \left (-1+\tan \left (\frac {1}{4} (e+f x)\right )\right )\right )-2 d \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (-90 c^2+30 c d-29 d^2+3 d^2 \cos (2 (e+f x))-2 (15 c-d) d \sin (e+f x)\right )\right )}{30 f \sqrt {a (1+\sin (e+f x))}} \] Input:
Integrate[(c + d*Sin[e + f*x])^3/Sqrt[a + a*Sin[e + f*x]],x]
Output:
-1/30*((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*((-60 - 60*I)*(-1)^(3/4)*(c - d)^3*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(e + f*x)/4])] - 2*d*(Cos[( e + f*x)/2] - Sin[(e + f*x)/2])*(-90*c^2 + 30*c*d - 29*d^2 + 3*d^2*Cos[2*( e + f*x)] - 2*(15*c - d)*d*Sin[e + f*x])))/(f*Sqrt[a*(1 + Sin[e + f*x])])
Time = 1.04 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.08, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {3042, 3257, 25, 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.
| \(\displaystyle \int \frac {(c+d \sin (e+f x))^3}{\sqrt {a \sin (e+f x)+a}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(c+d \sin (e+f x))^3}{\sqrt {a \sin (e+f x)+a}}dx\) |
| \(\Big \downarrow \) 3257 |
| \(\displaystyle -\frac {\int -\frac {(c+d \sin (e+f x)) \left (a \left (5 c^2-d c+4 d^2\right )+a (9 c-d) d \sin (e+f x)\right )}{\sqrt {\sin (e+f x) a+a}}dx}{5 a}-\frac {2 d \cos (e+f x) (c+d \sin (e+f x))^2}{5 f \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {(c+d \sin (e+f x)) \left (a \left (5 c^2-d c+4 d^2\right )+a (9 c-d) d \sin (e+f x)\right )}{\sqrt {\sin (e+f x) a+a}}dx}{5 a}-\frac {2 d \cos (e+f x) (c+d \sin (e+f x))^2}{5 f \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(c+d \sin (e+f x)) \left (a \left (5 c^2-d c+4 d^2\right )+a (9 c-d) d \sin (e+f x)\right )}{\sqrt {\sin (e+f x) a+a}}dx}{5 a}-\frac {2 d \cos (e+f x) (c+d \sin (e+f x))^2}{5 f \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {\int \frac {a (9 c-d) d^2 \sin ^2(e+f x)+\left (a c (9 c-d) d+a \left (5 c^2-d c+4 d^2\right ) d\right ) \sin (e+f x)+a c \left (5 c^2-d c+4 d^2\right )}{\sqrt {\sin (e+f x) a+a}}dx}{5 a}-\frac {2 d \cos (e+f x) (c+d \sin (e+f x))^2}{5 f \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a (9 c-d) d^2 \sin (e+f x)^2+\left (a c (9 c-d) d+a \left (5 c^2-d c+4 d^2\right ) d\right ) \sin (e+f x)+a c \left (5 c^2-d c+4 d^2\right )}{\sqrt {\sin (e+f x) a+a}}dx}{5 a}-\frac {2 d \cos (e+f x) (c+d \sin (e+f x))^2}{5 f \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {2 \int \frac {\left (15 c^3-3 d c^2+21 d^2 c-d^3\right ) a^2+2 d \left (21 c^2-12 d c+7 d^2\right ) \sin (e+f x) a^2}{2 \sqrt {\sin (e+f x) a+a}}dx}{3 a}-\frac {2 d^2 (9 c-d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}}{5 a}-\frac {2 d \cos (e+f x) (c+d \sin (e+f x))^2}{5 f \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\left (15 c^3-3 d c^2+21 d^2 c-d^3\right ) a^2+2 d \left (21 c^2-12 d c+7 d^2\right ) \sin (e+f x) a^2}{\sqrt {\sin (e+f x) a+a}}dx}{3 a}-\frac {2 d^2 (9 c-d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}}{5 a}-\frac {2 d \cos (e+f x) (c+d \sin (e+f x))^2}{5 f \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {\left (15 c^3-3 d c^2+21 d^2 c-d^3\right ) a^2+2 d \left (21 c^2-12 d c+7 d^2\right ) \sin (e+f x) a^2}{\sqrt {\sin (e+f x) a+a}}dx}{3 a}-\frac {2 d^2 (9 c-d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}}{5 a}-\frac {2 d \cos (e+f x) (c+d \sin (e+f x))^2}{5 f \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3230 |
| \(\displaystyle \frac {\frac {15 a^2 (c-d)^3 \int \frac {1}{\sqrt {\sin (e+f x) a+a}}dx-\frac {4 a^2 d \left (21 c^2-12 c d+7 d^2\right ) \cos (e+f x)}{f \sqrt {a \sin (e+f x)+a}}}{3 a}-\frac {2 d^2 (9 c-d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}}{5 a}-\frac {2 d \cos (e+f x) (c+d \sin (e+f x))^2}{5 f \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {15 a^2 (c-d)^3 \int \frac {1}{\sqrt {\sin (e+f x) a+a}}dx-\frac {4 a^2 d \left (21 c^2-12 c d+7 d^2\right ) \cos (e+f x)}{f \sqrt {a \sin (e+f x)+a}}}{3 a}-\frac {2 d^2 (9 c-d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}}{5 a}-\frac {2 d \cos (e+f x) (c+d \sin (e+f x))^2}{5 f \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3128 |
| \(\displaystyle \frac {\frac {-\frac {30 a^2 (c-d)^3 \int \frac {1}{2 a-\frac {a^2 \cos ^2(e+f x)}{\sin (e+f x) a+a}}d\frac {a \cos (e+f x)}{\sqrt {\sin (e+f x) a+a}}}{f}-\frac {4 a^2 d \left (21 c^2-12 c d+7 d^2\right ) \cos (e+f x)}{f \sqrt {a \sin (e+f x)+a}}}{3 a}-\frac {2 d^2 (9 c-d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}}{5 a}-\frac {2 d \cos (e+f x) (c+d \sin (e+f x))^2}{5 f \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {-\frac {15 \sqrt {2} a^{3/2} (c-d)^3 \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{f}-\frac {4 a^2 d \left (21 c^2-12 c d+7 d^2\right ) \cos (e+f x)}{f \sqrt {a \sin (e+f x)+a}}}{3 a}-\frac {2 d^2 (9 c-d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}}{5 a}-\frac {2 d \cos (e+f x) (c+d \sin (e+f x))^2}{5 f \sqrt {a \sin (e+f x)+a}}\) |
Input:
Int[(c + d*Sin[e + f*x])^3/Sqrt[a + a*Sin[e + f*x]],x]
Output:
(-2*d*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(5*f*Sqrt[a + a*Sin[e + f*x]]) + ((-2*(9*c - d)*d^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(3*f) + ((-15* Sqrt[2]*a^(3/2)*(c - d)^3*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/f - (4*a^2*d*(21*c^2 - 12*c*d + 7*d^2)*Cos[e + f*x])/( f*Sqrt[a + a*Sin[e + f*x]]))/(3*a))/(5*a)
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[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)/Sqrt[(a_) + (b_.)*sin[(e_. ) + (f_.)*(x_)]], x_Symbol] :> Simp[-2*d*Cos[e + f*x]*((c + d*Sin[e + f*x]) ^(n - 1)/(f*(2*n - 1)*Sqrt[a + b*Sin[e + f*x]])), x] - Simp[1/(b*(2*n - 1)) Int[((c + d*Sin[e + f*x])^(n - 2)/Sqrt[a + b*Sin[e + f*x]])*Simp[a*c*d - b*(2*d^2*(n - 1) + c^2*(2*n - 1)) + d*(a*d - b*c*(4*n - 3))*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] && GtQ[n, 1] && IntegerQ[2*n]
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]
Time = 4.43 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.60
| method | result | size |
| default | \(-\frac {\left (1+\sin \left (f x +e \right )\right ) \sqrt {-a \left (-1+\sin \left (f x +e \right )\right )}\, \left (15 a^{\frac {5}{2}} \operatorname {arctanh}\left (\frac {\sqrt {a -\sin \left (f x +e \right ) a}\, \sqrt {2}}{2 \sqrt {a}}\right ) \sqrt {2}\, c^{3}-45 a^{\frac {5}{2}} \operatorname {arctanh}\left (\frac {\sqrt {a -\sin \left (f x +e \right ) a}\, \sqrt {2}}{2 \sqrt {a}}\right ) \sqrt {2}\, c^{2} d +45 a^{\frac {5}{2}} \operatorname {arctanh}\left (\frac {\sqrt {a -\sin \left (f x +e \right ) a}\, \sqrt {2}}{2 \sqrt {a}}\right ) \sqrt {2}\, c \,d^{2}-15 a^{\frac {5}{2}} \operatorname {arctanh}\left (\frac {\sqrt {a -\sin \left (f x +e \right ) a}\, \sqrt {2}}{2 \sqrt {a}}\right ) \sqrt {2}\, d^{3}+6 \left (a -\sin \left (f x +e \right ) a \right )^{\frac {5}{2}} d^{3}-30 \left (a -\sin \left (f x +e \right ) a \right )^{\frac {3}{2}} a c \,d^{2}-10 \left (a -\sin \left (f x +e \right ) a \right )^{\frac {3}{2}} a \,d^{3}+90 c^{2} d \,a^{2} \sqrt {a -\sin \left (f x +e \right ) a}+30 d^{3} a^{2} \sqrt {a -\sin \left (f x +e \right ) a}\right )}{15 a^{3} \cos \left (f x +e \right ) \sqrt {a +\sin \left (f x +e \right ) a}\, f}\) | \(285\) |
| parts | \(-\frac {c^{3} \left (1+\sin \left (f x +e \right )\right ) \sqrt {-a \left (-1+\sin \left (f x +e \right )\right )}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a \left (-1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{\sqrt {a}\, \cos \left (f x +e \right ) \sqrt {a +\sin \left (f x +e \right ) a}\, f}+\frac {d^{3} \left (1+\sin \left (f x +e \right )\right ) \sqrt {-a \left (-1+\sin \left (f x +e \right )\right )}\, \left (15 a^{\frac {5}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -\sin \left (f x +e \right ) a}\, \sqrt {2}}{2 \sqrt {a}}\right )-6 \left (a -\sin \left (f x +e \right ) a \right )^{\frac {5}{2}}+10 a \left (a -\sin \left (f x +e \right ) a \right )^{\frac {3}{2}}-30 \sqrt {a -\sin \left (f x +e \right ) a}\, a^{2}\right )}{15 a^{3} \cos \left (f x +e \right ) \sqrt {a +\sin \left (f x +e \right ) a}\, f}-\frac {c \,d^{2} \left (1+\sin \left (f x +e \right )\right ) \sqrt {-a \left (-1+\sin \left (f x +e \right )\right )}\, \left (3 a^{\frac {3}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -\sin \left (f x +e \right ) a}\, \sqrt {2}}{2 \sqrt {a}}\right )-2 \left (a -\sin \left (f x +e \right ) a \right )^{\frac {3}{2}}\right )}{a^{2} \cos \left (f x +e \right ) \sqrt {a +\sin \left (f x +e \right ) a}\, f}+\frac {3 c^{2} d \left (1+\sin \left (f x +e \right )\right ) \sqrt {-a \left (-1+\sin \left (f x +e \right )\right )}\, \left (\sqrt {a}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -\sin \left (f x +e \right ) a}\, \sqrt {2}}{2 \sqrt {a}}\right )-2 \sqrt {a -\sin \left (f x +e \right ) a}\right )}{a \cos \left (f x +e \right ) \sqrt {a +\sin \left (f x +e \right ) a}\, f}\) | \(408\) |
Input:
int((c+d*sin(f*x+e))^3/(a+sin(f*x+e)*a)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/15*(1+sin(f*x+e))*(-a*(-1+sin(f*x+e)))^(1/2)*(15*a^(5/2)*arctanh(1/2*(a -sin(f*x+e)*a)^(1/2)*2^(1/2)/a^(1/2))*2^(1/2)*c^3-45*a^(5/2)*arctanh(1/2*( a-sin(f*x+e)*a)^(1/2)*2^(1/2)/a^(1/2))*2^(1/2)*c^2*d+45*a^(5/2)*arctanh(1/ 2*(a-sin(f*x+e)*a)^(1/2)*2^(1/2)/a^(1/2))*2^(1/2)*c*d^2-15*a^(5/2)*arctanh (1/2*(a-sin(f*x+e)*a)^(1/2)*2^(1/2)/a^(1/2))*2^(1/2)*d^3+6*(a-sin(f*x+e)*a )^(5/2)*d^3-30*(a-sin(f*x+e)*a)^(3/2)*a*c*d^2-10*(a-sin(f*x+e)*a)^(3/2)*a* d^3+90*c^2*d*a^2*(a-sin(f*x+e)*a)^(1/2)+30*d^3*a^2*(a-sin(f*x+e)*a)^(1/2)) /a^3/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. 387 vs. \(2 (157) = 314\).
Time = 0.09 (sec) , antiderivative size = 387, normalized size of antiderivative = 2.17 \[ \int \frac {(c+d \sin (e+f x))^3}{\sqrt {a+a \sin (e+f x)}} \, dx=-\frac {\frac {15 \, \sqrt {2} {\left (a c^{3} - 3 \, a c^{2} d + 3 \, a c d^{2} - a d^{3} + {\left (a c^{3} - 3 \, a c^{2} d + 3 \, a c d^{2} - a d^{3}\right )} \cos \left (f x + e\right ) + {\left (a c^{3} - 3 \, a c^{2} d + 3 \, a c d^{2} - a d^{3}\right )} \sin \left (f x + e\right )\right )} \log \left (-\frac {\cos \left (f x + e\right )^{2} - {\left (\cos \left (f x + e\right ) - 2\right )} \sin \left (f x + e\right ) + \frac {2 \, \sqrt {2} \sqrt {a \sin \left (f x + e\right ) + a} {\left (\cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right )}}{\sqrt {a}} + 3 \, \cos \left (f x + e\right ) + 2}{\cos \left (f x + e\right )^{2} - {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right )}{\sqrt {a}} - 4 \, {\left (3 \, d^{3} \cos \left (f x + e\right )^{3} - 45 \, c^{2} d + 30 \, c d^{2} - 17 \, d^{3} - {\left (15 \, c d^{2} - 4 \, d^{3}\right )} \cos \left (f x + e\right )^{2} - {\left (45 \, c^{2} d - 15 \, c d^{2} + 16 \, d^{3}\right )} \cos \left (f x + e\right ) - {\left (3 \, d^{3} \cos \left (f x + e\right )^{2} - 45 \, c^{2} d + 30 \, c d^{2} - 17 \, d^{3} + {\left (15 \, c d^{2} - d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{30 \, {\left (a f \cos \left (f x + e\right ) + a f \sin \left (f x + e\right ) + a f\right )}} \] Input:
integrate((c+d*sin(f*x+e))^3/(a+a*sin(f*x+e))^(1/2),x, algorithm="fricas")
Output:
-1/30*(15*sqrt(2)*(a*c^3 - 3*a*c^2*d + 3*a*c*d^2 - a*d^3 + (a*c^3 - 3*a*c^ 2*d + 3*a*c*d^2 - a*d^3)*cos(f*x + e) + (a*c^3 - 3*a*c^2*d + 3*a*c*d^2 - a *d^3)*sin(f*x + e))*log(-(cos(f*x + e)^2 - (cos(f*x + e) - 2)*sin(f*x + e) + 2*sqrt(2)*sqrt(a*sin(f*x + e) + a)*(cos(f*x + e) - sin(f*x + e) + 1)/sq rt(a) + 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(a) - 4*(3*d^3*cos(f*x + e)^3 - 45*c^2*d + 30 *c*d^2 - 17*d^3 - (15*c*d^2 - 4*d^3)*cos(f*x + e)^2 - (45*c^2*d - 15*c*d^2 + 16*d^3)*cos(f*x + e) - (3*d^3*cos(f*x + e)^2 - 45*c^2*d + 30*c*d^2 - 17 *d^3 + (15*c*d^2 - d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))/(a*f*cos(f*x + e) + a*f*sin(f*x + e) + a*f)
\[ \int \frac {(c+d \sin (e+f x))^3}{\sqrt {a+a \sin (e+f x)}} \, dx=\int \frac {\left (c + d \sin {\left (e + f x \right )}\right )^{3}}{\sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )}}\, dx \] Input:
integrate((c+d*sin(f*x+e))**3/(a+a*sin(f*x+e))**(1/2),x)
Output:
Integral((c + d*sin(e + f*x))**3/sqrt(a*(sin(e + f*x) + 1)), x)
\[ \int \frac {(c+d \sin (e+f x))^3}{\sqrt {a+a \sin (e+f x)}} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{3}}{\sqrt {a \sin \left (f x + e\right ) + a}} \,d x } \] Input:
integrate((c+d*sin(f*x+e))^3/(a+a*sin(f*x+e))^(1/2),x, algorithm="maxima")
Output:
integrate((d*sin(f*x + e) + c)^3/sqrt(a*sin(f*x + e) + a), x)
Exception generated. \[ \int \frac {(c+d \sin (e+f x))^3}{\sqrt {a+a \sin (e+f x)}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((c+d*sin(f*x+e))^3/(a+a*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. \[ \int \frac {(c+d \sin (e+f x))^3}{\sqrt {a+a \sin (e+f x)}} \, dx=\int \frac {{\left (c+d\,\sin \left (e+f\,x\right )\right )}^3}{\sqrt {a+a\,\sin \left (e+f\,x\right )}} \,d x \] Input:
int((c + d*sin(e + f*x))^3/(a + a*sin(e + f*x))^(1/2),x)
Output:
int((c + d*sin(e + f*x))^3/(a + a*sin(e + f*x))^(1/2), x)
\[ \int \frac {(c+d \sin (e+f x))^3}{\sqrt {a+a \sin (e+f x)}} \, dx=\frac {\sqrt {a}\, \left (\left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}}{\sin \left (f x +e \right )+1}d x \right ) c^{3}+\left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{3}}{\sin \left (f x +e \right )+1}d x \right ) d^{3}+3 \left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{2}}{\sin \left (f x +e \right )+1}d x \right ) c \,d^{2}+3 \left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )}{\sin \left (f x +e \right )+1}d x \right ) c^{2} d \right )}{a} \] Input:
int((c+d*sin(f*x+e))^3/(a+a*sin(f*x+e))^(1/2),x)
Output:
(sqrt(a)*(int(sqrt(sin(e + f*x) + 1)/(sin(e + f*x) + 1),x)*c**3 + int((sqr t(sin(e + f*x) + 1)*sin(e + f*x)**3)/(sin(e + f*x) + 1),x)*d**3 + 3*int((s qrt(sin(e + f*x) + 1)*sin(e + f*x)**2)/(sin(e + f*x) + 1),x)*c*d**2 + 3*in t((sqrt(sin(e + f*x) + 1)*sin(e + f*x))/(sin(e + f*x) + 1),x)*c**2*d))/a