Integrand size = 40, antiderivative size = 264 \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx=\frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{6 f (c-c \sin (e+f x))^{7/2}}-\frac {a (A+7 B) \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{12 c f (c-c \sin (e+f x))^{5/2}}+\frac {a^2 (A+7 B) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{4 c^2 f (c-c \sin (e+f x))^{3/2}}+\frac {a^4 (A+7 B) \cos (e+f x) \log (1-\sin (e+f x))}{c^3 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {a^3 (A+7 B) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{2 c^3 f \sqrt {c-c \sin (e+f x)}} \] Output:
1/6*(A+B)*cos(f*x+e)*(a+a*sin(f*x+e))^(7/2)/f/(c-c*sin(f*x+e))^(7/2)-1/12* a*(A+7*B)*cos(f*x+e)*(a+a*sin(f*x+e))^(5/2)/c/f/(c-c*sin(f*x+e))^(5/2)+1/4 *a^2*(A+7*B)*cos(f*x+e)*(a+a*sin(f*x+e))^(3/2)/c^2/f/(c-c*sin(f*x+e))^(3/2 )+a^4*(A+7*B)*cos(f*x+e)*ln(1-sin(f*x+e))/c^3/f/(a+a*sin(f*x+e))^(1/2)/(c- c*sin(f*x+e))^(1/2)+1/2*a^3*(A+7*B)*cos(f*x+e)*(a+a*sin(f*x+e))^(1/2)/c^3/ f/(c-c*sin(f*x+e))^(1/2)
Time = 14.03 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.92 \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (a (1+\sin (e+f x)))^{7/2} \left (8 (A+B)-6 (3 A+5 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2+18 (A+3 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^4+6 (A+7 B) \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^6+3 B \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^6 \sin (e+f x)\right )}{3 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{7/2}} \] Input:
Integrate[((a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(7/2),x]
Output:
((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(7/2)*(8*(A + B) - 6*(3*A + 5*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2 + 18*(A + 3*B )*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4 + 6*(A + 7*B)*Log[Cos[(e + f*x)/ 2] - Sin[(e + f*x)/2]]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^6 + 3*B*(Cos[ (e + f*x)/2] - Sin[(e + f*x)/2])^6*Sin[e + f*x]))/(3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(7/2))
Time = 1.40 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.96, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.325, Rules used = {3042, 3451, 3042, 3218, 3042, 3218, 3042, 3219, 3042, 3216, 3042, 3146, 16}
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.
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}}dx\) |
| \(\Big \downarrow \) 3451 |
| \(\displaystyle \frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{6 f (c-c \sin (e+f x))^{7/2}}-\frac {(A+7 B) \int \frac {(\sin (e+f x) a+a)^{7/2}}{(c-c \sin (e+f x))^{5/2}}dx}{6 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{6 f (c-c \sin (e+f x))^{7/2}}-\frac {(A+7 B) \int \frac {(\sin (e+f x) a+a)^{7/2}}{(c-c \sin (e+f x))^{5/2}}dx}{6 c}\) |
| \(\Big \downarrow \) 3218 |
| \(\displaystyle \frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{6 f (c-c \sin (e+f x))^{7/2}}-\frac {(A+7 B) \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{2 f (c-c \sin (e+f x))^{5/2}}-\frac {3 a \int \frac {(\sin (e+f x) a+a)^{5/2}}{(c-c \sin (e+f x))^{3/2}}dx}{2 c}\right )}{6 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{6 f (c-c \sin (e+f x))^{7/2}}-\frac {(A+7 B) \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{2 f (c-c \sin (e+f x))^{5/2}}-\frac {3 a \int \frac {(\sin (e+f x) a+a)^{5/2}}{(c-c \sin (e+f x))^{3/2}}dx}{2 c}\right )}{6 c}\) |
| \(\Big \downarrow \) 3218 |
| \(\displaystyle \frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{6 f (c-c \sin (e+f x))^{7/2}}-\frac {(A+7 B) \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{2 f (c-c \sin (e+f x))^{5/2}}-\frac {3 a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{f (c-c \sin (e+f x))^{3/2}}-\frac {2 a \int \frac {(\sin (e+f x) a+a)^{3/2}}{\sqrt {c-c \sin (e+f x)}}dx}{c}\right )}{2 c}\right )}{6 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{6 f (c-c \sin (e+f x))^{7/2}}-\frac {(A+7 B) \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{2 f (c-c \sin (e+f x))^{5/2}}-\frac {3 a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{f (c-c \sin (e+f x))^{3/2}}-\frac {2 a \int \frac {(\sin (e+f x) a+a)^{3/2}}{\sqrt {c-c \sin (e+f x)}}dx}{c}\right )}{2 c}\right )}{6 c}\) |
| \(\Big \downarrow \) 3219 |
| \(\displaystyle \frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{6 f (c-c \sin (e+f x))^{7/2}}-\frac {(A+7 B) \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{2 f (c-c \sin (e+f x))^{5/2}}-\frac {3 a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{f (c-c \sin (e+f x))^{3/2}}-\frac {2 a \left (2 a \int \frac {\sqrt {\sin (e+f x) a+a}}{\sqrt {c-c \sin (e+f x)}}dx-\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{f \sqrt {c-c \sin (e+f x)}}\right )}{c}\right )}{2 c}\right )}{6 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{6 f (c-c \sin (e+f x))^{7/2}}-\frac {(A+7 B) \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{2 f (c-c \sin (e+f x))^{5/2}}-\frac {3 a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{f (c-c \sin (e+f x))^{3/2}}-\frac {2 a \left (2 a \int \frac {\sqrt {\sin (e+f x) a+a}}{\sqrt {c-c \sin (e+f x)}}dx-\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{f \sqrt {c-c \sin (e+f x)}}\right )}{c}\right )}{2 c}\right )}{6 c}\) |
| \(\Big \downarrow \) 3216 |
| \(\displaystyle \frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{6 f (c-c \sin (e+f x))^{7/2}}-\frac {(A+7 B) \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{2 f (c-c \sin (e+f x))^{5/2}}-\frac {3 a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{f (c-c \sin (e+f x))^{3/2}}-\frac {2 a \left (\frac {2 a^2 c \cos (e+f x) \int \frac {\cos (e+f x)}{c-c \sin (e+f x)}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{f \sqrt {c-c \sin (e+f x)}}\right )}{c}\right )}{2 c}\right )}{6 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{6 f (c-c \sin (e+f x))^{7/2}}-\frac {(A+7 B) \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{2 f (c-c \sin (e+f x))^{5/2}}-\frac {3 a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{f (c-c \sin (e+f x))^{3/2}}-\frac {2 a \left (\frac {2 a^2 c \cos (e+f x) \int \frac {\cos (e+f x)}{c-c \sin (e+f x)}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{f \sqrt {c-c \sin (e+f x)}}\right )}{c}\right )}{2 c}\right )}{6 c}\) |
| \(\Big \downarrow \) 3146 |
| \(\displaystyle \frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{6 f (c-c \sin (e+f x))^{7/2}}-\frac {(A+7 B) \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{2 f (c-c \sin (e+f x))^{5/2}}-\frac {3 a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{f (c-c \sin (e+f x))^{3/2}}-\frac {2 a \left (-\frac {2 a^2 \cos (e+f x) \int \frac {1}{c-c \sin (e+f x)}d(-c \sin (e+f x))}{f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{f \sqrt {c-c \sin (e+f x)}}\right )}{c}\right )}{2 c}\right )}{6 c}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{6 f (c-c \sin (e+f x))^{7/2}}-\frac {(A+7 B) \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{2 f (c-c \sin (e+f x))^{5/2}}-\frac {3 a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{f (c-c \sin (e+f x))^{3/2}}-\frac {2 a \left (-\frac {2 a^2 \cos (e+f x) \log (c-c \sin (e+f x))}{f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{f \sqrt {c-c \sin (e+f x)}}\right )}{c}\right )}{2 c}\right )}{6 c}\) |
Input:
Int[((a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x]) ^(7/2),x]
Output:
((A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(6*f*(c - c*Sin[e + f*x] )^(7/2)) - ((A + 7*B)*((a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(2*f*(c - c*Sin[e + f*x])^(5/2)) - (3*a*((a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/ 2))/(f*(c - c*Sin[e + f*x])^(3/2)) - (2*a*((-2*a^2*Cos[e + f*x]*Log[c - c* Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (a* Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(f*Sqrt[c - c*Sin[e + f*x]])))/c))/ (2*c)))/(6*c)
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m _.), x_Symbol] :> Simp[1/(b^p*f) Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x )^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && I ntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] || !IntegerQ[m + 1/ 2])
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[a*c*(Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x ]]*Sqrt[c + d*Sin[e + f*x]])) Int[Cos[e + f*x]/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0 ]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + ( f_.)*(x_)])^(n_), x_Symbol] :> Simp[-2*b*Cos[e + f*x]*(a + b*Sin[e + f*x])^ (m - 1)*((c + d*Sin[e + f*x])^n/(f*(2*n + 1))), x] - Simp[b*((2*m - 1)/(d*( 2*n + 1))) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b ^2, 0] && IGtQ[m - 1/2, 0] && LtQ[n, -1] && !(ILtQ[m + n, 0] && GtQ[2*m + n + 1, 0])
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + ( f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[e + f*x]*(a + b*Sin[e + f*x])^ (m - 1)*((c + d*Sin[e + f*x])^n/(f*(m + n))), x] + Simp[a*((2*m - 1)/(m + n )) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n, x], x] /; Fre eQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && I GtQ[m - 1/2, 0] && !LtQ[n, -1] && !(IGtQ[n - 1/2, 0] && LtQ[n, m]) && !( ILtQ[m + n, 0] && GtQ[2*m + n + 1, 0])
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Sim p[(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^n/( a*f*(2*m + 1))), x] + Simp[(a*B*(m - n) + A*b*(m + n + 1))/(a*b*(2*m + 1)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[ {a, b, c, d, e, f, A, B, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0 ] && (LtQ[m, -2^(-1)] || (ILtQ[m + n, 0] && !SumSimplerQ[n, 1])) && NeQ[2* m + 1, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(705\) vs. \(2(236)=472\).
Time = 7.94 (sec) , antiderivative size = 706, normalized size of antiderivative = 2.67
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(706\) |
| parts | \(-\frac {A \sqrt {4}\, \left (12 \cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{6} \ln \left (\frac {2}{\cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )+1}\right )-12 \cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{6} \ln \left (-\cot \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )+\csc \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )-1\right )-12 \cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{6} \ln \left (-\cot \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )+\csc \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )+1\right )+11 \cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{6}-18 \cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{4}+9 \cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{2}-2\right ) a^{3} \sqrt {a \sin \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{2}}\, \sec \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{5} \csc \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )}{12 f \sqrt {c \cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{2}}\, c^{3}}-\frac {2 B \left (\left (\left (168 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}-168 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}-126 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-252 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+252 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+21\right ) \ln \left (\frac {2 \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )}{\cos \left (\frac {f x}{2}+\frac {e}{2}\right )+1}\right )+\left (\left (-168 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+168 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}+126 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )+252 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}-252 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-21\right ) \ln \left (\frac {2}{\cos \left (\frac {f x}{2}+\frac {e}{2}\right )+1}\right )+\cos \left (\frac {f x}{2}+\frac {e}{2}\right ) \left (\left (24 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}-24 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}+164 \sin \left (\frac {f x}{2}+\frac {e}{2}\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-102 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+21 \sin \left (\frac {f x}{2}+\frac {e}{2}\right )\right )\right ) \sqrt {\left (2 \sin \left (\frac {f x}{2}+\frac {e}{2}\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right ) a}\, a^{3}}{3 f \left (8 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}-12 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+8 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} \sin \left (\frac {f x}{2}+\frac {e}{2}\right )+2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-4 \sin \left (\frac {f x}{2}+\frac {e}{2}\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right ) \sqrt {-\left (2 \sin \left (\frac {f x}{2}+\frac {e}{2}\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right ) c}\, c^{3}}\) | \(706\) |
Input:
int((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(7/2),x,metho d=_RETURNVERBOSE)
Output:
-1/12*A/f*4^(1/2)*(12*cos(1/4*Pi+1/2*f*x+1/2*e)^6*ln(2/(cos(1/4*Pi+1/2*f*x +1/2*e)+1))-12*cos(1/4*Pi+1/2*f*x+1/2*e)^6*ln(-cot(1/4*Pi+1/2*f*x+1/2*e)+c sc(1/4*Pi+1/2*f*x+1/2*e)-1)-12*cos(1/4*Pi+1/2*f*x+1/2*e)^6*ln(-cot(1/4*Pi+ 1/2*f*x+1/2*e)+csc(1/4*Pi+1/2*f*x+1/2*e)+1)+11*cos(1/4*Pi+1/2*f*x+1/2*e)^6 -18*cos(1/4*Pi+1/2*f*x+1/2*e)^4+9*cos(1/4*Pi+1/2*f*x+1/2*e)^2-2)*a^3*(a*si n(1/4*Pi+1/2*f*x+1/2*e)^2)^(1/2)/(c*cos(1/4*Pi+1/2*f*x+1/2*e)^2)^(1/2)/c^3 *sec(1/4*Pi+1/2*f*x+1/2*e)^5*csc(1/4*Pi+1/2*f*x+1/2*e)-2/3*B/f*(((168*cos( 1/2*f*x+1/2*e)^5-168*cos(1/2*f*x+1/2*e)^3-126*cos(1/2*f*x+1/2*e))*sin(1/2* f*x+1/2*e)-252*cos(1/2*f*x+1/2*e)^4+252*cos(1/2*f*x+1/2*e)^2+21)*ln(-2*(co s(1/2*f*x+1/2*e)-sin(1/2*f*x+1/2*e))/(cos(1/2*f*x+1/2*e)+1))+((-168*cos(1/ 2*f*x+1/2*e)^5+168*cos(1/2*f*x+1/2*e)^3+126*cos(1/2*f*x+1/2*e))*sin(1/2*f* x+1/2*e)+252*cos(1/2*f*x+1/2*e)^4-252*cos(1/2*f*x+1/2*e)^2-21)*ln(2/(cos(1 /2*f*x+1/2*e)+1))+cos(1/2*f*x+1/2*e)*((24*cos(1/2*f*x+1/2*e)^5-24*cos(1/2* f*x+1/2*e)^3+164*sin(1/2*f*x+1/2*e)*cos(1/2*f*x+1/2*e)^2-102*cos(1/2*f*x+1 /2*e))*sin(1/2*f*x+1/2*e)^2+21*sin(1/2*f*x+1/2*e)))*((2*sin(1/2*f*x+1/2*e) *cos(1/2*f*x+1/2*e)+1)*a)^(1/2)*a^3/(8*cos(1/2*f*x+1/2*e)^6-12*cos(1/2*f*x +1/2*e)^4+8*cos(1/2*f*x+1/2*e)^3*sin(1/2*f*x+1/2*e)+2*cos(1/2*f*x+1/2*e)^2 -4*sin(1/2*f*x+1/2*e)*cos(1/2*f*x+1/2*e)+1)/(-(2*sin(1/2*f*x+1/2*e)*cos(1/ 2*f*x+1/2*e)-1)*c)^(1/2)/c^3
\[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx=\int { \frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(7/2),x , algorithm="fricas")
Output:
integral((B*a^3*cos(f*x + e)^4 - (3*A + 5*B)*a^3*cos(f*x + e)^2 + 4*(A + B )*a^3 - ((A + 3*B)*a^3*cos(f*x + e)^2 - 4*(A + B)*a^3)*sin(f*x + e))*sqrt( a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(c^4*cos(f*x + e)^4 - 8*c^4* cos(f*x + e)^2 + 8*c^4 + 4*(c^4*cos(f*x + e)^2 - 2*c^4)*sin(f*x + e)), x)
Timed out. \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx=\text {Timed out} \] Input:
integrate((a+a*sin(f*x+e))**(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))**(7/2) ,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 749 vs. \(2 (236) = 472\).
Time = 0.15 (sec) , antiderivative size = 749, normalized size of antiderivative = 2.84 \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx=\text {Too large to display} \] Input:
integrate((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(7/2),x , algorithm="maxima")
Output:
-1/3*(B*(42*a^(7/2)*log(sin(f*x + e)/(cos(f*x + e) + 1) - 1)/c^(7/2) - 21* a^(7/2)*log(sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1)/c^(7/2) + 2*(21*a^(7/ 2)*sin(f*x + e)/(cos(f*x + e) + 1) - 102*a^(7/2)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 227*a^(7/2)*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 228*a^(7/2) *sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 227*a^(7/2)*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 102*a^(7/2)*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 21*a^(7/2 )*sin(f*x + e)^7/(cos(f*x + e) + 1)^7)/(c^(7/2) - 6*c^(7/2)*sin(f*x + e)/( cos(f*x + e) + 1) + 16*c^(7/2)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 26*c^ (7/2)*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 30*c^(7/2)*sin(f*x + e)^4/(cos (f*x + e) + 1)^4 - 26*c^(7/2)*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 16*c^( 7/2)*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 6*c^(7/2)*sin(f*x + e)^7/(cos(f *x + e) + 1)^7 + c^(7/2)*sin(f*x + e)^8/(cos(f*x + e) + 1)^8)) + A*(6*a^(7 /2)*log(sin(f*x + e)/(cos(f*x + e) + 1) - 1)/c^(7/2) - 3*a^(7/2)*log(sin(f *x + e)^2/(cos(f*x + e) + 1)^2 + 1)/c^(7/2) + 4*(3*a^(7/2)*sqrt(c)*sin(f*x + e)/(cos(f*x + e) + 1) - 6*a^(7/2)*sqrt(c)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 22*a^(7/2)*sqrt(c)*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 6*a^(7/2 )*sqrt(c)*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 3*a^(7/2)*sqrt(c)*sin(f*x + e)^5/(cos(f*x + e) + 1)^5)/(c^4 - 6*c^4*sin(f*x + e)/(cos(f*x + e) + 1) + 15*c^4*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 20*c^4*sin(f*x + e)^3/(cos( f*x + e) + 1)^3 + 15*c^4*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 6*c^4*si...
Exception generated. \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(7/2),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. \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx=\int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{7/2}}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{7/2}} \,d x \] Input:
int(((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^(7/2))/(c - c*sin(e + f*x)) ^(7/2),x)
Output:
int(((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^(7/2))/(c - c*sin(e + f*x)) ^(7/2), x)
\[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx =\text {Too large to display} \] Input:
int((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(7/2),x)
Output:
(sqrt(c)*sqrt(a)*a**3*(int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sin(e + f*x)**4)/(sin(e + f*x)**4 - 4*sin(e + f*x)**3 + 6*sin(e + f*x)* *2 - 4*sin(e + f*x) + 1),x)*b + int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sin(e + f*x)**3)/(sin(e + f*x)**4 - 4*sin(e + f*x)**3 + 6*sin( e + f*x)**2 - 4*sin(e + f*x) + 1),x)*a + 3*int((sqrt(sin(e + f*x) + 1)*sqr t( - sin(e + f*x) + 1)*sin(e + f*x)**3)/(sin(e + f*x)**4 - 4*sin(e + f*x)* *3 + 6*sin(e + f*x)**2 - 4*sin(e + f*x) + 1),x)*b + 3*int((sqrt(sin(e + f* x) + 1)*sqrt( - sin(e + f*x) + 1)*sin(e + f*x)**2)/(sin(e + f*x)**4 - 4*si n(e + f*x)**3 + 6*sin(e + f*x)**2 - 4*sin(e + f*x) + 1),x)*a + 3*int((sqrt (sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sin(e + f*x)**2)/(sin(e + f*x )**4 - 4*sin(e + f*x)**3 + 6*sin(e + f*x)**2 - 4*sin(e + f*x) + 1),x)*b + 3*int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sin(e + f*x))/(sin (e + f*x)**4 - 4*sin(e + f*x)**3 + 6*sin(e + f*x)**2 - 4*sin(e + f*x) + 1) ,x)*a + int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sin(e + f*x) )/(sin(e + f*x)**4 - 4*sin(e + f*x)**3 + 6*sin(e + f*x)**2 - 4*sin(e + f*x ) + 1),x)*b + int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1))/(sin( e + f*x)**4 - 4*sin(e + f*x)**3 + 6*sin(e + f*x)**2 - 4*sin(e + f*x) + 1), x)*a))/c**4