Integrand size = 38, antiderivative size = 175 \[ \int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx=\frac {a^2 (A-11 B) \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{16 \sqrt {2} c^{7/2} f}+\frac {a^2 (A+B) c^2 \cos ^5(e+f x)}{6 f (c-c \sin (e+f x))^{11/2}}+\frac {a^2 (A-11 B) \cos ^3(e+f x)}{24 f (c-c \sin (e+f x))^{7/2}}-\frac {a^2 (A-11 B) \cos (e+f x)}{16 c^2 f (c-c \sin (e+f x))^{3/2}} \] Output:
1/32*a^2*(A-11*B)*arctanh(1/2*c^(1/2)*cos(f*x+e)*2^(1/2)/(c-c*sin(f*x+e))^ (1/2))*2^(1/2)/c^(7/2)/f+1/6*a^2*(A+B)*c^2*cos(f*x+e)^5/f/(c-c*sin(f*x+e)) ^(11/2)+1/24*a^2*(A-11*B)*cos(f*x+e)^3/f/(c-c*sin(f*x+e))^(7/2)-1/16*a^2*( A-11*B)*cos(f*x+e)/c^2/f/(c-c*sin(f*x+e))^(3/2)
Result contains complex when optimal does not.
Time = 12.59 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.95 \[ \int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx=\frac {a^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (32 (A+B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )-4 (7 A+19 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3+3 (A+21 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5-(3+3 i) \sqrt [4]{-1} (A-11 B) \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [4]{-1} \left (1+\tan \left (\frac {1}{4} (e+f x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^6+64 (A+B) \sin \left (\frac {1}{2} (e+f x)\right )-8 (7 A+19 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2 \sin \left (\frac {1}{2} (e+f x)\right )+6 (A+21 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^4 \sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))^2}{48 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4 (c-c \sin (e+f x))^{7/2}} \] Input:
Integrate[((a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x ])^(7/2),x]
Output:
(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(32*(A + B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]) - 4*(7*A + 19*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^ 3 + 3*(A + 21*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5 - (3 + 3*I)*(-1)^ (1/4)*(A - 11*B)*ArcTan[(1/2 + I/2)*(-1)^(1/4)*(1 + Tan[(e + f*x)/4])]*(Co s[(e + f*x)/2] - Sin[(e + f*x)/2])^6 + 64*(A + B)*Sin[(e + f*x)/2] - 8*(7* A + 19*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*Sin[(e + f*x)/2] + 6*(A + 21*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4*Sin[(e + f*x)/2])*(1 + Sin [e + f*x])^2)/(48*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*(c - c*Sin[e + f*x])^(7/2))
Time = 0.98 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.98, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.289, Rules used = {3042, 3446, 3042, 3338, 3042, 3159, 3042, 3159, 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.
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}}dx\) |
| \(\Big \downarrow \) 3446 |
| \(\displaystyle a^2 c^2 \int \frac {\cos ^4(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{11/2}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^2 c^2 \int \frac {\cos (e+f x)^4 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{11/2}}dx\) |
| \(\Big \downarrow \) 3338 |
| \(\displaystyle a^2 c^2 \left (\frac {(A-11 B) \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^{9/2}}dx}{12 c}+\frac {(A+B) \cos ^5(e+f x)}{6 f (c-c \sin (e+f x))^{11/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^2 c^2 \left (\frac {(A-11 B) \int \frac {\cos (e+f x)^4}{(c-c \sin (e+f x))^{9/2}}dx}{12 c}+\frac {(A+B) \cos ^5(e+f x)}{6 f (c-c \sin (e+f x))^{11/2}}\right )\) |
| \(\Big \downarrow \) 3159 |
| \(\displaystyle a^2 c^2 \left (\frac {(A-11 B) \left (\frac {\cos ^3(e+f x)}{2 c f (c-c \sin (e+f x))^{7/2}}-\frac {3 \int \frac {\cos ^2(e+f x)}{(c-c \sin (e+f x))^{5/2}}dx}{4 c^2}\right )}{12 c}+\frac {(A+B) \cos ^5(e+f x)}{6 f (c-c \sin (e+f x))^{11/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^2 c^2 \left (\frac {(A-11 B) \left (\frac {\cos ^3(e+f x)}{2 c f (c-c \sin (e+f x))^{7/2}}-\frac {3 \int \frac {\cos (e+f x)^2}{(c-c \sin (e+f x))^{5/2}}dx}{4 c^2}\right )}{12 c}+\frac {(A+B) \cos ^5(e+f x)}{6 f (c-c \sin (e+f x))^{11/2}}\right )\) |
| \(\Big \downarrow \) 3159 |
| \(\displaystyle a^2 c^2 \left (\frac {(A-11 B) \left (\frac {\cos ^3(e+f x)}{2 c f (c-c \sin (e+f x))^{7/2}}-\frac {3 \left (\frac {\cos (e+f x)}{c f (c-c \sin (e+f x))^{3/2}}-\frac {\int \frac {1}{\sqrt {c-c \sin (e+f x)}}dx}{2 c^2}\right )}{4 c^2}\right )}{12 c}+\frac {(A+B) \cos ^5(e+f x)}{6 f (c-c \sin (e+f x))^{11/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^2 c^2 \left (\frac {(A-11 B) \left (\frac {\cos ^3(e+f x)}{2 c f (c-c \sin (e+f x))^{7/2}}-\frac {3 \left (\frac {\cos (e+f x)}{c f (c-c \sin (e+f x))^{3/2}}-\frac {\int \frac {1}{\sqrt {c-c \sin (e+f x)}}dx}{2 c^2}\right )}{4 c^2}\right )}{12 c}+\frac {(A+B) \cos ^5(e+f x)}{6 f (c-c \sin (e+f x))^{11/2}}\right )\) |
| \(\Big \downarrow \) 3128 |
| \(\displaystyle a^2 c^2 \left (\frac {(A-11 B) \left (\frac {\cos ^3(e+f x)}{2 c f (c-c \sin (e+f x))^{7/2}}-\frac {3 \left (\frac {\int \frac {1}{2 c-\frac {c^2 \cos ^2(e+f x)}{c-c \sin (e+f x)}}d\left (-\frac {c \cos (e+f x)}{\sqrt {c-c \sin (e+f x)}}\right )}{c^2 f}+\frac {\cos (e+f x)}{c f (c-c \sin (e+f x))^{3/2}}\right )}{4 c^2}\right )}{12 c}+\frac {(A+B) \cos ^5(e+f x)}{6 f (c-c \sin (e+f x))^{11/2}}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle a^2 c^2 \left (\frac {(A-11 B) \left (\frac {\cos ^3(e+f x)}{2 c f (c-c \sin (e+f x))^{7/2}}-\frac {3 \left (\frac {\cos (e+f x)}{c f (c-c \sin (e+f x))^{3/2}}-\frac {\text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{\sqrt {2} c^{5/2} f}\right )}{4 c^2}\right )}{12 c}+\frac {(A+B) \cos ^5(e+f x)}{6 f (c-c \sin (e+f x))^{11/2}}\right )\) |
Input:
Int[((a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(7/ 2),x]
Output:
a^2*c^2*(((A + B)*Cos[e + f*x]^5)/(6*f*(c - c*Sin[e + f*x])^(11/2)) + ((A - 11*B)*(Cos[e + f*x]^3/(2*c*f*(c - c*Sin[e + f*x])^(7/2)) - (3*(-(ArcTanh [(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])]/(Sqrt[2]*c^(5/ 2)*f)) + Cos[e + f*x]/(c*f*(c - c*Sin[e + f*x])^(3/2))))/(4*c^2)))/(12*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[2*g*(g*Cos[e + f*x])^(p - 1)*((a + b*Sin[e + f *x])^(m + 1)/(b*f*(2*m + p + 1))), x] + Simp[g^2*((p - 1)/(b^2*(2*m + p + 1 ))) Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0] && LeQ[m, -2] && GtQ[p, 1] & & NeQ[2*m + p + 1, 0] && !ILtQ[m + p + 1, 0] && IntegersQ[2*m, 2*p]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(a*f*g*(2*m + p + 1) )), x] + Simp[(a*d*m + b*c*(m + p + 1))/(a*b*(2*m + p + 1)) Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && (LtQ[m, -1] || ILtQ[Simplify[m + p], 0 ]) && NeQ[2*m + p + 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] :> Si mp[a^m*c^m Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m)*(A + B*Sin [e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, 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]))
Leaf count of result is larger than twice the leaf count of optimal. \(353\) vs. \(2(152)=304\).
Time = 0.76 (sec) , antiderivative size = 354, normalized size of antiderivative = 2.02
| method | result | size |
| default | \(-\frac {a^{2} \left (-3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{3} \left (A -11 B \right ) \cos \left (f x +e \right )^{2} \sin \left (f x +e \right )+9 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{3} \left (A -11 B \right ) \cos \left (f x +e \right )^{2}+12 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{3} \left (A -11 B \right ) \sin \left (f x +e \right )+24 A \sqrt {c +c \sin \left (f x +e \right )}\, c^{\frac {5}{2}}-32 A \left (c +c \sin \left (f x +e \right )\right )^{\frac {3}{2}} c^{\frac {3}{2}}-6 A \left (c +c \sin \left (f x +e \right )\right )^{\frac {5}{2}} \sqrt {c}-264 B \sqrt {c +c \sin \left (f x +e \right )}\, c^{\frac {5}{2}}+352 B \left (c +c \sin \left (f x +e \right )\right )^{\frac {3}{2}} c^{\frac {3}{2}}-126 B \left (c +c \sin \left (f x +e \right )\right )^{\frac {5}{2}} \sqrt {c}-12 A \,\operatorname {arctanh}\left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}\, c^{3}+132 B \,\operatorname {arctanh}\left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}\, c^{3}\right ) \sqrt {c \left (1+\sin \left (f x +e \right )\right )}}{96 c^{\frac {13}{2}} \left (\sin \left (f x +e \right )-1\right )^{2} \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(354\) |
| parts | \(\text {Expression too large to display}\) | \(1004\) |
Input:
int((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(7/2),x,method=_R ETURNVERBOSE)
Output:
-1/96*a^2*(-3*2^(1/2)*arctanh(1/2*(c+c*sin(f*x+e))^(1/2)*2^(1/2)/c^(1/2))* c^3*(A-11*B)*cos(f*x+e)^2*sin(f*x+e)+9*2^(1/2)*arctanh(1/2*(c+c*sin(f*x+e) )^(1/2)*2^(1/2)/c^(1/2))*c^3*(A-11*B)*cos(f*x+e)^2+12*2^(1/2)*arctanh(1/2* (c+c*sin(f*x+e))^(1/2)*2^(1/2)/c^(1/2))*c^3*(A-11*B)*sin(f*x+e)+24*A*(c+c* sin(f*x+e))^(1/2)*c^(5/2)-32*A*(c+c*sin(f*x+e))^(3/2)*c^(3/2)-6*A*(c+c*sin (f*x+e))^(5/2)*c^(1/2)-264*B*(c+c*sin(f*x+e))^(1/2)*c^(5/2)+352*B*(c+c*sin (f*x+e))^(3/2)*c^(3/2)-126*B*(c+c*sin(f*x+e))^(5/2)*c^(1/2)-12*A*arctanh(1 /2*(c+c*sin(f*x+e))^(1/2)*2^(1/2)/c^(1/2))*2^(1/2)*c^3+132*B*arctanh(1/2*( c+c*sin(f*x+e))^(1/2)*2^(1/2)/c^(1/2))*2^(1/2)*c^3)*(c*(1+sin(f*x+e)))^(1/ 2)/c^(13/2)/(sin(f*x+e)-1)^2/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. 521 vs. \(2 (152) = 304\).
Time = 0.10 (sec) , antiderivative size = 521, normalized size of antiderivative = 2.98 \[ \int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx=-\frac {3 \, \sqrt {2} {\left ({\left (A - 11 \, B\right )} a^{2} \cos \left (f x + e\right )^{4} - 3 \, {\left (A - 11 \, B\right )} a^{2} \cos \left (f x + e\right )^{3} - 8 \, {\left (A - 11 \, B\right )} a^{2} \cos \left (f x + e\right )^{2} + 4 \, {\left (A - 11 \, B\right )} a^{2} \cos \left (f x + e\right ) + 8 \, {\left (A - 11 \, B\right )} a^{2} + {\left ({\left (A - 11 \, B\right )} a^{2} \cos \left (f x + e\right )^{3} + 4 \, {\left (A - 11 \, B\right )} a^{2} \cos \left (f x + e\right )^{2} - 4 \, {\left (A - 11 \, B\right )} a^{2} \cos \left (f x + e\right ) - 8 \, {\left (A - 11 \, B\right )} a^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {c} \log \left (-\frac {c \cos \left (f x + e\right )^{2} - 2 \, \sqrt {2} \sqrt {-c \sin \left (f x + e\right ) + c} \sqrt {c} {\left (\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right )} + 3 \, c \cos \left (f x + e\right ) + {\left (c \cos \left (f x + e\right ) - 2 \, c\right )} \sin \left (f x + e\right ) + 2 \, c}{\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 (3 \, {\left (A + 21 \, B\right )} a^{2} \cos \left (f x + e\right )^{3} + {\left (25 \, A + 13 \, B\right )} a^{2} \cos \left (f x + e\right )^{2} - 2 \, {\left (5 \, A + 41 \, B\right )} a^{2} \cos \left (f x + e\right ) - 32 \, {\left (A + B\right )} a^{2} + {\left (3 \, {\left (A + 21 \, B\right )} a^{2} \cos \left (f x + e\right )^{2} - 2 \, {\left (11 \, A - 25 \, B\right )} a^{2} \cos \left (f x + e\right ) - 32 \, {\left (A + B\right )} a^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{192 \, {\left (c^{4} f \cos \left (f x + e\right )^{4} - 3 \, c^{4} f \cos \left (f x + e\right )^{3} - 8 \, c^{4} f \cos \left (f x + e\right )^{2} + 4 \, c^{4} f \cos \left (f x + e\right ) + 8 \, c^{4} f + {\left (c^{4} f \cos \left (f x + e\right )^{3} + 4 \, c^{4} f \cos \left (f x + e\right )^{2} - 4 \, c^{4} f \cos \left (f x + e\right ) - 8 \, c^{4} f\right )} \sin \left (f x + e\right )\right )}} \] Input:
integrate((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(7/2),x, al gorithm="fricas")
Output:
-1/192*(3*sqrt(2)*((A - 11*B)*a^2*cos(f*x + e)^4 - 3*(A - 11*B)*a^2*cos(f* x + e)^3 - 8*(A - 11*B)*a^2*cos(f*x + e)^2 + 4*(A - 11*B)*a^2*cos(f*x + e) + 8*(A - 11*B)*a^2 + ((A - 11*B)*a^2*cos(f*x + e)^3 + 4*(A - 11*B)*a^2*co s(f*x + e)^2 - 4*(A - 11*B)*a^2*cos(f*x + e) - 8*(A - 11*B)*a^2)*sin(f*x + e))*sqrt(c)*log(-(c*cos(f*x + e)^2 - 2*sqrt(2)*sqrt(-c*sin(f*x + e) + c)* sqrt(c)*(cos(f*x + e) + sin(f*x + e) + 1) + 3*c*cos(f*x + e) + (c*cos(f*x + e) - 2*c)*sin(f*x + e) + 2*c)/(cos(f*x + e)^2 + (cos(f*x + e) + 2)*sin(f *x + e) - cos(f*x + e) - 2)) + 4*(3*(A + 21*B)*a^2*cos(f*x + e)^3 + (25*A + 13*B)*a^2*cos(f*x + e)^2 - 2*(5*A + 41*B)*a^2*cos(f*x + e) - 32*(A + B)* a^2 + (3*(A + 21*B)*a^2*cos(f*x + e)^2 - 2*(11*A - 25*B)*a^2*cos(f*x + e) - 32*(A + B)*a^2)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c))/(c^4*f*cos(f*x + e)^4 - 3*c^4*f*cos(f*x + e)^3 - 8*c^4*f*cos(f*x + e)^2 + 4*c^4*f*cos(f*x + e) + 8*c^4*f + (c^4*f*cos(f*x + e)^3 + 4*c^4*f*cos(f*x + e)^2 - 4*c^4*f *cos(f*x + e) - 8*c^4*f)*sin(f*x + e))
Timed out. \[ \int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx=\text {Timed out} \] Input:
integrate((a+a*sin(f*x+e))**2*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))**(7/2),x)
Output:
Timed out
\[ \int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx=\int { \frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{2}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(7/2),x, al gorithm="maxima")
Output:
integrate((B*sin(f*x + e) + A)*(a*sin(f*x + e) + a)^2/(-c*sin(f*x + e) + c )^(7/2), x)
Exception generated. \[ \int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(7/2),x, al gorithm="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. \[ \int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx=\int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^2}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{7/2}} \,d x \] Input:
int(((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^2)/(c - c*sin(e + f*x))^(7/ 2),x)
Output:
int(((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^2)/(c - c*sin(e + f*x))^(7/ 2), x)
\[ \int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx=\frac {\sqrt {c}\, a^{2} \left (\left (\int \frac {\sqrt {-\sin \left (f x +e \right )+1}}{\sin \left (f x +e \right )^{4}-4 \sin \left (f x +e \right )^{3}+6 \sin \left (f x +e \right )^{2}-4 \sin \left (f x +e \right )+1}d x \right ) a +\left (\int \frac {\sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{3}}{\sin \left (f x +e \right )^{4}-4 \sin \left (f x +e \right )^{3}+6 \sin \left (f x +e \right )^{2}-4 \sin \left (f x +e \right )+1}d x \right ) b +\left (\int \frac {\sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{2}}{\sin \left (f x +e \right )^{4}-4 \sin \left (f x +e \right )^{3}+6 \sin \left (f x +e \right )^{2}-4 \sin \left (f x +e \right )+1}d x \right ) a +2 \left (\int \frac {\sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{2}}{\sin \left (f x +e \right )^{4}-4 \sin \left (f x +e \right )^{3}+6 \sin \left (f x +e \right )^{2}-4 \sin \left (f x +e \right )+1}d x \right ) b +2 \left (\int \frac {\sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )}{\sin \left (f x +e \right )^{4}-4 \sin \left (f x +e \right )^{3}+6 \sin \left (f x +e \right )^{2}-4 \sin \left (f x +e \right )+1}d x \right ) a +\left (\int \frac {\sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )}{\sin \left (f x +e \right )^{4}-4 \sin \left (f x +e \right )^{3}+6 \sin \left (f x +e \right )^{2}-4 \sin \left (f x +e \right )+1}d x \right ) b \right )}{c^{4}} \] Input:
int((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(7/2),x)
Output:
(sqrt(c)*a**2*(int(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 + int((sqrt( - sin( e + f*x) + 1)*sin(e + f*x)**3)/(sin(e + f*x)**4 - 4*sin(e + f*x)**3 + 6*si n(e + f*x)**2 - 4*sin(e + f*x) + 1),x)*b + int((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)*a + 2*int((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 + 2*int((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)*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))/c**4