Integrand size = 36, antiderivative size = 163 \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx=-\frac {a (A-3 B) \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{32 \sqrt {2} c^{7/2} f}+\frac {a (A+B) \cos (e+f x)}{3 f (c-c \sin (e+f x))^{7/2}}-\frac {a (A+13 B) \cos (e+f x)}{24 c f (c-c \sin (e+f x))^{5/2}}-\frac {a (A-3 B) \cos (e+f x)}{32 c^2 f (c-c \sin (e+f x))^{3/2}} \] Output:
-1/64*a*(A-3*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+1/3*a*(A+B)*cos(f*x+e)/f/(c-c*sin(f*x+e))^(7/2)-1/2 4*a*(A+13*B)*cos(f*x+e)/c/f/(c-c*sin(f*x+e))^(5/2)-1/32*a*(A-3*B)*cos(f*x+ e)/c^2/f/(c-c*sin(f*x+e))^(3/2)
Time = 9.44 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.33 \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx=-\frac {a (-1+\sin (e+f x)) (1+\sin (e+f x)) \left (3 \sqrt {2} (A-3 B) \arctan \left (\frac {\sqrt {-c (1+\sin (e+f x))}}{\sqrt {2} \sqrt {c}}\right ) \sec (e+f x) \sqrt {-c (1+\sin (e+f x))}+\frac {\sqrt {c} \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (47 A-13 B+3 (A-3 B) \cos (2 (e+f x))+4 (5 A+17 B) \sin (e+f x))}{\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7}\right )}{192 c^{7/2} f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2 \sqrt {c-c \sin (e+f x)}} \] Input:
Integrate[((a + a*Sin[e + f*x])*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x]) ^(7/2),x]
Output:
-1/192*(a*(-1 + Sin[e + f*x])*(1 + Sin[e + f*x])*(3*Sqrt[2]*(A - 3*B)*ArcT an[Sqrt[-(c*(1 + Sin[e + f*x]))]/(Sqrt[2]*Sqrt[c])]*Sec[e + f*x]*Sqrt[-(c* (1 + Sin[e + f*x]))] + (Sqrt[c]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(47* A - 13*B + 3*(A - 3*B)*Cos[2*(e + f*x)] + 4*(5*A + 17*B)*Sin[e + f*x]))/(C os[(e + f*x)/2] - Sin[(e + f*x)/2])^7))/(c^(7/2)*f*(Cos[(e + f*x)/2] + Sin [(e + f*x)/2])^2*Sqrt[c - c*Sin[e + f*x]])
Time = 0.87 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.03, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 3446, 3042, 3336, 25, 3042, 3229, 3042, 3129, 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.
| \(\displaystyle \int \frac {(a \sin (e+f x)+a) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (e+f x)+a) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}}dx\) |
| \(\Big \downarrow \) 3446 |
| \(\displaystyle a c \int \frac {\cos ^2(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{9/2}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a c \int \frac {\cos (e+f x)^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{9/2}}dx\) |
| \(\Big \downarrow \) 3336 |
| \(\displaystyle a c \left (\frac {\int -\frac {(A+7 B) c+6 B \sin (e+f x) c}{(c-c \sin (e+f x))^{5/2}}dx}{6 c^3}+\frac {(A+B) \cos (e+f x)}{3 c f (c-c \sin (e+f x))^{7/2}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle a c \left (\frac {(A+B) \cos (e+f x)}{3 c f (c-c \sin (e+f x))^{7/2}}-\frac {\int \frac {(A+7 B) c+6 B \sin (e+f x) c}{(c-c \sin (e+f x))^{5/2}}dx}{6 c^3}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a c \left (\frac {(A+B) \cos (e+f x)}{3 c f (c-c \sin (e+f x))^{7/2}}-\frac {\int \frac {(A+7 B) c+6 B \sin (e+f x) c}{(c-c \sin (e+f x))^{5/2}}dx}{6 c^3}\right )\) |
| \(\Big \downarrow \) 3229 |
| \(\displaystyle a c \left (\frac {(A+B) \cos (e+f x)}{3 c f (c-c \sin (e+f x))^{7/2}}-\frac {\frac {3}{8} (A-3 B) \int \frac {1}{(c-c \sin (e+f x))^{3/2}}dx+\frac {c (A+13 B) \cos (e+f x)}{4 f (c-c \sin (e+f x))^{5/2}}}{6 c^3}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a c \left (\frac {(A+B) \cos (e+f x)}{3 c f (c-c \sin (e+f x))^{7/2}}-\frac {\frac {3}{8} (A-3 B) \int \frac {1}{(c-c \sin (e+f x))^{3/2}}dx+\frac {c (A+13 B) \cos (e+f x)}{4 f (c-c \sin (e+f x))^{5/2}}}{6 c^3}\right )\) |
| \(\Big \downarrow \) 3129 |
| \(\displaystyle a c \left (\frac {(A+B) \cos (e+f x)}{3 c f (c-c \sin (e+f x))^{7/2}}-\frac {\frac {3}{8} (A-3 B) \left (\frac {\int \frac {1}{\sqrt {c-c \sin (e+f x)}}dx}{4 c}+\frac {\cos (e+f x)}{2 f (c-c \sin (e+f x))^{3/2}}\right )+\frac {c (A+13 B) \cos (e+f x)}{4 f (c-c \sin (e+f x))^{5/2}}}{6 c^3}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a c \left (\frac {(A+B) \cos (e+f x)}{3 c f (c-c \sin (e+f x))^{7/2}}-\frac {\frac {3}{8} (A-3 B) \left (\frac {\int \frac {1}{\sqrt {c-c \sin (e+f x)}}dx}{4 c}+\frac {\cos (e+f x)}{2 f (c-c \sin (e+f x))^{3/2}}\right )+\frac {c (A+13 B) \cos (e+f x)}{4 f (c-c \sin (e+f x))^{5/2}}}{6 c^3}\right )\) |
| \(\Big \downarrow \) 3128 |
| \(\displaystyle a c \left (\frac {(A+B) \cos (e+f x)}{3 c f (c-c \sin (e+f x))^{7/2}}-\frac {\frac {3}{8} (A-3 B) \left (\frac {\cos (e+f x)}{2 f (c-c \sin (e+f x))^{3/2}}-\frac {\int \frac {1}{2 c-\frac {c^2 \cos ^2(e+f x)}{c-c \sin (e+f x)}}d\left (-\frac {c \cos (e+f x)}{\sqrt {c-c \sin (e+f x)}}\right )}{2 c f}\right )+\frac {c (A+13 B) \cos (e+f x)}{4 f (c-c \sin (e+f x))^{5/2}}}{6 c^3}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle a c \left (\frac {(A+B) \cos (e+f x)}{3 c f (c-c \sin (e+f x))^{7/2}}-\frac {\frac {3}{8} (A-3 B) \left (\frac {\text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{2 \sqrt {2} c^{3/2} f}+\frac {\cos (e+f x)}{2 f (c-c \sin (e+f x))^{3/2}}\right )+\frac {c (A+13 B) \cos (e+f x)}{4 f (c-c \sin (e+f x))^{5/2}}}{6 c^3}\right )\) |
Input:
Int[((a + a*Sin[e + f*x])*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(7/2) ,x]
Output:
a*c*(((A + B)*Cos[e + f*x])/(3*c*f*(c - c*Sin[e + f*x])^(7/2)) - (((A + 13 *B)*c*Cos[e + f*x])/(4*f*(c - c*Sin[e + f*x])^(5/2)) + (3*(A - 3*B)*(ArcTa nh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])]/(2*Sqrt[2]*c ^(3/2)*f) + Cos[e + f*x]/(2*f*(c - c*Sin[e + f*x])^(3/2))))/8)/(6*c^3))
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[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*Cos[c + d*x]*((a + b*Sin[c + d*x])^n/(a*d*(2*n + 1))), x] + Simp[(n + 1)/(a*(2*n + 1)) Int[(a + b*Sin[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*Cos[e + f*x]*((a + b*Sin[e + f* x])^m/(a*f*(2*m + 1))), x] + Simp[(a*d*m + b*c*(m + 1))/(a*b*(2*m + 1)) I nt[(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)]
Int[cos[(e_.) + (f_.)*(x_)]^2*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*( (c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[2*(b*c - a*d)*Cos [e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b^2*f*(2*m + 3))), x] + Simp[1/(b^ 3*(2*m + 3)) Int[(a + b*Sin[e + f*x])^(m + 2)*(b*c + 2*a*d*(m + 1) - b*d* (2*m + 3)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -3/2]
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. \(351\) vs. \(2(140)=280\).
Time = 0.60 (sec) , antiderivative size = 352, normalized size of antiderivative = 2.16
| method | result | size |
| default | \(-\frac {a \left (3 \,\operatorname {arctanh}\left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}\, c^{4} \left (A -3 B \right ) \cos \left (f x +e \right )^{2} \sin \left (f x +e \right )-9 \,\operatorname {arctanh}\left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}\, c^{4} \left (A -3 B \right ) \cos \left (f x +e \right )^{2}-12 \,\operatorname {arctanh}\left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}\, c^{4} \left (A -3 B \right ) \sin \left (f x +e \right )+6 A \left (c +c \sin \left (f x +e \right )\right )^{\frac {5}{2}} c^{\frac {3}{2}}-32 A \left (c +c \sin \left (f x +e \right )\right )^{\frac {3}{2}} c^{\frac {5}{2}}-24 A \sqrt {c +c \sin \left (f x +e \right )}\, c^{\frac {7}{2}}-18 B \left (c +c \sin \left (f x +e \right )\right )^{\frac {5}{2}} c^{\frac {3}{2}}-32 B \left (c +c \sin \left (f x +e \right )\right )^{\frac {3}{2}} c^{\frac {5}{2}}+72 B \sqrt {c +c \sin \left (f x +e \right )}\, c^{\frac {7}{2}}+12 A \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{4}-36 B \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{4}\right ) \sqrt {c \left (1+\sin \left (f x +e \right )\right )}}{192 c^{\frac {15}{2}} \left (\sin \left (f x +e \right )-1\right )^{2} \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(352\) |
| parts | \(\text {Expression too large to display}\) | \(747\) |
Input:
int((a+a*sin(f*x+e))*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(7/2),x,method=_RET URNVERBOSE)
Output:
-1/192*a*(3*arctanh(1/2*(c+c*sin(f*x+e))^(1/2)*2^(1/2)/c^(1/2))*2^(1/2)*c^ 4*(A-3*B)*cos(f*x+e)^2*sin(f*x+e)-9*arctanh(1/2*(c+c*sin(f*x+e))^(1/2)*2^( 1/2)/c^(1/2))*2^(1/2)*c^4*(A-3*B)*cos(f*x+e)^2-12*arctanh(1/2*(c+c*sin(f*x +e))^(1/2)*2^(1/2)/c^(1/2))*2^(1/2)*c^4*(A-3*B)*sin(f*x+e)+6*A*(c+c*sin(f* x+e))^(5/2)*c^(3/2)-32*A*(c+c*sin(f*x+e))^(3/2)*c^(5/2)-24*A*(c+c*sin(f*x+ e))^(1/2)*c^(7/2)-18*B*(c+c*sin(f*x+e))^(5/2)*c^(3/2)-32*B*(c+c*sin(f*x+e) )^(3/2)*c^(5/2)+72*B*(c+c*sin(f*x+e))^(1/2)*c^(7/2)+12*A*2^(1/2)*arctanh(1 /2*(c+c*sin(f*x+e))^(1/2)*2^(1/2)/c^(1/2))*c^4-36*B*2^(1/2)*arctanh(1/2*(c +c*sin(f*x+e))^(1/2)*2^(1/2)/c^(1/2))*c^4)*(c*(1+sin(f*x+e)))^(1/2)/c^(15/ 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. 490 vs. \(2 (140) = 280\).
Time = 0.11 (sec) , antiderivative size = 490, normalized size of antiderivative = 3.01 \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx=-\frac {3 \, \sqrt {2} {\left ({\left (A - 3 \, B\right )} a \cos \left (f x + e\right )^{4} - 3 \, {\left (A - 3 \, B\right )} a \cos \left (f x + e\right )^{3} - 8 \, {\left (A - 3 \, B\right )} a \cos \left (f x + e\right )^{2} + 4 \, {\left (A - 3 \, B\right )} a \cos \left (f x + e\right ) + 8 \, {\left (A - 3 \, B\right )} a + {\left ({\left (A - 3 \, B\right )} a \cos \left (f x + e\right )^{3} + 4 \, {\left (A - 3 \, B\right )} a \cos \left (f x + e\right )^{2} - 4 \, {\left (A - 3 \, B\right )} a \cos \left (f x + e\right ) - 8 \, {\left (A - 3 \, B\right )} a\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 - 3 \, B\right )} a \cos \left (f x + e\right )^{3} - {\left (7 \, A + 43 \, B\right )} a \cos \left (f x + e\right )^{2} + 2 \, {\left (11 \, A - B\right )} a \cos \left (f x + e\right ) + 32 \, {\left (A + B\right )} a + {\left (3 \, {\left (A - 3 \, B\right )} a \cos \left (f x + e\right )^{2} + 2 \, {\left (5 \, A + 17 \, B\right )} a \cos \left (f x + e\right ) + 32 \, {\left (A + B\right )} a\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{384 \, {\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))*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(7/2),x, algo rithm="fricas")
Output:
-1/384*(3*sqrt(2)*((A - 3*B)*a*cos(f*x + e)^4 - 3*(A - 3*B)*a*cos(f*x + e) ^3 - 8*(A - 3*B)*a*cos(f*x + e)^2 + 4*(A - 3*B)*a*cos(f*x + e) + 8*(A - 3* B)*a + ((A - 3*B)*a*cos(f*x + e)^3 + 4*(A - 3*B)*a*cos(f*x + e)^2 - 4*(A - 3*B)*a*cos(f*x + e) - 8*(A - 3*B)*a)*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 - 3*B)*a*cos(f*x + e)^3 - (7*A + 43*B)*a*cos(f*x + e)^2 + 2*(1 1*A - B)*a*cos(f*x + e) + 32*(A + B)*a + (3*(A - 3*B)*a*cos(f*x + e)^2 + 2 *(5*A + 17*B)*a*cos(f*x + e) + 32*(A + B)*a)*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. \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx=\text {Timed out} \] Input:
integrate((a+a*sin(f*x+e))*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))**(7/2),x)
Output:
Timed out
\[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx=\int { \frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate((a+a*sin(f*x+e))*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(7/2),x, algo rithm="maxima")
Output:
integrate((B*sin(f*x + e) + A)*(a*sin(f*x + e) + a)/(-c*sin(f*x + e) + c)^ (7/2), x)
Exception generated. \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+a*sin(f*x+e))*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(7/2),x, algo rithm="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. \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx=\int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,\left (a+a\,\sin \left (e+f\,x\right )\right )}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{7/2}} \,d x \] Input:
int(((A + B*sin(e + f*x))*(a + a*sin(e + f*x)))/(c - c*sin(e + f*x))^(7/2) ,x)
Output:
int(((A + B*sin(e + f*x))*(a + a*sin(e + f*x)))/(c - c*sin(e + f*x))^(7/2) , x)
\[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx=\frac {\sqrt {c}\, a \left (\left (\int \frac {\sqrt {-\sin \left (f x +e \right )+1}}{\sin \left (f x +e \right )^{4}-4 \sin \left (f x +e \right )^{3}+6 \sin \left (f x +e \right )^{2}-4 \sin \left (f x +e \right )+1}d x \right ) a +\left (\int \frac {\sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{2}}{\sin \left (f x +e \right )^{4}-4 \sin \left (f x +e \right )^{3}+6 \sin \left (f x +e \right )^{2}-4 \sin \left (f x +e \right )+1}d x \right ) b +\left (\int \frac {\sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )}{\sin \left (f x +e \right )^{4}-4 \sin \left (f x +e \right )^{3}+6 \sin \left (f x +e \right )^{2}-4 \sin \left (f x +e \right )+1}d x \right ) a +\left (\int \frac {\sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )}{\sin \left (f x +e \right )^{4}-4 \sin \left (f x +e \right )^{3}+6 \sin \left (f x +e \right )^{2}-4 \sin \left (f x +e \right )+1}d x \right ) b \right )}{c^{4}} \] Input:
int((a+a*sin(f*x+e))*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(7/2),x)
Output:
(sqrt(c)*a*(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)**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 + 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*si n(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