Integrand size = 41, antiderivative size = 117 \[ \int \frac {(a+b \sin (e+f x)) \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right )}{\sin ^{\frac {3}{2}}(e+f x)} \, dx=\frac {2 (b B-a (A-C)) E\left (\left .\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )\right |2\right )}{f}+\frac {2 (3 A b+3 a B+b C) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),2\right )}{3 f}-\frac {2 a A \cos (e+f x)}{f \sqrt {\sin (e+f x)}}-\frac {2 b C \cos (e+f x) \sqrt {\sin (e+f x)}}{3 f} \] Output:
-2*(b*B-a*(A-C))*EllipticE(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2))/f+2/3*(3*A*b +3*B*a+C*b)*InverseJacobiAM(1/2*e-1/4*Pi+1/2*f*x,2^(1/2))/f-2*a*A*cos(f*x+ e)/f/sin(f*x+e)^(1/2)-2/3*b*C*cos(f*x+e)*sin(f*x+e)^(1/2)/f
Time = 2.91 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.83 \[ \int \frac {(a+b \sin (e+f x)) \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right )}{\sin ^{\frac {3}{2}}(e+f x)} \, dx=-\frac {6 (b B+a (-A+C)) E\left (\left .\frac {1}{4} (-2 e+\pi -2 f x)\right |2\right )+2 (3 A b+3 a B+b C) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),2\right )+\frac {2 \cos (e+f x) (3 a A+b C \sin (e+f x))}{\sqrt {\sin (e+f x)}}}{3 f} \] Input:
Integrate[((a + b*Sin[e + f*x])*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2))/S in[e + f*x]^(3/2),x]
Output:
-1/3*(6*(b*B + a*(-A + C))*EllipticE[(-2*e + Pi - 2*f*x)/4, 2] + 2*(3*A*b + 3*a*B + b*C)*EllipticF[(-2*e + Pi - 2*f*x)/4, 2] + (2*Cos[e + f*x]*(3*a* A + b*C*Sin[e + f*x]))/Sqrt[Sin[e + f*x]])/f
Time = 0.67 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.03, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.268, Rules used = {3042, 3510, 27, 3042, 3502, 27, 3042, 3227, 3042, 3119, 3120}
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.
\(\displaystyle \int \frac {(a+b \sin (e+f x)) \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right )}{\sin ^{\frac {3}{2}}(e+f x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a+b \sin (e+f x)) \left (A+B \sin (e+f x)+C \sin (e+f x)^2\right )}{\sin (e+f x)^{3/2}}dx\) |
\(\Big \downarrow \) 3510 |
\(\displaystyle -2 \int -\frac {b C \sin ^2(e+f x)+(b B-a (A-C)) \sin (e+f x)+A b+a B}{2 \sqrt {\sin (e+f x)}}dx-\frac {2 a A \cos (e+f x)}{f \sqrt {\sin (e+f x)}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {b C \sin ^2(e+f x)+(b B-a (A-C)) \sin (e+f x)+A b+a B}{\sqrt {\sin (e+f x)}}dx-\frac {2 a A \cos (e+f x)}{f \sqrt {\sin (e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {b C \sin (e+f x)^2+(b B-a (A-C)) \sin (e+f x)+A b+a B}{\sqrt {\sin (e+f x)}}dx-\frac {2 a A \cos (e+f x)}{f \sqrt {\sin (e+f x)}}\) |
\(\Big \downarrow \) 3502 |
\(\displaystyle \frac {2}{3} \int \frac {3 A b+C b+3 a B+3 (b B-a (A-C)) \sin (e+f x)}{2 \sqrt {\sin (e+f x)}}dx-\frac {2 a A \cos (e+f x)}{f \sqrt {\sin (e+f x)}}-\frac {2 b C \sqrt {\sin (e+f x)} \cos (e+f x)}{3 f}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \int \frac {3 A b+C b+3 a B+3 (b B-a (A-C)) \sin (e+f x)}{\sqrt {\sin (e+f x)}}dx-\frac {2 a A \cos (e+f x)}{f \sqrt {\sin (e+f x)}}-\frac {2 b C \sqrt {\sin (e+f x)} \cos (e+f x)}{3 f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{3} \int \frac {3 A b+C b+3 a B+3 (b B-a (A-C)) \sin (e+f x)}{\sqrt {\sin (e+f x)}}dx-\frac {2 a A \cos (e+f x)}{f \sqrt {\sin (e+f x)}}-\frac {2 b C \sqrt {\sin (e+f x)} \cos (e+f x)}{3 f}\) |
\(\Big \downarrow \) 3227 |
\(\displaystyle \frac {1}{3} \left ((3 a B+3 A b+b C) \int \frac {1}{\sqrt {\sin (e+f x)}}dx+3 (b B-a (A-C)) \int \sqrt {\sin (e+f x)}dx\right )-\frac {2 a A \cos (e+f x)}{f \sqrt {\sin (e+f x)}}-\frac {2 b C \sqrt {\sin (e+f x)} \cos (e+f x)}{3 f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{3} \left ((3 a B+3 A b+b C) \int \frac {1}{\sqrt {\sin (e+f x)}}dx+3 (b B-a (A-C)) \int \sqrt {\sin (e+f x)}dx\right )-\frac {2 a A \cos (e+f x)}{f \sqrt {\sin (e+f x)}}-\frac {2 b C \sqrt {\sin (e+f x)} \cos (e+f x)}{3 f}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {1}{3} \left ((3 a B+3 A b+b C) \int \frac {1}{\sqrt {\sin (e+f x)}}dx+\frac {6 E\left (\left .\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )\right |2\right ) (b B-a (A-C))}{f}\right )-\frac {2 a A \cos (e+f x)}{f \sqrt {\sin (e+f x)}}-\frac {2 b C \sqrt {\sin (e+f x)} \cos (e+f x)}{3 f}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {1}{3} \left (\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),2\right ) (3 a B+3 A b+b C)}{f}+\frac {6 E\left (\left .\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )\right |2\right ) (b B-a (A-C))}{f}\right )-\frac {2 a A \cos (e+f x)}{f \sqrt {\sin (e+f x)}}-\frac {2 b C \sqrt {\sin (e+f x)} \cos (e+f x)}{3 f}\) |
Input:
Int[((a + b*Sin[e + f*x])*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2))/Sin[e + f*x]^(3/2),x]
Output:
((6*(b*B - a*(A - C))*EllipticE[(e - Pi/2 + f*x)/2, 2])/f + (2*(3*A*b + 3* a*B + b*C)*EllipticF[(e - Pi/2 + f*x)/2, 2])/f)/3 - (2*a*A*Cos[e + f*x])/( f*Sqrt[Sin[e + f*x]]) - (2*b*C*Cos[e + f*x]*Sqrt[Sin[e + f*x]])/(3*f)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
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]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f _.)*(x_)]^2), x_Symbol] :> Simp[(-(b*c - a*d))*(A*b^2 - a*b*B + a^2*C)*Cos[ e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b^2*f*(m + 1)*(a^2 - b^2))), x] - S imp[1/(b^2*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*( m + 1)*((b*B - a*C)*(b*c - a*d) - A*b*(a*c - b*d)) + (b*B*(a^2*d + b^2*d*(m + 1) - a*b*c*(m + 2)) + (b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1)) ))*Sin[e + f*x] - b*C*d*(m + 1)*(a^2 - b^2)*Sin[e + f*x]^2, x], x], x] /; F reeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(395\) vs. \(2(108)=216\).
Time = 0.85 (sec) , antiderivative size = 396, normalized size of antiderivative = 3.38
method | result | size |
parts | \(\frac {\left (A b +B a \right ) \sqrt {1+\sin \left (f x +e \right )}\, \sqrt {-2 \sin \left (f x +e \right )+2}\, \sqrt {-\sin \left (f x +e \right )}\, \operatorname {EllipticF}\left (\sqrt {1+\sin \left (f x +e \right )}, \frac {\sqrt {2}}{2}\right )}{\cos \left (f x +e \right ) \sqrt {\sin \left (f x +e \right )}\, f}-\frac {\left (B b +C a \right ) \sqrt {1+\sin \left (f x +e \right )}\, \sqrt {-2 \sin \left (f x +e \right )+2}\, \sqrt {-\sin \left (f x +e \right )}\, \left (2 \operatorname {EllipticE}\left (\sqrt {1+\sin \left (f x +e \right )}, \frac {\sqrt {2}}{2}\right )-\operatorname {EllipticF}\left (\sqrt {1+\sin \left (f x +e \right )}, \frac {\sqrt {2}}{2}\right )\right )}{\cos \left (f x +e \right ) \sqrt {\sin \left (f x +e \right )}\, f}+\frac {A a \left (2 \sqrt {1+\sin \left (f x +e \right )}\, \sqrt {-2 \sin \left (f x +e \right )+2}\, \sqrt {-\sin \left (f x +e \right )}\, \operatorname {EllipticE}\left (\sqrt {1+\sin \left (f x +e \right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {1+\sin \left (f x +e \right )}\, \sqrt {-2 \sin \left (f x +e \right )+2}\, \sqrt {-\sin \left (f x +e \right )}\, \operatorname {EllipticF}\left (\sqrt {1+\sin \left (f x +e \right )}, \frac {\sqrt {2}}{2}\right )-2 \cos \left (f x +e \right )^{2}\right )}{\cos \left (f x +e \right ) \sqrt {\sin \left (f x +e \right )}\, f}+\frac {C b \left (\frac {\sqrt {1+\sin \left (f x +e \right )}\, \sqrt {-2 \sin \left (f x +e \right )+2}\, \sqrt {-\sin \left (f x +e \right )}\, \operatorname {EllipticF}\left (\sqrt {1+\sin \left (f x +e \right )}, \frac {\sqrt {2}}{2}\right )}{3}-\frac {2 \cos \left (f x +e \right )^{2} \sin \left (f x +e \right )}{3}\right )}{\cos \left (f x +e \right ) \sqrt {\sin \left (f x +e \right )}\, f}\) | \(396\) |
default | \(\frac {-A \sqrt {1+\sin \left (f x +e \right )}\, \sqrt {-2 \sin \left (f x +e \right )+2}\, \sqrt {-\sin \left (f x +e \right )}\, \operatorname {EllipticF}\left (\sqrt {1+\sin \left (f x +e \right )}, \frac {\sqrt {2}}{2}\right ) a +A b \sqrt {1+\sin \left (f x +e \right )}\, \sqrt {-2 \sin \left (f x +e \right )+2}\, \sqrt {-\sin \left (f x +e \right )}\, \operatorname {EllipticF}\left (\sqrt {1+\sin \left (f x +e \right )}, \frac {\sqrt {2}}{2}\right )+2 A \sqrt {1+\sin \left (f x +e \right )}\, \sqrt {-2 \sin \left (f x +e \right )+2}\, \sqrt {-\sin \left (f x +e \right )}\, \operatorname {EllipticE}\left (\sqrt {1+\sin \left (f x +e \right )}, \frac {\sqrt {2}}{2}\right ) a +B a \sqrt {1+\sin \left (f x +e \right )}\, \sqrt {-2 \sin \left (f x +e \right )+2}\, \sqrt {-\sin \left (f x +e \right )}\, \operatorname {EllipticF}\left (\sqrt {1+\sin \left (f x +e \right )}, \frac {\sqrt {2}}{2}\right )+B b \sqrt {1+\sin \left (f x +e \right )}\, \sqrt {-2 \sin \left (f x +e \right )+2}\, \sqrt {-\sin \left (f x +e \right )}\, \operatorname {EllipticF}\left (\sqrt {1+\sin \left (f x +e \right )}, \frac {\sqrt {2}}{2}\right )-2 B b \sqrt {1+\sin \left (f x +e \right )}\, \sqrt {-2 \sin \left (f x +e \right )+2}\, \sqrt {-\sin \left (f x +e \right )}\, \operatorname {EllipticE}\left (\sqrt {1+\sin \left (f x +e \right )}, \frac {\sqrt {2}}{2}\right )+C a \sqrt {1+\sin \left (f x +e \right )}\, \sqrt {-2 \sin \left (f x +e \right )+2}\, \sqrt {-\sin \left (f x +e \right )}\, \operatorname {EllipticF}\left (\sqrt {1+\sin \left (f x +e \right )}, \frac {\sqrt {2}}{2}\right )+\frac {C \sqrt {1+\sin \left (f x +e \right )}\, \sqrt {-2 \sin \left (f x +e \right )+2}\, \sqrt {-\sin \left (f x +e \right )}\, \operatorname {EllipticF}\left (\sqrt {1+\sin \left (f x +e \right )}, \frac {\sqrt {2}}{2}\right ) b}{3}-2 C a \sqrt {1+\sin \left (f x +e \right )}\, \sqrt {-2 \sin \left (f x +e \right )+2}\, \sqrt {-\sin \left (f x +e \right )}\, \operatorname {EllipticE}\left (\sqrt {1+\sin \left (f x +e \right )}, \frac {\sqrt {2}}{2}\right )-\frac {2 C \cos \left (f x +e \right )^{2} \sin \left (f x +e \right ) b}{3}-2 A a \cos \left (f x +e \right )^{2}}{\cos \left (f x +e \right ) \sqrt {\sin \left (f x +e \right )}\, f}\) | \(516\) |
Input:
int((a+b*sin(f*x+e))*(A+B*sin(f*x+e)+C*sin(f*x+e)^2)/sin(f*x+e)^(3/2),x,me thod=_RETURNVERBOSE)
Output:
(A*b+B*a)*(1+sin(f*x+e))^(1/2)*(-2*sin(f*x+e)+2)^(1/2)*(-sin(f*x+e))^(1/2) *EllipticF((1+sin(f*x+e))^(1/2),1/2*2^(1/2))/cos(f*x+e)/sin(f*x+e)^(1/2)/f -(B*b+C*a)*(1+sin(f*x+e))^(1/2)*(-2*sin(f*x+e)+2)^(1/2)*(-sin(f*x+e))^(1/2 )*(2*EllipticE((1+sin(f*x+e))^(1/2),1/2*2^(1/2))-EllipticF((1+sin(f*x+e))^ (1/2),1/2*2^(1/2)))/cos(f*x+e)/sin(f*x+e)^(1/2)/f+A*a*(2*(1+sin(f*x+e))^(1 /2)*(-2*sin(f*x+e)+2)^(1/2)*(-sin(f*x+e))^(1/2)*EllipticE((1+sin(f*x+e))^( 1/2),1/2*2^(1/2))-(1+sin(f*x+e))^(1/2)*(-2*sin(f*x+e)+2)^(1/2)*(-sin(f*x+e ))^(1/2)*EllipticF((1+sin(f*x+e))^(1/2),1/2*2^(1/2))-2*cos(f*x+e)^2)/cos(f *x+e)/sin(f*x+e)^(1/2)/f+C*b*(1/3*(1+sin(f*x+e))^(1/2)*(-2*sin(f*x+e)+2)^( 1/2)*(-sin(f*x+e))^(1/2)*EllipticF((1+sin(f*x+e))^(1/2),1/2*2^(1/2))-2/3*c os(f*x+e)^2*sin(f*x+e))/cos(f*x+e)/sin(f*x+e)^(1/2)/f
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.88 \[ \int \frac {(a+b \sin (e+f x)) \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right )}{\sin ^{\frac {3}{2}}(e+f x)} \, dx=\frac {2 \, {\left (\sqrt {-\frac {1}{2} i} {\left (3 \, B a + {\left (3 \, A + C\right )} b\right )} \sin \left (f x + e\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + \sqrt {\frac {1}{2} i} {\left (3 \, B a + {\left (3 \, A + C\right )} b\right )} \sin \left (f x + e\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) - 3 \, \sqrt {-\frac {1}{2} i} {\left (i \, {\left (A - C\right )} a - i \, B b\right )} \sin \left (f x + e\right ) {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) - 3 \, \sqrt {\frac {1}{2} i} {\left (-i \, {\left (A - C\right )} a + i \, B b\right )} \sin \left (f x + e\right ) {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) - {\left (C b \cos \left (f x + e\right ) \sin \left (f x + e\right ) + 3 \, A a \cos \left (f x + e\right )\right )} \sqrt {\sin \left (f x + e\right )}\right )}}{3 \, f \sin \left (f x + e\right )} \] Input:
integrate((a+b*sin(f*x+e))*(A+B*sin(f*x+e)+C*sin(f*x+e)^2)/sin(f*x+e)^(3/2 ),x, algorithm="fricas")
Output:
2/3*(sqrt(-1/2*I)*(3*B*a + (3*A + C)*b)*sin(f*x + e)*weierstrassPInverse(4 , 0, cos(f*x + e) + I*sin(f*x + e)) + sqrt(1/2*I)*(3*B*a + (3*A + C)*b)*si n(f*x + e)*weierstrassPInverse(4, 0, cos(f*x + e) - I*sin(f*x + e)) - 3*sq rt(-1/2*I)*(I*(A - C)*a - I*B*b)*sin(f*x + e)*weierstrassZeta(4, 0, weiers trassPInverse(4, 0, cos(f*x + e) + I*sin(f*x + e))) - 3*sqrt(1/2*I)*(-I*(A - C)*a + I*B*b)*sin(f*x + e)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(f*x + e) - I*sin(f*x + e))) - (C*b*cos(f*x + e)*sin(f*x + e) + 3*A *a*cos(f*x + e))*sqrt(sin(f*x + e)))/(f*sin(f*x + e))
\[ \int \frac {(a+b \sin (e+f x)) \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right )}{\sin ^{\frac {3}{2}}(e+f x)} \, dx=\int \frac {\left (a + b \sin {\left (e + f x \right )}\right ) \left (A + B \sin {\left (e + f x \right )} + C \sin ^{2}{\left (e + f x \right )}\right )}{\sin ^{\frac {3}{2}}{\left (e + f x \right )}}\, dx \] Input:
integrate((a+b*sin(f*x+e))*(A+B*sin(f*x+e)+C*sin(f*x+e)**2)/sin(f*x+e)**(3 /2),x)
Output:
Integral((a + b*sin(e + f*x))*(A + B*sin(e + f*x) + C*sin(e + f*x)**2)/sin (e + f*x)**(3/2), x)
\[ \int \frac {(a+b \sin (e+f x)) \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right )}{\sin ^{\frac {3}{2}}(e+f x)} \, dx=\int { \frac {{\left (C \sin \left (f x + e\right )^{2} + B \sin \left (f x + e\right ) + A\right )} {\left (b \sin \left (f x + e\right ) + a\right )}}{\sin \left (f x + e\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((a+b*sin(f*x+e))*(A+B*sin(f*x+e)+C*sin(f*x+e)^2)/sin(f*x+e)^(3/2 ),x, algorithm="maxima")
Output:
integrate((C*sin(f*x + e)^2 + B*sin(f*x + e) + A)*(b*sin(f*x + e) + a)/sin (f*x + e)^(3/2), x)
\[ \int \frac {(a+b \sin (e+f x)) \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right )}{\sin ^{\frac {3}{2}}(e+f x)} \, dx=\int { \frac {{\left (C \sin \left (f x + e\right )^{2} + B \sin \left (f x + e\right ) + A\right )} {\left (b \sin \left (f x + e\right ) + a\right )}}{\sin \left (f x + e\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((a+b*sin(f*x+e))*(A+B*sin(f*x+e)+C*sin(f*x+e)^2)/sin(f*x+e)^(3/2 ),x, algorithm="giac")
Output:
integrate((C*sin(f*x + e)^2 + B*sin(f*x + e) + A)*(b*sin(f*x + e) + a)/sin (f*x + e)^(3/2), x)
Time = 39.26 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.44 \[ \int \frac {(a+b \sin (e+f x)) \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right )}{\sin ^{\frac {3}{2}}(e+f x)} \, dx=\frac {2\,B\,b\,\mathrm {E}\left (\frac {e}{2}-\frac {\pi }{4}+\frac {f\,x}{2}\middle |2\right )}{f}-\frac {2\,B\,a\,\mathrm {F}\left (\frac {\pi }{4}-\frac {e}{2}-\frac {f\,x}{2}\middle |2\right )}{f}-\frac {2\,A\,b\,\mathrm {F}\left (\frac {\pi }{4}-\frac {e}{2}-\frac {f\,x}{2}\middle |2\right )}{f}+\frac {2\,C\,a\,\mathrm {E}\left (\frac {e}{2}-\frac {\pi }{4}+\frac {f\,x}{2}\middle |2\right )}{f}-\frac {A\,a\,\cos \left (e+f\,x\right )\,{\left ({\sin \left (e+f\,x\right )}^2\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {5}{4};\ \frac {3}{2};\ {\cos \left (e+f\,x\right )}^2\right )}{f\,\sqrt {\sin \left (e+f\,x\right )}}-\frac {C\,b\,\cos \left (e+f\,x\right )\,{\sin \left (e+f\,x\right )}^{5/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{2};\ {\cos \left (e+f\,x\right )}^2\right )}{f\,{\left ({\sin \left (e+f\,x\right )}^2\right )}^{5/4}} \] Input:
int(((a + b*sin(e + f*x))*(A + B*sin(e + f*x) + C*sin(e + f*x)^2))/sin(e + f*x)^(3/2),x)
Output:
(2*B*b*ellipticE(e/2 - pi/4 + (f*x)/2, 2))/f - (2*B*a*ellipticF(pi/4 - e/2 - (f*x)/2, 2))/f - (2*A*b*ellipticF(pi/4 - e/2 - (f*x)/2, 2))/f + (2*C*a* ellipticE(e/2 - pi/4 + (f*x)/2, 2))/f - (A*a*cos(e + f*x)*(sin(e + f*x)^2) ^(1/4)*hypergeom([1/2, 5/4], 3/2, cos(e + f*x)^2))/(f*sin(e + f*x)^(1/2)) - (C*b*cos(e + f*x)*sin(e + f*x)^(5/2)*hypergeom([-1/4, 1/2], 3/2, cos(e + f*x)^2))/(f*(sin(e + f*x)^2)^(5/4))
\[ \int \frac {(a+b \sin (e+f x)) \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right )}{\sin ^{\frac {3}{2}}(e+f x)} \, dx=\left (\int \frac {\sqrt {\sin \left (f x +e \right )}}{\sin \left (f x +e \right )^{2}}d x \right ) a^{2}+2 \left (\int \frac {\sqrt {\sin \left (f x +e \right )}}{\sin \left (f x +e \right )}d x \right ) a b +\left (\int \sqrt {\sin \left (f x +e \right )}d x \right ) a c +\left (\int \sqrt {\sin \left (f x +e \right )}d x \right ) b^{2}+\left (\int \sqrt {\sin \left (f x +e \right )}\, \sin \left (f x +e \right )d x \right ) b c \] Input:
int((a+b*sin(f*x+e))*(A+B*sin(f*x+e)+C*sin(f*x+e)^2)/sin(f*x+e)^(3/2),x)
Output:
int(sqrt(sin(e + f*x))/sin(e + f*x)**2,x)*a**2 + 2*int(sqrt(sin(e + f*x))/ sin(e + f*x),x)*a*b + int(sqrt(sin(e + f*x)),x)*a*c + int(sqrt(sin(e + f*x )),x)*b**2 + int(sqrt(sin(e + f*x))*sin(e + f*x),x)*b*c