Integrand size = 25, antiderivative size = 118 \[ \int \frac {(a+b \cos (c+d x))^2}{(e \sin (c+d x))^{3/2}} \, dx=-\frac {2 (b+a \cos (c+d x)) (a+b \cos (c+d x))}{d e \sqrt {e \sin (c+d x)}}-\frac {2 \left (a^2+2 b^2\right ) E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d e^2 \sqrt {\sin (c+d x)}}-\frac {2 a b (e \sin (c+d x))^{3/2}}{d e^3} \] Output:
-2*(b+a*cos(d*x+c))*(a+b*cos(d*x+c))/d/e/(e*sin(d*x+c))^(1/2)+2*(a^2+2*b^2 )*EllipticE(cos(1/2*c+1/4*Pi+1/2*d*x),2^(1/2))*(e*sin(d*x+c))^(1/2)/d/e^2/ sin(d*x+c)^(1/2)-2*a*b*(e*sin(d*x+c))^(3/2)/d/e^3
Time = 1.08 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.64 \[ \int \frac {(a+b \cos (c+d x))^2}{(e \sin (c+d x))^{3/2}} \, dx=\frac {-4 a b-2 \left (a^2+b^2\right ) \cos (c+d x)+2 \left (a^2+2 b^2\right ) E\left (\left .\frac {1}{4} (-2 c+\pi -2 d x)\right |2\right ) \sqrt {\sin (c+d x)}}{d e \sqrt {e \sin (c+d x)}} \] Input:
Integrate[(a + b*Cos[c + d*x])^2/(e*Sin[c + d*x])^(3/2),x]
Output:
(-4*a*b - 2*(a^2 + b^2)*Cos[c + d*x] + 2*(a^2 + 2*b^2)*EllipticE[(-2*c + P i - 2*d*x)/4, 2]*Sqrt[Sin[c + d*x]])/(d*e*Sqrt[e*Sin[c + d*x]])
Time = 0.58 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.03, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3042, 3170, 27, 3042, 3148, 3042, 3121, 3042, 3119}
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.
| \(\displaystyle \int \frac {(a+b \cos (c+d x))^2}{(e \sin (c+d x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a-b \sin \left (c+d x-\frac {\pi }{2}\right )\right )^2}{\left (e \cos \left (c+d x-\frac {\pi }{2}\right )\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 3170 |
| \(\displaystyle -\frac {2 \int \frac {1}{2} \left (a^2+3 b \cos (c+d x) a+2 b^2\right ) \sqrt {e \sin (c+d x)}dx}{e^2}-\frac {2 (a \cos (c+d x)+b) (a+b \cos (c+d x))}{d e \sqrt {e \sin (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \left (a^2+3 b \cos (c+d x) a+2 b^2\right ) \sqrt {e \sin (c+d x)}dx}{e^2}-\frac {2 (a \cos (c+d x)+b) (a+b \cos (c+d x))}{d e \sqrt {e \sin (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )} \left (a^2-3 b \sin \left (c+d x-\frac {\pi }{2}\right ) a+2 b^2\right )dx}{e^2}-\frac {2 (a \cos (c+d x)+b) (a+b \cos (c+d x))}{d e \sqrt {e \sin (c+d x)}}\) |
| \(\Big \downarrow \) 3148 |
| \(\displaystyle -\frac {\left (a^2+2 b^2\right ) \int \sqrt {e \sin (c+d x)}dx+\frac {2 a b (e \sin (c+d x))^{3/2}}{d e}}{e^2}-\frac {2 (a \cos (c+d x)+b) (a+b \cos (c+d x))}{d e \sqrt {e \sin (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\left (a^2+2 b^2\right ) \int \sqrt {e \sin (c+d x)}dx+\frac {2 a b (e \sin (c+d x))^{3/2}}{d e}}{e^2}-\frac {2 (a \cos (c+d x)+b) (a+b \cos (c+d x))}{d e \sqrt {e \sin (c+d x)}}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle -\frac {\frac {\left (a^2+2 b^2\right ) \sqrt {e \sin (c+d x)} \int \sqrt {\sin (c+d x)}dx}{\sqrt {\sin (c+d x)}}+\frac {2 a b (e \sin (c+d x))^{3/2}}{d e}}{e^2}-\frac {2 (a \cos (c+d x)+b) (a+b \cos (c+d x))}{d e \sqrt {e \sin (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\left (a^2+2 b^2\right ) \sqrt {e \sin (c+d x)} \int \sqrt {\sin (c+d x)}dx}{\sqrt {\sin (c+d x)}}+\frac {2 a b (e \sin (c+d x))^{3/2}}{d e}}{e^2}-\frac {2 (a \cos (c+d x)+b) (a+b \cos (c+d x))}{d e \sqrt {e \sin (c+d x)}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle -\frac {\frac {2 \left (a^2+2 b^2\right ) E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}+\frac {2 a b (e \sin (c+d x))^{3/2}}{d e}}{e^2}-\frac {2 (a \cos (c+d x)+b) (a+b \cos (c+d x))}{d e \sqrt {e \sin (c+d x)}}\) |
Input:
Int[(a + b*Cos[c + d*x])^2/(e*Sin[c + d*x])^(3/2),x]
Output:
(-2*(b + a*Cos[c + d*x])*(a + b*Cos[c + d*x]))/(d*e*Sqrt[e*Sin[c + d*x]]) - ((2*(a^2 + 2*b^2)*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]]) /(d*Sqrt[Sin[c + d*x]]) + (2*a*b*(e*Sin[c + d*x])^(3/2))/(d*e))/e^2
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[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Sin[c + d*x]) ^n/Sin[c + d*x]^n Int[Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && Lt Q[-1, n, 1] && IntegerQ[2*n]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[(-b)*((g*Cos[e + f*x])^(p + 1)/(f*g*(p + 1))), x] + Simp[a Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x] && (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[(-(g*Cos[e + f*x])^(p + 1))*(a + b*Sin[e + f*x ])^(m - 1)*((b + a*Sin[e + f*x])/(f*g*(p + 1))), x] + Simp[1/(g^2*(p + 1)) Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 2)*(b^2*(m - 1) + a^2*(p + 2) + a*b*(m + p + 1)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g }, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && LtQ[p, -1] && (IntegersQ[2*m, 2* p] || IntegerQ[m])
Leaf count of result is larger than twice the leaf count of optimal. \(282\) vs. \(2(110)=220\).
Time = 3.36 (sec) , antiderivative size = 283, normalized size of antiderivative = 2.40
| method | result | size |
| default | \(\frac {2 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2+2 \sin \left (d x +c \right )}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticE}\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) a^{2}+4 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2+2 \sin \left (d x +c \right )}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticE}\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) b^{2}-\sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2+2 \sin \left (d x +c \right )}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) a^{2}-2 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2+2 \sin \left (d x +c \right )}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) b^{2}-2 a^{2} \cos \left (d x +c \right )^{2}-2 \cos \left (d x +c \right )^{2} b^{2}-4 a b \cos \left (d x +c \right )}{e \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d}\) | \(283\) |
| parts | \(\frac {a^{2} \left (2 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2+2 \sin \left (d x +c \right )}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticE}\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2+2 \sin \left (d x +c \right )}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-2 \cos \left (d x +c \right )^{2}\right )}{e \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d}+\frac {2 b^{2} \left (2 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2+2 \sin \left (d x +c \right )}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticE}\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2+2 \sin \left (d x +c \right )}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-\cos \left (d x +c \right )^{2}\right )}{e \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d}-\frac {4 a b}{\sqrt {e \sin \left (d x +c \right )}\, e d}\) | \(309\) |
Input:
int((a+cos(d*x+c)*b)^2/(e*sin(d*x+c))^(3/2),x,method=_RETURNVERBOSE)
Output:
1/e/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*(2*(1-sin(d*x+c))^(1/2)*(2+2*sin(d*x+c ))^(1/2)*sin(d*x+c)^(1/2)*EllipticE((1-sin(d*x+c))^(1/2),1/2*2^(1/2))*a^2+ 4*(1-sin(d*x+c))^(1/2)*(2+2*sin(d*x+c))^(1/2)*sin(d*x+c)^(1/2)*EllipticE(( 1-sin(d*x+c))^(1/2),1/2*2^(1/2))*b^2-(1-sin(d*x+c))^(1/2)*(2+2*sin(d*x+c)) ^(1/2)*sin(d*x+c)^(1/2)*EllipticF((1-sin(d*x+c))^(1/2),1/2*2^(1/2))*a^2-2* (1-sin(d*x+c))^(1/2)*(2+2*sin(d*x+c))^(1/2)*sin(d*x+c)^(1/2)*EllipticF((1- sin(d*x+c))^(1/2),1/2*2^(1/2))*b^2-2*a^2*cos(d*x+c)^2-2*cos(d*x+c)^2*b^2-4 *a*b*cos(d*x+c))/d
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.14 \[ \int \frac {(a+b \cos (c+d x))^2}{(e \sin (c+d x))^{3/2}} \, dx=-\frac {2 \, {\left ({\left (i \, a^{2} + 2 i \, b^{2}\right )} \sqrt {-\frac {1}{2} i \, e} \sin \left (d x + c\right ) {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + {\left (-i \, a^{2} - 2 i \, b^{2}\right )} \sqrt {\frac {1}{2} i \, e} \sin \left (d x + c\right ) {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + {\left (2 \, a b + {\left (a^{2} + b^{2}\right )} \cos \left (d x + c\right )\right )} \sqrt {e \sin \left (d x + c\right )}\right )}}{d e^{2} \sin \left (d x + c\right )} \] Input:
integrate((a+b*cos(d*x+c))^2/(e*sin(d*x+c))^(3/2),x, algorithm="fricas")
Output:
-2*((I*a^2 + 2*I*b^2)*sqrt(-1/2*I*e)*sin(d*x + c)*weierstrassZeta(4, 0, we ierstrassPInverse(4, 0, cos(d*x + c) + I*sin(d*x + c))) + (-I*a^2 - 2*I*b^ 2)*sqrt(1/2*I*e)*sin(d*x + c)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(d*x + c) - I*sin(d*x + c))) + (2*a*b + (a^2 + b^2)*cos(d*x + c))*s qrt(e*sin(d*x + c)))/(d*e^2*sin(d*x + c))
\[ \int \frac {(a+b \cos (c+d x))^2}{(e \sin (c+d x))^{3/2}} \, dx=\int \frac {\left (a + b \cos {\left (c + d x \right )}\right )^{2}}{\left (e \sin {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((a+b*cos(d*x+c))**2/(e*sin(d*x+c))**(3/2),x)
Output:
Integral((a + b*cos(c + d*x))**2/(e*sin(c + d*x))**(3/2), x)
\[ \int \frac {(a+b \cos (c+d x))^2}{(e \sin (c+d x))^{3/2}} \, dx=\int { \frac {{\left (b \cos \left (d x + c\right ) + a\right )}^{2}}{\left (e \sin \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^2/(e*sin(d*x+c))^(3/2),x, algorithm="maxima")
Output:
integrate((b*cos(d*x + c) + a)^2/(e*sin(d*x + c))^(3/2), x)
\[ \int \frac {(a+b \cos (c+d x))^2}{(e \sin (c+d x))^{3/2}} \, dx=\int { \frac {{\left (b \cos \left (d x + c\right ) + a\right )}^{2}}{\left (e \sin \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^2/(e*sin(d*x+c))^(3/2),x, algorithm="giac")
Output:
integrate((b*cos(d*x + c) + a)^2/(e*sin(d*x + c))^(3/2), x)
Timed out. \[ \int \frac {(a+b \cos (c+d x))^2}{(e \sin (c+d x))^{3/2}} \, dx=\int \frac {{\left (a+b\,\cos \left (c+d\,x\right )\right )}^2}{{\left (e\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \] Input:
int((a + b*cos(c + d*x))^2/(e*sin(c + d*x))^(3/2),x)
Output:
int((a + b*cos(c + d*x))^2/(e*sin(c + d*x))^(3/2), x)
\[ \int \frac {(a+b \cos (c+d x))^2}{(e \sin (c+d x))^{3/2}} \, dx=\frac {\sqrt {e}\, \left (-4 \sqrt {\sin \left (d x +c \right )}\, a b +\left (\int \frac {\sqrt {\sin \left (d x +c \right )}}{\sin \left (d x +c \right )^{2}}d x \right ) \sin \left (d x +c \right ) a^{2} d +\left (\int \frac {\sqrt {\sin \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}}{\sin \left (d x +c \right )^{2}}d x \right ) \sin \left (d x +c \right ) b^{2} d \right )}{\sin \left (d x +c \right ) d \,e^{2}} \] Input:
int((a+b*cos(d*x+c))^2/(e*sin(d*x+c))^(3/2),x)
Output:
(sqrt(e)*( - 4*sqrt(sin(c + d*x))*a*b + int(sqrt(sin(c + d*x))/sin(c + d*x )**2,x)*sin(c + d*x)*a**2*d + int((sqrt(sin(c + d*x))*cos(c + d*x)**2)/sin (c + d*x)**2,x)*sin(c + d*x)*b**2*d))/(sin(c + d*x)*d*e**2)