Integrand size = 25, antiderivative size = 160 \[ \int \frac {(a+b \sin (c+d x))^2}{(e \cos (c+d x))^{7/2}} \, dx=\frac {2 a b}{5 d e^3 \sqrt {e \cos (c+d x)}}-\frac {2 \left (3 a^2-2 b^2\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d e^4 \sqrt {\cos (c+d x)}}+\frac {2 \left (3 a^2-2 b^2\right ) \sin (c+d x)}{5 d e^3 \sqrt {e \cos (c+d x)}}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))}{5 d e (e \cos (c+d x))^{5/2}} \] Output:
2/5*a*b/d/e^3/(e*cos(d*x+c))^(1/2)-2/5*(3*a^2-2*b^2)*(e*cos(d*x+c))^(1/2)* EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/d/e^4/cos(d*x+c)^(1/2)+2/5*(3*a^2-2* b^2)*sin(d*x+c)/d/e^3/(e*cos(d*x+c))^(1/2)+2/5*(b+a*sin(d*x+c))*(a+b*sin(d *x+c))/d/e/(e*cos(d*x+c))^(5/2)
Time = 1.59 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.66 \[ \int \frac {(a+b \sin (c+d x))^2}{(e \cos (c+d x))^{7/2}} \, dx=\frac {8 a b-4 \left (3 a^2-2 b^2\right ) \cos ^{\frac {5}{2}}(c+d x) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+\left (7 a^2+2 b^2\right ) \sin (c+d x)+3 a^2 \sin (3 (c+d x))-2 b^2 \sin (3 (c+d x))}{10 d e (e \cos (c+d x))^{5/2}} \] Input:
Integrate[(a + b*Sin[c + d*x])^2/(e*Cos[c + d*x])^(7/2),x]
Output:
(8*a*b - 4*(3*a^2 - 2*b^2)*Cos[c + d*x]^(5/2)*EllipticE[(c + d*x)/2, 2] + (7*a^2 + 2*b^2)*Sin[c + d*x] + 3*a^2*Sin[3*(c + d*x)] - 2*b^2*Sin[3*(c + d *x)])/(10*d*e*(e*Cos[c + d*x])^(5/2))
Time = 0.69 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.96, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {3042, 3170, 27, 3042, 3148, 3042, 3116, 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.
| \(\displaystyle \int \frac {(a+b \sin (c+d x))^2}{(e \cos (c+d x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \sin (c+d x))^2}{(e \cos (c+d x))^{7/2}}dx\) |
| \(\Big \downarrow \) 3170 |
| \(\displaystyle \frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))}{5 d e (e \cos (c+d x))^{5/2}}-\frac {2 \int -\frac {3 a^2+b \sin (c+d x) a-2 b^2}{2 (e \cos (c+d x))^{3/2}}dx}{5 e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {3 a^2+b \sin (c+d x) a-2 b^2}{(e \cos (c+d x))^{3/2}}dx}{5 e^2}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))}{5 d e (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {3 a^2+b \sin (c+d x) a-2 b^2}{(e \cos (c+d x))^{3/2}}dx}{5 e^2}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))}{5 d e (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3148 |
| \(\displaystyle \frac {\left (3 a^2-2 b^2\right ) \int \frac {1}{(e \cos (c+d x))^{3/2}}dx+\frac {2 a b}{d e \sqrt {e \cos (c+d x)}}}{5 e^2}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))}{5 d e (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (3 a^2-2 b^2\right ) \int \frac {1}{\left (e \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx+\frac {2 a b}{d e \sqrt {e \cos (c+d x)}}}{5 e^2}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))}{5 d e (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle \frac {\left (3 a^2-2 b^2\right ) \left (\frac {2 \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}}-\frac {\int \sqrt {e \cos (c+d x)}dx}{e^2}\right )+\frac {2 a b}{d e \sqrt {e \cos (c+d x)}}}{5 e^2}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))}{5 d e (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (3 a^2-2 b^2\right ) \left (\frac {2 \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}}-\frac {\int \sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{e^2}\right )+\frac {2 a b}{d e \sqrt {e \cos (c+d x)}}}{5 e^2}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))}{5 d e (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {\left (3 a^2-2 b^2\right ) \left (\frac {2 \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}}-\frac {\sqrt {e \cos (c+d x)} \int \sqrt {\cos (c+d x)}dx}{e^2 \sqrt {\cos (c+d x)}}\right )+\frac {2 a b}{d e \sqrt {e \cos (c+d x)}}}{5 e^2}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))}{5 d e (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (3 a^2-2 b^2\right ) \left (\frac {2 \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}}-\frac {\sqrt {e \cos (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{e^2 \sqrt {\cos (c+d x)}}\right )+\frac {2 a b}{d e \sqrt {e \cos (c+d x)}}}{5 e^2}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))}{5 d e (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {\left (3 a^2-2 b^2\right ) \left (\frac {2 \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d e^2 \sqrt {\cos (c+d x)}}\right )+\frac {2 a b}{d e \sqrt {e \cos (c+d x)}}}{5 e^2}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))}{5 d e (e \cos (c+d x))^{5/2}}\) |
Input:
Int[(a + b*Sin[c + d*x])^2/(e*Cos[c + d*x])^(7/2),x]
Output:
(2*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x]))/(5*d*e*(e*Cos[c + d*x])^(5/2 )) + ((2*a*b)/(d*e*Sqrt[e*Cos[c + d*x]]) + (3*a^2 - 2*b^2)*((-2*Sqrt[e*Cos [c + d*x]]*EllipticE[(c + d*x)/2, 2])/(d*e^2*Sqrt[Cos[c + d*x]]) + (2*Sin[ c + d*x])/(d*e*Sqrt[e*Cos[c + d*x]])))/(5*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[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1))), x] + Simp[(n + 2)/(b^2*(n + 1)) I nt[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && IntegerQ[2*n]
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. \(563\) vs. \(2(144)=288\).
Time = 3.05 (sec) , antiderivative size = 564, normalized size of antiderivative = 3.52
| method | result | size |
| default | \(\frac {\frac {48 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} a^{2}}{5}-\frac {32 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} b^{2}}{5}-\frac {24 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a^{2}}{5}+\frac {16 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b^{2}}{5}-\frac {48 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a^{2}}{5}+\frac {32 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b^{2}}{5}+\frac {24 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a^{2}}{5}-\frac {16 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b^{2}}{5}+\frac {16 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a^{2}}{5}-\frac {4 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b^{2}}{5}-\frac {6 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{2}}{5}+\frac {4 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) b^{2}}{5}+\frac {4 a b \sin \left (\frac {d x}{2}+\frac {c}{2}\right )}{5}}{\left (4 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-4 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} e +e}\, e^{3} d}\) | \(564\) |
| parts | \(\text {Expression too large to display}\) | \(757\) |
Input:
int((a+b*sin(d*x+c))^2/(e*cos(d*x+c))^(7/2),x,method=_RETURNVERBOSE)
Output:
2/5/(4*sin(1/2*d*x+1/2*c)^4-4*sin(1/2*d*x+1/2*c)^2+1)/sin(1/2*d*x+1/2*c)/( -2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)/e^3*(24*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+ 1/2*c)^6*a^2-16*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6*b^2-12*(sin(1/2*d* x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1 /2*c),2^(1/2))*sin(1/2*d*x+1/2*c)^4*a^2+8*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2* sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*sin(1/ 2*d*x+1/2*c)^4*b^2-24*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^4*a^2+16*cos(1 /2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^4*b^2+12*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2* sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*sin(1/ 2*d*x+1/2*c)^2*a^2-8*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2- 1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*sin(1/2*d*x+1/2*c)^2*b^2+8* cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2*a^2-2*cos(1/2*d*x+1/2*c)*sin(1/2*d *x+1/2*c)^2*b^2-3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^ (1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a^2+2*(sin(1/2*d*x+1/2*c)^2)^( 1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2) )*b^2+2*a*b*sin(1/2*d*x+1/2*c))/d
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.01 \[ \int \frac {(a+b \sin (c+d x))^2}{(e \cos (c+d x))^{7/2}} \, dx=-\frac {2 \, {\left (\sqrt {\frac {1}{2}} {\left (3 i \, a^{2} - 2 i \, b^{2}\right )} \sqrt {e} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + \sqrt {\frac {1}{2}} {\left (-3 i \, a^{2} + 2 i \, b^{2}\right )} \sqrt {e} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - {\left (2 \, a b + {\left ({\left (3 \, a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + a^{2} + b^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}\right )}}{5 \, d e^{4} \cos \left (d x + c\right )^{3}} \] Input:
integrate((a+b*sin(d*x+c))^2/(e*cos(d*x+c))^(7/2),x, algorithm="fricas")
Output:
-2/5*(sqrt(1/2)*(3*I*a^2 - 2*I*b^2)*sqrt(e)*cos(d*x + c)^3*weierstrassZeta (-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + sqrt( 1/2)*(-3*I*a^2 + 2*I*b^2)*sqrt(e)*cos(d*x + c)^3*weierstrassZeta(-4, 0, we ierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - (2*a*b + ((3*a^ 2 - 2*b^2)*cos(d*x + c)^2 + a^2 + b^2)*sin(d*x + c))*sqrt(e*cos(d*x + c))) /(d*e^4*cos(d*x + c)^3)
Timed out. \[ \int \frac {(a+b \sin (c+d x))^2}{(e \cos (c+d x))^{7/2}} \, dx=\text {Timed out} \] Input:
integrate((a+b*sin(d*x+c))**2/(e*cos(d*x+c))**(7/2),x)
Output:
Timed out
\[ \int \frac {(a+b \sin (c+d x))^2}{(e \cos (c+d x))^{7/2}} \, dx=\int { \frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{2}}{\left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate((a+b*sin(d*x+c))^2/(e*cos(d*x+c))^(7/2),x, algorithm="maxima")
Output:
integrate((b*sin(d*x + c) + a)^2/(e*cos(d*x + c))^(7/2), x)
\[ \int \frac {(a+b \sin (c+d x))^2}{(e \cos (c+d x))^{7/2}} \, dx=\int { \frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{2}}{\left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate((a+b*sin(d*x+c))^2/(e*cos(d*x+c))^(7/2),x, algorithm="giac")
Output:
integrate((b*sin(d*x + c) + a)^2/(e*cos(d*x + c))^(7/2), x)
Timed out. \[ \int \frac {(a+b \sin (c+d x))^2}{(e \cos (c+d x))^{7/2}} \, dx=\int \frac {{\left (a+b\,\sin \left (c+d\,x\right )\right )}^2}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{7/2}} \,d x \] Input:
int((a + b*sin(c + d*x))^2/(e*cos(c + d*x))^(7/2),x)
Output:
int((a + b*sin(c + d*x))^2/(e*cos(c + d*x))^(7/2), x)
\[ \int \frac {(a+b \sin (c+d x))^2}{(e \cos (c+d x))^{7/2}} \, dx=\frac {\sqrt {e}\, \left (5 \cos \left (d x +c \right )^{3} \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{4}}d x \right ) a^{2} d +5 \cos \left (d x +c \right )^{3} \left (\int \frac {\sqrt {\cos \left (d x +c \right )}\, \sin \left (d x +c \right )^{2}}{\cos \left (d x +c \right )^{4}}d x \right ) b^{2} d +4 \sqrt {\cos \left (d x +c \right )}\, a b \right )}{5 \cos \left (d x +c \right )^{3} d \,e^{4}} \] Input:
int((a+b*sin(d*x+c))^2/(e*cos(d*x+c))^(7/2),x)
Output:
(sqrt(e)*(5*cos(c + d*x)**3*int(sqrt(cos(c + d*x))/cos(c + d*x)**4,x)*a**2 *d + 5*cos(c + d*x)**3*int((sqrt(cos(c + d*x))*sin(c + d*x)**2)/cos(c + d* x)**4,x)*b**2*d + 4*sqrt(cos(c + d*x))*a*b))/(5*cos(c + d*x)**3*d*e**4)