Integrand size = 25, antiderivative size = 127 \[ \int \frac {(a+a \sin (c+d x))^3}{(e \cos (c+d x))^{7/2}} \, dx=\frac {6 a^3 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d e^4 \sqrt {\cos (c+d x)}}+\frac {4 a^5 (e \cos (c+d x))^{3/2}}{5 d e^5 (a-a \sin (c+d x))^2}-\frac {6 a^6 (e \cos (c+d x))^{3/2}}{5 d e^5 \left (a^3-a^3 \sin (c+d x)\right )} \] Output:
6/5*a^3*(e*cos(d*x+c))^(1/2)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/d/e^4/c os(d*x+c)^(1/2)+4/5*a^5*(e*cos(d*x+c))^(3/2)/d/e^5/(a-a*sin(d*x+c))^2-6/5* a^6*(e*cos(d*x+c))^(3/2)/d/e^5/(a^3-a^3*sin(d*x+c))
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.05 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.52 \[ \int \frac {(a+a \sin (c+d x))^3}{(e \cos (c+d x))^{7/2}} \, dx=\frac {4\ 2^{3/4} a^3 \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {3}{4},-\frac {1}{4},\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{5/4}}{5 d e (e \cos (c+d x))^{5/2}} \] Input:
Integrate[(a + a*Sin[c + d*x])^3/(e*Cos[c + d*x])^(7/2),x]
Output:
(4*2^(3/4)*a^3*Hypergeometric2F1[-5/4, -3/4, -1/4, (1 - Sin[c + d*x])/2]*( 1 + Sin[c + d*x])^(5/4))/(5*d*e*(e*Cos[c + d*x])^(5/2))
Time = 0.65 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.02, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3042, 3149, 3042, 3159, 3042, 3162, 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.
| \(\displaystyle \int \frac {(a \sin (c+d x)+a)^3}{(e \cos (c+d x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (c+d x)+a)^3}{(e \cos (c+d x))^{7/2}}dx\) |
| \(\Big \downarrow \) 3149 |
| \(\displaystyle \frac {a^6 \int \frac {(e \cos (c+d x))^{5/2}}{(a-a \sin (c+d x))^3}dx}{e^6}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a^6 \int \frac {(e \cos (c+d x))^{5/2}}{(a-a \sin (c+d x))^3}dx}{e^6}\) |
| \(\Big \downarrow \) 3159 |
| \(\displaystyle \frac {a^6 \left (\frac {4 e (e \cos (c+d x))^{3/2}}{5 a d (a-a \sin (c+d x))^2}-\frac {3 e^2 \int \frac {\sqrt {e \cos (c+d x)}}{a-a \sin (c+d x)}dx}{5 a^2}\right )}{e^6}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a^6 \left (\frac {4 e (e \cos (c+d x))^{3/2}}{5 a d (a-a \sin (c+d x))^2}-\frac {3 e^2 \int \frac {\sqrt {e \cos (c+d x)}}{a-a \sin (c+d x)}dx}{5 a^2}\right )}{e^6}\) |
| \(\Big \downarrow \) 3162 |
| \(\displaystyle \frac {a^6 \left (\frac {4 e (e \cos (c+d x))^{3/2}}{5 a d (a-a \sin (c+d x))^2}-\frac {3 e^2 \left (\frac {2 (e \cos (c+d x))^{3/2}}{d e (a-a \sin (c+d x))}-\frac {\int \sqrt {e \cos (c+d x)}dx}{a}\right )}{5 a^2}\right )}{e^6}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a^6 \left (\frac {4 e (e \cos (c+d x))^{3/2}}{5 a d (a-a \sin (c+d x))^2}-\frac {3 e^2 \left (\frac {2 (e \cos (c+d x))^{3/2}}{d e (a-a \sin (c+d x))}-\frac {\int \sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a}\right )}{5 a^2}\right )}{e^6}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {a^6 \left (\frac {4 e (e \cos (c+d x))^{3/2}}{5 a d (a-a \sin (c+d x))^2}-\frac {3 e^2 \left (\frac {2 (e \cos (c+d x))^{3/2}}{d e (a-a \sin (c+d x))}-\frac {\sqrt {e \cos (c+d x)} \int \sqrt {\cos (c+d x)}dx}{a \sqrt {\cos (c+d x)}}\right )}{5 a^2}\right )}{e^6}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a^6 \left (\frac {4 e (e \cos (c+d x))^{3/2}}{5 a d (a-a \sin (c+d x))^2}-\frac {3 e^2 \left (\frac {2 (e \cos (c+d x))^{3/2}}{d e (a-a \sin (c+d x))}-\frac {\sqrt {e \cos (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a \sqrt {\cos (c+d x)}}\right )}{5 a^2}\right )}{e^6}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {a^6 \left (\frac {4 e (e \cos (c+d x))^{3/2}}{5 a d (a-a \sin (c+d x))^2}-\frac {3 e^2 \left (\frac {2 (e \cos (c+d x))^{3/2}}{d e (a-a \sin (c+d x))}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{a d \sqrt {\cos (c+d x)}}\right )}{5 a^2}\right )}{e^6}\) |
Input:
Int[(a + a*Sin[c + d*x])^3/(e*Cos[c + d*x])^(7/2),x]
Output:
(a^6*((4*e*(e*Cos[c + d*x])^(3/2))/(5*a*d*(a - a*Sin[c + d*x])^2) - (3*e^2 *((-2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(a*d*Sqrt[Cos[c + d* x]]) + (2*(e*Cos[c + d*x])^(3/2))/(d*e*(a - a*Sin[c + d*x]))))/(5*a^2)))/e ^6
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 _)])^(m_), x_Symbol] :> Simp[(a/g)^(2*m) Int[(g*Cos[e + f*x])^(2*m + p)/( a - b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2 , 0] && IntegerQ[m] && LtQ[p, -1] && GeQ[2*m + p, 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 _)]), x_Symbol] :> Simp[b*((g*Cos[e + f*x])^(p + 1)/(a*f*g*(p - 1)*(a + b*S in[e + f*x]))), x] + Simp[p/(a*(p - 1)) Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && !GeQ[p, 1] && Intege rQ[2*p]
Leaf count of result is larger than twice the leaf count of optimal. \(329\) vs. \(2(115)=230\).
Time = 2.22 (sec) , antiderivative size = 330, normalized size of antiderivative = 2.60
| method | result | size |
| default | \(-\frac {2 \left (24 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-12 \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-24 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+12 \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+20 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-3 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-20 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{5 \left (4 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-4 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} e +e}\, e^{3} d}\) | \(330\) |
| parts | \(\text {Expression too large to display}\) | \(797\) |
Input:
int((a+a*sin(d*x+c))^3/(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*sin(1/2*d*x+1/2*c)^6*cos(1/2*d *x+1/2*c)-12*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^ (1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*sin(1/2*d*x+1/2*c)^4-24*sin(1/2*d*x +1/2*c)^4*cos(1/2*d*x+1/2*c)+12*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin (1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*sin(1/2*d*x+1/2* c)^2+20*sin(1/2*d*x+1/2*c)^5-2*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-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))-20*sin(1/2*d*x+1/2*c)^3+sin(1/2*d*x+1/2*c))*a^3/d
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 294, normalized size of antiderivative = 2.31 \[ \int \frac {(a+a \sin (c+d x))^3}{(e \cos (c+d x))^{7/2}} \, dx=-\frac {2 \, {\left (3 \, \sqrt {\frac {1}{2}} {\left (-i \, a^{3} \cos \left (d x + c\right )^{2} + i \, a^{3} \cos \left (d x + c\right ) + 2 i \, a^{3} + {\left (-i \, a^{3} \cos \left (d x + c\right ) - 2 i \, a^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 3 \, \sqrt {\frac {1}{2}} {\left (i \, a^{3} \cos \left (d x + c\right )^{2} - i \, a^{3} \cos \left (d x + c\right ) - 2 i \, a^{3} + {\left (i \, a^{3} \cos \left (d x + c\right ) + 2 i \, a^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - {\left (3 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3} \cos \left (d x + c\right ) - 2 \, a^{3} - {\left (3 \, a^{3} \cos \left (d x + c\right ) + 2 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}\right )}}{5 \, {\left (d e^{4} \cos \left (d x + c\right )^{2} - d e^{4} \cos \left (d x + c\right ) - 2 \, d e^{4} + {\left (d e^{4} \cos \left (d x + c\right ) + 2 \, d e^{4}\right )} \sin \left (d x + c\right )\right )}} \] Input:
integrate((a+a*sin(d*x+c))^3/(e*cos(d*x+c))^(7/2),x, algorithm="fricas")
Output:
-2/5*(3*sqrt(1/2)*(-I*a^3*cos(d*x + c)^2 + I*a^3*cos(d*x + c) + 2*I*a^3 + (-I*a^3*cos(d*x + c) - 2*I*a^3)*sin(d*x + c))*sqrt(e)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 3*sqrt(1/2 )*(I*a^3*cos(d*x + c)^2 - I*a^3*cos(d*x + c) - 2*I*a^3 + (I*a^3*cos(d*x + c) + 2*I*a^3)*sin(d*x + c))*sqrt(e)*weierstrassZeta(-4, 0, weierstrassPInv erse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - (3*a^3*cos(d*x + c)^2 + a^3* cos(d*x + c) - 2*a^3 - (3*a^3*cos(d*x + c) + 2*a^3)*sin(d*x + c))*sqrt(e*c os(d*x + c)))/(d*e^4*cos(d*x + c)^2 - d*e^4*cos(d*x + c) - 2*d*e^4 + (d*e^ 4*cos(d*x + c) + 2*d*e^4)*sin(d*x + c))
Timed out. \[ \int \frac {(a+a \sin (c+d x))^3}{(e \cos (c+d x))^{7/2}} \, dx=\text {Timed out} \] Input:
integrate((a+a*sin(d*x+c))**3/(e*cos(d*x+c))**(7/2),x)
Output:
Timed out
\[ \int \frac {(a+a \sin (c+d x))^3}{(e \cos (c+d x))^{7/2}} \, dx=\int { \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{3}}{\left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate((a+a*sin(d*x+c))^3/(e*cos(d*x+c))^(7/2),x, algorithm="maxima")
Output:
integrate((a*sin(d*x + c) + a)^3/(e*cos(d*x + c))^(7/2), x)
\[ \int \frac {(a+a \sin (c+d x))^3}{(e \cos (c+d x))^{7/2}} \, dx=\int { \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{3}}{\left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate((a+a*sin(d*x+c))^3/(e*cos(d*x+c))^(7/2),x, algorithm="giac")
Output:
integrate((a*sin(d*x + c) + a)^3/(e*cos(d*x + c))^(7/2), x)
Timed out. \[ \int \frac {(a+a \sin (c+d x))^3}{(e \cos (c+d x))^{7/2}} \, dx=\int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^3}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{7/2}} \,d x \] Input:
int((a + a*sin(c + d*x))^3/(e*cos(c + d*x))^(7/2),x)
Output:
int((a + a*sin(c + d*x))^3/(e*cos(c + d*x))^(7/2), x)
\[ \int \frac {(a+a \sin (c+d x))^3}{(e \cos (c+d x))^{7/2}} \, dx=\frac {\sqrt {e}\, a^{3} \left (5 \cos \left (d x +c \right )^{3} \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{4}}d x \right ) d +5 \cos \left (d x +c \right )^{3} \left (\int \frac {\sqrt {\cos \left (d x +c \right )}\, \sin \left (d x +c \right )^{3}}{\cos \left (d x +c \right )^{4}}d x \right ) d +15 \cos \left (d x +c \right )^{3} \left (\int \frac {\sqrt {\cos \left (d x +c \right )}\, \sin \left (d x +c \right )^{2}}{\cos \left (d x +c \right )^{4}}d x \right ) d +6 \sqrt {\cos \left (d x +c \right )}\right )}{5 \cos \left (d x +c \right )^{3} d \,e^{4}} \] Input:
int((a+a*sin(d*x+c))^3/(e*cos(d*x+c))^(7/2),x)
Output:
(sqrt(e)*a**3*(5*cos(c + d*x)**3*int(sqrt(cos(c + d*x))/cos(c + d*x)**4,x) *d + 5*cos(c + d*x)**3*int((sqrt(cos(c + d*x))*sin(c + d*x)**3)/cos(c + d* x)**4,x)*d + 15*cos(c + d*x)**3*int((sqrt(cos(c + d*x))*sin(c + d*x)**2)/c os(c + d*x)**4,x)*d + 6*sqrt(cos(c + d*x))))/(5*cos(c + d*x)**3*d*e**4)