\(\int \frac {1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))} \, dx\) [242]

Optimal result

Integrand size = 25, antiderivative size = 143 \[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))} \, dx=-\frac {14 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 a d e^4 \sqrt {\cos (c+d x)}}+\frac {14 \sin (c+d x)}{45 a d e (e \cos (c+d x))^{5/2}}+\frac {14 \sin (c+d x)}{15 a d e^3 \sqrt {e \cos (c+d x)}}-\frac {2}{9 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))} \]

[Out]

14/45*sin(d*x+c)/a/d/e/(e*cos(d*x+c))^(5/2)-2/9/d/e/(e*cos(d*x+c))^(5/2)/(a+a*sin(d*x+c))+14/15*sin(d*x+c)/a/d
/e^3/(e*cos(d*x+c))^(1/2)-14/15*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2
^(1/2))*(e*cos(d*x+c))^(1/2)/a/d/e^4/cos(d*x+c)^(1/2)

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2762, 2716, 2721, 2719} \[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))} \, dx=-\frac {14 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{15 a d e^4 \sqrt {\cos (c+d x)}}+\frac {14 \sin (c+d x)}{15 a d e^3 \sqrt {e \cos (c+d x)}}+\frac {14 \sin (c+d x)}{45 a d e (e \cos (c+d x))^{5/2}}-\frac {2}{9 d e (a \sin (c+d x)+a) (e \cos (c+d x))^{5/2}} \]

[In]

Int[1/((e*Cos[c + d*x])^(7/2)*(a + a*Sin[c + d*x])),x]

[Out]

(-14*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(15*a*d*e^4*Sqrt[Cos[c + d*x]]) + (14*Sin[c + d*x])/(45*a
*d*e*(e*Cos[c + d*x])^(5/2)) + (14*Sin[c + d*x])/(15*a*d*e^3*Sqrt[e*Cos[c + d*x]]) - 2/(9*d*e*(e*Cos[c + d*x])
^(5/2)*(a + a*Sin[c + d*x]))

Rule 2716

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] + Dist[(n + 2)/(b^2*(n + 1)), Int[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1
] && IntegerQ[2*n]

Rule 2719

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]

Rule 2721

Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[(b*Sin[c + d*x])^n/Sin[c + d*x]^n, Int[Sin[c + d*x]
^n, x], x] /; FreeQ[{b, c, d}, x] && LtQ[-1, n, 1] && IntegerQ[2*n]

Rule 2762

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*Sin[e + f*x]))), x] + Dist[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] && IntegerQ[2*p]

Rubi steps \begin{align*} \text {integral}& = -\frac {2}{9 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))}+\frac {7 \int \frac {1}{(e \cos (c+d x))^{7/2}} \, dx}{9 a} \\ & = \frac {14 \sin (c+d x)}{45 a d e (e \cos (c+d x))^{5/2}}-\frac {2}{9 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))}+\frac {7 \int \frac {1}{(e \cos (c+d x))^{3/2}} \, dx}{15 a e^2} \\ & = \frac {14 \sin (c+d x)}{45 a d e (e \cos (c+d x))^{5/2}}+\frac {14 \sin (c+d x)}{15 a d e^3 \sqrt {e \cos (c+d x)}}-\frac {2}{9 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))}-\frac {7 \int \sqrt {e \cos (c+d x)} \, dx}{15 a e^4} \\ & = \frac {14 \sin (c+d x)}{45 a d e (e \cos (c+d x))^{5/2}}+\frac {14 \sin (c+d x)}{15 a d e^3 \sqrt {e \cos (c+d x)}}-\frac {2}{9 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))}-\frac {\left (7 \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{15 a e^4 \sqrt {\cos (c+d x)}} \\ & = -\frac {14 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 a d e^4 \sqrt {\cos (c+d x)}}+\frac {14 \sin (c+d x)}{45 a d e (e \cos (c+d x))^{5/2}}+\frac {14 \sin (c+d x)}{15 a d e^3 \sqrt {e \cos (c+d x)}}-\frac {2}{9 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))} \\ \end{align*}

Mathematica [C] (verified)

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 = 0.46 \[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))} \, dx=\frac {\operatorname {Hypergeometric2F1}\left (-\frac {5}{4},\frac {13}{4},-\frac {1}{4},\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{5/4}}{10 \sqrt [4]{2} a d e (e \cos (c+d x))^{5/2}} \]

[In]

Integrate[1/((e*Cos[c + d*x])^(7/2)*(a + a*Sin[c + d*x])),x]

[Out]

(Hypergeometric2F1[-5/4, 13/4, -1/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(5/4))/(10*2^(1/4)*a*d*e*(e*Cos[
c + d*x])^(5/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(487\) vs. \(2(151)=302\).

Time = 6.22 (sec) , antiderivative size = 488, normalized size of antiderivative = 3.41

method	result	size
default	\(\frac {\frac {448 \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )}{15}-\frac {224 E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15}-\frac {896 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15}+\frac {448 E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15}+\frac {2128 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )}{45}-\frac {112 \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\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}}\, \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {784 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )}{45}+\frac {112 \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\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}}\, \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15}+\frac {44 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )}{15}-\frac {14 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{15}-\frac {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )}{9}}{\left (16 \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-32 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+24 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1\right ) a \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, e^{3} d}\)	\(488\)

[In]

int(1/(e*cos(d*x+c))^(7/2)/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

2/45/(16*sin(1/2*d*x+1/2*c)^8-32*sin(1/2*d*x+1/2*c)^6+24*sin(1/2*d*x+1/2*c)^4-8*sin(1/2*d*x+1/2*c)^2+1)/a/sin(
1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)/e^3*(672*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)-336*Ellip
ticE(cos(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)^(1/2)*sin(1/2*d*x+1/2
*c)^8-1344*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8+672*EllipticE(cos(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)^(1/2)*sin(1/2*d*x+1/2*c)^6+1064*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)
-504*(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)^(1/2)*sin(1
/2*d*x+1/2*c)^4-392*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+168*(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)^(1/2)*sin(1/2*d*x+1/2*c)^2+66*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x
+1/2*c)-21*(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))
-5*sin(1/2*d*x+1/2*c))/d

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.11 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.48 \[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))} \, dx=-\frac {21 \, {\left (i \, \sqrt {2} \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) + i \, \sqrt {2} \cos \left (d x + c\right )^{3}\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, {\left (-i \, \sqrt {2} \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) - i \, \sqrt {2} \cos \left (d x + c\right )^{3}\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, {\left (21 \, \cos \left (d x + c\right )^{4} - 14 \, \cos \left (d x + c\right )^{2} - 7 \, {\left (3 \, \cos \left (d x + c\right )^{2} + 1\right )} \sin \left (d x + c\right ) - 2\right )} \sqrt {e \cos \left (d x + c\right )}}{45 \, {\left (a d e^{4} \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) + a d e^{4} \cos \left (d x + c\right )^{3}\right )}} \]

[In]

integrate(1/(e*cos(d*x+c))^(7/2)/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/45*(21*(I*sqrt(2)*cos(d*x + c)^3*sin(d*x + c) + I*sqrt(2)*cos(d*x + c)^3)*sqrt(e)*weierstrassZeta(-4, 0, we
ierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 21*(-I*sqrt(2)*cos(d*x + c)^3*sin(d*x + c) - I*sqrt
(2)*cos(d*x + c)^3)*sqrt(e)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)))
+ 2*(21*cos(d*x + c)^4 - 14*cos(d*x + c)^2 - 7*(3*cos(d*x + c)^2 + 1)*sin(d*x + c) - 2)*sqrt(e*cos(d*x + c)))/
(a*d*e^4*cos(d*x + c)^3*sin(d*x + c) + a*d*e^4*cos(d*x + c)^3)

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))} \, dx=\text {Timed out} \]

[In]

integrate(1/(e*cos(d*x+c))**(7/2)/(a+a*sin(d*x+c)),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))} \, dx=\int { \frac {1}{\left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}} {\left (a \sin \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

integrate(1/(e*cos(d*x+c))^(7/2)/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

integrate(1/((e*cos(d*x + c))^(7/2)*(a*sin(d*x + c) + a)), x)

Giac [F]

\[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))} \, dx=\int { \frac {1}{\left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}} {\left (a \sin \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

integrate(1/(e*cos(d*x+c))^(7/2)/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

integrate(1/((e*cos(d*x + c))^(7/2)*(a*sin(d*x + c) + a)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))} \, dx=\int \frac {1}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{7/2}\,\left (a+a\,\sin \left (c+d\,x\right )\right )} \,d x \]

[In]

int(1/((e*cos(c + d*x))^(7/2)*(a + a*sin(c + d*x))),x)

[Out]

int(1/((e*cos(c + d*x))^(7/2)*(a + a*sin(c + d*x))), x)