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

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

[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]
time = 0.09, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2762, 2716, 2721, 2719} \begin {gather*} -\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}} \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))} \, dx &=-\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] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 0.08, size = 66, normalized size = 0.46 \begin {gather*} \frac {\, _2F_1\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}} \end {gather*}

Antiderivative was successfully verified.

[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] Leaf count of result is larger than twice the leaf count of optimal. \(487\) vs. \(2(151)=302\).
time = 8.60, size = 488, normalized size = 3.41

method	result	size
default	\(-\frac {2 \left (336 \EllipticE \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 )-672 \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-672 \EllipticE \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 )+1344 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+504 \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 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1064 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-168 \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 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+392 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+21 \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}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-66 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+5 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{45 \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}\)	\(488\)

Verification of antiderivative is not currently implemented for this CAS.

[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*(336*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)^8-672*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+
1/2*c)-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+1344*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8+504*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)^4-1064*sin(1/2*d*x+1/2*c)^6*co
s(1/2*d*x+1/2*c)-168*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+392*sin(1/2*d*x+1/2*c)^4*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))-66*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c
)+5*sin(1/2*d*x+1/2*c))/d

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.11, size = 202, normalized size = 1.41 \begin {gather*} -\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 )} {\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 )} {\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 {\cos \left (d x + c\right )}}{45 \, {\left (a d \cos \left (d x + c\right )^{3} e^{\frac {7}{2}} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{3} e^{\frac {7}{2}}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)*weierstrassZeta(-4, 0, weierstras
sPInverse(-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)*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(cos(d*x + c)))/(a*d*cos(d*x + c)^
3*e^(7/2)*sin(d*x + c) + a*d*cos(d*x + c)^3*e^(7/2))

________________________________________________________________________________________

Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 5990 deep

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________