3.3.52 \(\int \frac {1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^2} \, dx\) [252]

Optimal. Leaf size=181 \[ -\frac {42 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{65 a^2 d e^4 \sqrt {\cos (c+d x)}}+\frac {14 \sin (c+d x)}{65 a^2 d e (e \cos (c+d x))^{5/2}}+\frac {42 \sin (c+d x)}{65 a^2 d e^3 \sqrt {e \cos (c+d x)}}-\frac {2}{13 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^2}-\frac {2}{13 d e (e \cos (c+d x))^{5/2} \left (a^2+a^2 \sin (c+d x)\right )} \]

[Out]

14/65*sin(d*x+c)/a^2/d/e/(e*cos(d*x+c))^(5/2)-2/13/d/e/(e*cos(d*x+c))^(5/2)/(a+a*sin(d*x+c))^2-2/13/d/e/(e*cos
(d*x+c))^(5/2)/(a^2+a^2*sin(d*x+c))+42/65*sin(d*x+c)/a^2/d/e^3/(e*cos(d*x+c))^(1/2)-42/65*(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^2/d/e^4/cos(d*x+c)^(1
/2)

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Rubi [A]
time = 0.13, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2760, 2762, 2716, 2721, 2719} \begin {gather*} -\frac {42 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{65 a^2 d e^4 \sqrt {\cos (c+d x)}}+\frac {42 \sin (c+d x)}{65 a^2 d e^3 \sqrt {e \cos (c+d x)}}+\frac {14 \sin (c+d x)}{65 a^2 d e (e \cos (c+d x))^{5/2}}-\frac {2}{13 d e \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{5/2}}-\frac {2}{13 d e (a \sin (c+d x)+a)^2 (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])^2),x]

[Out]

(-42*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(65*a^2*d*e^4*Sqrt[Cos[c + d*x]]) + (14*Sin[c + d*x])/(65
*a^2*d*e*(e*Cos[c + d*x])^(5/2)) + (42*Sin[c + d*x])/(65*a^2*d*e^3*Sqrt[e*Cos[c + d*x]]) - 2/(13*d*e*(e*Cos[c
+ d*x])^(5/2)*(a + a*Sin[c + d*x])^2) - 2/(13*d*e*(e*Cos[c + d*x])^(5/2)*(a^2 + a^2*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 2760

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[b*(g*C
os[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(a*f*g*(2*m + p + 1))), x] + Dist[(m + p + 1)/(a*(2*m + p + 1)),
Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^
2, 0] && LtQ[m, -1] && NeQ[2*m + p + 1, 0] && IntegersQ[2*m, 2*p]

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))^2} \, dx &=-\frac {2}{13 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^2}+\frac {9 \int \frac {1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))} \, dx}{13 a}\\ &=-\frac {2}{13 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^2}-\frac {2}{13 d e (e \cos (c+d x))^{5/2} \left (a^2+a^2 \sin (c+d x)\right )}+\frac {7 \int \frac {1}{(e \cos (c+d x))^{7/2}} \, dx}{13 a^2}\\ &=\frac {14 \sin (c+d x)}{65 a^2 d e (e \cos (c+d x))^{5/2}}-\frac {2}{13 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^2}-\frac {2}{13 d e (e \cos (c+d x))^{5/2} \left (a^2+a^2 \sin (c+d x)\right )}+\frac {21 \int \frac {1}{(e \cos (c+d x))^{3/2}} \, dx}{65 a^2 e^2}\\ &=\frac {14 \sin (c+d x)}{65 a^2 d e (e \cos (c+d x))^{5/2}}+\frac {42 \sin (c+d x)}{65 a^2 d e^3 \sqrt {e \cos (c+d x)}}-\frac {2}{13 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^2}-\frac {2}{13 d e (e \cos (c+d x))^{5/2} \left (a^2+a^2 \sin (c+d x)\right )}-\frac {21 \int \sqrt {e \cos (c+d x)} \, dx}{65 a^2 e^4}\\ &=\frac {14 \sin (c+d x)}{65 a^2 d e (e \cos (c+d x))^{5/2}}+\frac {42 \sin (c+d x)}{65 a^2 d e^3 \sqrt {e \cos (c+d x)}}-\frac {2}{13 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^2}-\frac {2}{13 d e (e \cos (c+d x))^{5/2} \left (a^2+a^2 \sin (c+d x)\right )}-\frac {\left (21 \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{65 a^2 e^4 \sqrt {\cos (c+d x)}}\\ &=-\frac {42 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{65 a^2 d e^4 \sqrt {\cos (c+d x)}}+\frac {14 \sin (c+d x)}{65 a^2 d e (e \cos (c+d x))^{5/2}}+\frac {42 \sin (c+d x)}{65 a^2 d e^3 \sqrt {e \cos (c+d x)}}-\frac {2}{13 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^2}-\frac {2}{13 d e (e \cos (c+d x))^{5/2} \left (a^2+a^2 \sin (c+d x)\right )}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 0.11, size = 66, normalized size = 0.36 \begin {gather*} \frac {\, _2F_1\left (-\frac {5}{4},\frac {17}{4};-\frac {1}{4};\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{5/4}}{20 \sqrt [4]{2} a^2 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])^2),x]

[Out]

(Hypergeometric2F1[-5/4, 17/4, -1/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(5/4))/(20*2^(1/4)*a^2*d*e*(e*Co
s[c + d*x])^(5/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(669\) vs. \(2(185)=370\).
time = 12.48, size = 670, normalized size = 3.70

method	result	size
default	\(-\frac {2 \left (1344 \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 ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2688 \left (\sin ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-4032 \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 ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8064 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+5040 \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 )-10304 \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-3360 \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 )+7168 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1260 \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 )-2896 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-252 \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 )+656 \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 )-86 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+10 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{65 \left (64 \left (\sin ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-192 \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+240 \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-160 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+60 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-12 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1\right ) a^{2} \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}\)	\(670\)

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))^2,x,method=_RETURNVERBOSE)

[Out]

-2/65/(64*sin(1/2*d*x+1/2*c)^12-192*sin(1/2*d*x+1/2*c)^10+240*sin(1/2*d*x+1/2*c)^8-160*sin(1/2*d*x+1/2*c)^6+60
*sin(1/2*d*x+1/2*c)^4-12*sin(1/2*d*x+1/2*c)^2+1)/a^2/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)/e^
3*(1344*(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)*si
n(1/2*d*x+1/2*c)^12-2688*sin(1/2*d*x+1/2*c)^14*cos(1/2*d*x+1/2*c)-4032*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*Ellipt
icE(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)^10+8064*cos(1/2*d*x+1/2*c)*sin
(1/2*d*x+1/2*c)^12+5040*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)^8-10304*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)-3360*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+7168*cos(1/2
*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8+1260*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*si
n(1/2*d*x+1/2*c)^2-1)^(1/2)*sin(1/2*d*x+1/2*c)^4-2896*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)-252*EllipticE(co
s(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+6
56*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)*El
lipticE(cos(1/2*d*x+1/2*c),2^(1/2))-86*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+10*sin(1/2*d*x+1/2*c))/d

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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))^2,x, algorithm="maxima")

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.13, size = 258, normalized size = 1.43 \begin {gather*} \frac {21 \, {\left (-i \, \sqrt {2} \cos \left (d x + c\right )^{5} + 2 i \, \sqrt {2} \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) + 2 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 )^{5} - 2 i \, \sqrt {2} \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) - 2 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 (42 \, \cos \left (d x + c\right )^{4} - 28 \, \cos \left (d x + c\right )^{2} + {\left (21 \, \cos \left (d x + c\right )^{4} - 35 \, \cos \left (d x + c\right )^{2} - 9\right )} \sin \left (d x + c\right ) - 4\right )} \sqrt {\cos \left (d x + c\right )}}{65 \, {\left (a^{2} d \cos \left (d x + c\right )^{5} e^{\frac {7}{2}} - 2 \, a^{2} d \cos \left (d x + c\right )^{3} e^{\frac {7}{2}} \sin \left (d x + c\right ) - 2 \, a^{2} 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))^2,x, algorithm="fricas")

[Out]

1/65*(21*(-I*sqrt(2)*cos(d*x + c)^5 + 2*I*sqrt(2)*cos(d*x + c)^3*sin(d*x + c) + 2*I*sqrt(2)*cos(d*x + c)^3)*we
ierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 21*(I*sqrt(2)*cos(d*x + c)^5
- 2*I*sqrt(2)*cos(d*x + c)^3*sin(d*x + c) - 2*I*sqrt(2)*cos(d*x + c)^3)*weierstrassZeta(-4, 0, weierstrassPIn
verse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) + 2*(42*cos(d*x + c)^4 - 28*cos(d*x + c)^2 + (21*cos(d*x + c)^4 -
35*cos(d*x + c)^2 - 9)*sin(d*x + c) - 4)*sqrt(cos(d*x + c)))/(a^2*d*cos(d*x + c)^5*e^(7/2) - 2*a^2*d*cos(d*x
+ c)^3*e^(7/2)*sin(d*x + c) - 2*a^2*d*cos(d*x + c)^3*e^(7/2))

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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))**2,x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3063 deep

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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))^2,x, algorithm="giac")

[Out]

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

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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 )}^2} \,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))^2),x)

[Out]

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

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