3.3.50 \(\int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^2} \, dx\) [250]

Optimal. Leaf size=150 \[ -\frac {2 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 a^2 d e^2 \sqrt {\cos (c+d x)}}+\frac {2 \sin (c+d x)}{3 a^2 d e \sqrt {e \cos (c+d x)}}-\frac {2}{9 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^2}-\frac {2}{9 d e \sqrt {e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )} \]

[Out]

2/3*sin(d*x+c)/a^2/d/e/(e*cos(d*x+c))^(1/2)-2/9/d/e/(a+a*sin(d*x+c))^2/(e*cos(d*x+c))^(1/2)-2/9/d/e/(a^2+a^2*s
in(d*x+c))/(e*cos(d*x+c))^(1/2)-2/3*(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^2/cos(d*x+c)^(1/2)

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Rubi [A]
time = 0.12, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 5, 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 {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{3 a^2 d e^2 \sqrt {\cos (c+d x)}}+\frac {2 \sin (c+d x)}{3 a^2 d e \sqrt {e \cos (c+d x)}}-\frac {2}{9 d e \left (a^2 \sin (c+d x)+a^2\right ) \sqrt {e \cos (c+d x)}}-\frac {2}{9 d e (a \sin (c+d x)+a)^2 \sqrt {e \cos (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^2),x]

[Out]

(-2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(3*a^2*d*e^2*Sqrt[Cos[c + d*x]]) + (2*Sin[c + d*x])/(3*a^2
*d*e*Sqrt[e*Cos[c + d*x]]) - 2/(9*d*e*Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^2) - 2/(9*d*e*Sqrt[e*Cos[c + d
*x]]*(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))^{3/2} (a+a \sin (c+d x))^2} \, dx &=-\frac {2}{9 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^2}+\frac {5 \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))} \, dx}{9 a}\\ &=-\frac {2}{9 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^2}-\frac {2}{9 d e \sqrt {e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )}+\frac {\int \frac {1}{(e \cos (c+d x))^{3/2}} \, dx}{3 a^2}\\ &=\frac {2 \sin (c+d x)}{3 a^2 d e \sqrt {e \cos (c+d x)}}-\frac {2}{9 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^2}-\frac {2}{9 d e \sqrt {e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )}-\frac {\int \sqrt {e \cos (c+d x)} \, dx}{3 a^2 e^2}\\ &=\frac {2 \sin (c+d x)}{3 a^2 d e \sqrt {e \cos (c+d x)}}-\frac {2}{9 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^2}-\frac {2}{9 d e \sqrt {e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )}-\frac {\sqrt {e \cos (c+d x)} \int \sqrt {\cos (c+d x)} \, dx}{3 a^2 e^2 \sqrt {\cos (c+d x)}}\\ &=-\frac {2 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 a^2 d e^2 \sqrt {\cos (c+d x)}}+\frac {2 \sin (c+d x)}{3 a^2 d e \sqrt {e \cos (c+d x)}}-\frac {2}{9 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^2}-\frac {2}{9 d e \sqrt {e \cos (c+d x)} \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 = 0.06, size = 66, normalized size = 0.44 \begin {gather*} \frac {\, _2F_1\left (-\frac {1}{4},\frac {13}{4};\frac {3}{4};\frac {1}{2} (1-\sin (c+d x))\right ) \sqrt [4]{1+\sin (c+d x)}}{2 \sqrt [4]{2} a^2 d e \sqrt {e \cos (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^2),x]

[Out]

(Hypergeometric2F1[-1/4, 13/4, 3/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(1/4))/(2*2^(1/4)*a^2*d*e*Sqrt[e*
Cos[c + d*x]])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(487\) vs. \(2(158)=316\).
time = 8.80, size = 488, normalized size = 3.25

method	result	size
default	\(-\frac {2 \left (48 \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 )-96 \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-96 \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 )+192 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+72 \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 )-152 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-24 \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 )+56 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \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 \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 )-12 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\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^{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 d}\)	\(488\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*cos(d*x+c))^(3/2)/(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

-2/9/(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^2/si
n(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)/e*(48*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)^8-96*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2
*c)-96*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+192*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8+72*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)^4-152*sin(1/2*d*x+1/2*c)^6*cos(1/2*d
*x+1/2*c)-24*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+56*sin(1/2*d*x+1/2*c)^4*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))-12*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+2*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))^(3/2)/(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

e^(-3/2)*integrate(1/((a*sin(d*x + c) + a)^2*cos(d*x + c)^(3/2)), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.11, size = 226, normalized size = 1.51 \begin {gather*} \frac {3 \, {\left (-i \, \sqrt {2} \cos \left (d x + c\right )^{3} + 2 i \, \sqrt {2} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 2 i \, \sqrt {2} \cos \left (d x + c\right )\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 ) + 3 \, {\left (i \, \sqrt {2} \cos \left (d x + c\right )^{3} - 2 i \, \sqrt {2} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 2 i \, \sqrt {2} \cos \left (d x + c\right )\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 (6 \, \cos \left (d x + c\right )^{2} + {\left (3 \, \cos \left (d x + c\right )^{2} - 5\right )} \sin \left (d x + c\right ) - 4\right )} \sqrt {\cos \left (d x + c\right )}}{9 \, {\left (a^{2} d \cos \left (d x + c\right )^{3} e^{\frac {3}{2}} - 2 \, a^{2} d \cos \left (d x + c\right ) e^{\frac {3}{2}} \sin \left (d x + c\right ) - 2 \, a^{2} d \cos \left (d x + c\right ) e^{\frac {3}{2}}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*cos(d*x+c))^(3/2)/(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

1/9*(3*(-I*sqrt(2)*cos(d*x + c)^3 + 2*I*sqrt(2)*cos(d*x + c)*sin(d*x + c) + 2*I*sqrt(2)*cos(d*x + c))*weierstr
assZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 3*(I*sqrt(2)*cos(d*x + c)^3 - 2*I*
sqrt(2)*cos(d*x + c)*sin(d*x + c) - 2*I*sqrt(2)*cos(d*x + c))*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0
, cos(d*x + c) - I*sin(d*x + c))) + 2*(6*cos(d*x + c)^2 + (3*cos(d*x + c)^2 - 5)*sin(d*x + c) - 4)*sqrt(cos(d*
x + c)))/(a^2*d*cos(d*x + c)^3*e^(3/2) - 2*a^2*d*cos(d*x + c)*e^(3/2)*sin(d*x + c) - 2*a^2*d*cos(d*x + c)*e^(3
/2))

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*cos(d*x+c))**(3/2)/(a+a*sin(d*x+c))**2,x)

[Out]

Timed out

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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))^(3/2)/(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

integrate(e^(-3/2)/((a*sin(d*x + c) + a)^2*cos(d*x + c)^(3/2)), 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 )}^{3/2}\,{\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))^(3/2)*(a + a*sin(c + d*x))^2),x)

[Out]

int(1/((e*cos(c + d*x))^(3/2)*(a + a*sin(c + d*x))^2), x)

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