3.3.13 \(\int \frac {(a+a \sin (c+d x))^2}{(e \cos (c+d x))^{11/2}} \, dx\) [213]

Optimal. Leaf size=145 \[ -\frac {2 a^2 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d e^6 \sqrt {\cos (c+d x)}}+\frac {2 a^2 \sin (c+d x)}{9 d e^3 (e \cos (c+d x))^{5/2}}+\frac {2 a^2 \sin (c+d x)}{3 d e^5 \sqrt {e \cos (c+d x)}}+\frac {4 \left (a^2+a^2 \sin (c+d x)\right )}{9 d e (e \cos (c+d x))^{9/2}} \]

[Out]

2/9*a^2*sin(d*x+c)/d/e^3/(e*cos(d*x+c))^(5/2)+4/9*(a^2+a^2*sin(d*x+c))/d/e/(e*cos(d*x+c))^(9/2)+2/3*a^2*sin(d*
x+c)/d/e^5/(e*cos(d*x+c))^(1/2)-2/3*a^2*(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)/d/e^6/cos(d*x+c)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.08, antiderivative size = 145, 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 = {2755, 2716, 2721, 2719} \begin {gather*} -\frac {2 a^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{3 d e^6 \sqrt {\cos (c+d x)}}+\frac {2 a^2 \sin (c+d x)}{3 d e^5 \sqrt {e \cos (c+d x)}}+\frac {2 a^2 \sin (c+d x)}{9 d e^3 (e \cos (c+d x))^{5/2}}+\frac {4 \left (a^2 \sin (c+d x)+a^2\right )}{9 d e (e \cos (c+d x))^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[c + d*x])^2/(e*Cos[c + d*x])^(11/2),x]

[Out]

(-2*a^2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(3*d*e^6*Sqrt[Cos[c + d*x]]) + (2*a^2*Sin[c + d*x])/(9
*d*e^3*(e*Cos[c + d*x])^(5/2)) + (2*a^2*Sin[c + d*x])/(3*d*e^5*Sqrt[e*Cos[c + d*x]]) + (4*(a^2 + a^2*Sin[c + d
*x]))/(9*d*e*(e*Cos[c + d*x])^(9/2))

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 2755

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

Rubi steps

\begin {align*} \int \frac {(a+a \sin (c+d x))^2}{(e \cos (c+d x))^{11/2}} \, dx &=\frac {4 \left (a^2+a^2 \sin (c+d x)\right )}{9 d e (e \cos (c+d x))^{9/2}}+\frac {\left (5 a^2\right ) \int \frac {1}{(e \cos (c+d x))^{7/2}} \, dx}{9 e^2}\\ &=\frac {2 a^2 \sin (c+d x)}{9 d e^3 (e \cos (c+d x))^{5/2}}+\frac {4 \left (a^2+a^2 \sin (c+d x)\right )}{9 d e (e \cos (c+d x))^{9/2}}+\frac {a^2 \int \frac {1}{(e \cos (c+d x))^{3/2}} \, dx}{3 e^4}\\ &=\frac {2 a^2 \sin (c+d x)}{9 d e^3 (e \cos (c+d x))^{5/2}}+\frac {2 a^2 \sin (c+d x)}{3 d e^5 \sqrt {e \cos (c+d x)}}+\frac {4 \left (a^2+a^2 \sin (c+d x)\right )}{9 d e (e \cos (c+d x))^{9/2}}-\frac {a^2 \int \sqrt {e \cos (c+d x)} \, dx}{3 e^6}\\ &=\frac {2 a^2 \sin (c+d x)}{9 d e^3 (e \cos (c+d x))^{5/2}}+\frac {2 a^2 \sin (c+d x)}{3 d e^5 \sqrt {e \cos (c+d x)}}+\frac {4 \left (a^2+a^2 \sin (c+d x)\right )}{9 d e (e \cos (c+d x))^{9/2}}-\frac {\left (a^2 \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{3 e^6 \sqrt {\cos (c+d x)}}\\ &=-\frac {2 a^2 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d e^6 \sqrt {\cos (c+d x)}}+\frac {2 a^2 \sin (c+d x)}{9 d e^3 (e \cos (c+d x))^{5/2}}+\frac {2 a^2 \sin (c+d x)}{3 d e^5 \sqrt {e \cos (c+d x)}}+\frac {4 \left (a^2+a^2 \sin (c+d x)\right )}{9 d e (e \cos (c+d x))^{9/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 0.10, size = 66, normalized size = 0.46 \begin {gather*} \frac {2^{3/4} a^2 \, _2F_1\left (-\frac {9}{4},\frac {5}{4};-\frac {5}{4};\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{9/4}}{9 d e (e \cos (c+d x))^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[c + d*x])^2/(e*Cos[c + d*x])^(11/2),x]

[Out]

(2^(3/4)*a^2*Hypergeometric2F1[-9/4, 5/4, -5/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(9/4))/(9*d*e*(e*Cos[
c + d*x])^(9/2))

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(487\) vs. \(2(153)=306\).
time = 6.98, size = 488, normalized size = 3.37

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 ) a^{2}}{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 ) \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^{5} d}\)	\(488\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sin(d*x+c))^2/(e*cos(d*x+c))^(11/2),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)/sin(1/
2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)/e^5*(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))*a^2/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((a+a*sin(d*x+c))^2/(e*cos(d*x+c))^(11/2),x, algorithm="maxima")

[Out]

e^(-11/2)*integrate((a*sin(d*x + c) + a)^2/cos(d*x + c)^(11/2), x)

________________________________________________________________________________________

Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.11, size = 250, normalized size = 1.72 \begin {gather*} -\frac {3 \, {\left (i \, \sqrt {2} a^{2} \cos \left (d x + c\right )^{3} + 2 i \, \sqrt {2} a^{2} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 2 i \, \sqrt {2} a^{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} a^{2} \cos \left (d x + c\right )^{3} - 2 i \, \sqrt {2} a^{2} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 2 i \, \sqrt {2} a^{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 \, a^{2} \cos \left (d x + c\right )^{2} - 4 \, a^{2} - {\left (3 \, a^{2} \cos \left (d x + c\right )^{2} - 5 \, a^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )}}{9 \, {\left (d \cos \left (d x + c\right )^{3} e^{\frac {11}{2}} + 2 \, d \cos \left (d x + c\right ) e^{\frac {11}{2}} \sin \left (d x + c\right ) - 2 \, d \cos \left (d x + c\right ) e^{\frac {11}{2}}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(d*x+c))^2/(e*cos(d*x+c))^(11/2),x, algorithm="fricas")

[Out]

-1/9*(3*(I*sqrt(2)*a^2*cos(d*x + c)^3 + 2*I*sqrt(2)*a^2*cos(d*x + c)*sin(d*x + c) - 2*I*sqrt(2)*a^2*cos(d*x +
c))*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 3*(-I*sqrt(2)*a^2*cos(
d*x + c)^3 - 2*I*sqrt(2)*a^2*cos(d*x + c)*sin(d*x + c) + 2*I*sqrt(2)*a^2*cos(d*x + c))*weierstrassZeta(-4, 0,
weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) + 2*(6*a^2*cos(d*x + c)^2 - 4*a^2 - (3*a^2*cos(d*x
+ c)^2 - 5*a^2)*sin(d*x + c))*sqrt(cos(d*x + c)))/(d*cos(d*x + c)^3*e^(11/2) + 2*d*cos(d*x + c)*e^(11/2)*sin(d
*x + c) - 2*d*cos(d*x + c)*e^(11/2))

________________________________________________________________________________________

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((a+a*sin(d*x+c))**2/(e*cos(d*x+c))**(11/2),x)

[Out]

Timed out

________________________________________________________________________________________

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((a+a*sin(d*x+c))^2/(e*cos(d*x+c))^(11/2),x, algorithm="giac")

[Out]

integrate((a*sin(d*x + c) + a)^2*e^(-11/2)/cos(d*x + c)^(11/2), x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^2}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{11/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*sin(c + d*x))^2/(e*cos(c + d*x))^(11/2),x)

[Out]

int((a + a*sin(c + d*x))^2/(e*cos(c + d*x))^(11/2), x)

________________________________________________________________________________________