3.6.63 \(\int \frac {(a+b \sin (c+d x))^3}{(e \cos (c+d x))^{9/2}} \, dx\) [563]

Optimal. Leaf size=188 \[ \frac {2 b \left (5 a^2-4 b^2\right ) \sqrt {e \cos (c+d x)}}{21 d e^5}+\frac {2 a \left (5 a^2-6 b^2\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d e^4 \sqrt {e \cos (c+d x)}}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{7 d e (e \cos (c+d x))^{7/2}}+\frac {2 (a+b \sin (c+d x)) \left (a b+\left (5 a^2-4 b^2\right ) \sin (c+d x)\right )}{21 d e^3 (e \cos (c+d x))^{3/2}} \]

[Out]

2/7*(b+a*sin(d*x+c))*(a+b*sin(d*x+c))^2/d/e/(e*cos(d*x+c))^(7/2)+2/21*(a+b*sin(d*x+c))*(a*b+(5*a^2-4*b^2)*sin(
d*x+c))/d/e^3/(e*cos(d*x+c))^(3/2)+2/21*a*(5*a^2-6*b^2)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*Ellipt
icF(sin(1/2*d*x+1/2*c),2^(1/2))*cos(d*x+c)^(1/2)/d/e^4/(e*cos(d*x+c))^(1/2)+2/21*b*(5*a^2-4*b^2)*(e*cos(d*x+c)
)^(1/2)/d/e^5

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Rubi [A]
time = 0.17, antiderivative size = 188, 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 = {2770, 2940, 2748, 2721, 2720} \begin {gather*} \frac {2 b \left (5 a^2-4 b^2\right ) \sqrt {e \cos (c+d x)}}{21 d e^5}+\frac {2 a \left (5 a^2-6 b^2\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d e^4 \sqrt {e \cos (c+d x)}}+\frac {2 \left (\left (5 a^2-4 b^2\right ) \sin (c+d x)+a b\right ) (a+b \sin (c+d x))}{21 d e^3 (e \cos (c+d x))^{3/2}}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{7 d e (e \cos (c+d x))^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Sin[c + d*x])^3/(e*Cos[c + d*x])^(9/2),x]

[Out]

(2*b*(5*a^2 - 4*b^2)*Sqrt[e*Cos[c + d*x]])/(21*d*e^5) + (2*a*(5*a^2 - 6*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c +
d*x)/2, 2])/(21*d*e^4*Sqrt[e*Cos[c + d*x]]) + (2*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^2)/(7*d*e*(e*Cos[c
+ d*x])^(7/2)) + (2*(a + b*Sin[c + d*x])*(a*b + (5*a^2 - 4*b^2)*Sin[c + d*x]))/(21*d*e^3*(e*Cos[c + d*x])^(3/
2))

Rule 2720

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(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 2748

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-b)*((g*Co
s[e + f*x])^(p + 1)/(f*g*(p + 1))), x] + Dist[a, Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x
] && (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])

Rule 2770

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

Rule 2940

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

Rubi steps

\begin {align*} \int \frac {(a+b \sin (c+d x))^3}{(e \cos (c+d x))^{9/2}} \, dx &=\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{7 d e (e \cos (c+d x))^{7/2}}-\frac {2 \int \frac {(a+b \sin (c+d x)) \left (-\frac {5 a^2}{2}+2 b^2-\frac {1}{2} a b \sin (c+d x)\right )}{(e \cos (c+d x))^{5/2}} \, dx}{7 e^2}\\ &=\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{7 d e (e \cos (c+d x))^{7/2}}+\frac {2 (a+b \sin (c+d x)) \left (a b+\left (5 a^2-4 b^2\right ) \sin (c+d x)\right )}{21 d e^3 (e \cos (c+d x))^{3/2}}+\frac {4 \int \frac {\frac {1}{4} a \left (5 a^2-6 b^2\right )-\frac {1}{4} b \left (5 a^2-4 b^2\right ) \sin (c+d x)}{\sqrt {e \cos (c+d x)}} \, dx}{21 e^4}\\ &=\frac {2 b \left (5 a^2-4 b^2\right ) \sqrt {e \cos (c+d x)}}{21 d e^5}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{7 d e (e \cos (c+d x))^{7/2}}+\frac {2 (a+b \sin (c+d x)) \left (a b+\left (5 a^2-4 b^2\right ) \sin (c+d x)\right )}{21 d e^3 (e \cos (c+d x))^{3/2}}+\frac {\left (a \left (5 a^2-6 b^2\right )\right ) \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx}{21 e^4}\\ &=\frac {2 b \left (5 a^2-4 b^2\right ) \sqrt {e \cos (c+d x)}}{21 d e^5}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{7 d e (e \cos (c+d x))^{7/2}}+\frac {2 (a+b \sin (c+d x)) \left (a b+\left (5 a^2-4 b^2\right ) \sin (c+d x)\right )}{21 d e^3 (e \cos (c+d x))^{3/2}}+\frac {\left (a \left (5 a^2-6 b^2\right ) \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{21 e^4 \sqrt {e \cos (c+d x)}}\\ &=\frac {2 b \left (5 a^2-4 b^2\right ) \sqrt {e \cos (c+d x)}}{21 d e^5}+\frac {2 a \left (5 a^2-6 b^2\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d e^4 \sqrt {e \cos (c+d x)}}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{7 d e (e \cos (c+d x))^{7/2}}+\frac {2 (a+b \sin (c+d x)) \left (a b+\left (5 a^2-4 b^2\right ) \sin (c+d x)\right )}{21 d e^3 (e \cos (c+d x))^{3/2}}\\ \end {align*}

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Mathematica [A]
time = 0.67, size = 140, normalized size = 0.74 \begin {gather*} \frac {\sqrt {e \cos (c+d x)} \sec ^4(c+d x) \left (36 a^2 b-2 b^3-14 b^3 \cos (2 (c+d x))+4 a \left (5 a^2-6 b^2\right ) \cos ^{\frac {7}{2}}(c+d x) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+17 a^3 \sin (c+d x)+30 a b^2 \sin (c+d x)+5 a^3 \sin (3 (c+d x))-6 a b^2 \sin (3 (c+d x))\right )}{42 d e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sin[c + d*x])^3/(e*Cos[c + d*x])^(9/2),x]

[Out]

(Sqrt[e*Cos[c + d*x]]*Sec[c + d*x]^4*(36*a^2*b - 2*b^3 - 14*b^3*Cos[2*(c + d*x)] + 4*a*(5*a^2 - 6*b^2)*Cos[c +
d*x]^(7/2)*EllipticF[(c + d*x)/2, 2] + 17*a^3*Sin[c + d*x] + 30*a*b^2*Sin[c + d*x] + 5*a^3*Sin[3*(c + d*x)] -
6*a*b^2*Sin[3*(c + d*x)]))/(42*d*e^5)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(749\) vs. \(2(196)=392\).
time = 21.66, size = 750, normalized size = 3.99

method	result	size
default	\(\text {Expression too large to display}\)	\(750\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sin(d*x+c))^3/(e*cos(d*x+c))^(9/2),x,method=_RETURNVERBOSE)

[Out]

-2/21/(8*sin(1/2*d*x+1/2*c)^6-12*sin(1/2*d*x+1/2*c)^4+6*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^4*(40*EllipticF(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)*a^3*sin(1/2*d*x+1/2*c)^6-48*EllipticF(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)*a*b^2*sin(1/2*d*x+1/2*c)^6-60*EllipticF(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)*a^3*sin(1/2*d*x+1/2*c)^4+72*EllipticF(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)*a*b^2*sin(1/2*d*x+1/2*c)^4+40*a
^3*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6-48*a*b^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6+30*(sin(1/2*d*x+1/
2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*a^3*sin(1/2*d*x+1/2*c)^2-
36*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*a*b^2*s
in(1/2*d*x+1/2*c)^2-40*a^3*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^4+48*a*b^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2
*c)^4-28*b^3*sin(1/2*d*x+1/2*c)^5-5*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(co
s(1/2*d*x+1/2*c),2^(1/2))*a^3+6*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/
2*d*x+1/2*c),2^(1/2))*a*b^2+16*a^3*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2+6*a*b^2*cos(1/2*d*x+1/2*c)*sin(1/2*
d*x+1/2*c)^2+28*b^3*sin(1/2*d*x+1/2*c)^3+9*a^2*b*sin(1/2*d*x+1/2*c)-4*b^3*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((a+b*sin(d*x+c))^3/(e*cos(d*x+c))^(9/2),x, algorithm="maxima")

[Out]

e^(-9/2)*integrate((b*sin(d*x + c) + a)^3/cos(d*x + c)^(9/2), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.11, size = 176, normalized size = 0.94 \begin {gather*} \frac {{\left (\sqrt {2} {\left (-5 i \, a^{3} + 6 i \, a b^{2}\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + \sqrt {2} {\left (5 i \, a^{3} - 6 i \, a b^{2}\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 2 \, {\left (7 \, b^{3} \cos \left (d x + c\right )^{2} - 9 \, a^{2} b - 3 \, b^{3} - {\left (3 \, a^{3} + 9 \, a b^{2} + {\left (5 \, a^{3} - 6 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )}\right )} e^{\left (-\frac {9}{2}\right )}}{21 \, d \cos \left (d x + c\right )^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(d*x+c))^3/(e*cos(d*x+c))^(9/2),x, algorithm="fricas")

[Out]

1/21*(sqrt(2)*(-5*I*a^3 + 6*I*a*b^2)*cos(d*x + c)^4*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))
+ sqrt(2)*(5*I*a^3 - 6*I*a*b^2)*cos(d*x + c)^4*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 2*(
7*b^3*cos(d*x + c)^2 - 9*a^2*b - 3*b^3 - (3*a^3 + 9*a*b^2 + (5*a^3 - 6*a*b^2)*cos(d*x + c)^2)*sin(d*x + c))*sq
rt(cos(d*x + c)))*e^(-9/2)/(d*cos(d*x + c)^4)

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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((a+b*sin(d*x+c))**3/(e*cos(d*x+c))**(9/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 4847 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((a+b*sin(d*x+c))^3/(e*cos(d*x+c))^(9/2),x, algorithm="giac")

[Out]

integrate((b*sin(d*x + c) + a)^3*e^(-9/2)/cos(d*x + c)^(9/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,\sin \left (c+d\,x\right )\right )}^3}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{9/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*sin(c + d*x))^3/(e*cos(c + d*x))^(9/2),x)

[Out]

int((a + b*sin(c + d*x))^3/(e*cos(c + d*x))^(9/2), x)

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