3.3.19 \(\int \frac {\sin ^4(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx\) [219]

Optimal. Leaf size=102 \[ \frac {24 \sqrt {d \cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{5 b d^4 \sqrt {\cos (a+b x)}}-\frac {12 \sin (a+b x)}{5 b d^3 \sqrt {d \cos (a+b x)}}+\frac {2 \sin ^3(a+b x)}{5 b d (d \cos (a+b x))^{5/2}} \]

[Out]

2/5*sin(b*x+a)^3/b/d/(d*cos(b*x+a))^(5/2)-12/5*sin(b*x+a)/b/d^3/(d*cos(b*x+a))^(1/2)+24/5*(cos(1/2*a+1/2*b*x)^
2)^(1/2)/cos(1/2*a+1/2*b*x)*EllipticE(sin(1/2*a+1/2*b*x),2^(1/2))*(d*cos(b*x+a))^(1/2)/b/d^4/cos(b*x+a)^(1/2)

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Rubi [A]
time = 0.08, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2646, 2721, 2719} \begin {gather*} \frac {24 E\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {d \cos (a+b x)}}{5 b d^4 \sqrt {\cos (a+b x)}}-\frac {12 \sin (a+b x)}{5 b d^3 \sqrt {d \cos (a+b x)}}+\frac {2 \sin ^3(a+b x)}{5 b d (d \cos (a+b x))^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*x]^4/(d*Cos[a + b*x])^(7/2),x]

[Out]

(24*Sqrt[d*Cos[a + b*x]]*EllipticE[(a + b*x)/2, 2])/(5*b*d^4*Sqrt[Cos[a + b*x]]) - (12*Sin[a + b*x])/(5*b*d^3*
Sqrt[d*Cos[a + b*x]]) + (2*Sin[a + b*x]^3)/(5*b*d*(d*Cos[a + b*x])^(5/2))

Rule 2646

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(-a)*(a*Sin[e
+ f*x])^(m - 1)*((b*Cos[e + f*x])^(n + 1)/(b*f*(n + 1))), x] + Dist[a^2*((m - 1)/(b^2*(n + 1))), Int[(a*Sin[e
+ f*x])^(m - 2)*(b*Cos[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f}, x] && GtQ[m, 1] && LtQ[n, -1] && (Int
egersQ[2*m, 2*n] || EqQ[m + n, 0])

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]

Rubi steps

\begin {align*} \int \frac {\sin ^4(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx &=\frac {2 \sin ^3(a+b x)}{5 b d (d \cos (a+b x))^{5/2}}-\frac {6 \int \frac {\sin ^2(a+b x)}{(d \cos (a+b x))^{3/2}} \, dx}{5 d^2}\\ &=-\frac {12 \sin (a+b x)}{5 b d^3 \sqrt {d \cos (a+b x)}}+\frac {2 \sin ^3(a+b x)}{5 b d (d \cos (a+b x))^{5/2}}+\frac {12 \int \sqrt {d \cos (a+b x)} \, dx}{5 d^4}\\ &=-\frac {12 \sin (a+b x)}{5 b d^3 \sqrt {d \cos (a+b x)}}+\frac {2 \sin ^3(a+b x)}{5 b d (d \cos (a+b x))^{5/2}}+\frac {\left (12 \sqrt {d \cos (a+b x)}\right ) \int \sqrt {\cos (a+b x)} \, dx}{5 d^4 \sqrt {\cos (a+b x)}}\\ &=\frac {24 \sqrt {d \cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{5 b d^4 \sqrt {\cos (a+b x)}}-\frac {12 \sin (a+b x)}{5 b d^3 \sqrt {d \cos (a+b x)}}+\frac {2 \sin ^3(a+b x)}{5 b d (d \cos (a+b x))^{5/2}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 0.04, size = 65, normalized size = 0.64 \begin {gather*} \frac {\cos ^3(a+b x) \sqrt [4]{\cos ^2(a+b x)} \, _2F_1\left (\frac {9}{4},\frac {5}{2};\frac {7}{2};\sin ^2(a+b x)\right ) \sin ^5(a+b x)}{5 b (d \cos (a+b x))^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b*x]^4/(d*Cos[a + b*x])^(7/2),x]

[Out]

(Cos[a + b*x]^3*(Cos[a + b*x]^2)^(1/4)*Hypergeometric2F1[9/4, 5/2, 7/2, Sin[a + b*x]^2]*Sin[a + b*x]^5)/(5*b*(
d*Cos[a + b*x])^(7/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(365\) vs. \(2(114)=228\).
time = 0.32, size = 366, normalized size = 3.59

method	result	size
default	\(-\frac {8 \sqrt {d \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \left (12 \EllipticE \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-14 \left (\sin ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )-12 \EllipticE \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+14 \cos \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+3 \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \EllipticE \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )-3 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) d +\left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) d}}{5 d^{4} \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{3} \left (8 \left (\sin ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-12 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+6 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \sqrt {d \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right )}\, b}\)	\(366\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(b*x+a)^4/(d*cos(b*x+a))^(7/2),x,method=_RETURNVERBOSE)

[Out]

-8/5*(d*(2*cos(1/2*b*x+1/2*a)^2-1)*sin(1/2*b*x+1/2*a)^2)^(1/2)/d^4/sin(1/2*b*x+1/2*a)^3/(8*sin(1/2*b*x+1/2*a)^
6-12*sin(1/2*b*x+1/2*a)^4+6*sin(1/2*b*x+1/2*a)^2-1)*(12*EllipticE(cos(1/2*b*x+1/2*a),2^(1/2))*(2*sin(1/2*b*x+1
/2*a)^2-1)^(1/2)*(sin(1/2*b*x+1/2*a)^2)^(1/2)*sin(1/2*b*x+1/2*a)^4-14*sin(1/2*b*x+1/2*a)^6*cos(1/2*b*x+1/2*a)-
12*EllipticE(cos(1/2*b*x+1/2*a),2^(1/2))*(2*sin(1/2*b*x+1/2*a)^2-1)^(1/2)*(sin(1/2*b*x+1/2*a)^2)^(1/2)*sin(1/2
*b*x+1/2*a)^2+14*cos(1/2*b*x+1/2*a)*sin(1/2*b*x+1/2*a)^4+3*(2*sin(1/2*b*x+1/2*a)^2-1)^(1/2)*(sin(1/2*b*x+1/2*a
)^2)^(1/2)*EllipticE(cos(1/2*b*x+1/2*a),2^(1/2))-3*sin(1/2*b*x+1/2*a)^2*cos(1/2*b*x+1/2*a))*(-2*sin(1/2*b*x+1/
2*a)^4*d+sin(1/2*b*x+1/2*a)^2*d)^(1/2)/(d*(2*cos(1/2*b*x+1/2*a)^2-1))^(1/2)/b

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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(sin(b*x+a)^4/(d*cos(b*x+a))^(7/2),x, algorithm="maxima")

[Out]

integrate(sin(b*x + a)^4/(d*cos(b*x + a))^(7/2), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.10, size = 120, normalized size = 1.18 \begin {gather*} -\frac {2 \, {\left (-6 i \, \sqrt {2} \sqrt {d} \cos \left (b x + a\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) + 6 i \, \sqrt {2} \sqrt {d} \cos \left (b x + a\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right ) + \sqrt {d \cos \left (b x + a\right )} {\left (7 \, \cos \left (b x + a\right )^{2} - 1\right )} \sin \left (b x + a\right )\right )}}{5 \, b d^{4} \cos \left (b x + a\right )^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x+a)^4/(d*cos(b*x+a))^(7/2),x, algorithm="fricas")

[Out]

-2/5*(-6*I*sqrt(2)*sqrt(d)*cos(b*x + a)^3*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(b*x + a) + I*s
in(b*x + a))) + 6*I*sqrt(2)*sqrt(d)*cos(b*x + a)^3*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(b*x +
a) - I*sin(b*x + a))) + sqrt(d*cos(b*x + a))*(7*cos(b*x + a)^2 - 1)*sin(b*x + a))/(b*d^4*cos(b*x + a)^3)

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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(sin(b*x+a)**4/(d*cos(b*x+a))**(7/2),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(sin(b*x+a)^4/(d*cos(b*x+a))^(7/2),x, algorithm="giac")

[Out]

integrate(sin(b*x + a)^4/(d*cos(b*x + a))^(7/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\sin \left (a+b\,x\right )}^4}{{\left (d\,\cos \left (a+b\,x\right )\right )}^{7/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a + b*x)^4/(d*cos(a + b*x))^(7/2),x)

[Out]

int(sin(a + b*x)^4/(d*cos(a + b*x))^(7/2), x)

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