\(\int \frac {\sin ^2(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx\) [202]

Optimal result

Integrand size = 21, antiderivative size = 100 \[ \int \frac {\sin ^2(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx=\frac {4 \sqrt {d \cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{5 b d^4 \sqrt {\cos (a+b x)}}+\frac {2 \sin (a+b x)}{5 b d (d \cos (a+b x))^{5/2}}-\frac {4 \sin (a+b x)}{5 b d^3 \sqrt {d \cos (a+b x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2646, 2716, 2721, 2719} \[ \int \frac {\sin ^2(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx=\frac {4 E\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {d \cos (a+b x)}}{5 b d^4 \sqrt {\cos (a+b x)}}-\frac {4 \sin (a+b x)}{5 b d^3 \sqrt {d \cos (a+b x)}}+\frac {2 \sin (a+b x)}{5 b d (d \cos (a+b x))^{5/2}} \]

[In]

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

[Out]

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

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 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]

Rubi steps \begin{align*} \text {integral}& = \frac {2 \sin (a+b x)}{5 b d (d \cos (a+b x))^{5/2}}-\frac {2 \int \frac {1}{(d \cos (a+b x))^{3/2}} \, dx}{5 d^2} \\ & = \frac {2 \sin (a+b x)}{5 b d (d \cos (a+b x))^{5/2}}-\frac {4 \sin (a+b x)}{5 b d^3 \sqrt {d \cos (a+b x)}}+\frac {2 \int \sqrt {d \cos (a+b x)} \, dx}{5 d^4} \\ & = \frac {2 \sin (a+b x)}{5 b d (d \cos (a+b x))^{5/2}}-\frac {4 \sin (a+b x)}{5 b d^3 \sqrt {d \cos (a+b x)}}+\frac {\left (2 \sqrt {d \cos (a+b x)}\right ) \int \sqrt {\cos (a+b x)} \, dx}{5 d^4 \sqrt {\cos (a+b x)}} \\ & = \frac {4 \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 {2 \sin (a+b x)}{5 b d (d \cos (a+b x))^{5/2}}-\frac {4 \sin (a+b x)}{5 b d^3 \sqrt {d \cos (a+b x)}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.05 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.59 \[ \int \frac {\sin ^2(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx=\frac {\sqrt [4]{\cos ^2(a+b x)} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {9}{4},\frac {5}{2},\sin ^2(a+b x)\right ) \sin ^3(2 (a+b x))}{24 b (d \cos (a+b x))^{7/2}} \]

[In]

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

[Out]

((Cos[a + b*x]^2)^(1/4)*Hypergeometric2F1[3/2, 9/4, 5/2, Sin[a + b*x]^2]*Sin[2*(a + b*x)]^3)/(24*b*(d*Cos[a +
b*x])^(7/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(364\) vs. \(2(112)=224\).

Time = 1.03 (sec) , antiderivative size = 365, normalized size of antiderivative = 3.65

method	result	size
default	\(\frac {4 \sqrt {d \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \left (8 \left (\sin ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )-4 \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, E\left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-8 \cos \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+4 \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, E\left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )-\sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, E\left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )\right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) d +d \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}}{5 d^{4} \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{3} \left (8 \left (\sin ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-12 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+6 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \sqrt {d \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right )}\, b}\)	\(365\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.10 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.20 \[ \int \frac {\sin ^2(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx=-\frac {2 \, {\left (-i \, \sqrt {2} \sqrt {d} \cos \left (b x + a\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) + i \, \sqrt {2} \sqrt {d} \cos \left (b x + a\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right ) + \sqrt {d \cos \left (b x + a\right )} {\left (2 \, \cos \left (b x + a\right )^{2} - 1\right )} \sin \left (b x + a\right )\right )}}{5 \, b d^{4} \cos \left (b x + a\right )^{3}} \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {\sin ^2(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {\sin ^2(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx=\int { \frac {\sin \left (b x + a\right )^{2}}{\left (d \cos \left (b x + a\right )\right )^{\frac {7}{2}}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\sin ^2(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx=\int { \frac {\sin \left (b x + a\right )^{2}}{\left (d \cos \left (b x + a\right )\right )^{\frac {7}{2}}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sin ^2(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx=\int \frac {{\sin \left (a+b\,x\right )}^2}{{\left (d\,\cos \left (a+b\,x\right )\right )}^{7/2}} \,d x \]

[In]

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

[Out]

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