3.120 \(\int \frac{\sin ^3(e+f x)}{(a+b \sec ^2(e+f x))^{5/2}} \, dx\)

Optimal. Leaf size=146 \[ -\frac{8 b (a+2 b) \sec (e+f x)}{3 a^4 f \sqrt{a+b \sec ^2(e+f x)}}-\frac{4 b (a+2 b) \sec (e+f x)}{3 a^3 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac{(a+2 b) \cos (e+f x)}{a^2 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}+\frac{\cos ^3(e+f x)}{3 a f \left (a+b \sec ^2(e+f x)\right )^{3/2}} \]

[Out]

-(((a + 2*b)*Cos[e + f*x])/(a^2*f*(a + b*Sec[e + f*x]^2)^(3/2))) + Cos[e + f*x]^3/(3*a*f*(a + b*Sec[e + f*x]^2
)^(3/2)) - (4*b*(a + 2*b)*Sec[e + f*x])/(3*a^3*f*(a + b*Sec[e + f*x]^2)^(3/2)) - (8*b*(a + 2*b)*Sec[e + f*x])/
(3*a^4*f*Sqrt[a + b*Sec[e + f*x]^2])

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Rubi [A]  time = 0.138049, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4134, 453, 271, 192, 191} \[ -\frac{8 b (a+2 b) \sec (e+f x)}{3 a^4 f \sqrt{a+b \sec ^2(e+f x)}}-\frac{4 b (a+2 b) \sec (e+f x)}{3 a^3 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac{(a+2 b) \cos (e+f x)}{a^2 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}+\frac{\cos ^3(e+f x)}{3 a f \left (a+b \sec ^2(e+f x)\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sin[e + f*x]^3/(a + b*Sec[e + f*x]^2)^(5/2),x]

[Out]

-(((a + 2*b)*Cos[e + f*x])/(a^2*f*(a + b*Sec[e + f*x]^2)^(3/2))) + Cos[e + f*x]^3/(3*a*f*(a + b*Sec[e + f*x]^2
)^(3/2)) - (4*b*(a + 2*b)*Sec[e + f*x])/(3*a^3*f*(a + b*Sec[e + f*x]^2)^(3/2)) - (8*b*(a + 2*b)*Sec[e + f*x])/
(3*a^4*f*Sqrt[a + b*Sec[e + f*x]^2])

Rule 4134

Int[((a_) + (b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_.)*sin[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> With
[{ff = FreeFactors[Cos[e + f*x], x]}, Dist[1/(f*ff^m), Subst[Int[((-1 + ff^2*x^2)^((m - 1)/2)*(a + b*(c*ff*x)^
n)^p)/x^(m + 1), x], x, Sec[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[(m - 1)/2] && (Gt
Q[m, 0] || EqQ[n, 2] || EqQ[n, 4])

Rule 453

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 192

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p + 1
], 0] && NeQ[p, -1]

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

\begin{align*} \int \frac{\sin ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{-1+x^2}{x^4 \left (a+b x^2\right )^{5/2}} \, dx,x,\sec (e+f x)\right )}{f}\\ &=\frac{\cos ^3(e+f x)}{3 a f \left (a+b \sec ^2(e+f x)\right )^{3/2}}+\frac{(a+2 b) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (a+b x^2\right )^{5/2}} \, dx,x,\sec (e+f x)\right )}{a f}\\ &=-\frac{(a+2 b) \cos (e+f x)}{a^2 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}+\frac{\cos ^3(e+f x)}{3 a f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac{(4 b (a+2 b)) \operatorname{Subst}\left (\int \frac{1}{\left (a+b x^2\right )^{5/2}} \, dx,x,\sec (e+f x)\right )}{a^2 f}\\ &=-\frac{(a+2 b) \cos (e+f x)}{a^2 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}+\frac{\cos ^3(e+f x)}{3 a f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac{4 b (a+2 b) \sec (e+f x)}{3 a^3 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac{(8 b (a+2 b)) \operatorname{Subst}\left (\int \frac{1}{\left (a+b x^2\right )^{3/2}} \, dx,x,\sec (e+f x)\right )}{3 a^3 f}\\ &=-\frac{(a+2 b) \cos (e+f x)}{a^2 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}+\frac{\cos ^3(e+f x)}{3 a f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac{4 b (a+2 b) \sec (e+f x)}{3 a^3 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac{8 b (a+2 b) \sec (e+f x)}{3 a^4 f \sqrt{a+b \sec ^2(e+f x)}}\\ \end{align*}

Mathematica [A]  time = 2.52247, size = 129, normalized size = 0.88 \[ -\frac{\sec ^5(e+f x) (a \cos (2 (e+f x))+a+2 b) \left (3 a \left (11 a^2+96 a b+128 b^2\right ) \cos (2 (e+f x))+6 a^2 (a+4 b) \cos (4 (e+f x))+264 a^2 b+a^3 (-\cos (6 (e+f x)))+26 a^3+640 a b^2+512 b^3\right )}{192 a^4 f \left (a+b \sec ^2(e+f x)\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[e + f*x]^3/(a + b*Sec[e + f*x]^2)^(5/2),x]

[Out]

-((a + 2*b + a*Cos[2*(e + f*x)])*(26*a^3 + 264*a^2*b + 640*a*b^2 + 512*b^3 + 3*a*(11*a^2 + 96*a*b + 128*b^2)*C
os[2*(e + f*x)] + 6*a^2*(a + 4*b)*Cos[4*(e + f*x)] - a^3*Cos[6*(e + f*x)])*Sec[e + f*x]^5)/(192*a^4*f*(a + b*S
ec[e + f*x]^2)^(5/2))

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Maple [A]  time = 1.063, size = 159, normalized size = 1.1 \begin{align*} -{\frac{a\sqrt{4} \left ( a+b \right ) ^{5} \left ( b+a \left ( \cos \left ( fx+e \right ) \right ) ^{2} \right ) \left ( \left ( \cos \left ( fx+e \right ) \right ) ^{6}{a}^{3}-3\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}{a}^{3}-6\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}{a}^{2}b-12\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}{a}^{2}b-24\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}a{b}^{2}-8\,a{b}^{2}-16\,{b}^{3} \right ) }{6\,f \left ( \cos \left ( fx+e \right ) \right ) ^{5}} \left ( \sqrt{-ab}+a \right ) ^{-5} \left ( \sqrt{-ab}-a \right ) ^{-5} \left ({\frac{b+a \left ( \cos \left ( fx+e \right ) \right ) ^{2}}{ \left ( \cos \left ( fx+e \right ) \right ) ^{2}}} \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(f*x+e)^3/(a+b*sec(f*x+e)^2)^(5/2),x)

[Out]

-1/6/f*a/((-a*b)^(1/2)+a)^5/((-a*b)^(1/2)-a)^5*4^(1/2)*(a+b)^5*(b+a*cos(f*x+e)^2)*(cos(f*x+e)^6*a^3-3*cos(f*x+
e)^4*a^3-6*cos(f*x+e)^4*a^2*b-12*cos(f*x+e)^2*a^2*b-24*cos(f*x+e)^2*a*b^2-8*a*b^2-16*b^3)/cos(f*x+e)^5/((b+a*c
os(f*x+e)^2)/cos(f*x+e)^2)^(5/2)

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Maxima [A]  time = 0.989938, size = 263, normalized size = 1.8 \begin{align*} -\frac{\frac{3 \, \sqrt{a + \frac{b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right )}{a^{3}} - \frac{{\left (a + \frac{b}{\cos \left (f x + e\right )^{2}}\right )}^{\frac{3}{2}} \cos \left (f x + e\right )^{3} - 9 \, \sqrt{a + \frac{b}{\cos \left (f x + e\right )^{2}}} b \cos \left (f x + e\right )}{a^{4}} + \frac{6 \,{\left (a + \frac{b}{\cos \left (f x + e\right )^{2}}\right )} b \cos \left (f x + e\right )^{2} - b^{2}}{{\left (a + \frac{b}{\cos \left (f x + e\right )^{2}}\right )}^{\frac{3}{2}} a^{3} \cos \left (f x + e\right )^{3}} + \frac{9 \,{\left (a + \frac{b}{\cos \left (f x + e\right )^{2}}\right )} b^{2} \cos \left (f x + e\right )^{2} - b^{3}}{{\left (a + \frac{b}{\cos \left (f x + e\right )^{2}}\right )}^{\frac{3}{2}} a^{4} \cos \left (f x + e\right )^{3}}}{3 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x+e)^3/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm="maxima")

[Out]

-1/3*(3*sqrt(a + b/cos(f*x + e)^2)*cos(f*x + e)/a^3 - ((a + b/cos(f*x + e)^2)^(3/2)*cos(f*x + e)^3 - 9*sqrt(a
+ b/cos(f*x + e)^2)*b*cos(f*x + e))/a^4 + (6*(a + b/cos(f*x + e)^2)*b*cos(f*x + e)^2 - b^2)/((a + b/cos(f*x +
e)^2)^(3/2)*a^3*cos(f*x + e)^3) + (9*(a + b/cos(f*x + e)^2)*b^2*cos(f*x + e)^2 - b^3)/((a + b/cos(f*x + e)^2)^
(3/2)*a^4*cos(f*x + e)^3))/f

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Fricas [A]  time = 1.10187, size = 321, normalized size = 2.2 \begin{align*} \frac{{\left (a^{3} \cos \left (f x + e\right )^{7} - 3 \,{\left (a^{3} + 2 \, a^{2} b\right )} \cos \left (f x + e\right )^{5} - 12 \,{\left (a^{2} b + 2 \, a b^{2}\right )} \cos \left (f x + e\right )^{3} - 8 \,{\left (a b^{2} + 2 \, b^{3}\right )} \cos \left (f x + e\right )\right )} \sqrt{\frac{a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{3 \,{\left (a^{6} f \cos \left (f x + e\right )^{4} + 2 \, a^{5} b f \cos \left (f x + e\right )^{2} + a^{4} b^{2} f\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x+e)^3/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm="fricas")

[Out]

1/3*(a^3*cos(f*x + e)^7 - 3*(a^3 + 2*a^2*b)*cos(f*x + e)^5 - 12*(a^2*b + 2*a*b^2)*cos(f*x + e)^3 - 8*(a*b^2 +
2*b^3)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(a^6*f*cos(f*x + e)^4 + 2*a^5*b*f*cos(f*x + e
)^2 + a^4*b^2*f)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x+e)**3/(a+b*sec(f*x+e)**2)**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (f x + e\right )^{3}}{{\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x+e)^3/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm="giac")

[Out]

integrate(sin(f*x + e)^3/(b*sec(f*x + e)^2 + a)^(5/2), x)