3.2.21 \(\int \frac {\sin (e+f x)}{(a+b \sec ^2(e+f x))^{5/2}} \, dx\) [121]

Optimal. Leaf size=97 \[ -\frac {\cos (e+f x)}{a f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {4 b \sec (e+f x)}{3 a^2 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {8 b \sec (e+f x)}{3 a^3 f \sqrt {a+b \sec ^2(e+f x)}} \]

[Out]

-cos(f*x+e)/a/f/(a+b*sec(f*x+e)^2)^(3/2)-4/3*b*sec(f*x+e)/a^2/f/(a+b*sec(f*x+e)^2)^(3/2)-8/3*b*sec(f*x+e)/a^3/
f/(a+b*sec(f*x+e)^2)^(1/2)

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Rubi [A]
time = 0.05, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {4219, 277, 198, 197} \begin {gather*} -\frac {8 b \sec (e+f x)}{3 a^3 f \sqrt {a+b \sec ^2(e+f x)}}-\frac {4 b \sec (e+f x)}{3 a^2 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\cos (e+f x)}{a f \left (a+b \sec ^2(e+f x)\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 197

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]

Rule 198

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 277

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 4219

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])

Rubi steps

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

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Mathematica [A]
time = 1.42, size = 88, normalized size = 0.91 \begin {gather*} -\frac {(a+2 b+a \cos (2 (e+f x))) \left ((3 a+8 b)^2+12 a (a+4 b) \cos (2 (e+f x))+3 a^2 \cos (4 (e+f x))\right ) \sec ^5(e+f x)}{48 a^3 f \left (a+b \sec ^2(e+f x)\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/48*((a + 2*b + a*Cos[2*(e + f*x)])*((3*a + 8*b)^2 + 12*a*(a + 4*b)*Cos[2*(e + f*x)] + 3*a^2*Cos[4*(e + f*x)
])*Sec[e + f*x]^5)/(a^3*f*(a + b*Sec[e + f*x]^2)^(5/2))

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Maple [A]
time = 0.03, size = 90, normalized size = 0.93

method	result	size
derivativedivides	\(\frac {-\frac {1}{a \sec \left (f x +e \right ) \left (a +b \left (\sec ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}}}-\frac {4 b \left (\frac {\sec \left (f x +e \right )}{3 a \left (a +b \left (\sec ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}}}+\frac {2 \sec \left (f x +e \right )}{3 a^{2} \sqrt {a +b \left (\sec ^{2}\left (f x +e \right )\right )}}\right )}{a}}{f}\)	\(90\)
default	\(\frac {-\frac {1}{a \sec \left (f x +e \right ) \left (a +b \left (\sec ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}}}-\frac {4 b \left (\frac {\sec \left (f x +e \right )}{3 a \left (a +b \left (\sec ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}}}+\frac {2 \sec \left (f x +e \right )}{3 a^{2} \sqrt {a +b \left (\sec ^{2}\left (f x +e \right )\right )}}\right )}{a}}{f}\)	\(90\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/f*(-1/a/sec(f*x+e)/(a+b*sec(f*x+e)^2)^(3/2)-4*b/a*(1/3*sec(f*x+e)/a/(a+b*sec(f*x+e)^2)^(3/2)+2/3/a^2*sec(f*x
+e)/(a+b*sec(f*x+e)^2)^(1/2)))

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Maxima [A]
time = 0.27, size = 92, normalized size = 0.95 \begin {gather*} -\frac {\frac {3 \, \sqrt {a + \frac {b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right )}{a^{3}} + \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}}}{3 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x+e)/(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 + (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))/f

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Fricas [A]
time = 4.20, size = 108, normalized size = 1.11 \begin {gather*} -\frac {{\left (3 \, a^{2} \cos \left (f x + e\right )^{5} + 12 \, a b \cos \left (f x + e\right )^{3} + 8 \, b^{2} \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^{5} f \cos \left (f x + e\right )^{4} + 2 \, a^{4} b f \cos \left (f x + e\right )^{2} + a^{3} b^{2} f\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sin {\left (e + f x \right )}}{\left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.61, size = 70, normalized size = 0.72 \begin {gather*} -\frac {3 \, \sqrt {a \cos \left (f x + e\right )^{2} + b} + \frac {6 \, {\left (a \cos \left (f x + e\right )^{2} + b\right )} b - b^{2}}{{\left (a \cos \left (f x + e\right )^{2} + b\right )}^{\frac {3}{2}}}}{3 \, a^{3} f \mathrm {sgn}\left (\cos \left (f x + e\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 17.23, size = 2500, normalized size = 25.77 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((a + b/(exp(- e*1i - f*x*1i)/2 + exp(e*1i + f*x*1i)/2)^2)^(1/2)*(exp(e*3i + f*x*3i)*(((2*a + 4*b)*(((2*a + 4*
b)*((((32*a*b^2 + 30*a^2*b + 3*a^3)/(48*a^3*b*f*(a + b)) - ((2*a + 4*b)*(8*a*b^2 + 8*a^2*b + a^3))/(48*a^4*b*f
*(a + b)))*(2*a + 4*b))/a + (8*a*b^2 + 8*a^2*b + a^3)/(48*a^3*b*f*(a + b)) - (16*a*b + a^2 + 20*b^2)/(24*a^2*b
*f*(a + b))))/a - (a + 6*b)/(24*a*b*f*(a + b)) - (32*a*b^2 + 30*a^2*b + 3*a^3)/(48*a^3*b*f*(a + b)) + ((2*a +
4*b)*(8*a*b^2 + 8*a^2*b + a^3))/(48*a^4*b*f*(a + b))))/a - (((32*a*b^2 + 30*a^2*b + 3*a^3)/(48*a^3*b*f*(a + b)
) - ((2*a + 4*b)*(8*a*b^2 + 8*a^2*b + a^3))/(48*a^4*b*f*(a + b)))*(2*a + 4*b))/a - (8*a*b^2 + 8*a^2*b + a^3)/(
48*a^3*b*f*(a + b)) + (16*a*b + a^2 + 20*b^2)/(24*a^2*b*f*(a + b)) + (32*a*b^2 + 40*a^2*b + 3*a^3)/(48*a^3*b*f
*(a + b))) + exp(e*1i + f*x*1i)*(((2*a + 4*b)*((((32*a*b^2 + 30*a^2*b + 3*a^3)/(48*a^3*b*f*(a + b)) - ((2*a +
4*b)*(8*a*b^2 + 8*a^2*b + a^3))/(48*a^4*b*f*(a + b)))*(2*a + 4*b))/a + (8*a*b^2 + 8*a^2*b + a^3)/(48*a^3*b*f*(
a + b)) - (16*a*b + a^2 + 20*b^2)/(24*a^2*b*f*(a + b))))/a + (48*a*b^2 + 18*a^2*b + a^3 + 32*b^3)/(48*a^3*b*f*
(a + b)) - (a + 6*b)/(24*a*b*f*(a + b)) - (32*a*b^2 + 30*a^2*b + 3*a^3)/(48*a^3*b*f*(a + b)) + ((2*a + 4*b)*(8
*a*b^2 + 8*a^2*b + a^3))/(48*a^4*b*f*(a + b))))*(2*exp(e*2i + f*x*2i) + exp(e*4i + f*x*4i) + 1))/((exp(e*2i +
f*x*2i) + 1)*(a + exp(e*2i + f*x*2i)*(2*a + 4*b) + a*exp(e*4i + f*x*4i))) - ((a + b/(exp(- e*1i - f*x*1i)/2 +
exp(e*1i + f*x*1i)/2)^2)^(1/2)*(exp(e*1i + f*x*1i)*(((a*2i + b*4i)*(((a*2i + b*4i)*(((a + 2*b)*(a*2i + b*4i)*(
8*a*b + a^2 + 8*b^2)*1i)/(48*f*(2*a*b + a^2)*(a*b^2 + a^2*b)) - ((a*1i + ((8*a*b + a^2 + 8*b^2)^2*1i)/(4*(a*b^
2 + a^2*b)))*1i)/(12*f*(2*a*b + a^2)) + (a*(a + 2*b)*(8*a*b + a^2 + 8*b^2))/(12*f*(2*a*b + a^2)*(a*b^2 + a^2*b
)))*1i)/a - ((a*1i + ((8*a*b + a^2 + 8*b^2)^2*1i)/(4*(a*b^2 + a^2*b)))*1i)/(3*f*(2*a*b + a^2)) + (a*(a + 2*b)*
(8*a*b + a^2 + 8*b^2))/(12*f*(2*a*b + a^2)*(a*b^2 + a^2*b)))*1i)/a - ((a*1i + ((8*a*b + a^2 + 8*b^2)^2*1i)/(4*
(a*b^2 + a^2*b)))*2i)/(3*f*(2*a*b + a^2)) + ((a*2i + b*4i)*(((a*2i + b*4i)*(((a + 2*b)*(a*2i + b*4i)*(8*a*b +
a^2 + 8*b^2)*1i)/(48*f*(2*a*b + a^2)*(a*b^2 + a^2*b)) - ((a*1i + ((8*a*b + a^2 + 8*b^2)^2*1i)/(4*(a*b^2 + a^2*
b)))*1i)/(12*f*(2*a*b + a^2)) + (a*(a + 2*b)*(8*a*b + a^2 + 8*b^2))/(12*f*(2*a*b + a^2)*(a*b^2 + a^2*b)))*1i)/
a + ((a*2i + b*4i)*(((a*1i + ((8*a*b + a^2 + 8*b^2)^2*1i)/(4*(a*b^2 + a^2*b)))*1i)/(3*f*(2*a*b + a^2)) - ((a*2
i + b*4i)*(((a*2i + b*4i)*(((a + 2*b)*(a*2i + b*4i)*(8*a*b + a^2 + 8*b^2)*1i)/(48*f*(2*a*b + a^2)*(a*b^2 + a^2
*b)) - ((a*1i + ((8*a*b + a^2 + 8*b^2)^2*1i)/(4*(a*b^2 + a^2*b)))*1i)/(12*f*(2*a*b + a^2)) + (a*(a + 2*b)*(8*a
*b + a^2 + 8*b^2))/(12*f*(2*a*b + a^2)*(a*b^2 + a^2*b)))*1i)/a - ((a*1i + ((8*a*b + a^2 + 8*b^2)^2*1i)/(4*(a*b
^2 + a^2*b)))*1i)/(3*f*(2*a*b + a^2)) + (a*(a + 2*b)*(8*a*b + a^2 + 8*b^2))/(12*f*(2*a*b + a^2)*(a*b^2 + a^2*b
)))*1i)/a + ((a + 2*b)*(a*2i + b*4i)*(8*a*b + a^2 + 8*b^2)*1i)/(48*f*(2*a*b + a^2)*(a*b^2 + a^2*b)) + (a*(a +
2*b)*(8*a*b + a^2 + 8*b^2))/(12*f*(2*a*b + a^2)*(a*b^2 + a^2*b)))*1i)/a - ((a*1i + ((8*a*b + a^2 + 8*b^2)^2*1i
)/(4*(a*b^2 + a^2*b)))*1i)/(3*f*(2*a*b + a^2)) + (3*a*(a + 2*b)*(8*a*b + a^2 + 8*b^2))/(16*f*(2*a*b + a^2)*(a*
b^2 + a^2*b)))*1i)/a - ((a + 2*b)*(a*2i + b*4i)*(8*a*b + a^2 + 8*b^2)*1i)/(48*f*(2*a*b + a^2)*(a*b^2 + a^2*b))
) + exp(e*3i + f*x*3i)*(((a*2i + b*4i)*(((a + 2*b)*(a*2i + b*4i)*(8*a*b + a^2 + 8*b^2)*1i)/(48*f*(2*a*b + a^2)
*(a*b^2 + a^2*b)) - ((a*1i + ((8*a*b + a^2 + 8*b^2)^2*1i)/(4*(a*b^2 + a^2*b)))*1i)/(12*f*(2*a*b + a^2)) + (a*(
a + 2*b)*(8*a*b + a^2 + 8*b^2))/(12*f*(2*a*b + a^2)*(a*b^2 + a^2*b)))*1i)/a + ((a*2i + b*4i)*(((a*1i + ((8*a*b
+ a^2 + 8*b^2)^2*1i)/(4*(a*b^2 + a^2*b)))*1i)/(3*f*(2*a*b + a^2)) - ((a*2i + b*4i)*(((a*2i + b*4i)*(((a + 2*b
)*(a*2i + b*4i)*(8*a*b + a^2 + 8*b^2)*1i)/(48*f*(2*a*b + a^2)*(a*b^2 + a^2*b)) - ((a*1i + ((8*a*b + a^2 + 8*b^
2)^2*1i)/(4*(a*b^2 + a^2*b)))*1i)/(12*f*(2*a*b + a^2)) + (a*(a + 2*b)*(8*a*b + a^2 + 8*b^2))/(12*f*(2*a*b + a^
2)*(a*b^2 + a^2*b)))*1i)/a - ((a*1i + ((8*a*b + a^2 + 8*b^2)^2*1i)/(4*(a*b^2 + a^2*b)))*1i)/(3*f*(2*a*b + a^2)
) + (a*(a + 2*b)*(8*a*b + a^2 + 8*b^2))/(12*f*(2*a*b + a^2)*(a*b^2 + a^2*b)))*1i)/a + ((a + 2*b)*(a*2i + b*4i)
*(8*a*b + a^2 + 8*b^2)*1i)/(48*f*(2*a*b + a^2)*(a*b^2 + a^2*b)) + (a*(a + 2*b)*(8*a*b + a^2 + 8*b^2))/(12*f*(2
*a*b + a^2)*(a*b^2 + a^2*b)))*1i)/a - ((a*2i + b*4i)*(((a*2i + b*4i)*(((a*2i + b*4i)*(((a + 2*b)*(a*2i + b*4i)
*(8*a*b + a^2 + 8*b^2)*1i)/(48*f*(2*a*b + a^2)*(a*b^2 + a^2*b)) - ((a*1i + ((8*a*b + a^2 + 8*b^2)^2*1i)/(4*(a*
b^2 + a^2*b)))*1i)/(12*f*(2*a*b + a^2)) + (a*(a + 2*b)*(8*a*b + a^2 + 8*b^2))/(12*f*(2*a*b + a^2)*(a*b^2 + a^2
*b)))*1i)/a - ((a*1i + ((8*a*b + a^2 + 8*b^2)^2*1i)/(4*(a*b^2 + a^2*b)))*1i)/(3*f*(2*a*b + a^2)) + (a*(a + 2*b
)*(8*a*b + a^2 + 8*b^2))/(12*f*(2*a*b + a^2)*(a*b^2 + a^2*b)))*1i)/a - ((a*1i + ((8*a*b + a^2 + 8*b^2)^2*1i)/(
4*(a*b^2 + a^2*b)))*3i)/(4*f*(2*a*b + a^2)) + ((a*2i + b*4i)*(((a*2i + b*4i)*(((a + 2*b)*(a*2i + b*4i)*(8*a*b
+ a^2 + 8*b^2)*1i)/(48*f*(2*a*b + a^2)*(a*b^2 +...

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