3.33 \(\int \frac{(c-c \sec (e+f x))^3}{(a+a \sec (e+f x))^3} \, dx\)

Optimal. Leaf size=96 \[ -\frac{26 c^3 \tan (e+f x)}{15 a^3 f (\sec (e+f x)+1)}+\frac{4 c^3 \tan (e+f x)}{15 a^3 f (\sec (e+f x)+1)^2}-\frac{8 c^3 \tan (e+f x)}{5 a^3 f (\sec (e+f x)+1)^3}+\frac{c^3 x}{a^3} \]

[Out]

(c^3*x)/a^3 - (8*c^3*Tan[e + f*x])/(5*a^3*f*(1 + Sec[e + f*x])^3) + (4*c^3*Tan[e + f*x])/(15*a^3*f*(1 + Sec[e
+ f*x])^2) - (26*c^3*Tan[e + f*x])/(15*a^3*f*(1 + Sec[e + f*x]))

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Rubi [A]  time = 0.420562, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 9, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.346, Rules used = {3903, 3777, 3922, 3919, 3794, 3796, 3797, 3799, 4000} \[ -\frac{26 c^3 \tan (e+f x)}{15 a^3 f (\sec (e+f x)+1)}+\frac{4 c^3 \tan (e+f x)}{15 a^3 f (\sec (e+f x)+1)^2}-\frac{8 c^3 \tan (e+f x)}{5 a^3 f (\sec (e+f x)+1)^3}+\frac{c^3 x}{a^3} \]

Antiderivative was successfully verified.

[In]

Int[(c - c*Sec[e + f*x])^3/(a + a*Sec[e + f*x])^3,x]

[Out]

(c^3*x)/a^3 - (8*c^3*Tan[e + f*x])/(5*a^3*f*(1 + Sec[e + f*x])^3) + (4*c^3*Tan[e + f*x])/(15*a^3*f*(1 + Sec[e
+ f*x])^2) - (26*c^3*Tan[e + f*x])/(15*a^3*f*(1 + Sec[e + f*x]))

Rule 3903

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_), x_Symbol] :> Dis
t[c^n, Int[ExpandTrig[(1 + (d*csc[e + f*x])/c)^n, (a + b*csc[e + f*x])^m, x], x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0] && ILtQ[n, 0] && LtQ[m + n, 2]

Rule 3777

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> -Simp[(Cot[c + d*x]*(a + b*Csc[c + d*x])^n)/(d*(
2*n + 1)), x] + Dist[1/(a^2*(2*n + 1)), Int[(a + b*Csc[c + d*x])^(n + 1)*(a*(2*n + 1) - b*(n + 1)*Csc[c + d*x]
), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && LeQ[n, -1] && IntegerQ[2*n]

Rule 3922

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)), x_Symbol] :> -Simp[((b
*c - a*d)*Cot[e + f*x]*(a + b*Csc[e + f*x])^m)/(b*f*(2*m + 1)), x] + Dist[1/(a^2*(2*m + 1)), Int[(a + b*Csc[e
+ f*x])^(m + 1)*Simp[a*c*(2*m + 1) - (b*c - a*d)*(m + 1)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f},
x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && EqQ[a^2 - b^2, 0] && IntegerQ[2*m]

Rule 3919

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[(c*x)/a,
x] - Dist[(b*c - a*d)/a, Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[
b*c - a*d, 0]

Rule 3794

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> -Simp[Cot[e + f*x]/(f*(b + a*
Csc[e + f*x])), x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]

Rule 3796

Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(b*Cot[e + f*x]*(a
+ b*Csc[e + f*x])^m)/(a*f*(2*m + 1)), x] + Dist[(m + 1)/(a*(2*m + 1)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(
m + 1), x], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)] && IntegerQ[2*m]

Rule 3797

Int[csc[(e_.) + (f_.)*(x_)]^2*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> -Simp[(Cot[e + f*x]*(a
+ b*Csc[e + f*x])^m)/(f*(2*m + 1)), x] + Dist[m/(b*(2*m + 1)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1),
x], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)]

Rule 3799

Int[csc[(e_.) + (f_.)*(x_)]^3*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(b*Cot[e + f*x]*(
a + b*Csc[e + f*x])^m)/(a*f*(2*m + 1)), x] - Dist[1/(a^2*(2*m + 1)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m
+ 1)*(a*m - b*(2*m + 1)*Csc[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)
]

Rule 4000

Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_))
, x_Symbol] :> Simp[((A*b - a*B)*Cot[e + f*x]*(a + b*Csc[e + f*x])^m)/(a*f*(2*m + 1)), x] + Dist[(a*B*m + A*b*
(m + 1))/(a*b*(2*m + 1)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, A, B, e, f}, x
] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && NeQ[a*B*m + A*b*(m + 1), 0] && LtQ[m, -2^(-1)]

Rubi steps

\begin{align*} \int \frac{(c-c \sec (e+f x))^3}{(a+a \sec (e+f x))^3} \, dx &=\frac{\int \left (\frac{c^3}{(1+\sec (e+f x))^3}-\frac{3 c^3 \sec (e+f x)}{(1+\sec (e+f x))^3}+\frac{3 c^3 \sec ^2(e+f x)}{(1+\sec (e+f x))^3}-\frac{c^3 \sec ^3(e+f x)}{(1+\sec (e+f x))^3}\right ) \, dx}{a^3}\\ &=\frac{c^3 \int \frac{1}{(1+\sec (e+f x))^3} \, dx}{a^3}-\frac{c^3 \int \frac{\sec ^3(e+f x)}{(1+\sec (e+f x))^3} \, dx}{a^3}-\frac{\left (3 c^3\right ) \int \frac{\sec (e+f x)}{(1+\sec (e+f x))^3} \, dx}{a^3}+\frac{\left (3 c^3\right ) \int \frac{\sec ^2(e+f x)}{(1+\sec (e+f x))^3} \, dx}{a^3}\\ &=-\frac{8 c^3 \tan (e+f x)}{5 a^3 f (1+\sec (e+f x))^3}-\frac{c^3 \int \frac{-5+2 \sec (e+f x)}{(1+\sec (e+f x))^2} \, dx}{5 a^3}-\frac{c^3 \int \frac{\sec (e+f x) (-3+5 \sec (e+f x))}{(1+\sec (e+f x))^2} \, dx}{5 a^3}-\frac{\left (6 c^3\right ) \int \frac{\sec (e+f x)}{(1+\sec (e+f x))^2} \, dx}{5 a^3}+\frac{\left (9 c^3\right ) \int \frac{\sec (e+f x)}{(1+\sec (e+f x))^2} \, dx}{5 a^3}\\ &=-\frac{8 c^3 \tan (e+f x)}{5 a^3 f (1+\sec (e+f x))^3}+\frac{4 c^3 \tan (e+f x)}{15 a^3 f (1+\sec (e+f x))^2}+\frac{c^3 \int \frac{15-7 \sec (e+f x)}{1+\sec (e+f x)} \, dx}{15 a^3}-\frac{\left (2 c^3\right ) \int \frac{\sec (e+f x)}{1+\sec (e+f x)} \, dx}{5 a^3}-\frac{\left (7 c^3\right ) \int \frac{\sec (e+f x)}{1+\sec (e+f x)} \, dx}{15 a^3}+\frac{\left (3 c^3\right ) \int \frac{\sec (e+f x)}{1+\sec (e+f x)} \, dx}{5 a^3}\\ &=\frac{c^3 x}{a^3}-\frac{8 c^3 \tan (e+f x)}{5 a^3 f (1+\sec (e+f x))^3}+\frac{4 c^3 \tan (e+f x)}{15 a^3 f (1+\sec (e+f x))^2}-\frac{4 c^3 \tan (e+f x)}{15 a^3 f (1+\sec (e+f x))}-\frac{\left (22 c^3\right ) \int \frac{\sec (e+f x)}{1+\sec (e+f x)} \, dx}{15 a^3}\\ &=\frac{c^3 x}{a^3}-\frac{8 c^3 \tan (e+f x)}{5 a^3 f (1+\sec (e+f x))^3}+\frac{4 c^3 \tan (e+f x)}{15 a^3 f (1+\sec (e+f x))^2}-\frac{26 c^3 \tan (e+f x)}{15 a^3 f (1+\sec (e+f x))}\\ \end{align*}

Mathematica [A]  time = 0.0785299, size = 90, normalized size = 0.94 \[ -\frac{c^3 \left (\frac{2 \tan ^5\left (\frac{e}{2}+\frac{f x}{2}\right )}{5 f}-\frac{2 \tan ^3\left (\frac{e}{2}+\frac{f x}{2}\right )}{3 f}-\frac{2 \tan ^{-1}\left (\tan \left (\frac{e}{2}+\frac{f x}{2}\right )\right )}{f}+\frac{2 \tan \left (\frac{e}{2}+\frac{f x}{2}\right )}{f}\right )}{a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(c - c*Sec[e + f*x])^3/(a + a*Sec[e + f*x])^3,x]

[Out]

-((c^3*((-2*ArcTan[Tan[e/2 + (f*x)/2]])/f + (2*Tan[e/2 + (f*x)/2])/f - (2*Tan[e/2 + (f*x)/2]^3)/(3*f) + (2*Tan
[e/2 + (f*x)/2]^5)/(5*f)))/a^3)

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Maple [A]  time = 0.102, size = 87, normalized size = 0.9 \begin{align*} -{\frac{2\,{c}^{3}}{5\,f{a}^{3}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{5}}+{\frac{2\,{c}^{3}}{3\,f{a}^{3}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{3}}-2\,{\frac{{c}^{3}\tan \left ( 1/2\,fx+e/2 \right ) }{f{a}^{3}}}+2\,{\frac{{c}^{3}\arctan \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) }{f{a}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c-c*sec(f*x+e))^3/(a+a*sec(f*x+e))^3,x)

[Out]

-2/5/f*c^3/a^3*tan(1/2*f*x+1/2*e)^5+2/3/f*c^3/a^3*tan(1/2*f*x+1/2*e)^3-2/f*c^3/a^3*tan(1/2*f*x+1/2*e)+2/f*c^3/
a^3*arctan(tan(1/2*f*x+1/2*e))

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Maxima [B]  time = 1.64579, size = 374, normalized size = 3.9 \begin{align*} -\frac{c^{3}{\left (\frac{\frac{105 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac{20 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac{3 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}}{a^{3}} - \frac{120 \, \arctan \left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a^{3}}\right )} + \frac{c^{3}{\left (\frac{15 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac{10 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac{3 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{a^{3}} + \frac{3 \, c^{3}{\left (\frac{15 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac{10 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac{3 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{a^{3}} - \frac{9 \, c^{3}{\left (\frac{5 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac{\sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{a^{3}}}{60 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c*sec(f*x+e))^3/(a+a*sec(f*x+e))^3,x, algorithm="maxima")

[Out]

-1/60*(c^3*((105*sin(f*x + e)/(cos(f*x + e) + 1) - 20*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 3*sin(f*x + e)^5/(
cos(f*x + e) + 1)^5)/a^3 - 120*arctan(sin(f*x + e)/(cos(f*x + e) + 1))/a^3) + c^3*(15*sin(f*x + e)/(cos(f*x +
e) + 1) + 10*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5)/a^3 + 3*c^3*(15*sin(
f*x + e)/(cos(f*x + e) + 1) - 10*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5)/
a^3 - 9*c^3*(5*sin(f*x + e)/(cos(f*x + e) + 1) - sin(f*x + e)^5/(cos(f*x + e) + 1)^5)/a^3)/f

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Fricas [A]  time = 1.02357, size = 340, normalized size = 3.54 \begin{align*} \frac{15 \, c^{3} f x \cos \left (f x + e\right )^{3} + 45 \, c^{3} f x \cos \left (f x + e\right )^{2} + 45 \, c^{3} f x \cos \left (f x + e\right ) + 15 \, c^{3} f x - 2 \,{\left (23 \, c^{3} \cos \left (f x + e\right )^{2} + 24 \, c^{3} \cos \left (f x + e\right ) + 13 \, c^{3}\right )} \sin \left (f x + e\right )}{15 \,{\left (a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} + 3 \, a^{3} f \cos \left (f x + e\right ) + a^{3} f\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c*sec(f*x+e))^3/(a+a*sec(f*x+e))^3,x, algorithm="fricas")

[Out]

1/15*(15*c^3*f*x*cos(f*x + e)^3 + 45*c^3*f*x*cos(f*x + e)^2 + 45*c^3*f*x*cos(f*x + e) + 15*c^3*f*x - 2*(23*c^3
*cos(f*x + e)^2 + 24*c^3*cos(f*x + e) + 13*c^3)*sin(f*x + e))/(a^3*f*cos(f*x + e)^3 + 3*a^3*f*cos(f*x + e)^2 +
3*a^3*f*cos(f*x + e) + a^3*f)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{c^{3} \left (\int \frac{3 \sec{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec{\left (e + f x \right )} + 1}\, dx + \int - \frac{3 \sec ^{2}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec{\left (e + f x \right )} + 1}\, dx + \int \frac{\sec ^{3}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec{\left (e + f x \right )} + 1}\, dx + \int - \frac{1}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec{\left (e + f x \right )} + 1}\, dx\right )}{a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c*sec(f*x+e))**3/(a+a*sec(f*x+e))**3,x)

[Out]

-c**3*(Integral(3*sec(e + f*x)/(sec(e + f*x)**3 + 3*sec(e + f*x)**2 + 3*sec(e + f*x) + 1), x) + Integral(-3*se
c(e + f*x)**2/(sec(e + f*x)**3 + 3*sec(e + f*x)**2 + 3*sec(e + f*x) + 1), x) + Integral(sec(e + f*x)**3/(sec(e
+ f*x)**3 + 3*sec(e + f*x)**2 + 3*sec(e + f*x) + 1), x) + Integral(-1/(sec(e + f*x)**3 + 3*sec(e + f*x)**2 +
3*sec(e + f*x) + 1), x))/a**3

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Giac [A]  time = 1.48101, size = 113, normalized size = 1.18 \begin{align*} \frac{\frac{15 \,{\left (f x + e\right )} c^{3}}{a^{3}} - \frac{2 \,{\left (3 \, a^{12} c^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} - 5 \, a^{12} c^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 15 \, a^{12} c^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )\right )}}{a^{15}}}{15 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c*sec(f*x+e))^3/(a+a*sec(f*x+e))^3,x, algorithm="giac")

[Out]

1/15*(15*(f*x + e)*c^3/a^3 - 2*(3*a^12*c^3*tan(1/2*f*x + 1/2*e)^5 - 5*a^12*c^3*tan(1/2*f*x + 1/2*e)^3 + 15*a^1
2*c^3*tan(1/2*f*x + 1/2*e))/a^15)/f