3.1.58 \(\int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))} \, dx\) [58]

Optimal. Leaf size=78 \[ -\frac {2 \cot ^5(e+f x)}{5 a^3 c f}+\frac {\csc (e+f x)}{a^3 c f}-\frac {\csc ^3(e+f x)}{a^3 c f}+\frac {2 \csc ^5(e+f x)}{5 a^3 c f} \]

[Out]

-2/5*cot(f*x+e)^5/a^3/c/f+csc(f*x+e)/a^3/c/f-csc(f*x+e)^3/a^3/c/f+2/5*csc(f*x+e)^5/a^3/c/f

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Rubi [A]
time = 0.14, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {4043, 2686, 200, 2687, 30, 14} \begin {gather*} -\frac {2 \cot ^5(e+f x)}{5 a^3 c f}+\frac {2 \csc ^5(e+f x)}{5 a^3 c f}-\frac {\csc ^3(e+f x)}{a^3 c f}+\frac {\csc (e+f x)}{a^3 c f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 200

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

Rule 2686

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 2687

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rule 4043

Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)
)^(n_.), x_Symbol] :> Dist[((-a)*c)^m, Int[ExpandTrig[csc[e + f*x]*cot[e + f*x]^(2*m), (c + d*csc[e + f*x])^(n
- 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] && IntegersQ[m,
n] && GeQ[n - m, 0] && GtQ[m*n, 0]

Rubi steps

\begin {align*} \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))} \, dx &=-\frac {\int \left (c^2 \cot ^5(e+f x) \csc (e+f x)-2 c^2 \cot ^4(e+f x) \csc ^2(e+f x)+c^2 \cot ^3(e+f x) \csc ^3(e+f x)\right ) \, dx}{a^3 c^3}\\ &=-\frac {\int \cot ^5(e+f x) \csc (e+f x) \, dx}{a^3 c}-\frac {\int \cot ^3(e+f x) \csc ^3(e+f x) \, dx}{a^3 c}+\frac {2 \int \cot ^4(e+f x) \csc ^2(e+f x) \, dx}{a^3 c}\\ &=\frac {\text {Subst}\left (\int x^2 \left (-1+x^2\right ) \, dx,x,\csc (e+f x)\right )}{a^3 c f}+\frac {\text {Subst}\left (\int \left (-1+x^2\right )^2 \, dx,x,\csc (e+f x)\right )}{a^3 c f}+\frac {2 \text {Subst}\left (\int x^4 \, dx,x,-\cot (e+f x)\right )}{a^3 c f}\\ &=-\frac {2 \cot ^5(e+f x)}{5 a^3 c f}+\frac {\text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\csc (e+f x)\right )}{a^3 c f}+\frac {\text {Subst}\left (\int \left (-x^2+x^4\right ) \, dx,x,\csc (e+f x)\right )}{a^3 c f}\\ &=-\frac {2 \cot ^5(e+f x)}{5 a^3 c f}+\frac {\csc (e+f x)}{a^3 c f}-\frac {\csc ^3(e+f x)}{a^3 c f}+\frac {2 \csc ^5(e+f x)}{5 a^3 c f}\\ \end {align*}

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Mathematica [A]
time = 0.80, size = 109, normalized size = 1.40 \begin {gather*} -\frac {\csc (e) \csc ^5(e+f x) \sin ^4\left (\frac {1}{2} (e+f x)\right ) (-40 \sin (e)+65 \sin (e+f x)+52 \sin (2 (e+f x))+13 \sin (3 (e+f x))-40 \sin (2 e+f x)-12 \sin (e+2 f x)-20 \sin (3 e+2 f x)-8 \sin (2 e+3 f x))}{20 a^3 c f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/20*(Csc[e]*Csc[e + f*x]^5*Sin[(e + f*x)/2]^4*(-40*Sin[e] + 65*Sin[e + f*x] + 52*Sin[2*(e + f*x)] + 13*Sin[3
*(e + f*x)] - 40*Sin[2*e + f*x] - 12*Sin[e + 2*f*x] - 20*Sin[3*e + 2*f*x] - 8*Sin[2*e + 3*f*x]))/(a^3*c*f)

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Maple [A]
time = 0.15, size = 61, normalized size = 0.78

method	result	size
derivativedivides	\(\frac {\frac {\left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5}-\left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}}{8 f \,a^{3} c}\)	\(61\)
default	\(\frac {\frac {\left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5}-\left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}}{8 f \,a^{3} c}\)	\(61\)
risch	\(\frac {2 i \left (5 \,{\mathrm e}^{5 i \left (f x +e \right )}+10 \,{\mathrm e}^{4 i \left (f x +e \right )}+10 \,{\mathrm e}^{3 i \left (f x +e \right )}-3 \,{\mathrm e}^{i \left (f x +e \right )}-2\right )}{5 f \,a^{3} c \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{5} \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}\)	\(85\)
norman	\(\frac {\frac {1}{8 a c f}+\frac {3 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 a c f}-\frac {\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )}{8 a c f}+\frac {\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )}{40 a c f}}{a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}\)	\(94\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8/f/a^3/c*(1/5*tan(1/2*f*x+1/2*e)^5-tan(1/2*f*x+1/2*e)^3+3*tan(1/2*f*x+1/2*e)+1/tan(1/2*f*x+1/2*e))

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Maxima [A]
time = 0.28, size = 103, normalized size = 1.32 \begin {gather*} \frac {\frac {\frac {15 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {5 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {\sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}}{a^{3} c} + \frac {5 \, {\left (\cos \left (f x + e\right ) + 1\right )}}{a^{3} c \sin \left (f x + e\right )}}{40 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/40*((15*sin(f*x + e)/(cos(f*x + e) + 1) - 5*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + sin(f*x + e)^5/(cos(f*x +
e) + 1)^5)/(a^3*c) + 5*(cos(f*x + e) + 1)/(a^3*c*sin(f*x + e)))/f

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Fricas [A]
time = 2.69, size = 82, normalized size = 1.05 \begin {gather*} -\frac {2 \, \cos \left (f x + e\right )^{3} - \cos \left (f x + e\right )^{2} - 4 \, \cos \left (f x + e\right ) - 2}{5 \, {\left (a^{3} c f \cos \left (f x + e\right )^{2} + 2 \, a^{3} c f \cos \left (f x + e\right ) + a^{3} c f\right )} \sin \left (f x + e\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*(2*cos(f*x + e)^3 - cos(f*x + e)^2 - 4*cos(f*x + e) - 2)/((a^3*c*f*cos(f*x + e)^2 + 2*a^3*c*f*cos(f*x + e
) + a^3*c*f)*sin(f*x + e))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \frac {\int \frac {\sec {\left (e + f x \right )}}{\sec ^{4}{\left (e + f x \right )} + 2 \sec ^{3}{\left (e + f x \right )} - 2 \sec {\left (e + f x \right )} - 1}\, dx}{a^{3} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(sec(e + f*x)/(sec(e + f*x)**4 + 2*sec(e + f*x)**3 - 2*sec(e + f*x) - 1), x)/(a**3*c)

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Giac [A]
time = 0.53, size = 87, normalized size = 1.12 \begin {gather*} \frac {\frac {5}{a^{3} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )} + \frac {a^{12} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 5 \, a^{12} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 15 \, a^{12} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{a^{15} c^{5}}}{40 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/40*(5/(a^3*c*tan(1/2*f*x + 1/2*e)) + (a^12*c^4*tan(1/2*f*x + 1/2*e)^5 - 5*a^12*c^4*tan(1/2*f*x + 1/2*e)^3 +
15*a^12*c^4*tan(1/2*f*x + 1/2*e))/(a^15*c^5))/f

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Mupad [B]
time = 1.62, size = 74, normalized size = 0.95 \begin {gather*} -\frac {16\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6-28\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+8\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-1}{40\,a^3\,c\,f\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(e + f*x)*(a + a/cos(e + f*x))^3*(c - c/cos(e + f*x))),x)

[Out]

-(8*cos(e/2 + (f*x)/2)^2 - 28*cos(e/2 + (f*x)/2)^4 + 16*cos(e/2 + (f*x)/2)^6 - 1)/(40*a^3*c*f*cos(e/2 + (f*x)/
2)^5*sin(e/2 + (f*x)/2))

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