3.1.38 \(\int \frac {1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^3} \, dx\) [38]

Optimal. Leaf size=67 \[ \frac {x}{a^3 c^3}+\frac {\cot (e+f x)}{a^3 c^3 f}-\frac {\cot ^3(e+f x)}{3 a^3 c^3 f}+\frac {\cot ^5(e+f x)}{5 a^3 c^3 f} \]

[Out]

x/a^3/c^3+cot(f*x+e)/a^3/c^3/f-1/3*cot(f*x+e)^3/a^3/c^3/f+1/5*cot(f*x+e)^5/a^3/c^3/f

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Rubi [A]
time = 0.06, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {3989, 3554, 8} \begin {gather*} \frac {\cot ^5(e+f x)}{5 a^3 c^3 f}-\frac {\cot ^3(e+f x)}{3 a^3 c^3 f}+\frac {\cot (e+f x)}{a^3 c^3 f}+\frac {x}{a^3 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^3),x]

[Out]

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3554

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d*x])^(n - 1)/(d*(n - 1))), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 3989

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_.), x_Symbol] :> Di
st[((-a)*c)^m, Int[Cot[e + f*x]^(2*m)*(c + d*Csc[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x]
&& EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && RationalQ[n] &&  !(IntegerQ[n] && GtQ[m - n, 0])

Rubi steps

\begin {align*} \int \frac {1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^3} \, dx &=-\frac {\int \cot ^6(e+f x) \, dx}{a^3 c^3}\\ &=\frac {\cot ^5(e+f x)}{5 a^3 c^3 f}+\frac {\int \cot ^4(e+f x) \, dx}{a^3 c^3}\\ &=-\frac {\cot ^3(e+f x)}{3 a^3 c^3 f}+\frac {\cot ^5(e+f x)}{5 a^3 c^3 f}-\frac {\int \cot ^2(e+f x) \, dx}{a^3 c^3}\\ &=\frac {\cot (e+f x)}{a^3 c^3 f}-\frac {\cot ^3(e+f x)}{3 a^3 c^3 f}+\frac {\cot ^5(e+f x)}{5 a^3 c^3 f}+\frac {\int 1 \, dx}{a^3 c^3}\\ &=\frac {x}{a^3 c^3}+\frac {\cot (e+f x)}{a^3 c^3 f}-\frac {\cot ^3(e+f x)}{3 a^3 c^3 f}+\frac {\cot ^5(e+f x)}{5 a^3 c^3 f}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.08, size = 39, normalized size = 0.58 \begin {gather*} \frac {\cot ^5(e+f x) \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};-\tan ^2(e+f x)\right )}{5 a^3 c^3 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^3),x]

[Out]

(Cot[e + f*x]^5*Hypergeometric2F1[-5/2, 1, -3/2, -Tan[e + f*x]^2])/(5*a^3*c^3*f)

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Maple [A]
time = 0.13, size = 48, normalized size = 0.72

method	result	size
default	\(-\frac {-\frac {\left (\cot ^{5}\left (f x +e \right )\right )}{5}+\frac {\left (\cot ^{3}\left (f x +e \right )\right )}{3}-\cot \left (f x +e \right )-f x -e}{c^{3} a^{3} f}\)	\(48\)
risch	\(\frac {x}{a^{3} c^{3}}+\frac {2 i \left (45 \,{\mathrm e}^{8 i \left (f x +e \right )}-90 \,{\mathrm e}^{6 i \left (f x +e \right )}+140 \,{\mathrm e}^{4 i \left (f x +e \right )}-70 \,{\mathrm e}^{2 i \left (f x +e \right )}+23\right )}{15 f \,c^{3} a^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{5} \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{5}}\)	\(94\)
norman	\(\frac {\frac {x \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c a}+\frac {1}{160 a c f}-\frac {7 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{96 a c f}+\frac {11 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{16 a c f}-\frac {11 \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{16 a c f}+\frac {7 \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{96 a c f}-\frac {\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )}{160 a c f}}{a^{2} c^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}\)	\(160\)
derivativedivides	error in RationalFunction: argument is not a rational function\	N/A

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+a*sec(f*x+e))^3/(c-c*sec(f*x+e))^3,x,method=_RETURNVERBOSE)

[Out]

-1/c^3/a^3/f*(-1/5*cot(f*x+e)^5+1/3*cot(f*x+e)^3-cot(f*x+e)-f*x-e)

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Maxima [A]
time = 0.49, size = 60, normalized size = 0.90 \begin {gather*} \frac {\frac {15 \, {\left (f x + e\right )}}{a^{3} c^{3}} + \frac {15 \, \tan \left (f x + e\right )^{4} - 5 \, \tan \left (f x + e\right )^{2} + 3}{a^{3} c^{3} \tan \left (f x + e\right )^{5}}}{15 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*sec(f*x+e))^3/(c-c*sec(f*x+e))^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 4.07, size = 127, normalized size = 1.90 \begin {gather*} \frac {23 \, \cos \left (f x + e\right )^{5} - 35 \, \cos \left (f x + e\right )^{3} + 15 \, {\left (f x \cos \left (f x + e\right )^{4} - 2 \, f x \cos \left (f x + e\right )^{2} + f x\right )} \sin \left (f x + e\right ) + 15 \, \cos \left (f x + e\right )}{15 \, {\left (a^{3} c^{3} f \cos \left (f x + e\right )^{4} - 2 \, a^{3} c^{3} f \cos \left (f x + e\right )^{2} + a^{3} c^{3} f\right )} \sin \left (f x + e\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*sec(f*x+e))^3/(c-c*sec(f*x+e))^3,x, algorithm="fricas")

[Out]

1/15*(23*cos(f*x + e)^5 - 35*cos(f*x + e)^3 + 15*(f*x*cos(f*x + e)^4 - 2*f*x*cos(f*x + e)^2 + f*x)*sin(f*x + e
) + 15*cos(f*x + e))/((a^3*c^3*f*cos(f*x + e)^4 - 2*a^3*c^3*f*cos(f*x + e)^2 + a^3*c^3*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 {1}{\sec ^{6}{\left (e + f x \right )} - 3 \sec ^{4}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} - 1}\, dx}{a^{3} c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*sec(f*x+e))**3/(c-c*sec(f*x+e))**3,x)

[Out]

-Integral(1/(sec(e + f*x)**6 - 3*sec(e + f*x)**4 + 3*sec(e + f*x)**2 - 1), x)/(a**3*c**3)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (63) = 126\).
time = 0.55, size = 129, normalized size = 1.93 \begin {gather*} \frac {\frac {480 \, {\left (f x + e\right )}}{a^{3} c^{3}} + \frac {330 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 35 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 3}{a^{3} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5}} - \frac {3 \, a^{12} c^{12} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 35 \, a^{12} c^{12} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 330 \, a^{12} c^{12} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{a^{15} c^{15}}}{480 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*sec(f*x+e))^3/(c-c*sec(f*x+e))^3,x, algorithm="giac")

[Out]

1/480*(480*(f*x + e)/(a^3*c^3) + (330*tan(1/2*f*x + 1/2*e)^4 - 35*tan(1/2*f*x + 1/2*e)^2 + 3)/(a^3*c^3*tan(1/2
*f*x + 1/2*e)^5) - (3*a^12*c^12*tan(1/2*f*x + 1/2*e)^5 - 35*a^12*c^12*tan(1/2*f*x + 1/2*e)^3 + 330*a^12*c^12*t
an(1/2*f*x + 1/2*e))/(a^15*c^15))/f

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Mupad [B]
time = 1.57, size = 94, normalized size = 1.40 \begin {gather*} \frac {\frac {5\,\cos \left (e+f\,x\right )}{24}-\frac {5\,\cos \left (3\,e+3\,f\,x\right )}{48}+\frac {23\,\cos \left (5\,e+5\,f\,x\right )}{240}-\frac {5\,\sin \left (3\,e+3\,f\,x\right )\,\left (e+f\,x\right )}{16}+\frac {\sin \left (5\,e+5\,f\,x\right )\,\left (e+f\,x\right )}{16}+\frac {5\,\sin \left (e+f\,x\right )\,\left (e+f\,x\right )}{8}}{a^3\,c^3\,f\,{\sin \left (e+f\,x\right )}^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((5*cos(e + f*x))/24 - (5*cos(3*e + 3*f*x))/48 + (23*cos(5*e + 5*f*x))/240 - (5*sin(3*e + 3*f*x)*(e + f*x))/16
+ (sin(5*e + 5*f*x)*(e + f*x))/16 + (5*sin(e + f*x)*(e + f*x))/8)/(a^3*c^3*f*sin(e + f*x)^5)

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