3.1.10 \(\int \frac {(a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^5} \, dx\) [10]

Optimal. Leaf size=164 \[ \frac {a^2 x}{c^5}-\frac {4 a^2 \tan (e+f x)}{9 c^5 f (1-\sec (e+f x))^5}-\frac {16 a^2 \tan (e+f x)}{63 c^5 f (1-\sec (e+f x))^4}-\frac {37 a^2 \tan (e+f x)}{105 c^5 f (1-\sec (e+f x))^3}-\frac {179 a^2 \tan (e+f x)}{315 c^5 f (1-\sec (e+f x))^2}-\frac {494 a^2 \tan (e+f x)}{315 c^5 f (1-\sec (e+f x))} \]

[Out]

a^2*x/c^5-4/9*a^2*tan(f*x+e)/c^5/f/(1-sec(f*x+e))^5-16/63*a^2*tan(f*x+e)/c^5/f/(1-sec(f*x+e))^4-37/105*a^2*tan
(f*x+e)/c^5/f/(1-sec(f*x+e))^3-179/315*a^2*tan(f*x+e)/c^5/f/(1-sec(f*x+e))^2-494/315*a^2*tan(f*x+e)/c^5/f/(1-s
ec(f*x+e))

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Rubi [A]
time = 0.42, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {3988, 3862, 4007, 4004, 3879, 3881, 3882} \begin {gather*} -\frac {494 a^2 \tan (e+f x)}{315 c^5 f (1-\sec (e+f x))}-\frac {179 a^2 \tan (e+f x)}{315 c^5 f (1-\sec (e+f x))^2}-\frac {37 a^2 \tan (e+f x)}{105 c^5 f (1-\sec (e+f x))^3}-\frac {16 a^2 \tan (e+f x)}{63 c^5 f (1-\sec (e+f x))^4}-\frac {4 a^2 \tan (e+f x)}{9 c^5 f (1-\sec (e+f x))^5}+\frac {a^2 x}{c^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*x)/c^5 - (4*a^2*Tan[e + f*x])/(9*c^5*f*(1 - Sec[e + f*x])^5) - (16*a^2*Tan[e + f*x])/(63*c^5*f*(1 - Sec[e
+ f*x])^4) - (37*a^2*Tan[e + f*x])/(105*c^5*f*(1 - Sec[e + f*x])^3) - (179*a^2*Tan[e + f*x])/(315*c^5*f*(1 -
Sec[e + f*x])^2) - (494*a^2*Tan[e + f*x])/(315*c^5*f*(1 - Sec[e + f*x]))

Rule 3862

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 3879

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 3881

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 3882

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 3988

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/c)*csc[e + f*x])^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 4004

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 4007

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]

Rubi steps

\begin {align*} \int \frac {(a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^5} \, dx &=\frac {\int \left (\frac {a^2}{(1-\sec (e+f x))^5}+\frac {2 a^2 \sec (e+f x)}{(1-\sec (e+f x))^5}+\frac {a^2 \sec ^2(e+f x)}{(1-\sec (e+f x))^5}\right ) \, dx}{c^5}\\ &=\frac {a^2 \int \frac {1}{(1-\sec (e+f x))^5} \, dx}{c^5}+\frac {a^2 \int \frac {\sec ^2(e+f x)}{(1-\sec (e+f x))^5} \, dx}{c^5}+\frac {\left (2 a^2\right ) \int \frac {\sec (e+f x)}{(1-\sec (e+f x))^5} \, dx}{c^5}\\ &=-\frac {4 a^2 \tan (e+f x)}{9 c^5 f (1-\sec (e+f x))^5}-\frac {a^2 \int \frac {-9-4 \sec (e+f x)}{(1-\sec (e+f x))^4} \, dx}{9 c^5}-\frac {\left (5 a^2\right ) \int \frac {\sec (e+f x)}{(1-\sec (e+f x))^4} \, dx}{9 c^5}+\frac {\left (8 a^2\right ) \int \frac {\sec (e+f x)}{(1-\sec (e+f x))^4} \, dx}{9 c^5}\\ &=-\frac {4 a^2 \tan (e+f x)}{9 c^5 f (1-\sec (e+f x))^5}-\frac {16 a^2 \tan (e+f x)}{63 c^5 f (1-\sec (e+f x))^4}+\frac {a^2 \int \frac {63+39 \sec (e+f x)}{(1-\sec (e+f x))^3} \, dx}{63 c^5}-\frac {\left (5 a^2\right ) \int \frac {\sec (e+f x)}{(1-\sec (e+f x))^3} \, dx}{21 c^5}+\frac {\left (8 a^2\right ) \int \frac {\sec (e+f x)}{(1-\sec (e+f x))^3} \, dx}{21 c^5}\\ &=-\frac {4 a^2 \tan (e+f x)}{9 c^5 f (1-\sec (e+f x))^5}-\frac {16 a^2 \tan (e+f x)}{63 c^5 f (1-\sec (e+f x))^4}-\frac {37 a^2 \tan (e+f x)}{105 c^5 f (1-\sec (e+f x))^3}-\frac {a^2 \int \frac {-315-204 \sec (e+f x)}{(1-\sec (e+f x))^2} \, dx}{315 c^5}-\frac {\left (2 a^2\right ) \int \frac {\sec (e+f x)}{(1-\sec (e+f x))^2} \, dx}{21 c^5}+\frac {\left (16 a^2\right ) \int \frac {\sec (e+f x)}{(1-\sec (e+f x))^2} \, dx}{105 c^5}\\ &=-\frac {4 a^2 \tan (e+f x)}{9 c^5 f (1-\sec (e+f x))^5}-\frac {16 a^2 \tan (e+f x)}{63 c^5 f (1-\sec (e+f x))^4}-\frac {37 a^2 \tan (e+f x)}{105 c^5 f (1-\sec (e+f x))^3}-\frac {179 a^2 \tan (e+f x)}{315 c^5 f (1-\sec (e+f x))^2}+\frac {a^2 \int \frac {945+519 \sec (e+f x)}{1-\sec (e+f x)} \, dx}{945 c^5}-\frac {\left (2 a^2\right ) \int \frac {\sec (e+f x)}{1-\sec (e+f x)} \, dx}{63 c^5}+\frac {\left (16 a^2\right ) \int \frac {\sec (e+f x)}{1-\sec (e+f x)} \, dx}{315 c^5}\\ &=\frac {a^2 x}{c^5}-\frac {4 a^2 \tan (e+f x)}{9 c^5 f (1-\sec (e+f x))^5}-\frac {16 a^2 \tan (e+f x)}{63 c^5 f (1-\sec (e+f x))^4}-\frac {37 a^2 \tan (e+f x)}{105 c^5 f (1-\sec (e+f x))^3}-\frac {179 a^2 \tan (e+f x)}{315 c^5 f (1-\sec (e+f x))^2}-\frac {2 a^2 \tan (e+f x)}{105 c^5 f (1-\sec (e+f x))}+\frac {\left (488 a^2\right ) \int \frac {\sec (e+f x)}{1-\sec (e+f x)} \, dx}{315 c^5}\\ &=\frac {a^2 x}{c^5}-\frac {4 a^2 \tan (e+f x)}{9 c^5 f (1-\sec (e+f x))^5}-\frac {16 a^2 \tan (e+f x)}{63 c^5 f (1-\sec (e+f x))^4}-\frac {37 a^2 \tan (e+f x)}{105 c^5 f (1-\sec (e+f x))^3}-\frac {179 a^2 \tan (e+f x)}{315 c^5 f (1-\sec (e+f x))^2}-\frac {494 a^2 \tan (e+f x)}{315 c^5 f (1-\sec (e+f x))}\\ \end {align*}

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Mathematica [A]
time = 0.93, size = 283, normalized size = 1.73 \begin {gather*} \frac {a^2 \csc \left (\frac {e}{2}\right ) \csc ^9\left (\frac {1}{2} (e+f x)\right ) \left (39690 f x \cos \left (\frac {f x}{2}\right )-39690 f x \cos \left (e+\frac {f x}{2}\right )-26460 f x \cos \left (e+\frac {3 f x}{2}\right )+26460 f x \cos \left (2 e+\frac {3 f x}{2}\right )+11340 f x \cos \left (2 e+\frac {5 f x}{2}\right )-11340 f x \cos \left (3 e+\frac {5 f x}{2}\right )-2835 f x \cos \left (3 e+\frac {7 f x}{2}\right )+2835 f x \cos \left (4 e+\frac {7 f x}{2}\right )+315 f x \cos \left (4 e+\frac {9 f x}{2}\right )-315 f x \cos \left (5 e+\frac {9 f x}{2}\right )-135198 \sin \left (\frac {f x}{2}\right )-117810 \sin \left (e+\frac {f x}{2}\right )+100002 \sin \left (e+\frac {3 f x}{2}\right )+68670 \sin \left (2 e+\frac {3 f x}{2}\right )-48978 \sin \left (2 e+\frac {5 f x}{2}\right )-23310 \sin \left (3 e+\frac {5 f x}{2}\right )+13662 \sin \left (3 e+\frac {7 f x}{2}\right )+4410 \sin \left (4 e+\frac {7 f x}{2}\right )-2008 \sin \left (4 e+\frac {9 f x}{2}\right )\right )}{161280 c^5 f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*Csc[e/2]*Csc[(e + f*x)/2]^9*(39690*f*x*Cos[(f*x)/2] - 39690*f*x*Cos[e + (f*x)/2] - 26460*f*x*Cos[e + (3*f
*x)/2] + 26460*f*x*Cos[2*e + (3*f*x)/2] + 11340*f*x*Cos[2*e + (5*f*x)/2] - 11340*f*x*Cos[3*e + (5*f*x)/2] - 28
35*f*x*Cos[3*e + (7*f*x)/2] + 2835*f*x*Cos[4*e + (7*f*x)/2] + 315*f*x*Cos[4*e + (9*f*x)/2] - 315*f*x*Cos[5*e +
(9*f*x)/2] - 135198*Sin[(f*x)/2] - 117810*Sin[e + (f*x)/2] + 100002*Sin[e + (3*f*x)/2] + 68670*Sin[2*e + (3*f
*x)/2] - 48978*Sin[2*e + (5*f*x)/2] - 23310*Sin[3*e + (5*f*x)/2] + 13662*Sin[3*e + (7*f*x)/2] + 4410*Sin[4*e +
(7*f*x)/2] - 2008*Sin[4*e + (9*f*x)/2]))/(161280*c^5*f)

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Maple [A]
time = 0.17, size = 90, normalized size = 0.55

method	result	size
derivativedivides	\(\frac {a^{2} \left (8 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {1}{9 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}-\frac {4}{7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}+\frac {7}{5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}-\frac {8}{3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}+\frac {8}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}\right )}{4 f \,c^{5}}\)	\(90\)
default	\(\frac {a^{2} \left (8 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {1}{9 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}-\frac {4}{7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}+\frac {7}{5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}-\frac {8}{3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}+\frac {8}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}\right )}{4 f \,c^{5}}\)	\(90\)
risch	\(\frac {a^{2} x}{c^{5}}+\frac {2 i a^{2} \left (2205 \,{\mathrm e}^{8 i \left (f x +e \right )}-11655 \,{\mathrm e}^{7 i \left (f x +e \right )}+34335 \,{\mathrm e}^{6 i \left (f x +e \right )}-58905 \,{\mathrm e}^{5 i \left (f x +e \right )}+67599 \,{\mathrm e}^{4 i \left (f x +e \right )}-50001 \,{\mathrm e}^{3 i \left (f x +e \right )}+24489 \,{\mathrm e}^{2 i \left (f x +e \right )}-6831 \,{\mathrm e}^{i \left (f x +e \right )}+1004\right )}{315 f \,c^{5} \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{9}}\)	\(125\)
norman	\(\frac {\frac {a^{2} x \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c}-\frac {a^{2}}{36 c f}+\frac {43 a^{2} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{252 c f}-\frac {69 a^{2} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{140 c f}+\frac {61 a^{2} \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{60 c f}-\frac {8 a^{2} \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 c f}+\frac {2 a^{2} \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}-\frac {a^{2} x \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c}}{\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )-1\right ) c^{4} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}\)	\(192\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4/f*a^2/c^5*(8*arctan(tan(1/2*f*x+1/2*e))+1/9/tan(1/2*f*x+1/2*e)^9-4/7/tan(1/2*f*x+1/2*e)^7+7/5/tan(1/2*f*x+
1/2*e)^5-8/3/tan(1/2*f*x+1/2*e)^3+8/tan(1/2*f*x+1/2*e))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 365 vs. \(2 (154) = 308\).
time = 0.52, size = 365, normalized size = 2.23 \begin {gather*} \frac {a^{2} {\left (\frac {10080 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{c^{5}} - \frac {{\left (\frac {270 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {1008 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {2730 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {9765 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - 35\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{9}}{c^{5} \sin \left (f x + e\right )^{9}}\right )} - \frac {2 \, a^{2} {\left (\frac {180 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {378 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {420 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {315 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - 35\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{9}}{c^{5} \sin \left (f x + e\right )^{9}} - \frac {5 \, a^{2} {\left (\frac {18 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {42 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {63 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - 7\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{9}}{c^{5} \sin \left (f x + e\right )^{9}}}{5040 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(f*x+e))^2/(c-c*sec(f*x+e))^5,x, algorithm="maxima")

[Out]

1/5040*(a^2*(10080*arctan(sin(f*x + e)/(cos(f*x + e) + 1))/c^5 - (270*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 10
08*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 2730*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 9765*sin(f*x + e)^8/(cos(f
*x + e) + 1)^8 - 35)*(cos(f*x + e) + 1)^9/(c^5*sin(f*x + e)^9)) - 2*a^2*(180*sin(f*x + e)^2/(cos(f*x + e) + 1)
^2 - 378*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 420*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 315*sin(f*x + e)^8/(c
os(f*x + e) + 1)^8 - 35)*(cos(f*x + e) + 1)^9/(c^5*sin(f*x + e)^9) - 5*a^2*(18*sin(f*x + e)^2/(cos(f*x + e) +
1)^2 - 42*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 63*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 7)*(cos(f*x + e) + 1)
^9/(c^5*sin(f*x + e)^9))/f

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Fricas [A]
time = 3.22, size = 227, normalized size = 1.38 \begin {gather*} \frac {1004 \, a^{2} \cos \left (f x + e\right )^{5} - 1811 \, a^{2} \cos \left (f x + e\right )^{4} + 797 \, a^{2} \cos \left (f x + e\right )^{3} + 1457 \, a^{2} \cos \left (f x + e\right )^{2} - 1661 \, a^{2} \cos \left (f x + e\right ) + 494 \, a^{2} + 315 \, {\left (a^{2} f x \cos \left (f x + e\right )^{4} - 4 \, a^{2} f x \cos \left (f x + e\right )^{3} + 6 \, a^{2} f x \cos \left (f x + e\right )^{2} - 4 \, a^{2} f x \cos \left (f x + e\right ) + a^{2} f x\right )} \sin \left (f x + e\right )}{315 \, {\left (c^{5} f \cos \left (f x + e\right )^{4} - 4 \, c^{5} f \cos \left (f x + e\right )^{3} + 6 \, c^{5} f \cos \left (f x + e\right )^{2} - 4 \, c^{5} f \cos \left (f x + e\right ) + c^{5} f\right )} \sin \left (f x + e\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/315*(1004*a^2*cos(f*x + e)^5 - 1811*a^2*cos(f*x + e)^4 + 797*a^2*cos(f*x + e)^3 + 1457*a^2*cos(f*x + e)^2 -
1661*a^2*cos(f*x + e) + 494*a^2 + 315*(a^2*f*x*cos(f*x + e)^4 - 4*a^2*f*x*cos(f*x + e)^3 + 6*a^2*f*x*cos(f*x +
e)^2 - 4*a^2*f*x*cos(f*x + e) + a^2*f*x)*sin(f*x + e))/((c^5*f*cos(f*x + e)^4 - 4*c^5*f*cos(f*x + e)^3 + 6*c^
5*f*cos(f*x + e)^2 - 4*c^5*f*cos(f*x + e) + c^5*f)*sin(f*x + e))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \frac {a^{2} \left (\int \frac {2 \sec {\left (e + f x \right )}}{\sec ^{5}{\left (e + f x \right )} - 5 \sec ^{4}{\left (e + f x \right )} + 10 \sec ^{3}{\left (e + f x \right )} - 10 \sec ^{2}{\left (e + f x \right )} + 5 \sec {\left (e + f x \right )} - 1}\, dx + \int \frac {\sec ^{2}{\left (e + f x \right )}}{\sec ^{5}{\left (e + f x \right )} - 5 \sec ^{4}{\left (e + f x \right )} + 10 \sec ^{3}{\left (e + f x \right )} - 10 \sec ^{2}{\left (e + f x \right )} + 5 \sec {\left (e + f x \right )} - 1}\, dx + \int \frac {1}{\sec ^{5}{\left (e + f x \right )} - 5 \sec ^{4}{\left (e + f x \right )} + 10 \sec ^{3}{\left (e + f x \right )} - 10 \sec ^{2}{\left (e + f x \right )} + 5 \sec {\left (e + f x \right )} - 1}\, dx\right )}{c^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(f*x+e))**2/(c-c*sec(f*x+e))**5,x)

[Out]

-a**2*(Integral(2*sec(e + f*x)/(sec(e + f*x)**5 - 5*sec(e + f*x)**4 + 10*sec(e + f*x)**3 - 10*sec(e + f*x)**2
+ 5*sec(e + f*x) - 1), x) + Integral(sec(e + f*x)**2/(sec(e + f*x)**5 - 5*sec(e + f*x)**4 + 10*sec(e + f*x)**3
- 10*sec(e + f*x)**2 + 5*sec(e + f*x) - 1), x) + Integral(1/(sec(e + f*x)**5 - 5*sec(e + f*x)**4 + 10*sec(e +
f*x)**3 - 10*sec(e + f*x)**2 + 5*sec(e + f*x) - 1), x))/c**5

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Giac [A]
time = 0.55, size = 104, normalized size = 0.63 \begin {gather*} \frac {\frac {1260 \, {\left (f x + e\right )} a^{2}}{c^{5}} + \frac {2520 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 840 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 441 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 180 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 35 \, a^{2}}{c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9}}}{1260 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1260*(1260*(f*x + e)*a^2/c^5 + (2520*a^2*tan(1/2*f*x + 1/2*e)^8 - 840*a^2*tan(1/2*f*x + 1/2*e)^6 + 441*a^2*t
an(1/2*f*x + 1/2*e)^4 - 180*a^2*tan(1/2*f*x + 1/2*e)^2 + 35*a^2)/(c^5*tan(1/2*f*x + 1/2*e)^9))/f

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Mupad [B]
time = 1.54, size = 146, normalized size = 0.89 \begin {gather*} \frac {a^2\,\left (\frac {{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9}{36}-\frac {{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2}{7}+\frac {7\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4}{20}-\frac {2\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6}{3}+2\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+\left (e+f\,x\right )\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9\right )}{c^5\,f\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a/cos(e + f*x))^2/(c - c/cos(e + f*x))^5,x)

[Out]

(a^2*(cos(e/2 + (f*x)/2)^9/36 + 2*cos(e/2 + (f*x)/2)*sin(e/2 + (f*x)/2)^8 + sin(e/2 + (f*x)/2)^9*(e + f*x) - (
2*cos(e/2 + (f*x)/2)^3*sin(e/2 + (f*x)/2)^6)/3 + (7*cos(e/2 + (f*x)/2)^5*sin(e/2 + (f*x)/2)^4)/20 - (cos(e/2 +
(f*x)/2)^7*sin(e/2 + (f*x)/2)^2)/7))/(c^5*f*sin(e/2 + (f*x)/2)^9)

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