\(\int \frac {(a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^4} \, dx\) [19]

Optimal result

Integrand size = 26, antiderivative size = 133 \[ \int \frac {(a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^4} \, dx=\frac {a^3 x}{c^4}-\frac {8 a^3 \tan (e+f x)}{7 c^4 f (1-\sec (e+f x))^4}+\frac {4 a^3 \tan (e+f x)}{35 c^4 f (1-\sec (e+f x))^3}-\frac {62 a^3 \tan (e+f x)}{105 c^4 f (1-\sec (e+f x))^2}-\frac {167 a^3 \tan (e+f x)}{105 c^4 f (1-\sec (e+f x))} \]

[Out]

a^3*x/c^4-8/7*a^3*tan(f*x+e)/c^4/f/(1-sec(f*x+e))^4+4/35*a^3*tan(f*x+e)/c^4/f/(1-sec(f*x+e))^3-62/105*a^3*tan(
f*x+e)/c^4/f/(1-sec(f*x+e))^2-167/105*a^3*tan(f*x+e)/c^4/f/(1-sec(f*x+e))

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {3988, 3862, 4007, 4004, 3879, 3881, 3882, 3884, 4085} \[ \int \frac {(a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^4} \, dx=-\frac {167 a^3 \tan (e+f x)}{105 c^4 f (1-\sec (e+f x))}-\frac {62 a^3 \tan (e+f x)}{105 c^4 f (1-\sec (e+f x))^2}+\frac {4 a^3 \tan (e+f x)}{35 c^4 f (1-\sec (e+f x))^3}-\frac {8 a^3 \tan (e+f x)}{7 c^4 f (1-\sec (e+f x))^4}+\frac {a^3 x}{c^4} \]

[In]

Int[(a + a*Sec[e + f*x])^3/(c - c*Sec[e + f*x])^4,x]

[Out]

(a^3*x)/c^4 - (8*a^3*Tan[e + f*x])/(7*c^4*f*(1 - Sec[e + f*x])^4) + (4*a^3*Tan[e + f*x])/(35*c^4*f*(1 - Sec[e
+ f*x])^3) - (62*a^3*Tan[e + f*x])/(105*c^4*f*(1 - Sec[e + f*x])^2) - (167*a^3*Tan[e + f*x])/(105*c^4*f*(1 - S
ec[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 3884

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 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]

Rule 4085

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*} \text {integral}& = \frac {\int \left (\frac {a^3}{(1-\sec (e+f x))^4}+\frac {3 a^3 \sec (e+f x)}{(1-\sec (e+f x))^4}+\frac {3 a^3 \sec ^2(e+f x)}{(1-\sec (e+f x))^4}+\frac {a^3 \sec ^3(e+f x)}{(1-\sec (e+f x))^4}\right ) \, dx}{c^4} \\ & = \frac {a^3 \int \frac {1}{(1-\sec (e+f x))^4} \, dx}{c^4}+\frac {a^3 \int \frac {\sec ^3(e+f x)}{(1-\sec (e+f x))^4} \, dx}{c^4}+\frac {\left (3 a^3\right ) \int \frac {\sec (e+f x)}{(1-\sec (e+f x))^4} \, dx}{c^4}+\frac {\left (3 a^3\right ) \int \frac {\sec ^2(e+f x)}{(1-\sec (e+f x))^4} \, dx}{c^4} \\ & = -\frac {8 a^3 \tan (e+f x)}{7 c^4 f (1-\sec (e+f x))^4}-\frac {a^3 \int \frac {-7-3 \sec (e+f x)}{(1-\sec (e+f x))^3} \, dx}{7 c^4}+\frac {a^3 \int \frac {(-4-7 \sec (e+f x)) \sec (e+f x)}{(1-\sec (e+f x))^3} \, dx}{7 c^4}+\frac {\left (9 a^3\right ) \int \frac {\sec (e+f x)}{(1-\sec (e+f x))^3} \, dx}{7 c^4}-\frac {\left (12 a^3\right ) \int \frac {\sec (e+f x)}{(1-\sec (e+f x))^3} \, dx}{7 c^4} \\ & = -\frac {8 a^3 \tan (e+f x)}{7 c^4 f (1-\sec (e+f x))^4}+\frac {4 a^3 \tan (e+f x)}{35 c^4 f (1-\sec (e+f x))^3}+\frac {a^3 \int \frac {35+20 \sec (e+f x)}{(1-\sec (e+f x))^2} \, dx}{35 c^4}+\frac {\left (13 a^3\right ) \int \frac {\sec (e+f x)}{(1-\sec (e+f x))^2} \, dx}{35 c^4}+\frac {\left (18 a^3\right ) \int \frac {\sec (e+f x)}{(1-\sec (e+f x))^2} \, dx}{35 c^4}-\frac {\left (24 a^3\right ) \int \frac {\sec (e+f x)}{(1-\sec (e+f x))^2} \, dx}{35 c^4} \\ & = -\frac {8 a^3 \tan (e+f x)}{7 c^4 f (1-\sec (e+f x))^4}+\frac {4 a^3 \tan (e+f x)}{35 c^4 f (1-\sec (e+f x))^3}-\frac {62 a^3 \tan (e+f x)}{105 c^4 f (1-\sec (e+f x))^2}-\frac {a^3 \int \frac {-105-55 \sec (e+f x)}{1-\sec (e+f x)} \, dx}{105 c^4}+\frac {\left (13 a^3\right ) \int \frac {\sec (e+f x)}{1-\sec (e+f x)} \, dx}{105 c^4}+\frac {\left (6 a^3\right ) \int \frac {\sec (e+f x)}{1-\sec (e+f x)} \, dx}{35 c^4}-\frac {\left (8 a^3\right ) \int \frac {\sec (e+f x)}{1-\sec (e+f x)} \, dx}{35 c^4} \\ & = \frac {a^3 x}{c^4}-\frac {8 a^3 \tan (e+f x)}{7 c^4 f (1-\sec (e+f x))^4}+\frac {4 a^3 \tan (e+f x)}{35 c^4 f (1-\sec (e+f x))^3}-\frac {62 a^3 \tan (e+f x)}{105 c^4 f (1-\sec (e+f x))^2}-\frac {a^3 \tan (e+f x)}{15 c^4 f (1-\sec (e+f x))}+\frac {\left (32 a^3\right ) \int \frac {\sec (e+f x)}{1-\sec (e+f x)} \, dx}{21 c^4} \\ & = \frac {a^3 x}{c^4}-\frac {8 a^3 \tan (e+f x)}{7 c^4 f (1-\sec (e+f x))^4}+\frac {4 a^3 \tan (e+f x)}{35 c^4 f (1-\sec (e+f x))^3}-\frac {62 a^3 \tan (e+f x)}{105 c^4 f (1-\sec (e+f x))^2}-\frac {167 a^3 \tan (e+f x)}{105 c^4 f (1-\sec (e+f x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.46 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.21 \[ \int \frac {(a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^4} \, dx=\frac {a^{5/2} \tan (e+f x) \left (\sqrt {a} \sqrt {c} \left (-337+276 \sec (e+f x)+50 \sec ^2(e+f x)-396 \sec ^3(e+f x)+167 \sec ^4(e+f x)\right )-840 \text {arctanh}\left (\frac {\sqrt {-a c \tan ^2(e+f x)}}{\sqrt {a} \sqrt {c}}\right ) \sec ^3(e+f x) \sin ^6\left (\frac {1}{2} (e+f x)\right ) \sqrt {-a c \tan ^2(e+f x)}\right )}{105 c^{9/2} f (-1+\sec (e+f x))^4 (1+\sec (e+f x))} \]

[In]

Integrate[(a + a*Sec[e + f*x])^3/(c - c*Sec[e + f*x])^4,x]

[Out]

(a^(5/2)*Tan[e + f*x]*(Sqrt[a]*Sqrt[c]*(-337 + 276*Sec[e + f*x] + 50*Sec[e + f*x]^2 - 396*Sec[e + f*x]^3 + 167
*Sec[e + f*x]^4) - 840*ArcTanh[Sqrt[-(a*c*Tan[e + f*x]^2)]/(Sqrt[a]*Sqrt[c])]*Sec[e + f*x]^3*Sin[(e + f*x)/2]^
6*Sqrt[-(a*c*Tan[e + f*x]^2)]))/(105*c^(9/2)*f*(-1 + Sec[e + f*x])^4*(1 + Sec[e + f*x]))

Maple [A] (verified)

Time = 0.64 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.50

method	result	size
parallelrisch	\(-\frac {a^{3} \left (15 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}-42 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+70 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}-105 f x -210 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{105 c^{4} f}\)	\(67\)
derivativedivides	\(\frac {a^{3} \left (-\frac {1}{7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}+\frac {2}{5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}-\frac {2}{3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}+\frac {2}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}+2 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )\right )}{f \,c^{4}}\)	\(76\)
default	\(\frac {a^{3} \left (-\frac {1}{7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}+\frac {2}{5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}-\frac {2}{3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}+\frac {2}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}+2 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )\right )}{f \,c^{4}}\)	\(76\)
risch	\(\frac {a^{3} x}{c^{4}}+\frac {2 i a^{3} \left (735 \,{\mathrm e}^{6 i \left (f x +e \right )}-2520 \,{\mathrm e}^{5 i \left (f x +e \right )}+5635 \,{\mathrm e}^{4 i \left (f x +e \right )}-6160 \,{\mathrm e}^{3 i \left (f x +e \right )}+4557 \,{\mathrm e}^{2 i \left (f x +e \right )}-1624 \,{\mathrm e}^{i \left (f x +e \right )}+337\right )}{105 f \,c^{4} \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{7}}\)	\(103\)
norman	\(\frac {\frac {a^{3} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{c}+\frac {a^{3} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}{c}-\frac {a^{3}}{7 c f}+\frac {24 a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{35 c f}-\frac {169 a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{105 c f}+\frac {56 a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{15 c f}-\frac {14 a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{3 c f}+\frac {2 a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}}{c f}-\frac {2 a^{3} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{c}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{2} c^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}\)	\(211\)

[In]

int((a+a*sec(f*x+e))^3/(c-c*sec(f*x+e))^4,x,method=_RETURNVERBOSE)

[Out]

-1/105*a^3*(15*cot(1/2*f*x+1/2*e)^7-42*cot(1/2*f*x+1/2*e)^5+70*cot(1/2*f*x+1/2*e)^3-105*f*x-210*cot(1/2*f*x+1/
2*e))/c^4/f

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.29 \[ \int \frac {(a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^4} \, dx=\frac {337 \, a^{3} \cos \left (f x + e\right )^{4} - 276 \, a^{3} \cos \left (f x + e\right )^{3} - 50 \, a^{3} \cos \left (f x + e\right )^{2} + 396 \, a^{3} \cos \left (f x + e\right ) - 167 \, a^{3} + 105 \, {\left (a^{3} f x \cos \left (f x + e\right )^{3} - 3 \, a^{3} f x \cos \left (f x + e\right )^{2} + 3 \, a^{3} f x \cos \left (f x + e\right ) - a^{3} f x\right )} \sin \left (f x + e\right )}{105 \, {\left (c^{4} f \cos \left (f x + e\right )^{3} - 3 \, c^{4} f \cos \left (f x + e\right )^{2} + 3 \, c^{4} f \cos \left (f x + e\right ) - c^{4} f\right )} \sin \left (f x + e\right )} \]

[In]

integrate((a+a*sec(f*x+e))^3/(c-c*sec(f*x+e))^4,x, algorithm="fricas")

[Out]

1/105*(337*a^3*cos(f*x + e)^4 - 276*a^3*cos(f*x + e)^3 - 50*a^3*cos(f*x + e)^2 + 396*a^3*cos(f*x + e) - 167*a^
3 + 105*(a^3*f*x*cos(f*x + e)^3 - 3*a^3*f*x*cos(f*x + e)^2 + 3*a^3*f*x*cos(f*x + e) - a^3*f*x)*sin(f*x + e))/(
(c^4*f*cos(f*x + e)^3 - 3*c^4*f*cos(f*x + e)^2 + 3*c^4*f*cos(f*x + e) - c^4*f)*sin(f*x + e))

Sympy [F]

\[ \int \frac {(a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^4} \, dx=\frac {a^{3} \left (\int \frac {3 \sec {\left (e + f x \right )}}{\sec ^{4}{\left (e + f x \right )} - 4 \sec ^{3}{\left (e + f x \right )} + 6 \sec ^{2}{\left (e + f x \right )} - 4 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {3 \sec ^{2}{\left (e + f x \right )}}{\sec ^{4}{\left (e + f x \right )} - 4 \sec ^{3}{\left (e + f x \right )} + 6 \sec ^{2}{\left (e + f x \right )} - 4 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {\sec ^{3}{\left (e + f x \right )}}{\sec ^{4}{\left (e + f x \right )} - 4 \sec ^{3}{\left (e + f x \right )} + 6 \sec ^{2}{\left (e + f x \right )} - 4 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {1}{\sec ^{4}{\left (e + f x \right )} - 4 \sec ^{3}{\left (e + f x \right )} + 6 \sec ^{2}{\left (e + f x \right )} - 4 \sec {\left (e + f x \right )} + 1}\, dx\right )}{c^{4}} \]

[In]

integrate((a+a*sec(f*x+e))**3/(c-c*sec(f*x+e))**4,x)

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 383 vs. \(2 (117) = 234\).

Time = 0.34 (sec) , antiderivative size = 383, normalized size of antiderivative = 2.88 \[ \int \frac {(a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^4} \, dx=\frac {5 \, a^{3} {\left (\frac {336 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{c^{4}} + \frac {{\left (\frac {21 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {77 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {315 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - 3\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{7}}{c^{4} \sin \left (f x + e\right )^{7}}\right )} + \frac {3 \, a^{3} {\left (\frac {21 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {35 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {105 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - 15\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{7}}{c^{4} \sin \left (f x + e\right )^{7}} + \frac {9 \, a^{3} {\left (\frac {21 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {35 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {35 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - 5\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{7}}{c^{4} \sin \left (f x + e\right )^{7}} - \frac {a^{3} {\left (\frac {21 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {35 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {105 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + 15\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{7}}{c^{4} \sin \left (f x + e\right )^{7}}}{840 \, f} \]

[In]

integrate((a+a*sec(f*x+e))^3/(c-c*sec(f*x+e))^4,x, algorithm="maxima")

[Out]

1/840*(5*a^3*(336*arctan(sin(f*x + e)/(cos(f*x + e) + 1))/c^4 + (21*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 77*s
in(f*x + e)^4/(cos(f*x + e) + 1)^4 + 315*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 3)*(cos(f*x + e) + 1)^7/(c^4*si
n(f*x + e)^7)) + 3*a^3*(21*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 35*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 105*
sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 15)*(cos(f*x + e) + 1)^7/(c^4*sin(f*x + e)^7) + 9*a^3*(21*sin(f*x + e)^2
/(cos(f*x + e) + 1)^2 - 35*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 35*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 5)*(
cos(f*x + e) + 1)^7/(c^4*sin(f*x + e)^7) - a^3*(21*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 35*sin(f*x + e)^4/(co
s(f*x + e) + 1)^4 - 105*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 15)*(cos(f*x + e) + 1)^7/(c^4*sin(f*x + e)^7))/f

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.66 \[ \int \frac {(a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^4} \, dx=\frac {\frac {105 \, {\left (f x + e\right )} a^{3}}{c^{4}} + \frac {210 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 70 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 42 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 15 \, a^{3}}{c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7}}}{105 \, f} \]

[In]

integrate((a+a*sec(f*x+e))^3/(c-c*sec(f*x+e))^4,x, algorithm="giac")

[Out]

1/105*(105*(f*x + e)*a^3/c^4 + (210*a^3*tan(1/2*f*x + 1/2*e)^6 - 70*a^3*tan(1/2*f*x + 1/2*e)^4 + 42*a^3*tan(1/
2*f*x + 1/2*e)^2 - 15*a^3)/(c^4*tan(1/2*f*x + 1/2*e)^7))/f

Mupad [B] (verification not implemented)

Time = 14.80 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.92 \[ \int \frac {(a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^4} \, dx=\frac {a^3\,\left (-\frac {{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7}{7}+\frac {2\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2}{5}-\frac {2\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4}{3}+2\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+\left (e+f\,x\right )\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\right )}{c^4\,f\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7} \]

[In]

int((a + a/cos(e + f*x))^3/(c - c/cos(e + f*x))^4,x)

[Out]

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