3.1.31 \(\int \frac {(c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^3} \, dx\) [31]

Optimal. Leaf size=162 \[ \frac {c^5 x}{a^3}+\frac {8 c^5 \tanh ^{-1}(\sin (e+f x))}{a^3 f}+\frac {32 c^5 \cot (e+f x)}{a^3 f}+\frac {128 c^5 \cot ^3(e+f x)}{3 a^3 f}+\frac {128 c^5 \cot ^5(e+f x)}{5 a^3 f}-\frac {16 c^5 \csc (e+f x)}{a^3 f}+\frac {64 c^5 \csc ^3(e+f x)}{3 a^3 f}-\frac {128 c^5 \csc ^5(e+f x)}{5 a^3 f}-\frac {c^5 \tan (e+f x)}{a^3 f} \]

[Out]

c^5*x/a^3+8*c^5*arctanh(sin(f*x+e))/a^3/f+32*c^5*cot(f*x+e)/a^3/f+128/3*c^5*cot(f*x+e)^3/a^3/f+128/5*c^5*cot(f
*x+e)^5/a^3/f-16*c^5*csc(f*x+e)/a^3/f+64/3*c^5*csc(f*x+e)^3/a^3/f-128/5*c^5*csc(f*x+e)^5/a^3/f-c^5*tan(f*x+e)/
a^3/f

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Rubi [A]
time = 0.32, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 29, number of rules used = 15, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.577, Rules used = {3989, 3971, 3554, 8, 2686, 200, 2687, 30, 14, 3852, 2701, 308, 213, 2700, 276} \begin {gather*} -\frac {c^5 \tan (e+f x)}{a^3 f}+\frac {128 c^5 \cot ^5(e+f x)}{5 a^3 f}+\frac {128 c^5 \cot ^3(e+f x)}{3 a^3 f}+\frac {32 c^5 \cot (e+f x)}{a^3 f}-\frac {128 c^5 \csc ^5(e+f x)}{5 a^3 f}+\frac {64 c^5 \csc ^3(e+f x)}{3 a^3 f}-\frac {16 c^5 \csc (e+f x)}{a^3 f}+\frac {8 c^5 \tanh ^{-1}(\sin (e+f x))}{a^3 f}+\frac {c^5 x}{a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c^5*x)/a^3 + (8*c^5*ArcTanh[Sin[e + f*x]])/(a^3*f) + (32*c^5*Cot[e + f*x])/(a^3*f) + (128*c^5*Cot[e + f*x]^3)
/(3*a^3*f) + (128*c^5*Cot[e + f*x]^5)/(5*a^3*f) - (16*c^5*Csc[e + f*x])/(a^3*f) + (64*c^5*Csc[e + f*x]^3)/(3*a
^3*f) - (128*c^5*Csc[e + f*x]^5)/(5*a^3*f) - (c^5*Tan[e + f*x])/(a^3*f)

Rule 8

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

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 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 276

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

Rule 308

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

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 2700

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

Rule 2701

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

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 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 3971

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Int[ExpandI
ntegrand[(e*Cot[c + d*x])^m, (a + b*Csc[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0]

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 {(c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^3} \, dx &=-\frac {\int \cot ^6(e+f x) (c-c \sec (e+f x))^8 \, dx}{a^3 c^3}\\ &=-\frac {\int \left (c^8 \cot ^6(e+f x)-8 c^8 \cot ^5(e+f x) \csc (e+f x)+28 c^8 \cot ^4(e+f x) \csc ^2(e+f x)-56 c^8 \cot ^3(e+f x) \csc ^3(e+f x)+70 c^8 \cot ^2(e+f x) \csc ^4(e+f x)-56 c^8 \cot (e+f x) \csc ^5(e+f x)+28 c^8 \csc ^6(e+f x)-8 c^8 \csc ^6(e+f x) \sec (e+f x)+c^8 \csc ^6(e+f x) \sec ^2(e+f x)\right ) \, dx}{a^3 c^3}\\ &=-\frac {c^5 \int \cot ^6(e+f x) \, dx}{a^3}-\frac {c^5 \int \csc ^6(e+f x) \sec ^2(e+f x) \, dx}{a^3}+\frac {\left (8 c^5\right ) \int \cot ^5(e+f x) \csc (e+f x) \, dx}{a^3}+\frac {\left (8 c^5\right ) \int \csc ^6(e+f x) \sec (e+f x) \, dx}{a^3}-\frac {\left (28 c^5\right ) \int \cot ^4(e+f x) \csc ^2(e+f x) \, dx}{a^3}-\frac {\left (28 c^5\right ) \int \csc ^6(e+f x) \, dx}{a^3}+\frac {\left (56 c^5\right ) \int \cot ^3(e+f x) \csc ^3(e+f x) \, dx}{a^3}+\frac {\left (56 c^5\right ) \int \cot (e+f x) \csc ^5(e+f x) \, dx}{a^3}-\frac {\left (70 c^5\right ) \int \cot ^2(e+f x) \csc ^4(e+f x) \, dx}{a^3}\\ &=\frac {c^5 \cot ^5(e+f x)}{5 a^3 f}+\frac {c^5 \int \cot ^4(e+f x) \, dx}{a^3}-\frac {c^5 \text {Subst}\left (\int \frac {\left (1+x^2\right )^3}{x^6} \, dx,x,\tan (e+f x)\right )}{a^3 f}-\frac {\left (8 c^5\right ) \text {Subst}\left (\int \frac {x^6}{-1+x^2} \, dx,x,\csc (e+f x)\right )}{a^3 f}-\frac {\left (8 c^5\right ) \text {Subst}\left (\int \left (-1+x^2\right )^2 \, dx,x,\csc (e+f x)\right )}{a^3 f}-\frac {\left (28 c^5\right ) \text {Subst}\left (\int x^4 \, dx,x,-\cot (e+f x)\right )}{a^3 f}+\frac {\left (28 c^5\right ) \text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (e+f x)\right )}{a^3 f}-\frac {\left (56 c^5\right ) \text {Subst}\left (\int x^4 \, dx,x,\csc (e+f x)\right )}{a^3 f}-\frac {\left (56 c^5\right ) \text {Subst}\left (\int x^2 \left (-1+x^2\right ) \, dx,x,\csc (e+f x)\right )}{a^3 f}-\frac {\left (70 c^5\right ) \text {Subst}\left (\int x^2 \left (1+x^2\right ) \, dx,x,-\cot (e+f x)\right )}{a^3 f}\\ &=\frac {28 c^5 \cot (e+f x)}{a^3 f}+\frac {55 c^5 \cot ^3(e+f x)}{3 a^3 f}+\frac {57 c^5 \cot ^5(e+f x)}{5 a^3 f}-\frac {56 c^5 \csc ^5(e+f x)}{5 a^3 f}-\frac {c^5 \int \cot ^2(e+f x) \, dx}{a^3}-\frac {c^5 \text {Subst}\left (\int \left (1+\frac {1}{x^6}+\frac {3}{x^4}+\frac {3}{x^2}\right ) \, dx,x,\tan (e+f x)\right )}{a^3 f}-\frac {\left (8 c^5\right ) \text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\csc (e+f x)\right )}{a^3 f}-\frac {\left (8 c^5\right ) \text {Subst}\left (\int \left (1+x^2+x^4+\frac {1}{-1+x^2}\right ) \, dx,x,\csc (e+f x)\right )}{a^3 f}-\frac {\left (56 c^5\right ) \text {Subst}\left (\int \left (-x^2+x^4\right ) \, dx,x,\csc (e+f x)\right )}{a^3 f}-\frac {\left (70 c^5\right ) \text {Subst}\left (\int \left (x^2+x^4\right ) \, dx,x,-\cot (e+f x)\right )}{a^3 f}\\ &=\frac {32 c^5 \cot (e+f x)}{a^3 f}+\frac {128 c^5 \cot ^3(e+f x)}{3 a^3 f}+\frac {128 c^5 \cot ^5(e+f x)}{5 a^3 f}-\frac {16 c^5 \csc (e+f x)}{a^3 f}+\frac {64 c^5 \csc ^3(e+f x)}{3 a^3 f}-\frac {128 c^5 \csc ^5(e+f x)}{5 a^3 f}-\frac {c^5 \tan (e+f x)}{a^3 f}+\frac {c^5 \int 1 \, dx}{a^3}-\frac {\left (8 c^5\right ) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (e+f x)\right )}{a^3 f}\\ &=\frac {c^5 x}{a^3}+\frac {8 c^5 \tanh ^{-1}(\sin (e+f x))}{a^3 f}+\frac {32 c^5 \cot (e+f x)}{a^3 f}+\frac {128 c^5 \cot ^3(e+f x)}{3 a^3 f}+\frac {128 c^5 \cot ^5(e+f x)}{5 a^3 f}-\frac {16 c^5 \csc (e+f x)}{a^3 f}+\frac {64 c^5 \csc ^3(e+f x)}{3 a^3 f}-\frac {128 c^5 \csc ^5(e+f x)}{5 a^3 f}-\frac {c^5 \tan (e+f x)}{a^3 f}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(557\) vs. \(2(162)=324\).
time = 6.17, size = 557, normalized size = 3.44 \begin {gather*} -\frac {c^5 (-1+\cos (e+f x))^5 \cot \left (\frac {1}{2} (e+f x)\right ) \csc ^4\left (\frac {1}{2} (e+f x)\right ) \sec \left (\frac {e}{2}\right ) \left (-60 \cos (e) \cot ^7\left (\frac {1}{2} (e+f x)\right ) \left (f x-8 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )+8 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )\right ) \sec \left (\frac {e}{2}\right )+48 \cot \left (\frac {1}{2} (e+f x)\right ) \csc ^4\left (\frac {1}{2} (e+f x)\right ) \sec ^2\left (\frac {e}{2}\right ) \left (\sin \left (\frac {e}{2}\right )-\sin \left (\frac {3 e}{2}\right )\right )+8 (-7+\cos (e+f x)) \cot ^3\left (\frac {1}{2} (e+f x)\right ) \csc ^4\left (\frac {1}{2} (e+f x)\right ) \sec ^2\left (\frac {e}{2}\right ) \left (\sin \left (\frac {e}{2}\right )-\sin \left (\frac {3 e}{2}\right )\right )+1016 \cot ^6\left (\frac {1}{2} (e+f x)\right ) \csc \left (\frac {1}{2} (e+f x)\right ) \sin \left (\frac {f x}{2}\right )+(-140+76 \cos (e)+131 \cos (f x)-210 \cos (e+f x)-84 \cos (2 (e+f x))-14 \cos (3 (e+f x))+131 \cos (2 e+f x)+66 \cos (e+2 f x)+66 \cos (3 e+2 f x)+21 \cos (2 e+3 f x)+21 \cos (4 e+3 f x)) \csc ^7\left (\frac {1}{2} (e+f x)\right ) \sec ^2\left (\frac {e}{2}\right ) \sin \left (\frac {f x}{2}\right )+2 \cot ^5\left (\frac {1}{2} (e+f x)\right ) \sec \left (\frac {e}{2}\right ) \left (30 \cos (e) \left (f x-8 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )+8 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )\right )-(\cos (e)+15 (-1+\cos (f x)+\cos (e+f x))) \csc ^2\left (\frac {1}{2} (e+f x)\right ) \tan \left (\frac {e}{2}\right )\right )\right )}{240 a^3 f (1+\cos (e+f x))^3 \left (-1+\cot \left (\frac {1}{2} (e+f x)\right )\right ) \left (1+\cot \left (\frac {1}{2} (e+f x)\right )\right ) \left (-1+\tan \left (\frac {e}{2}\right )\right ) \left (1+\tan \left (\frac {e}{2}\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/240*(c^5*(-1 + Cos[e + f*x])^5*Cot[(e + f*x)/2]*Csc[(e + f*x)/2]^4*Sec[e/2]*(-60*Cos[e]*Cot[(e + f*x)/2]^7*
(f*x - 8*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 8*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]])*Sec[e/2] + 48*
Cot[(e + f*x)/2]*Csc[(e + f*x)/2]^4*Sec[e/2]^2*(Sin[e/2] - Sin[(3*e)/2]) + 8*(-7 + Cos[e + f*x])*Cot[(e + f*x)
/2]^3*Csc[(e + f*x)/2]^4*Sec[e/2]^2*(Sin[e/2] - Sin[(3*e)/2]) + 1016*Cot[(e + f*x)/2]^6*Csc[(e + f*x)/2]*Sin[(
f*x)/2] + (-140 + 76*Cos[e] + 131*Cos[f*x] - 210*Cos[e + f*x] - 84*Cos[2*(e + f*x)] - 14*Cos[3*(e + f*x)] + 13
1*Cos[2*e + f*x] + 66*Cos[e + 2*f*x] + 66*Cos[3*e + 2*f*x] + 21*Cos[2*e + 3*f*x] + 21*Cos[4*e + 3*f*x])*Csc[(e
+ f*x)/2]^7*Sec[e/2]^2*Sin[(f*x)/2] + 2*Cot[(e + f*x)/2]^5*Sec[e/2]*(30*Cos[e]*(f*x - 8*Log[Cos[(e + f*x)/2]
- Sin[(e + f*x)/2]] + 8*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]) - (Cos[e] + 15*(-1 + Cos[f*x] + Cos[e + f*x]
))*Csc[(e + f*x)/2]^2*Tan[e/2])))/(a^3*f*(1 + Cos[e + f*x])^3*(-1 + Cot[(e + f*x)/2])*(1 + Cot[(e + f*x)/2])*(
-1 + Tan[e/2])*(1 + Tan[e/2]))

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Maple [A]
time = 0.17, size = 118, normalized size = 0.73

method	result	size
derivativedivides	\(\frac {8 c^{5} \left (-\frac {\left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5}-\frac {\left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}-2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+\frac {1}{8 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-8}-\ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )+\frac {\arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4}+\ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )+\frac {1}{8 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+8}\right )}{f \,a^{3}}\)	\(118\)
default	\(\frac {8 c^{5} \left (-\frac {\left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5}-\frac {\left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}-2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+\frac {1}{8 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-8}-\ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )+\frac {\arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4}+\ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )+\frac {1}{8 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+8}\right )}{f \,a^{3}}\)	\(118\)
risch	\(\frac {c^{5} x}{a^{3}}-\frac {2 i c^{5} \left (240 \,{\mathrm e}^{6 i \left (f x +e \right )}+735 \,{\mathrm e}^{5 i \left (f x +e \right )}+1835 \,{\mathrm e}^{4 i \left (f x +e \right )}+1750 \,{\mathrm e}^{3 i \left (f x +e \right )}+1894 \,{\mathrm e}^{2 i \left (f x +e \right )}+955 \,{\mathrm e}^{i \left (f x +e \right )}+239\right )}{15 f \,a^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{5} \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}+\frac {8 c^{5} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{a^{3} f}-\frac {8 c^{5} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}{a^{3} f}\)	\(164\)
norman	\(\frac {\frac {c^{5} x}{a}+\frac {c^{5} x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}-\frac {4 c^{5} x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {6 c^{5} x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}-\frac {4 c^{5} x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}-\frac {18 c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a f}+\frac {202 c^{5} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a f}-\frac {1394 c^{5} \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 a f}+\frac {282 c^{5} \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5 a f}-\frac {224 c^{5} \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 a f}+\frac {56 c^{5} \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 a f}-\frac {8 c^{5} \left (\tan ^{13}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5 a f}}{\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4} a^{2}}-\frac {8 c^{5} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{a^{3} f}+\frac {8 c^{5} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{a^{3} f}\)	\(307\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c-c*sec(f*x+e))^5/(a+a*sec(f*x+e))^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 610 vs. \(2 (162) = 324\).
time = 0.50, size = 610, normalized size = 3.77 \begin {gather*} -\frac {3 \, c^{5} {\left (\frac {40 \, \sin \left (f x + e\right )}{{\left (a^{3} - \frac {a^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (f x + e\right ) + 1\right )}} + \frac {\frac {85 \, \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 {\sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}}{a^{3}} - \frac {60 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a^{3}} + \frac {60 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a^{3}}\right )} + 5 \, c^{5} {\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 {60 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a^{3}} + \frac {60 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a^{3}}\right )} + c^{5} {\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 {10 \, c^{5} {\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 {5 \, c^{5} {\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 {30 \, c^{5} {\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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c*sec(f*x+e))^5/(a+a*sec(f*x+e))^3,x, algorithm="maxima")

[Out]

-1/60*(3*c^5*(40*sin(f*x + e)/((a^3 - a^3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2)*(cos(f*x + e) + 1)) + (85*sin(f
*x + e)/(cos(f*x + e) + 1) + 10*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + sin(f*x + e)^5/(cos(f*x + e) + 1)^5)/a^3
- 60*log(sin(f*x + e)/(cos(f*x + e) + 1) + 1)/a^3 + 60*log(sin(f*x + e)/(cos(f*x + e) + 1) - 1)/a^3) + 5*c^5*
((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 - 60*log(sin(f*x + e)/(cos(f*x + e) + 1) + 1)/a^3 + 60*log(sin(f*x + e)/(cos(f*x + e) + 1) - 1)/
a^3) + c^5*((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) + 10*c^5*(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 + 5*c^5*(15*s
in(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 - 30*c^5*(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 = 2.62, size = 311, normalized size = 1.92 \begin {gather*} \frac {15 \, c^{5} f x \cos \left (f x + e\right )^{4} + 45 \, c^{5} f x \cos \left (f x + e\right )^{3} + 45 \, c^{5} f x \cos \left (f x + e\right )^{2} + 15 \, c^{5} f x \cos \left (f x + e\right ) + 60 \, {\left (c^{5} \cos \left (f x + e\right )^{4} + 3 \, c^{5} \cos \left (f x + e\right )^{3} + 3 \, c^{5} \cos \left (f x + e\right )^{2} + c^{5} \cos \left (f x + e\right )\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) - 60 \, {\left (c^{5} \cos \left (f x + e\right )^{4} + 3 \, c^{5} \cos \left (f x + e\right )^{3} + 3 \, c^{5} \cos \left (f x + e\right )^{2} + c^{5} \cos \left (f x + e\right )\right )} \log \left (-\sin \left (f x + e\right ) + 1\right ) - {\left (239 \, c^{5} \cos \left (f x + e\right )^{3} + 477 \, c^{5} \cos \left (f x + e\right )^{2} + 349 \, c^{5} \cos \left (f x + e\right ) + 15 \, c^{5}\right )} \sin \left (f x + e\right )}{15 \, {\left (a^{3} f \cos \left (f x + e\right )^{4} + 3 \, a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} + a^{3} f \cos \left (f x + e\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c*sec(f*x+e))^5/(a+a*sec(f*x+e))^3,x, algorithm="fricas")

[Out]

1/15*(15*c^5*f*x*cos(f*x + e)^4 + 45*c^5*f*x*cos(f*x + e)^3 + 45*c^5*f*x*cos(f*x + e)^2 + 15*c^5*f*x*cos(f*x +
e) + 60*(c^5*cos(f*x + e)^4 + 3*c^5*cos(f*x + e)^3 + 3*c^5*cos(f*x + e)^2 + c^5*cos(f*x + e))*log(sin(f*x + e
) + 1) - 60*(c^5*cos(f*x + e)^4 + 3*c^5*cos(f*x + e)^3 + 3*c^5*cos(f*x + e)^2 + c^5*cos(f*x + e))*log(-sin(f*x
+ e) + 1) - (239*c^5*cos(f*x + e)^3 + 477*c^5*cos(f*x + e)^2 + 349*c^5*cos(f*x + e) + 15*c^5)*sin(f*x + e))/(
a^3*f*cos(f*x + e)^4 + 3*a^3*f*cos(f*x + e)^3 + 3*a^3*f*cos(f*x + e)^2 + a^3*f*cos(f*x + e))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c*sec(f*x+e))**5/(a+a*sec(f*x+e))**3,x)

[Out]

-c**5*(Integral(5*sec(e + f*x)/(sec(e + f*x)**3 + 3*sec(e + f*x)**2 + 3*sec(e + f*x) + 1), x) + Integral(-10*s
ec(e + f*x)**2/(sec(e + f*x)**3 + 3*sec(e + f*x)**2 + 3*sec(e + f*x) + 1), x) + Integral(10*sec(e + f*x)**3/(s
ec(e + f*x)**3 + 3*sec(e + f*x)**2 + 3*sec(e + f*x) + 1), x) + Integral(-5*sec(e + f*x)**4/(sec(e + f*x)**3 +
3*sec(e + f*x)**2 + 3*sec(e + f*x) + 1), x) + Integral(sec(e + f*x)**5/(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 = 0.56, size = 154, normalized size = 0.95 \begin {gather*} \frac {\frac {15 \, {\left (f x + e\right )} c^{5}}{a^{3}} + \frac {120 \, c^{5} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{a^{3}} - \frac {120 \, c^{5} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{a^{3}} + \frac {30 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )} a^{3}} - \frac {8 \, {\left (3 \, a^{12} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 5 \, a^{12} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 30 \, a^{12} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{15}}}{15 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c*sec(f*x+e))^5/(a+a*sec(f*x+e))^3,x, algorithm="giac")

[Out]

1/15*(15*(f*x + e)*c^5/a^3 + 120*c^5*log(abs(tan(1/2*f*x + 1/2*e) + 1))/a^3 - 120*c^5*log(abs(tan(1/2*f*x + 1/
2*e) - 1))/a^3 + 30*c^5*tan(1/2*f*x + 1/2*e)/((tan(1/2*f*x + 1/2*e)^2 - 1)*a^3) - 8*(3*a^12*c^5*tan(1/2*f*x +
1/2*e)^5 + 5*a^12*c^5*tan(1/2*f*x + 1/2*e)^3 + 30*a^12*c^5*tan(1/2*f*x + 1/2*e))/a^15)/f

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Mupad [B]
time = 1.49, size = 134, normalized size = 0.83 \begin {gather*} \frac {c^5\,x}{a^3}-\frac {16\,c^5\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{a^3\,f}-\frac {8\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{3\,a^3\,f}-\frac {8\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5}{5\,a^3\,f}+\frac {16\,c^5\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{a^3\,f}+\frac {2\,c^5\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{f\,\left (a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-a^3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c - c/cos(e + f*x))^5/(a + a/cos(e + f*x))^3,x)

[Out]

(c^5*x)/a^3 - (16*c^5*tan(e/2 + (f*x)/2))/(a^3*f) - (8*c^5*tan(e/2 + (f*x)/2)^3)/(3*a^3*f) - (8*c^5*tan(e/2 +
(f*x)/2)^5)/(5*a^3*f) + (16*c^5*atanh(tan(e/2 + (f*x)/2)))/(a^3*f) + (2*c^5*tan(e/2 + (f*x)/2))/(f*(a^3*tan(e/
2 + (f*x)/2)^2 - a^3))

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