3.1.22 \(\int \frac {(c-c \sec (e+f x))^4}{(a+a \sec (e+f x))^2} \, dx\) [22]

Optimal. Leaf size=102 \[ \frac {c^4 x}{a^2}-\frac {6 c^4 \tanh ^{-1}(\sin (e+f x))}{a^2 f}-\frac {16 c^4 \cot (e+f x)}{a^2 f}-\frac {32 c^4 \cot ^3(e+f x)}{3 a^2 f}+\frac {32 c^4 \csc ^3(e+f x)}{3 a^2 f}+\frac {c^4 \tan (e+f x)}{a^2 f} \]

[Out]

c^4*x/a^2-6*c^4*arctanh(sin(f*x+e))/a^2/f-16*c^4*cot(f*x+e)/a^2/f-32/3*c^4*cot(f*x+e)^3/a^2/f+32/3*c^4*csc(f*x
+e)^3/a^2/f+c^4*tan(f*x+e)/a^2/f

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Rubi [A]
time = 0.22, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 13, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3989, 3971, 3554, 8, 2686, 2687, 30, 3852, 2701, 308, 213, 2700, 276} \begin {gather*} \frac {c^4 \tan (e+f x)}{a^2 f}-\frac {32 c^4 \cot ^3(e+f x)}{3 a^2 f}-\frac {16 c^4 \cot (e+f x)}{a^2 f}+\frac {32 c^4 \csc ^3(e+f x)}{3 a^2 f}-\frac {6 c^4 \tanh ^{-1}(\sin (e+f x))}{a^2 f}+\frac {c^4 x}{a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c - c*Sec[e + f*x])^4/(a + a*Sec[e + f*x])^2,x]

[Out]

(c^4*x)/a^2 - (6*c^4*ArcTanh[Sin[e + f*x]])/(a^2*f) - (16*c^4*Cot[e + f*x])/(a^2*f) - (32*c^4*Cot[e + f*x]^3)/
(3*a^2*f) + (32*c^4*Csc[e + f*x]^3)/(3*a^2*f) + (c^4*Tan[e + f*x])/(a^2*f)

Rule 8

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

Rule 30

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

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))^4}{(a+a \sec (e+f x))^2} \, dx &=\frac {\int \cot ^4(e+f x) (c-c \sec (e+f x))^6 \, dx}{a^2 c^2}\\ &=\frac {\int \left (c^6 \cot ^4(e+f x)-6 c^6 \cot ^3(e+f x) \csc (e+f x)+15 c^6 \cot ^2(e+f x) \csc ^2(e+f x)-20 c^6 \cot (e+f x) \csc ^3(e+f x)+15 c^6 \csc ^4(e+f x)-6 c^6 \csc ^4(e+f x) \sec (e+f x)+c^6 \csc ^4(e+f x) \sec ^2(e+f x)\right ) \, dx}{a^2 c^2}\\ &=\frac {c^4 \int \cot ^4(e+f x) \, dx}{a^2}+\frac {c^4 \int \csc ^4(e+f x) \sec ^2(e+f x) \, dx}{a^2}-\frac {\left (6 c^4\right ) \int \cot ^3(e+f x) \csc (e+f x) \, dx}{a^2}-\frac {\left (6 c^4\right ) \int \csc ^4(e+f x) \sec (e+f x) \, dx}{a^2}+\frac {\left (15 c^4\right ) \int \cot ^2(e+f x) \csc ^2(e+f x) \, dx}{a^2}+\frac {\left (15 c^4\right ) \int \csc ^4(e+f x) \, dx}{a^2}-\frac {\left (20 c^4\right ) \int \cot (e+f x) \csc ^3(e+f x) \, dx}{a^2}\\ &=-\frac {c^4 \cot ^3(e+f x)}{3 a^2 f}-\frac {c^4 \int \cot ^2(e+f x) \, dx}{a^2}+\frac {c^4 \text {Subst}\left (\int \frac {\left (1+x^2\right )^2}{x^4} \, dx,x,\tan (e+f x)\right )}{a^2 f}+\frac {\left (6 c^4\right ) \text {Subst}\left (\int \frac {x^4}{-1+x^2} \, dx,x,\csc (e+f x)\right )}{a^2 f}+\frac {\left (6 c^4\right ) \text {Subst}\left (\int \left (-1+x^2\right ) \, dx,x,\csc (e+f x)\right )}{a^2 f}+\frac {\left (15 c^4\right ) \text {Subst}\left (\int x^2 \, dx,x,-\cot (e+f x)\right )}{a^2 f}-\frac {\left (15 c^4\right ) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (e+f x)\right )}{a^2 f}+\frac {\left (20 c^4\right ) \text {Subst}\left (\int x^2 \, dx,x,\csc (e+f x)\right )}{a^2 f}\\ &=-\frac {14 c^4 \cot (e+f x)}{a^2 f}-\frac {31 c^4 \cot ^3(e+f x)}{3 a^2 f}-\frac {6 c^4 \csc (e+f x)}{a^2 f}+\frac {26 c^4 \csc ^3(e+f x)}{3 a^2 f}+\frac {c^4 \int 1 \, dx}{a^2}+\frac {c^4 \text {Subst}\left (\int \left (1+\frac {1}{x^4}+\frac {2}{x^2}\right ) \, dx,x,\tan (e+f x)\right )}{a^2 f}+\frac {\left (6 c^4\right ) \text {Subst}\left (\int \left (1+x^2+\frac {1}{-1+x^2}\right ) \, dx,x,\csc (e+f x)\right )}{a^2 f}\\ &=\frac {c^4 x}{a^2}-\frac {16 c^4 \cot (e+f x)}{a^2 f}-\frac {32 c^4 \cot ^3(e+f x)}{3 a^2 f}+\frac {32 c^4 \csc ^3(e+f x)}{3 a^2 f}+\frac {c^4 \tan (e+f x)}{a^2 f}+\frac {\left (6 c^4\right ) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (e+f x)\right )}{a^2 f}\\ &=\frac {c^4 x}{a^2}-\frac {6 c^4 \tanh ^{-1}(\sin (e+f x))}{a^2 f}-\frac {16 c^4 \cot (e+f x)}{a^2 f}-\frac {32 c^4 \cot ^3(e+f x)}{3 a^2 f}+\frac {32 c^4 \csc ^3(e+f x)}{3 a^2 f}+\frac {c^4 \tan (e+f x)}{a^2 f}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(753\) vs. \(2(102)=204\).
time = 6.28, size = 753, normalized size = 7.38 \begin {gather*} \frac {x \cos ^2(e+f x) \cot ^4\left (\frac {e}{2}+\frac {f x}{2}\right ) \csc ^4\left (\frac {e}{2}+\frac {f x}{2}\right ) (c-c \sec (e+f x))^4}{4 (a+a \sec (e+f x))^2}+\frac {3 \cos ^2(e+f x) \cot ^4\left (\frac {e}{2}+\frac {f x}{2}\right ) \csc ^4\left (\frac {e}{2}+\frac {f x}{2}\right ) \log \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )-\sin \left (\frac {e}{2}+\frac {f x}{2}\right )\right ) (c-c \sec (e+f x))^4}{2 f (a+a \sec (e+f x))^2}-\frac {3 \cos ^2(e+f x) \cot ^4\left (\frac {e}{2}+\frac {f x}{2}\right ) \csc ^4\left (\frac {e}{2}+\frac {f x}{2}\right ) \log \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )+\sin \left (\frac {e}{2}+\frac {f x}{2}\right )\right ) (c-c \sec (e+f x))^4}{2 f (a+a \sec (e+f x))^2}+\frac {4 \cos ^2(e+f x) \cot ^3\left (\frac {e}{2}+\frac {f x}{2}\right ) \csc ^5\left (\frac {e}{2}+\frac {f x}{2}\right ) \sec \left (\frac {e}{2}\right ) (c-c \sec (e+f x))^4 \sin \left (\frac {f x}{2}\right )}{3 f (a+a \sec (e+f x))^2}+\frac {2 \cos ^2(e+f x) \cot \left (\frac {e}{2}+\frac {f x}{2}\right ) \csc ^7\left (\frac {e}{2}+\frac {f x}{2}\right ) \sec \left (\frac {e}{2}\right ) (c-c \sec (e+f x))^4 \sin \left (\frac {f x}{2}\right )}{3 f (a+a \sec (e+f x))^2}+\frac {\cos ^2(e+f x) \cot ^4\left (\frac {e}{2}+\frac {f x}{2}\right ) \csc ^4\left (\frac {e}{2}+\frac {f x}{2}\right ) (c-c \sec (e+f x))^4 \sin \left (\frac {f x}{2}\right )}{4 f (a+a \sec (e+f x))^2 \left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )-\sin \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}+\frac {\cos ^2(e+f x) \cot ^4\left (\frac {e}{2}+\frac {f x}{2}\right ) \csc ^4\left (\frac {e}{2}+\frac {f x}{2}\right ) (c-c \sec (e+f x))^4 \sin \left (\frac {f x}{2}\right )}{4 f (a+a \sec (e+f x))^2 \left (\cos \left (\frac {e}{2}\right )+\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )+\sin \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}+\frac {2 \cos ^2(e+f x) \cot ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \csc ^6\left (\frac {e}{2}+\frac {f x}{2}\right ) (c-c \sec (e+f x))^4 \tan \left (\frac {e}{2}\right )}{3 f (a+a \sec (e+f x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c - c*Sec[e + f*x])^4/(a + a*Sec[e + f*x])^2,x]

[Out]

(x*Cos[e + f*x]^2*Cot[e/2 + (f*x)/2]^4*Csc[e/2 + (f*x)/2]^4*(c - c*Sec[e + f*x])^4)/(4*(a + a*Sec[e + f*x])^2)
+ (3*Cos[e + f*x]^2*Cot[e/2 + (f*x)/2]^4*Csc[e/2 + (f*x)/2]^4*Log[Cos[e/2 + (f*x)/2] - Sin[e/2 + (f*x)/2]]*(c
- c*Sec[e + f*x])^4)/(2*f*(a + a*Sec[e + f*x])^2) - (3*Cos[e + f*x]^2*Cot[e/2 + (f*x)/2]^4*Csc[e/2 + (f*x)/2]
^4*Log[Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2]]*(c - c*Sec[e + f*x])^4)/(2*f*(a + a*Sec[e + f*x])^2) + (4*Cos[
e + f*x]^2*Cot[e/2 + (f*x)/2]^3*Csc[e/2 + (f*x)/2]^5*Sec[e/2]*(c - c*Sec[e + f*x])^4*Sin[(f*x)/2])/(3*f*(a + a
*Sec[e + f*x])^2) + (2*Cos[e + f*x]^2*Cot[e/2 + (f*x)/2]*Csc[e/2 + (f*x)/2]^7*Sec[e/2]*(c - c*Sec[e + f*x])^4*
Sin[(f*x)/2])/(3*f*(a + a*Sec[e + f*x])^2) + (Cos[e + f*x]^2*Cot[e/2 + (f*x)/2]^4*Csc[e/2 + (f*x)/2]^4*(c - c*
Sec[e + f*x])^4*Sin[(f*x)/2])/(4*f*(a + a*Sec[e + f*x])^2*(Cos[e/2] - Sin[e/2])*(Cos[e/2 + (f*x)/2] - Sin[e/2
+ (f*x)/2])) + (Cos[e + f*x]^2*Cot[e/2 + (f*x)/2]^4*Csc[e/2 + (f*x)/2]^4*(c - c*Sec[e + f*x])^4*Sin[(f*x)/2])/
(4*f*(a + a*Sec[e + f*x])^2*(Cos[e/2] + Sin[e/2])*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])) + (2*Cos[e + f*x]
^2*Cot[e/2 + (f*x)/2]^2*Csc[e/2 + (f*x)/2]^6*(c - c*Sec[e + f*x])^4*Tan[e/2])/(3*f*(a + a*Sec[e + f*x])^2)

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Maple [A]
time = 0.14, size = 105, normalized size = 1.03

method	result	size
derivativedivides	\(\frac {8 c^{4} \left (\frac {\left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-\frac {1}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}+\frac {3 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{4}+\frac {\arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4}-\frac {1}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}-\frac {3 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{4}\right )}{f \,a^{2}}\)	\(105\)
default	\(\frac {8 c^{4} \left (\frac {\left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-\frac {1}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}+\frac {3 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{4}+\frac {\arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4}-\frac {1}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}-\frac {3 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{4}\right )}{f \,a^{2}}\)	\(105\)
risch	\(\frac {c^{4} x}{a^{2}}+\frac {2 i c^{4} \left (51 \,{\mathrm e}^{3 i \left (f x +e \right )}+25 \,{\mathrm e}^{2 i \left (f x +e \right )}+57 \,{\mathrm e}^{i \left (f x +e \right )}+19\right )}{3 f \,a^{2} \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{3}}+\frac {6 c^{4} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}{a^{2} f}-\frac {6 c^{4} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{a^{2} f}\)	\(131\)
norman	\(\frac {\frac {c^{4} x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}-\frac {c^{4} x}{a}-\frac {10 c^{4} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a f}+\frac {76 c^{4} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a f}-\frac {18 c^{4} \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}+\frac {8 c^{4} \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a f}+\frac {3 c^{4} x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}-\frac {3 c^{4} x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}}{\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3} a}+\frac {6 c^{4} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{a^{2} f}-\frac {6 c^{4} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{a^{2} f}\)	\(222\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c-c*sec(f*x+e))^4/(a+a*sec(f*x+e))^2,x,method=_RETURNVERBOSE)

[Out]

8/f*c^4/a^2*(1/3*tan(1/2*f*x+1/2*e)^3+tan(1/2*f*x+1/2*e)-1/8/(tan(1/2*f*x+1/2*e)-1)+3/4*ln(tan(1/2*f*x+1/2*e)-
1)+1/4*arctan(tan(1/2*f*x+1/2*e))-1/8/(tan(1/2*f*x+1/2*e)+1)-3/4*ln(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. 447 vs. \(2 (103) = 206\).
time = 0.50, size = 447, normalized size = 4.38 \begin {gather*} \frac {c^{4} {\left (\frac {\frac {15 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}{a^{2}} - \frac {12 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a^{2}} + \frac {12 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a^{2}} + \frac {12 \, \sin \left (f x + e\right )}{{\left (a^{2} - \frac {a^{2} \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 )}}\right )} + 4 \, c^{4} {\left (\frac {\frac {9 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}{a^{2}} - \frac {6 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a^{2}} + \frac {6 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a^{2}}\right )} - c^{4} {\left (\frac {\frac {9 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}{a^{2}} - \frac {12 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a^{2}}\right )} + \frac {6 \, c^{4} {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{a^{2}} - \frac {4 \, c^{4} {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{a^{2}}}{6 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c*sec(f*x+e))^4/(a+a*sec(f*x+e))^2,x, algorithm="maxima")

[Out]

1/6*(c^4*((15*sin(f*x + e)/(cos(f*x + e) + 1) + sin(f*x + e)^3/(cos(f*x + e) + 1)^3)/a^2 - 12*log(sin(f*x + e)
/(cos(f*x + e) + 1) + 1)/a^2 + 12*log(sin(f*x + e)/(cos(f*x + e) + 1) - 1)/a^2 + 12*sin(f*x + e)/((a^2 - a^2*s
in(f*x + e)^2/(cos(f*x + e) + 1)^2)*(cos(f*x + e) + 1))) + 4*c^4*((9*sin(f*x + e)/(cos(f*x + e) + 1) + sin(f*x
+ e)^3/(cos(f*x + e) + 1)^3)/a^2 - 6*log(sin(f*x + e)/(cos(f*x + e) + 1) + 1)/a^2 + 6*log(sin(f*x + e)/(cos(f
*x + e) + 1) - 1)/a^2) - c^4*((9*sin(f*x + e)/(cos(f*x + e) + 1) - sin(f*x + e)^3/(cos(f*x + e) + 1)^3)/a^2 -
12*arctan(sin(f*x + e)/(cos(f*x + e) + 1))/a^2) + 6*c^4*(3*sin(f*x + e)/(cos(f*x + e) + 1) + sin(f*x + e)^3/(c
os(f*x + e) + 1)^3)/a^2 - 4*c^4*(3*sin(f*x + e)/(cos(f*x + e) + 1) - sin(f*x + e)^3/(cos(f*x + e) + 1)^3)/a^2)
/f

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 237 vs. \(2 (103) = 206\).
time = 2.28, size = 237, normalized size = 2.32 \begin {gather*} \frac {3 \, c^{4} f x \cos \left (f x + e\right )^{3} + 6 \, c^{4} f x \cos \left (f x + e\right )^{2} + 3 \, c^{4} f x \cos \left (f x + e\right ) - 9 \, {\left (c^{4} \cos \left (f x + e\right )^{3} + 2 \, c^{4} \cos \left (f x + e\right )^{2} + c^{4} \cos \left (f x + e\right )\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) + 9 \, {\left (c^{4} \cos \left (f x + e\right )^{3} + 2 \, c^{4} \cos \left (f x + e\right )^{2} + c^{4} \cos \left (f x + e\right )\right )} \log \left (-\sin \left (f x + e\right ) + 1\right ) + {\left (19 \, c^{4} \cos \left (f x + e\right )^{2} + 38 \, c^{4} \cos \left (f x + e\right ) + 3 \, c^{4}\right )} \sin \left (f x + e\right )}{3 \, {\left (a^{2} f \cos \left (f x + e\right )^{3} + 2 \, a^{2} f \cos \left (f x + e\right )^{2} + a^{2} 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))^4/(a+a*sec(f*x+e))^2,x, algorithm="fricas")

[Out]

1/3*(3*c^4*f*x*cos(f*x + e)^3 + 6*c^4*f*x*cos(f*x + e)^2 + 3*c^4*f*x*cos(f*x + e) - 9*(c^4*cos(f*x + e)^3 + 2*
c^4*cos(f*x + e)^2 + c^4*cos(f*x + e))*log(sin(f*x + e) + 1) + 9*(c^4*cos(f*x + e)^3 + 2*c^4*cos(f*x + e)^2 +
c^4*cos(f*x + e))*log(-sin(f*x + e) + 1) + (19*c^4*cos(f*x + e)^2 + 38*c^4*cos(f*x + e) + 3*c^4)*sin(f*x + e))
/(a^2*f*cos(f*x + e)^3 + 2*a^2*f*cos(f*x + e)^2 + a^2*f*cos(f*x + e))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {c^{4} \left (\int \left (- \frac {4 \sec {\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\right )\, dx + \int \frac {6 \sec ^{2}{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx + \int \left (- \frac {4 \sec ^{3}{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\right )\, dx + \int \frac {\sec ^{4}{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {1}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx\right )}{a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c*sec(f*x+e))**4/(a+a*sec(f*x+e))**2,x)

[Out]

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

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Giac [A]
time = 0.51, size = 134, normalized size = 1.31 \begin {gather*} \frac {\frac {3 \, {\left (f x + e\right )} c^{4}}{a^{2}} - \frac {18 \, c^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{a^{2}} + \frac {18 \, c^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{a^{2}} - \frac {6 \, c^{4} \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^{2}} + \frac {8 \, {\left (a^{4} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 3 \, a^{4} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{6}}}{3 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c*sec(f*x+e))^4/(a+a*sec(f*x+e))^2,x, algorithm="giac")

[Out]

1/3*(3*(f*x + e)*c^4/a^2 - 18*c^4*log(abs(tan(1/2*f*x + 1/2*e) + 1))/a^2 + 18*c^4*log(abs(tan(1/2*f*x + 1/2*e)
- 1))/a^2 - 6*c^4*tan(1/2*f*x + 1/2*e)/((tan(1/2*f*x + 1/2*e)^2 - 1)*a^2) + 8*(a^4*c^4*tan(1/2*f*x + 1/2*e)^3
+ 3*a^4*c^4*tan(1/2*f*x + 1/2*e))/a^6)/f

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Mupad [B]
time = 1.47, size = 112, normalized size = 1.10 \begin {gather*} \frac {c^4\,x}{a^2}+\frac {8\,c^4\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{a^2\,f}+\frac {8\,c^4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{3\,a^2\,f}-\frac {12\,c^4\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{a^2\,f}-\frac {2\,c^4\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{f\,\left (a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-a^2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c - c/cos(e + f*x))^4/(a + a/cos(e + f*x))^2,x)

[Out]

(c^4*x)/a^2 + (8*c^4*tan(e/2 + (f*x)/2))/(a^2*f) + (8*c^4*tan(e/2 + (f*x)/2)^3)/(3*a^2*f) - (12*c^4*atanh(tan(
e/2 + (f*x)/2)))/(a^2*f) - (2*c^4*tan(e/2 + (f*x)/2))/(f*(a^2*tan(e/2 + (f*x)/2)^2 - a^2))

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