3.1.54 \(\int \frac {\sec (e+f x) (c-c \sec (e+f x))^4}{(a+a \sec (e+f x))^3} \, dx\) [54]

Optimal. Leaf size=164 \[ -\frac {7 c^4 \tanh ^{-1}(\sin (e+f x))}{a^3 f}+\frac {7 c^4 \tan (e+f x)}{a^3 f}+\frac {2 c (c-c \sec (e+f x))^3 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}-\frac {14 \left (c^2-c^2 \sec (e+f x)\right )^2 \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac {14 \left (c^4-c^4 \sec (e+f x)\right ) \tan (e+f x)}{3 f \left (a^3+a^3 \sec (e+f x)\right )} \]

[Out]

-7*c^4*arctanh(sin(f*x+e))/a^3/f+7*c^4*tan(f*x+e)/a^3/f+2/5*c*(c-c*sec(f*x+e))^3*tan(f*x+e)/f/(a+a*sec(f*x+e))
^3-14/15*(c^2-c^2*sec(f*x+e))^2*tan(f*x+e)/a/f/(a+a*sec(f*x+e))^2+14/3*(c^4-c^4*sec(f*x+e))*tan(f*x+e)/f/(a^3+
a^3*sec(f*x+e))

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Rubi [A]
time = 0.19, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {4042, 3872, 3855, 3852, 8} \begin {gather*} \frac {7 c^4 \tan (e+f x)}{a^3 f}-\frac {7 c^4 \tanh ^{-1}(\sin (e+f x))}{a^3 f}+\frac {14 \tan (e+f x) \left (c^4-c^4 \sec (e+f x)\right )}{3 f \left (a^3 \sec (e+f x)+a^3\right )}-\frac {14 \tan (e+f x) \left (c^2-c^2 \sec (e+f x)\right )^2}{15 a f (a \sec (e+f x)+a)^2}+\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^3}{5 f (a \sec (e+f x)+a)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-7*c^4*ArcTanh[Sin[e + f*x]])/(a^3*f) + (7*c^4*Tan[e + f*x])/(a^3*f) + (2*c*(c - c*Sec[e + f*x])^3*Tan[e + f*
x])/(5*f*(a + a*Sec[e + f*x])^3) - (14*(c^2 - c^2*Sec[e + f*x])^2*Tan[e + f*x])/(15*a*f*(a + a*Sec[e + f*x])^2
) + (14*(c^4 - c^4*Sec[e + f*x])*Tan[e + f*x])/(3*f*(a^3 + a^3*Sec[e + f*x]))

Rule 8

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

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 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3872

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[a, Int[(d*
Csc[e + f*x])^n, x], x] + Dist[b/d, Int[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]

Rule 4042

Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))
^(n_.), x_Symbol] :> Simp[2*a*c*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((c + d*Csc[e + f*x])^(n - 1)/(b*f*(2*m +
1))), x] - Dist[d*((2*n - 1)/(b*(2*m + 1))), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(c + d*Csc[e + f*x]
)^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IGtQ[n, 0] && L
tQ[m, -2^(-1)] && IntegerQ[2*m]

Rubi steps

\begin {align*} \int \frac {\sec (e+f x) (c-c \sec (e+f x))^4}{(a+a \sec (e+f x))^3} \, dx &=\frac {2 c (c-c \sec (e+f x))^3 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}-\frac {(7 c) \int \frac {\sec (e+f x) (c-c \sec (e+f x))^3}{(a+a \sec (e+f x))^2} \, dx}{5 a}\\ &=\frac {2 c (c-c \sec (e+f x))^3 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}-\frac {14 \left (c^2-c^2 \sec (e+f x)\right )^2 \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac {\left (7 c^2\right ) \int \frac {\sec (e+f x) (c-c \sec (e+f x))^2}{a+a \sec (e+f x)} \, dx}{3 a^2}\\ &=\frac {2 c (c-c \sec (e+f x))^3 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}-\frac {14 \left (c^2-c^2 \sec (e+f x)\right )^2 \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac {14 \left (c^4-c^4 \sec (e+f x)\right ) \tan (e+f x)}{3 f \left (a^3+a^3 \sec (e+f x)\right )}-\frac {\left (7 c^3\right ) \int \sec (e+f x) (c-c \sec (e+f x)) \, dx}{a^3}\\ &=\frac {2 c (c-c \sec (e+f x))^3 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}-\frac {14 \left (c^2-c^2 \sec (e+f x)\right )^2 \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac {14 \left (c^4-c^4 \sec (e+f x)\right ) \tan (e+f x)}{3 f \left (a^3+a^3 \sec (e+f x)\right )}-\frac {\left (7 c^4\right ) \int \sec (e+f x) \, dx}{a^3}+\frac {\left (7 c^4\right ) \int \sec ^2(e+f x) \, dx}{a^3}\\ &=-\frac {7 c^4 \tanh ^{-1}(\sin (e+f x))}{a^3 f}+\frac {2 c (c-c \sec (e+f x))^3 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}-\frac {14 \left (c^2-c^2 \sec (e+f x)\right )^2 \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac {14 \left (c^4-c^4 \sec (e+f x)\right ) \tan (e+f x)}{3 f \left (a^3+a^3 \sec (e+f x)\right )}-\frac {\left (7 c^4\right ) \text {Subst}(\int 1 \, dx,x,-\tan (e+f x))}{a^3 f}\\ &=-\frac {7 c^4 \tanh ^{-1}(\sin (e+f x))}{a^3 f}+\frac {7 c^4 \tan (e+f x)}{a^3 f}+\frac {2 c (c-c \sec (e+f x))^3 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}-\frac {14 \left (c^2-c^2 \sec (e+f x)\right )^2 \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac {14 \left (c^4-c^4 \sec (e+f x)\right ) \tan (e+f x)}{3 f \left (a^3+a^3 \sec (e+f x)\right )}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(826\) vs. \(2(164)=328\).
time = 6.33, size = 826, normalized size = 5.04 \begin {gather*} \frac {7 \cos (e+f x) \cot ^6\left (\frac {e}{2}+\frac {f x}{2}\right ) \csc ^2\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))^3}-\frac {7 \cos (e+f x) \cot ^6\left (\frac {e}{2}+\frac {f x}{2}\right ) \csc ^2\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))^3}+\frac {76 \cos (e+f x) \cot ^5\left (\frac {e}{2}+\frac {f x}{2}\right ) \csc ^3\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 )}{15 f (a+a \sec (e+f x))^3}+\frac {8 \cos (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 )}{15 f (a+a \sec (e+f x))^3}+\frac {2 \cos (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 )}{5 f (a+a \sec (e+f x))^3}+\frac {\cos (e+f x) \cot ^6\left (\frac {e}{2}+\frac {f x}{2}\right ) \csc ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) (c-c \sec (e+f x))^4 \sin \left (\frac {f x}{2}\right )}{2 f (a+a \sec (e+f x))^3 \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 (e+f x) \cot ^6\left (\frac {e}{2}+\frac {f x}{2}\right ) \csc ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) (c-c \sec (e+f x))^4 \sin \left (\frac {f x}{2}\right )}{2 f (a+a \sec (e+f x))^3 \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 {8 \cos (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 \tan \left (\frac {e}{2}\right )}{15 f (a+a \sec (e+f x))^3}+\frac {2 \cos (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 )}{5 f (a+a \sec (e+f x))^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(7*Cos[e + f*x]*Cot[e/2 + (f*x)/2]^6*Csc[e/2 + (f*x)/2]^2*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])^3) - (7*Cos[e + f*x]*Cot[e/2 + (f*x)/2]^6*Csc[e/2 + (f*x)/2]^2*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])^3) + (76*Cos[e + f*
x]*Cot[e/2 + (f*x)/2]^5*Csc[e/2 + (f*x)/2]^3*Sec[e/2]*(c - c*Sec[e + f*x])^4*Sin[(f*x)/2])/(15*f*(a + a*Sec[e
+ f*x])^3) + (8*Cos[e + f*x]*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])/(15*f*(a + a*Sec[e + f*x])^3) + (2*Cos[e + f*x]*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])/(5*f*(a + a*Sec[e + f*x])^3) + (Cos[e + f*x]*Cot[e/2 + (f*x)/2]^6*Csc[e/2 + (f*
x)/2]^2*(c - c*Sec[e + f*x])^4*Sin[(f*x)/2])/(2*f*(a + a*Sec[e + f*x])^3*(Cos[e/2] - Sin[e/2])*(Cos[e/2 + (f*x
)/2] - Sin[e/2 + (f*x)/2])) + (Cos[e + f*x]*Cot[e/2 + (f*x)/2]^6*Csc[e/2 + (f*x)/2]^2*(c - c*Sec[e + f*x])^4*S
in[(f*x)/2])/(2*f*(a + a*Sec[e + f*x])^3*(Cos[e/2] + Sin[e/2])*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])) + (8
*Cos[e + f*x]*Cot[e/2 + (f*x)/2]^4*Csc[e/2 + (f*x)/2]^4*(c - c*Sec[e + f*x])^4*Tan[e/2])/(15*f*(a + a*Sec[e +
f*x])^3) + (2*Cos[e + f*x]*Cot[e/2 + (f*x)/2]^2*Csc[e/2 + (f*x)/2]^6*(c - c*Sec[e + f*x])^4*Tan[e/2])/(5*f*(a
+ a*Sec[e + f*x])^3)

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Maple [A]
time = 0.20, size = 108, normalized size = 0.66

method	result	size
derivativedivides	\(\frac {4 c^{4} \left (\frac {\left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5}+\frac {2 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}+3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-\frac {1}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}+\frac {7 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{4}-\frac {1}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}-\frac {7 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{4}\right )}{f \,a^{3}}\)	\(108\)
default	\(\frac {4 c^{4} \left (\frac {\left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5}+\frac {2 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}+3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-\frac {1}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}+\frac {7 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{4}-\frac {1}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}-\frac {7 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{4}\right )}{f \,a^{3}}\)	\(108\)
risch	\(\frac {2 i c^{4} \left (120 \,{\mathrm e}^{6 i \left (f x +e \right )}+495 \,{\mathrm e}^{5 i \left (f x +e \right )}+1235 \,{\mathrm e}^{4 i \left (f x +e \right )}+1270 \,{\mathrm e}^{3 i \left (f x +e \right )}+1342 \,{\mathrm e}^{2 i \left (f x +e \right )}+715 \,{\mathrm e}^{i \left (f x +e \right )}+167\right )}{15 f \,a^{3} \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{5}}+\frac {7 c^{4} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}{a^{3} f}-\frac {7 c^{4} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{a^{3} f}\)	\(156\)
norman	\(\frac {\frac {14 c^{4} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a f}-\frac {154 c^{4} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a f}+\frac {1022 c^{4} \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 a f}-\frac {186 c^{4} \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5 a f}+\frac {92 c^{4} \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 a f}-\frac {8 c^{4} \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 a f}+\frac {4 c^{4} \left (\tan ^{13}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5 a f}}{\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4} a^{2}}+\frac {7 c^{4} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{a^{3} f}-\frac {7 c^{4} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{a^{3} f}\)	\(220\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/f*c^4/a^3*(1/5*tan(1/2*f*x+1/2*e)^5+2/3*tan(1/2*f*x+1/2*e)^3+3*tan(1/2*f*x+1/2*e)-1/4/(tan(1/2*f*x+1/2*e)-1)
+7/4*ln(tan(1/2*f*x+1/2*e)-1)-1/4/(tan(1/2*f*x+1/2*e)+1)-7/4*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. 510 vs. \(2 (172) = 344\).
time = 0.29, size = 510, normalized size = 3.11 \begin {gather*} \frac {3 \, c^{4} {\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 )} + 4 \, c^{4} {\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 )} + \frac {6 \, c^{4} {\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 {c^{4} {\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 {12 \, c^{4} {\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(sec(f*x+e)*(c-c*sec(f*x+e))^4/(a+a*sec(f*x+e))^3,x, algorithm="maxima")

[Out]

1/60*(3*c^4*(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) + 4*c^4*(
(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) + 6*c^4*(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/(c
os(f*x + e) + 1)^5)/a^3 + c^4*(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 - 12*c^4*(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 = 2.39, size = 249, normalized size = 1.52 \begin {gather*} -\frac {105 \, {\left (c^{4} \cos \left (f x + e\right )^{4} + 3 \, c^{4} \cos \left (f x + e\right )^{3} + 3 \, 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 ) - 105 \, {\left (c^{4} \cos \left (f x + e\right )^{4} + 3 \, c^{4} \cos \left (f x + e\right )^{3} + 3 \, 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 ) - 2 \, {\left (167 \, c^{4} \cos \left (f x + e\right )^{3} + 381 \, c^{4} \cos \left (f x + e\right )^{2} + 277 \, c^{4} \cos \left (f x + e\right ) + 15 \, c^{4}\right )} \sin \left (f x + e\right )}{30 \, {\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(sec(f*x+e)*(c-c*sec(f*x+e))^4/(a+a*sec(f*x+e))^3,x, algorithm="fricas")

[Out]

-1/30*(105*(c^4*cos(f*x + e)^4 + 3*c^4*cos(f*x + e)^3 + 3*c^4*cos(f*x + e)^2 + c^4*cos(f*x + e))*log(sin(f*x +
e) + 1) - 105*(c^4*cos(f*x + e)^4 + 3*c^4*cos(f*x + e)^3 + 3*c^4*cos(f*x + e)^2 + c^4*cos(f*x + e))*log(-sin(
f*x + e) + 1) - 2*(167*c^4*cos(f*x + e)^3 + 381*c^4*cos(f*x + e)^2 + 277*c^4*cos(f*x + e) + 15*c^4)*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^{4} \left (\int \frac {\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 {4 \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 {6 \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 {4 \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\right )}{a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.61, size = 141, normalized size = 0.86 \begin {gather*} -\frac {\frac {105 \, c^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{a^{3}} - \frac {105 \, c^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{a^{3}} + \frac {30 \, 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^{3}} - \frac {4 \, {\left (3 \, a^{12} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 10 \, a^{12} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 45 \, a^{12} c^{4} \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(sec(f*x+e)*(c-c*sec(f*x+e))^4/(a+a*sec(f*x+e))^3,x, algorithm="giac")

[Out]

-1/15*(105*c^4*log(abs(tan(1/2*f*x + 1/2*e) + 1))/a^3 - 105*c^4*log(abs(tan(1/2*f*x + 1/2*e) - 1))/a^3 + 30*c^
4*tan(1/2*f*x + 1/2*e)/((tan(1/2*f*x + 1/2*e)^2 - 1)*a^3) - 4*(3*a^12*c^4*tan(1/2*f*x + 1/2*e)^5 + 10*a^12*c^4
*tan(1/2*f*x + 1/2*e)^3 + 45*a^12*c^4*tan(1/2*f*x + 1/2*e))/a^15)/f

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Mupad [B]
time = 1.63, size = 126, normalized size = 0.77 \begin {gather*} \frac {12\,c^4\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{a^3\,f}+\frac {8\,c^4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{3\,a^3\,f}+\frac {4\,c^4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5}{5\,a^3\,f}-\frac {14\,c^4\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{a^3\,f}-\frac {2\,c^4\,\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))^4/(cos(e + f*x)*(a + a/cos(e + f*x))^3),x)

[Out]

(12*c^4*tan(e/2 + (f*x)/2))/(a^3*f) + (8*c^4*tan(e/2 + (f*x)/2)^3)/(3*a^3*f) + (4*c^4*tan(e/2 + (f*x)/2)^5)/(5
*a^3*f) - (14*c^4*atanh(tan(e/2 + (f*x)/2)))/(a^3*f) - (2*c^4*tan(e/2 + (f*x)/2))/(f*(a^3*tan(e/2 + (f*x)/2)^2
- a^3))

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