\(\int \frac {(c-c \sec (e+f x))^4}{\sqrt {a+a \sec (e+f x)}} \, dx\) [65]

Optimal result

Integrand size = 28, antiderivative size = 185 \[ \int \frac {(c-c \sec (e+f x))^4}{\sqrt {a+a \sec (e+f x)}} \, dx=\frac {2 c^4 \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{\sqrt {a} f}-\frac {16 \sqrt {2} c^4 \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a+a \sec (e+f x)}}\right )}{\sqrt {a} f}+\frac {14 c^4 \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)}}-\frac {2 a c^4 \tan ^3(e+f x)}{f (a+a \sec (e+f x))^{3/2}}+\frac {2 a^2 c^4 \tan ^5(e+f x)}{5 f (a+a \sec (e+f x))^{5/2}} \]

[Out]

2*c^4*arctan(a^(1/2)*tan(f*x+e)/(a+a*sec(f*x+e))^(1/2))/f/a^(1/2)-16*c^4*arctan(1/2*a^(1/2)*tan(f*x+e)*2^(1/2)
/(a+a*sec(f*x+e))^(1/2))*2^(1/2)/f/a^(1/2)+14*c^4*tan(f*x+e)/f/(a+a*sec(f*x+e))^(1/2)-2*a*c^4*tan(f*x+e)^3/f/(
a+a*sec(f*x+e))^(3/2)+2/5*a^2*c^4*tan(f*x+e)^5/f/(a+a*sec(f*x+e))^(5/2)

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3989, 3972, 490, 596, 536, 209} \[ \int \frac {(c-c \sec (e+f x))^4}{\sqrt {a+a \sec (e+f x)}} \, dx=\frac {2 a^2 c^4 \tan ^5(e+f x)}{5 f (a \sec (e+f x)+a)^{5/2}}+\frac {2 c^4 \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a}}\right )}{\sqrt {a} f}-\frac {16 \sqrt {2} c^4 \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a \sec (e+f x)+a}}\right )}{\sqrt {a} f}-\frac {2 a c^4 \tan ^3(e+f x)}{f (a \sec (e+f x)+a)^{3/2}}+\frac {14 c^4 \tan (e+f x)}{f \sqrt {a \sec (e+f x)+a}} \]

[In]

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

[Out]

(2*c^4*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(Sqrt[a]*f) - (16*Sqrt[2]*c^4*ArcTan[(Sqrt[a]*
Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(Sqrt[a]*f) + (14*c^4*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f
*x]]) - (2*a*c^4*Tan[e + f*x]^3)/(f*(a + a*Sec[e + f*x])^(3/2)) + (2*a^2*c^4*Tan[e + f*x]^5)/(5*f*(a + a*Sec[e
+ f*x])^(5/2))

Rule 209

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

Rule 490

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[e^(2*n -
1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*d*(m + n*(p + q) + 1))), x] - Dist[e^(2*n)
/(b*d*(m + n*(p + q) + 1)), Int[(e*x)^(m - 2*n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*c*(m - 2*n + 1) + (a*d*(m +
n*(q - 1) + 1) + b*c*(m + n*(p - 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d
, 0] && IGtQ[n, 0] && GtQ[m - n + 1, n] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 536

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

Rule 596

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[f*g^(n - 1)*(g*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*d*(m + n*(p + q +
1) + 1))), x] - Dist[g^n/(b*d*(m + n*(p + q + 1) + 1)), Int[(g*x)^(m - n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*
f*c*(m - n + 1) + (a*f*d*(m + n*q + 1) + b*(f*c*(m + n*p + 1) - e*d*(m + n*(p + q + 1) + 1)))*x^n, x], x], x]
/; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 0] && GtQ[m, n - 1]

Rule 3972

Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_.), x_Symbol] :> Dist[-2*(a^(m/2 +
n + 1/2)/d), Subst[Int[x^m*((2 + a*x^2)^(m/2 + n - 1/2)/(1 + a*x^2)), x], x, Cot[c + d*x]/Sqrt[a + b*Csc[c +
d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m/2] && IntegerQ[n - 1/2]

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*} \text {integral}& = \left (a^4 c^4\right ) \int \frac {\tan ^8(e+f x)}{(a+a \sec (e+f x))^{9/2}} \, dx \\ & = -\frac {\left (2 a^4 c^4\right ) \text {Subst}\left (\int \frac {x^8}{\left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{f} \\ & = \frac {2 a^2 c^4 \tan ^5(e+f x)}{5 f (a+a \sec (e+f x))^{5/2}}+\frac {\left (2 a^2 c^4\right ) \text {Subst}\left (\int \frac {x^4 \left (10+15 a x^2\right )}{\left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{5 f} \\ & = -\frac {2 a c^4 \tan ^3(e+f x)}{f (a+a \sec (e+f x))^{3/2}}+\frac {2 a^2 c^4 \tan ^5(e+f x)}{5 f (a+a \sec (e+f x))^{5/2}}-\frac {\left (2 c^4\right ) \text {Subst}\left (\int \frac {x^2 \left (90 a+105 a^2 x^2\right )}{\left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{15 f} \\ & = \frac {14 c^4 \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)}}-\frac {2 a c^4 \tan ^3(e+f x)}{f (a+a \sec (e+f x))^{3/2}}+\frac {2 a^2 c^4 \tan ^5(e+f x)}{5 f (a+a \sec (e+f x))^{5/2}}+\frac {\left (2 c^4\right ) \text {Subst}\left (\int \frac {210 a^2+225 a^3 x^2}{\left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{15 a^2 f} \\ & = \frac {14 c^4 \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)}}-\frac {2 a c^4 \tan ^3(e+f x)}{f (a+a \sec (e+f x))^{3/2}}+\frac {2 a^2 c^4 \tan ^5(e+f x)}{5 f (a+a \sec (e+f x))^{5/2}}-\frac {\left (2 c^4\right ) \text {Subst}\left (\int \frac {1}{1+a x^2} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{f}+\frac {\left (32 c^4\right ) \text {Subst}\left (\int \frac {1}{2+a x^2} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{f} \\ & = \frac {2 c^4 \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{\sqrt {a} f}-\frac {16 \sqrt {2} c^4 \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a+a \sec (e+f x)}}\right )}{\sqrt {a} f}+\frac {14 c^4 \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)}}-\frac {2 a c^4 \tan ^3(e+f x)}{f (a+a \sec (e+f x))^{3/2}}+\frac {2 a^2 c^4 \tan ^5(e+f x)}{5 f (a+a \sec (e+f x))^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.43 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.83 \[ \int \frac {(c-c \sec (e+f x))^4}{\sqrt {a+a \sec (e+f x)}} \, dx=\frac {c^4 \cot \left (\frac {1}{2} (e+f x)\right ) \left (100-155 \cos (e+f x)+96 \cos (2 (e+f x))-41 \cos (3 (e+f x))+20 \arctan \left (\sqrt {-1+\sec (e+f x)}\right ) \cos ^3(e+f x) \sqrt {-1+\sec (e+f x)}-160 \sqrt {2} \arctan \left (\frac {\sqrt {-1+\sec (e+f x)}}{\sqrt {2}}\right ) \cos ^3(e+f x) \sqrt {-1+\sec (e+f x)}\right ) \sec ^3(e+f x)}{10 f \sqrt {a (1+\sec (e+f x))}} \]

[In]

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

[Out]

(c^4*Cot[(e + f*x)/2]*(100 - 155*Cos[e + f*x] + 96*Cos[2*(e + f*x)] - 41*Cos[3*(e + f*x)] + 20*ArcTan[Sqrt[-1
+ Sec[e + f*x]]]*Cos[e + f*x]^3*Sqrt[-1 + Sec[e + f*x]] - 160*Sqrt[2]*ArcTan[Sqrt[-1 + Sec[e + f*x]]/Sqrt[2]]*
Cos[e + f*x]^3*Sqrt[-1 + Sec[e + f*x]])*Sec[e + f*x]^3)/(10*f*Sqrt[a*(1 + Sec[e + f*x])])

Maple [A] (warning: unable to verify)

Time = 6.49 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.52

method	result	size
default	\(\frac {c^{4} \left (5 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{\sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\right ) \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1\right )^{\frac {5}{2}}-80 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )+\sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\right ) \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1\right )^{\frac {5}{2}}+98 \left (1-\cos \left (f x +e \right )\right )^{5} \csc \left (f x +e \right )^{5}-160 \left (1-\cos \left (f x +e \right )\right )^{3} \csc \left (f x +e \right )^{3}+70 \csc \left (f x +e \right )-70 \cot \left (f x +e \right )\right ) \sqrt {-\frac {2 a}{\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}}{5 f a \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right )^{2} \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )-1\right )^{2}}\)	\(282\)
parts	\(-\frac {c^{4} \sqrt {a \left (\sec \left (f x +e \right )+1\right )}\, \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \left (\sqrt {2}\, \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )+\sqrt {\cot \left (f x +e \right )^{2}-2 \csc \left (f x +e \right ) \cot \left (f x +e \right )+\csc \left (f x +e \right )^{2}-1}\right )-2 \,\operatorname {arctanh}\left (\frac {\sin \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}}\right )\right )}{f a}-\frac {c^{4} \sqrt {-\frac {2 a}{\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\, \left (15 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )+\sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\right ) \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1\right )^{\frac {5}{2}}-34 \left (1-\cos \left (f x +e \right )\right )^{5} \csc \left (f x +e \right )^{5}+40 \left (1-\cos \left (f x +e \right )\right )^{3} \csc \left (f x +e \right )^{3}-30 \csc \left (f x +e \right )+30 \cot \left (f x +e \right )\right )}{15 f a \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1\right )^{2}}-\frac {4 c^{4} \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {2}\, \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )+\sqrt {\cot \left (f x +e \right )^{2}-2 \csc \left (f x +e \right ) \cot \left (f x +e \right )+\csc \left (f x +e \right )^{2}-1}\right ) \sqrt {a \left (\sec \left (f x +e \right )+1\right )}}{f a}-\frac {6 c^{4} \sqrt {a \left (\sec \left (f x +e \right )+1\right )}\, \left (\sqrt {2}\, \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )+\sqrt {\cot \left (f x +e \right )^{2}-2 \csc \left (f x +e \right ) \cot \left (f x +e \right )+\csc \left (f x +e \right )^{2}-1}\right )+2 \cot \left (f x +e \right )-2 \csc \left (f x +e \right )\right )}{f a}-\frac {4 c^{4} \sqrt {-\frac {2 a}{\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\, \left (3 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )+\sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\right ) \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1\right )^{\frac {3}{2}}-4 \left (1-\cos \left (f x +e \right )\right )^{3} \csc \left (f x +e \right )^{3}\right )}{3 f a \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1\right )}\)	\(702\)

[In]

int((c-c*sec(f*x+e))^4/(a+a*sec(f*x+e))^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/5*c^4/f/a*(5*2^(1/2)*arctanh(2^(1/2)/((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(1/2)*(-cot(f*x+e)+csc(f*x+e)))*((1-c
os(f*x+e))^2*csc(f*x+e)^2-1)^(5/2)-80*ln(csc(f*x+e)-cot(f*x+e)+((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(1/2))*((1-co
s(f*x+e))^2*csc(f*x+e)^2-1)^(5/2)+98*(1-cos(f*x+e))^5*csc(f*x+e)^5-160*(1-cos(f*x+e))^3*csc(f*x+e)^3+70*csc(f*
x+e)-70*cot(f*x+e))*(-2*a/((1-cos(f*x+e))^2*csc(f*x+e)^2-1))^(1/2)/(-cot(f*x+e)+csc(f*x+e)+1)^2/(-cot(f*x+e)+c
sc(f*x+e)-1)^2

Fricas [A] (verification not implemented)

none

Time = 0.70 (sec) , antiderivative size = 552, normalized size of antiderivative = 2.98 \[ \int \frac {(c-c \sec (e+f x))^4}{\sqrt {a+a \sec (e+f x)}} \, dx=\left [\frac {40 \, \sqrt {2} {\left (a c^{4} \cos \left (f x + e\right )^{3} + a c^{4} \cos \left (f x + e\right )^{2}\right )} \sqrt {-\frac {1}{a}} \log \left (\frac {2 \, \sqrt {2} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {-\frac {1}{a}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + 3 \, \cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1}\right ) - 5 \, {\left (c^{4} \cos \left (f x + e\right )^{3} + c^{4} \cos \left (f x + e\right )^{2}\right )} \sqrt {-a} \log \left (\frac {2 \, a \cos \left (f x + e\right )^{2} + 2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + a \cos \left (f x + e\right ) - a}{\cos \left (f x + e\right ) + 1}\right ) + 2 \, {\left (41 \, c^{4} \cos \left (f x + e\right )^{2} - 7 \, c^{4} \cos \left (f x + e\right ) + c^{4}\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )}{5 \, {\left (a f \cos \left (f x + e\right )^{3} + a f \cos \left (f x + e\right )^{2}\right )}}, -\frac {2 \, {\left (5 \, {\left (c^{4} \cos \left (f x + e\right )^{3} + c^{4} \cos \left (f x + e\right )^{2}\right )} \sqrt {a} \arctan \left (\frac {\sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{\sqrt {a} \sin \left (f x + e\right )}\right ) - {\left (41 \, c^{4} \cos \left (f x + e\right )^{2} - 7 \, c^{4} \cos \left (f x + e\right ) + c^{4}\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) - \frac {40 \, \sqrt {2} {\left (a c^{4} \cos \left (f x + e\right )^{3} + a c^{4} \cos \left (f x + e\right )^{2}\right )} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{\sqrt {a} \sin \left (f x + e\right )}\right )}{\sqrt {a}}\right )}}{5 \, {\left (a f \cos \left (f x + e\right )^{3} + a f \cos \left (f x + e\right )^{2}\right )}}\right ] \]

[In]

integrate((c-c*sec(f*x+e))^4/(a+a*sec(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

[1/5*(40*sqrt(2)*(a*c^4*cos(f*x + e)^3 + a*c^4*cos(f*x + e)^2)*sqrt(-1/a)*log((2*sqrt(2)*sqrt((a*cos(f*x + e)
+ a)/cos(f*x + e))*sqrt(-1/a)*cos(f*x + e)*sin(f*x + e) + 3*cos(f*x + e)^2 + 2*cos(f*x + e) - 1)/(cos(f*x + e)
^2 + 2*cos(f*x + e) + 1)) - 5*(c^4*cos(f*x + e)^3 + c^4*cos(f*x + e)^2)*sqrt(-a)*log((2*a*cos(f*x + e)^2 + 2*s
qrt(-a)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*cos(f*x + e)*sin(f*x + e) + a*cos(f*x + e) - a)/(cos(f*x + e)
+ 1)) + 2*(41*c^4*cos(f*x + e)^2 - 7*c^4*cos(f*x + e) + c^4)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sin(f*x +
e))/(a*f*cos(f*x + e)^3 + a*f*cos(f*x + e)^2), -2/5*(5*(c^4*cos(f*x + e)^3 + c^4*cos(f*x + e)^2)*sqrt(a)*arct
an(sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*cos(f*x + e)/(sqrt(a)*sin(f*x + e))) - (41*c^4*cos(f*x + e)^2 - 7*c
^4*cos(f*x + e) + c^4)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sin(f*x + e) - 40*sqrt(2)*(a*c^4*cos(f*x + e)^3
+ a*c^4*cos(f*x + e)^2)*arctan(sqrt(2)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*cos(f*x + e)/(sqrt(a)*sin(f*x
+ e)))/sqrt(a))/(a*f*cos(f*x + e)^3 + a*f*cos(f*x + e)^2)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

integrate((c-c*sec(f*x+e))^4/(a+a*sec(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {(c-c \sec (e+f x))^4}{\sqrt {a+a \sec (e+f x)}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((c-c*sec(f*x+e))^4/(a+a*sec(f*x+e))^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:index.cc index_m i_lex_is_greater Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {(c-c \sec (e+f x))^4}{\sqrt {a+a \sec (e+f x)}} \, dx=\int \frac {{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^4}{\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}} \,d x \]

[In]

int((c - c/cos(e + f*x))^4/(a + a/cos(e + f*x))^(1/2),x)

[Out]

int((c - c/cos(e + f*x))^4/(a + a/cos(e + f*x))^(1/2), x)