3.147 \(\int \sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))^4 \, dx\)

Optimal. Leaf size=271 \[ \frac {2 a^{3/2} c^4 \tan (e+f x) \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a}}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}-\frac {2 d^4 \tan (e+f x) (a-a \sec (e+f x))^3}{7 a^2 f \sqrt {a \sec (e+f x)+a}}-\frac {2 d^2 \left (6 c^2+8 c d+3 d^2\right ) \tan (e+f x) (a-a \sec (e+f x))}{3 f \sqrt {a \sec (e+f x)+a}}+\frac {2 a d (2 c+d) \left (2 c^2+2 c d+d^2\right ) \tan (e+f x)}{f \sqrt {a \sec (e+f x)+a}}+\frac {2 d^3 (4 c+3 d) \tan (e+f x) (a-a \sec (e+f x))^2}{5 a f \sqrt {a \sec (e+f x)+a}} \]

[Out]

2*a*d*(2*c+d)*(2*c^2+2*c*d+d^2)*tan(f*x+e)/f/(a+a*sec(f*x+e))^(1/2)-2/3*d^2*(6*c^2+8*c*d+3*d^2)*(a-a*sec(f*x+e
))*tan(f*x+e)/f/(a+a*sec(f*x+e))^(1/2)+2/5*d^3*(4*c+3*d)*(a-a*sec(f*x+e))^2*tan(f*x+e)/a/f/(a+a*sec(f*x+e))^(1
/2)-2/7*d^4*(a-a*sec(f*x+e))^3*tan(f*x+e)/a^2/f/(a+a*sec(f*x+e))^(1/2)+2*a^(3/2)*c^4*arctanh((a-a*sec(f*x+e))^
(1/2)/a^(1/2))*tan(f*x+e)/f/(a-a*sec(f*x+e))^(1/2)/(a+a*sec(f*x+e))^(1/2)

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Rubi [A]  time = 0.17, antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {3940, 88, 63, 206} \[ \frac {2 a^{3/2} c^4 \tan (e+f x) \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a}}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}-\frac {2 d^4 \tan (e+f x) (a-a \sec (e+f x))^3}{7 a^2 f \sqrt {a \sec (e+f x)+a}}-\frac {2 d^2 \left (6 c^2+8 c d+3 d^2\right ) \tan (e+f x) (a-a \sec (e+f x))}{3 f \sqrt {a \sec (e+f x)+a}}+\frac {2 a d (2 c+d) \left (2 c^2+2 c d+d^2\right ) \tan (e+f x)}{f \sqrt {a \sec (e+f x)+a}}+\frac {2 d^3 (4 c+3 d) \tan (e+f x) (a-a \sec (e+f x))^2}{5 a f \sqrt {a \sec (e+f x)+a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a*d*(2*c + d)*(2*c^2 + 2*c*d + d^2)*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) + (2*a^(3/2)*c^4*ArcTanh[Sqr
t[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (2*d^2*(6
*c^2 + 8*c*d + 3*d^2)*(a - a*Sec[e + f*x])*Tan[e + f*x])/(3*f*Sqrt[a + a*Sec[e + f*x]]) + (2*d^3*(4*c + 3*d)*(
a - a*Sec[e + f*x])^2*Tan[e + f*x])/(5*a*f*Sqrt[a + a*Sec[e + f*x]]) - (2*d^4*(a - a*Sec[e + f*x])^3*Tan[e + f
*x])/(7*a^2*f*Sqrt[a + a*Sec[e + f*x]])

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 206

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

Rule 3940

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

Rubi steps

\begin {align*} \int \sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))^4 \, dx &=-\frac {\left (a^2 \tan (e+f x)\right ) \operatorname {Subst}\left (\int \frac {(c+d x)^4}{x \sqrt {a-a x}} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=-\frac {\left (a^2 \tan (e+f x)\right ) \operatorname {Subst}\left (\int \left (\frac {d (2 c+d) \left (2 c^2+2 c d+d^2\right )}{\sqrt {a-a x}}+\frac {c^4}{x \sqrt {a-a x}}-\frac {d^2 \left (6 c^2+8 c d+3 d^2\right ) \sqrt {a-a x}}{a}+\frac {d^3 (4 c+3 d) (a-a x)^{3/2}}{a^2}-\frac {d^4 (a-a x)^{5/2}}{a^3}\right ) \, dx,x,\sec (e+f x)\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {2 a d (2 c+d) \left (2 c^2+2 c d+d^2\right ) \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)}}-\frac {2 d^2 \left (6 c^2+8 c d+3 d^2\right ) (a-a \sec (e+f x)) \tan (e+f x)}{3 f \sqrt {a+a \sec (e+f x)}}+\frac {2 d^3 (4 c+3 d) (a-a \sec (e+f x))^2 \tan (e+f x)}{5 a f \sqrt {a+a \sec (e+f x)}}-\frac {2 d^4 (a-a \sec (e+f x))^3 \tan (e+f x)}{7 a^2 f \sqrt {a+a \sec (e+f x)}}-\frac {\left (a^2 c^4 \tan (e+f x)\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a-a x}} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {2 a d (2 c+d) \left (2 c^2+2 c d+d^2\right ) \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)}}-\frac {2 d^2 \left (6 c^2+8 c d+3 d^2\right ) (a-a \sec (e+f x)) \tan (e+f x)}{3 f \sqrt {a+a \sec (e+f x)}}+\frac {2 d^3 (4 c+3 d) (a-a \sec (e+f x))^2 \tan (e+f x)}{5 a f \sqrt {a+a \sec (e+f x)}}-\frac {2 d^4 (a-a \sec (e+f x))^3 \tan (e+f x)}{7 a^2 f \sqrt {a+a \sec (e+f x)}}+\frac {\left (2 a c^4 \tan (e+f x)\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {x^2}{a}} \, dx,x,\sqrt {a-a \sec (e+f x)}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {2 a d (2 c+d) \left (2 c^2+2 c d+d^2\right ) \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)}}+\frac {2 a^{3/2} c^4 \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a}}\right ) \tan (e+f x)}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {2 d^2 \left (6 c^2+8 c d+3 d^2\right ) (a-a \sec (e+f x)) \tan (e+f x)}{3 f \sqrt {a+a \sec (e+f x)}}+\frac {2 d^3 (4 c+3 d) (a-a \sec (e+f x))^2 \tan (e+f x)}{5 a f \sqrt {a+a \sec (e+f x)}}-\frac {2 d^4 (a-a \sec (e+f x))^3 \tan (e+f x)}{7 a^2 f \sqrt {a+a \sec (e+f x)}}\\ \end {align*}

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Mathematica [C]  time = 14.30, size = 587, normalized size = 2.17 \[ \frac {\cos ^4(e+f x) \sec \left (\frac {1}{2} (e+f x)\right ) \sqrt {a (\sec (e+f x)+1)} (c+d \sec (e+f x))^4 \left (\frac {4}{105} \sec (e+f x) \left (105 c^2 d^2 \sin \left (\frac {1}{2} (e+f x)\right )+56 c d^3 \sin \left (\frac {1}{2} (e+f x)\right )+12 d^4 \sin \left (\frac {1}{2} (e+f x)\right )\right )+\frac {8}{105} d \left (105 c^3+105 c^2 d+56 c d^2+12 d^3\right ) \sin \left (\frac {1}{2} (e+f x)\right )+\frac {4}{35} \sec ^2(e+f x) \left (14 c d^3 \sin \left (\frac {1}{2} (e+f x)\right )+3 d^4 \sin \left (\frac {1}{2} (e+f x)\right )\right )+\frac {2}{7} d^4 \sin \left (\frac {1}{2} (e+f x)\right ) \sec ^3(e+f x)\right )}{f (c \cos (e+f x)+d)^4}-\frac {8 \left (-3-2 \sqrt {2}\right ) c^4 \cos ^4\left (\frac {1}{4} (e+f x)\right ) \sqrt {\frac {\left (10-7 \sqrt {2}\right ) \cos \left (\frac {1}{2} (e+f x)\right )-5 \sqrt {2}+7}{\cos \left (\frac {1}{2} (e+f x)\right )+1}} \sqrt {\frac {-\left (\left (\sqrt {2}-2\right ) \cos \left (\frac {1}{2} (e+f x)\right )\right )+\sqrt {2}-1}{\cos \left (\frac {1}{2} (e+f x)\right )+1}} \left (\left (\sqrt {2}-2\right ) \cos \left (\frac {1}{2} (e+f x)\right )-\sqrt {2}+1\right ) \cos ^3(e+f x) \sqrt {-\tan ^2\left (\frac {1}{4} (e+f x)\right )-2 \sqrt {2}+3} \sec \left (\frac {1}{2} (e+f x)\right ) \sqrt {a (\sec (e+f x)+1)} \sqrt {\left (\left (2+\sqrt {2}\right ) \cos \left (\frac {1}{2} (e+f x)\right )-\sqrt {2}-1\right ) \sec ^2\left (\frac {1}{4} (e+f x)\right )} (c+d \sec (e+f x))^4 \left (F\left (\sin ^{-1}\left (\frac {\tan \left (\frac {1}{4} (e+f x)\right )}{\sqrt {3-2 \sqrt {2}}}\right )|17-12 \sqrt {2}\right )-2 \Pi \left (-3+2 \sqrt {2};\sin ^{-1}\left (\frac {\tan \left (\frac {1}{4} (e+f x)\right )}{\sqrt {3-2 \sqrt {2}}}\right )|17-12 \sqrt {2}\right )\right )}{f (c \cos (e+f x)+d)^4} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Cos[e + f*x]^4*Sec[(e + f*x)/2]*Sqrt[a*(1 + Sec[e + f*x])]*(c + d*Sec[e + f*x])^4*((8*d*(105*c^3 + 105*c^2*d
+ 56*c*d^2 + 12*d^3)*Sin[(e + f*x)/2])/105 + (2*d^4*Sec[e + f*x]^3*Sin[(e + f*x)/2])/7 + (4*Sec[e + f*x]^2*(14
*c*d^3*Sin[(e + f*x)/2] + 3*d^4*Sin[(e + f*x)/2]))/35 + (4*Sec[e + f*x]*(105*c^2*d^2*Sin[(e + f*x)/2] + 56*c*d
^3*Sin[(e + f*x)/2] + 12*d^4*Sin[(e + f*x)/2]))/105))/(f*(d + c*Cos[e + f*x])^4) - (8*(-3 - 2*Sqrt[2])*c^4*Cos
[(e + f*x)/4]^4*Sqrt[(7 - 5*Sqrt[2] + (10 - 7*Sqrt[2])*Cos[(e + f*x)/2])/(1 + Cos[(e + f*x)/2])]*Sqrt[(-1 + Sq
rt[2] - (-2 + Sqrt[2])*Cos[(e + f*x)/2])/(1 + Cos[(e + f*x)/2])]*(1 - Sqrt[2] + (-2 + Sqrt[2])*Cos[(e + f*x)/2
])*Cos[e + f*x]^3*(EllipticF[ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - 2*EllipticPi[-3
+ 2*Sqrt[2], ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]])*Sqrt[(-1 - Sqrt[2] + (2 + Sqrt[2]
)*Cos[(e + f*x)/2])*Sec[(e + f*x)/4]^2]*Sec[(e + f*x)/2]*Sqrt[a*(1 + Sec[e + f*x])]*(c + d*Sec[e + f*x])^4*Sqr
t[3 - 2*Sqrt[2] - Tan[(e + f*x)/4]^2])/(f*(d + c*Cos[e + f*x])^4)

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fricas [A]  time = 0.51, size = 472, normalized size = 1.74 \[ \left [\frac {105 \, {\left (c^{4} \cos \left (f x + e\right )^{4} + c^{4} \cos \left (f x + e\right )^{3}\right )} \sqrt {-a} \log \left (\frac {2 \, a \cos \left (f x + e\right )^{2} - 2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + a \cos \left (f x + e\right ) - a}{\cos \left (f x + e\right ) + 1}\right ) + 2 \, {\left (15 \, d^{4} + 4 \, {\left (105 \, c^{3} d + 105 \, c^{2} d^{2} + 56 \, c d^{3} + 12 \, d^{4}\right )} \cos \left (f x + e\right )^{3} + 2 \, {\left (105 \, c^{2} d^{2} + 56 \, c d^{3} + 12 \, d^{4}\right )} \cos \left (f x + e\right )^{2} + 6 \, {\left (14 \, c d^{3} + 3 \, d^{4}\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )}{105 \, {\left (f \cos \left (f x + e\right )^{4} + f \cos \left (f x + e\right )^{3}\right )}}, -\frac {2 \, {\left (105 \, {\left (c^{4} \cos \left (f x + e\right )^{4} + c^{4} \cos \left (f x + e\right )^{3}\right )} \sqrt {a} \arctan \left (\frac {\sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{\sqrt {a} \sin \left (f x + e\right )}\right ) - {\left (15 \, d^{4} + 4 \, {\left (105 \, c^{3} d + 105 \, c^{2} d^{2} + 56 \, c d^{3} + 12 \, d^{4}\right )} \cos \left (f x + e\right )^{3} + 2 \, {\left (105 \, c^{2} d^{2} + 56 \, c d^{3} + 12 \, d^{4}\right )} \cos \left (f x + e\right )^{2} + 6 \, {\left (14 \, c d^{3} + 3 \, d^{4}\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )\right )}}{105 \, {\left (f \cos \left (f x + e\right )^{4} + f \cos \left (f x + e\right )^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/105*(105*(c^4*cos(f*x + e)^4 + c^4*cos(f*x + e)^3)*sqrt(-a)*log((2*a*cos(f*x + e)^2 - 2*sqrt(-a)*sqrt((a*co
s(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*(15*d^4
+ 4*(105*c^3*d + 105*c^2*d^2 + 56*c*d^3 + 12*d^4)*cos(f*x + e)^3 + 2*(105*c^2*d^2 + 56*c*d^3 + 12*d^4)*cos(f*x
+ e)^2 + 6*(14*c*d^3 + 3*d^4)*cos(f*x + e))*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sin(f*x + e))/(f*cos(f*x
+ e)^4 + f*cos(f*x + e)^3), -2/105*(105*(c^4*cos(f*x + e)^4 + c^4*cos(f*x + e)^3)*sqrt(a)*arctan(sqrt((a*cos(f
*x + e) + a)/cos(f*x + e))*cos(f*x + e)/(sqrt(a)*sin(f*x + e))) - (15*d^4 + 4*(105*c^3*d + 105*c^2*d^2 + 56*c*
d^3 + 12*d^4)*cos(f*x + e)^3 + 2*(105*c^2*d^2 + 56*c*d^3 + 12*d^4)*cos(f*x + e)^2 + 6*(14*c*d^3 + 3*d^4)*cos(f
*x + e))*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sin(f*x + e))/(f*cos(f*x + e)^4 + f*cos(f*x + e)^3)]

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (2*pi/x/2)>(-2*pi/
x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check si
gn: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/
x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check si
gn: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/
x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check si
gn: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/
x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)2*(2*(((-1/216090000*(27783000*sqrt(2)*a^4*d^4*sign(cos(f*x+exp(1)))[phone]*sqrt(2)*a^4*c*d^3*sign(cos(
f*x+exp(1)))[phone]*sqrt(2)*a^4*c^2*d^2*sign(cos(f*x+exp(1)))[phone]*sqrt(2)*a^4*c^3*d*sign(cos(f*x+exp(
1))))*tan(1/2*(f*x+exp(1)))^2-1/147000*(-102900*sqrt(2)*a^4*d^4*sign(cos(f*x+exp(1)))-333200*sqrt(2)*a^4*c*d^3
*sign(cos(f*x+exp(1)))-735000*sqrt(2)*a^4*c^2*d^2*sign(cos(f*x+exp(1)))-882000*sqrt(2)*a^4*c^3*d*sign(cos(f*x+
exp(1)))))*tan(1/2*(f*x+exp(1)))^2+1/3087000*(-1543500*sqrt(2)*a^4*d^4*sign(cos(f*x+exp(1)))-10290000*sqrt(2)*
a^4*c*d^3*sign(cos(f*x+exp(1)))-21609000*sqrt(2)*a^4*c^2*d^2*sign(cos(f*x+exp(1)))-18522000*sqrt(2)*a^4*c^3*d*
sign(cos(f*x+exp(1)))))*tan(1/2*(f*x+exp(1)))^2+1/14700*(7350*sqrt(2)*a^4*d^4*sign(cos(f*x+exp(1)))+29400*sqrt
(2)*a^4*c*d^3*sign(cos(f*x+exp(1)))+44100*sqrt(2)*a^4*c^2*d^2*sign(cos(f*x+exp(1)))+29400*sqrt(2)*a^4*c^3*d*si
gn(cos(f*x+exp(1)))))/sqrt(-a*tan(1/2*(f*x+exp(1)))^2+a)/(-a*tan(1/2*(f*x+exp(1)))^2+a)^3*tan(1/2*(f*x+exp(1))
)-1/2*a*sqrt(-a)*c^4*sign(cos(f*x+exp(1)))*ln(abs(2*(sqrt(-a*tan(1/2*(f*x+exp(1)))^2+a)-sqrt(-a)*tan(1/2*(f*x+
exp(1))))^2-4*sqrt(2)*abs(a)-6*a)/abs(2*(sqrt(-a*tan(1/2*(f*x+exp(1)))^2+a)-sqrt(-a)*tan(1/2*(f*x+exp(1))))^2+
4*sqrt(2)*abs(a)-6*a))/abs(a))/f

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maple [B]  time = 2.13, size = 546, normalized size = 2.01 \[ \frac {\sqrt {\frac {a \left (1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}}\, \left (105 \sqrt {2}\, \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right ) \left (-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}\right )^{\frac {7}{2}} \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sin \left (f x +e \right ) \sqrt {2}}{2 \cos \left (f x +e \right )}\right ) c^{4}+315 \sqrt {2}\, \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) \left (-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}\right )^{\frac {7}{2}} \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sin \left (f x +e \right ) \sqrt {2}}{2 \cos \left (f x +e \right )}\right ) c^{4}+315 \sqrt {2}\, \cos \left (f x +e \right ) \sin \left (f x +e \right ) \left (-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}\right )^{\frac {7}{2}} \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sin \left (f x +e \right ) \sqrt {2}}{2 \cos \left (f x +e \right )}\right ) c^{4}+105 \sqrt {2}\, \left (-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}\right )^{\frac {7}{2}} \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sin \left (f x +e \right ) \sqrt {2}}{2 \cos \left (f x +e \right )}\right ) c^{4} \sin \left (f x +e \right )-6720 \left (\cos ^{4}\left (f x +e \right )\right ) c^{3} d -6720 \left (\cos ^{4}\left (f x +e \right )\right ) c^{2} d^{2}-3584 \left (\cos ^{4}\left (f x +e \right )\right ) c \,d^{3}-768 \left (\cos ^{4}\left (f x +e \right )\right ) d^{4}+6720 \left (\cos ^{3}\left (f x +e \right )\right ) c^{3} d +3360 \left (\cos ^{3}\left (f x +e \right )\right ) c^{2} d^{2}+1792 \left (\cos ^{3}\left (f x +e \right )\right ) c \,d^{3}+384 \left (\cos ^{3}\left (f x +e \right )\right ) d^{4}+3360 \left (\cos ^{2}\left (f x +e \right )\right ) c^{2} d^{2}+448 \left (\cos ^{2}\left (f x +e \right )\right ) c \,d^{3}+96 \left (\cos ^{2}\left (f x +e \right )\right ) d^{4}+1344 \cos \left (f x +e \right ) c \,d^{3}+48 \cos \left (f x +e \right ) d^{4}+240 d^{4}\right )}{840 f \cos \left (f x +e \right )^{3} \sin \left (f x +e \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/840/f*(a*(1+cos(f*x+e))/cos(f*x+e))^(1/2)*(105*2^(1/2)*cos(f*x+e)^3*sin(f*x+e)*(-2*cos(f*x+e)/(1+cos(f*x+e))
)^(7/2)*arctanh(1/2*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*sin(f*x+e)/cos(f*x+e)*2^(1/2))*c^4+315*2^(1/2)*cos(f*
x+e)^2*sin(f*x+e)*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(7/2)*arctanh(1/2*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*sin(f*
x+e)/cos(f*x+e)*2^(1/2))*c^4+315*2^(1/2)*cos(f*x+e)*sin(f*x+e)*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(7/2)*arctanh(1/
2*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*sin(f*x+e)/cos(f*x+e)*2^(1/2))*c^4+105*2^(1/2)*(-2*cos(f*x+e)/(1+cos(f*
x+e)))^(7/2)*arctanh(1/2*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*sin(f*x+e)/cos(f*x+e)*2^(1/2))*c^4*sin(f*x+e)-67
20*cos(f*x+e)^4*c^3*d-6720*cos(f*x+e)^4*c^2*d^2-3584*cos(f*x+e)^4*c*d^3-768*cos(f*x+e)^4*d^4+6720*cos(f*x+e)^3
*c^3*d+3360*cos(f*x+e)^3*c^2*d^2+1792*cos(f*x+e)^3*c*d^3+384*cos(f*x+e)^3*d^4+3360*cos(f*x+e)^2*c^2*d^2+448*co
s(f*x+e)^2*c*d^3+96*cos(f*x+e)^2*d^4+1344*cos(f*x+e)*c*d^3+48*cos(f*x+e)*d^4+240*d^4)/cos(f*x+e)^3/sin(f*x+e)

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,{\left (c+\frac {d}{\cos \left (e+f\,x\right )}\right )}^4 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a \left (\sec {\left (e + f x \right )} + 1\right )} \left (c + d \sec {\left (e + f x \right )}\right )^{4}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a*(sec(e + f*x) + 1))*(c + d*sec(e + f*x))**4, x)

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