3.2.47 \(\int \sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))^4 \, dx\) [147]

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

[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 = 0.12, 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 = {4025, 90, 65, 212} \begin {gather*} \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}} \end {gather*}

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 65

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 90

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 212

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

Rule 4025

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 ) \text {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 ) \text {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 ) \text {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 ) \text {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] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 14.46, size = 587, normalized size = 2.17 \begin {gather*} \frac {\cos ^4(e+f x) \sec \left (\frac {1}{2} (e+f x)\right ) \sqrt {a (1+\sec (e+f x))} (c+d \sec (e+f x))^4 \left (\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 {2}{7} d^4 \sec ^3(e+f x) \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 {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 )\right )}{f (d+c \cos (e+f x))^4}-\frac {8 \left (-3-2 \sqrt {2}\right ) c^4 \cos ^4\left (\frac {1}{4} (e+f x)\right ) \sqrt {\frac {7-5 \sqrt {2}+\left (10-7 \sqrt {2}\right ) \cos \left (\frac {1}{2} (e+f x)\right )}{1+\cos \left (\frac {1}{2} (e+f x)\right )}} \sqrt {\frac {-1+\sqrt {2}-\left (-2+\sqrt {2}\right ) \cos \left (\frac {1}{2} (e+f x)\right )}{1+\cos \left (\frac {1}{2} (e+f x)\right )}} \left (1-\sqrt {2}+\left (-2+\sqrt {2}\right ) \cos \left (\frac {1}{2} (e+f x)\right )\right ) \cos ^3(e+f x) \left (F\left (\text {ArcSin}\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};\text {ArcSin}\left (\frac {\tan \left (\frac {1}{4} (e+f x)\right )}{\sqrt {3-2 \sqrt {2}}}\right )|17-12 \sqrt {2}\right )\right ) \sqrt {\left (-1-\sqrt {2}+\left (2+\sqrt {2}\right ) \cos \left (\frac {1}{2} (e+f x)\right )\right ) \sec ^2\left (\frac {1}{4} (e+f x)\right )} \sec \left (\frac {1}{2} (e+f x)\right ) \sqrt {a (1+\sec (e+f x))} (c+d \sec (e+f x))^4 \sqrt {3-2 \sqrt {2}-\tan ^2\left (\frac {1}{4} (e+f x)\right )}}{f (d+c \cos (e+f x))^4} \end {gather*}

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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(545\) vs. \(2(247)=494\).
time = 0.30, size = 546, normalized size = 2.01

method	result	size
default	\(-\frac {\sqrt {\frac {a \left (\cos \left (f x +e \right )+1\right )}{\cos \left (f x +e \right )}}\, \left (-105 \left (\cos ^{3}\left (f x +e \right )\right ) \left (-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}\right )^{\frac {7}{2}} \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right ) \sqrt {2}}{2 \cos \left (f x +e \right )}\right ) \sqrt {2}\, \sin \left (f x +e \right ) c^{4}-315 \left (\cos ^{2}\left (f x +e \right )\right ) \left (-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}\right )^{\frac {7}{2}} \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right ) \sqrt {2}}{2 \cos \left (f x +e \right )}\right ) \sqrt {2}\, \sin \left (f x +e \right ) c^{4}-315 \cos \left (f x +e \right ) \left (-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}\right )^{\frac {7}{2}} \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right ) \sqrt {2}}{2 \cos \left (f x +e \right )}\right ) \sqrt {2}\, \sin \left (f x +e \right ) c^{4}-105 \left (-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}\right )^{\frac {7}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \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 )}\)	\(546\)

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,method=_RETURNVERBOSE)

[Out]

-1/840/f*(a*(cos(f*x+e)+1)/cos(f*x+e))^(1/2)*(-105*cos(f*x+e)^3*(-2*cos(f*x+e)/(cos(f*x+e)+1))^(7/2)*arctanh(1
/2*(-2*cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)/cos(f*x+e)*2^(1/2))*2^(1/2)*sin(f*x+e)*c^4-315*cos(f*x+e)^2
*(-2*cos(f*x+e)/(cos(f*x+e)+1))^(7/2)*arctanh(1/2*(-2*cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)/cos(f*x+e)*2
^(1/2))*2^(1/2)*sin(f*x+e)*c^4-315*cos(f*x+e)*(-2*cos(f*x+e)/(cos(f*x+e)+1))^(7/2)*arctanh(1/2*(-2*cos(f*x+e)/
(cos(f*x+e)+1))^(1/2)*sin(f*x+e)/cos(f*x+e)*2^(1/2))*2^(1/2)*sin(f*x+e)*c^4-105*(-2*cos(f*x+e)/(cos(f*x+e)+1))
^(7/2)*2^(1/2)*arctanh(1/2*(-2*cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)/cos(f*x+e)*2^(1/2))*c^4*sin(f*x+e)+
6720*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*
cos(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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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]

-1/210*(16*(cos(2*f*x + 2*e)^2 + sin(2*f*x + 2*e)^2 + 2*cos(2*f*x + 2*e) + 1)^(1/4)*(7*(15*c^3*d*sin(6*f*x + 6
*e) + 5*(9*c^3*d + 3*c^2*d^2 + 4*c*d^3)*sin(4*f*x + 4*e) + (45*c^3*d + 30*c^2*d^2 + 28*c*d^3 + 6*d^4)*sin(2*f*
x + 2*e))*cos(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e) + 1)) - (105*c^3*d*cos(6*f*x + 6*e) + 105*c^3*d +
105*c^2*d^2 + 56*c*d^3 + 12*d^4 + 35*(9*c^3*d + 3*c^2*d^2 + 4*c*d^3)*cos(4*f*x + 4*e) + 7*(45*c^3*d + 30*c^2*
d^2 + 28*c*d^3 + 6*d^4)*cos(2*f*x + 2*e))*sin(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e) + 1)))*sqrt(a) +
105*((c^4*cos(2*f*x + 2*e)^4 + c^4*sin(2*f*x + 2*e)^4 + 4*c^4*cos(2*f*x + 2*e)^3 + 6*c^4*cos(2*f*x + 2*e)^2 +
4*c^4*cos(2*f*x + 2*e) + c^4 + 2*(c^4*cos(2*f*x + 2*e)^2 + 2*c^4*cos(2*f*x + 2*e) + c^4)*sin(2*f*x + 2*e)^2)*a
rctan2((cos(2*f*x + 2*e)^2 + sin(2*f*x + 2*e)^2 + 2*cos(2*f*x + 2*e) + 1)^(1/4)*sin(1/2*arctan2(sin(2*f*x + 2*
e), cos(2*f*x + 2*e) + 1)), (cos(2*f*x + 2*e)^2 + sin(2*f*x + 2*e)^2 + 2*cos(2*f*x + 2*e) + 1)^(1/4)*cos(1/2*a
rctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e) + 1)) + 1) - (c^4*cos(2*f*x + 2*e)^4 + c^4*sin(2*f*x + 2*e)^4 + 4*c^
4*cos(2*f*x + 2*e)^3 + 6*c^4*cos(2*f*x + 2*e)^2 + 4*c^4*cos(2*f*x + 2*e) + c^4 + 2*(c^4*cos(2*f*x + 2*e)^2 + 2
*c^4*cos(2*f*x + 2*e) + c^4)*sin(2*f*x + 2*e)^2)*arctan2((cos(2*f*x + 2*e)^2 + sin(2*f*x + 2*e)^2 + 2*cos(2*f*
x + 2*e) + 1)^(1/4)*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e) + 1)), (cos(2*f*x + 2*e)^2 + sin(2*f*x
+ 2*e)^2 + 2*cos(2*f*x + 2*e) + 1)^(1/4)*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e) + 1)) - 1) - 2*(c^
4*f*cos(2*f*x + 2*e)^4 + c^4*f*sin(2*f*x + 2*e)^4 + 4*c^4*f*cos(2*f*x + 2*e)^3 + 6*c^4*f*cos(2*f*x + 2*e)^2 +
4*c^4*f*cos(2*f*x + 2*e) + c^4*f + 2*(c^4*f*cos(2*f*x + 2*e)^2 + 2*c^4*f*cos(2*f*x + 2*e) + c^4*f)*sin(2*f*x +
2*e)^2)*integrate((((cos(10*f*x + 10*e)*cos(2*f*x + 2*e) + 4*cos(8*f*x + 8*e)*cos(2*f*x + 2*e) + 6*cos(6*f*x
+ 6*e)*cos(2*f*x + 2*e) + 4*cos(4*f*x + 4*e)*cos(2*f*x + 2*e) + cos(2*f*x + 2*e)^2 + sin(10*f*x + 10*e)*sin(2*
f*x + 2*e) + 4*sin(8*f*x + 8*e)*sin(2*f*x + 2*e) + 6*sin(6*f*x + 6*e)*sin(2*f*x + 2*e) + 4*sin(4*f*x + 4*e)*si
n(2*f*x + 2*e) + sin(2*f*x + 2*e)^2)*cos(9/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + (cos(2*f*x + 2*e)*
sin(10*f*x + 10*e) + 4*cos(2*f*x + 2*e)*sin(8*f*x + 8*e) + 6*cos(2*f*x + 2*e)*sin(6*f*x + 6*e) + 4*cos(2*f*x +
2*e)*sin(4*f*x + 4*e) - cos(10*f*x + 10*e)*sin(2*f*x + 2*e) - 4*cos(8*f*x + 8*e)*sin(2*f*x + 2*e) - 6*cos(6*f
*x + 6*e)*sin(2*f*x + 2*e) - 4*cos(4*f*x + 4*e)*sin(2*f*x + 2*e))*sin(9/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x
+ 2*e))))*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e) + 1)) - ((cos(2*f*x + 2*e)*sin(10*f*x + 10*e) + 4
*cos(2*f*x + 2*e)*sin(8*f*x + 8*e) + 6*cos(2*f*x + 2*e)*sin(6*f*x + 6*e) + 4*cos(2*f*x + 2*e)*sin(4*f*x + 4*e)
- cos(10*f*x + 10*e)*sin(2*f*x + 2*e) - 4*cos(8*f*x + 8*e)*sin(2*f*x + 2*e) - 6*cos(6*f*x + 6*e)*sin(2*f*x +
2*e) - 4*cos(4*f*x + 4*e)*sin(2*f*x + 2*e))*cos(9/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - (cos(10*f*x
+ 10*e)*cos(2*f*x + 2*e) + 4*cos(8*f*x + 8*e)*cos(2*f*x + 2*e) + 6*cos(6*f*x + 6*e)*cos(2*f*x + 2*e) + 4*cos(
4*f*x + 4*e)*cos(2*f*x + 2*e) + cos(2*f*x + 2*e)^2 + sin(10*f*x + 10*e)*sin(2*f*x + 2*e) + 4*sin(8*f*x + 8*e)*
sin(2*f*x + 2*e) + 6*sin(6*f*x + 6*e)*sin(2*f*x + 2*e) + 4*sin(4*f*x + 4*e)*sin(2*f*x + 2*e) + sin(2*f*x + 2*e
)^2)*sin(9/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)
+ 1)))/(((2*(4*cos(8*f*x + 8*e) + 6*cos(6*f*x + 6*e) + 4*cos(4*f*x + 4*e) + cos(2*f*x + 2*e))*cos(10*f*x + 10*
e) + cos(10*f*x + 10*e)^2 + 8*(6*cos(6*f*x + 6*e) + 4*cos(4*f*x + 4*e) + cos(2*f*x + 2*e))*cos(8*f*x + 8*e) +
16*cos(8*f*x + 8*e)^2 + 12*(4*cos(4*f*x + 4*e) + cos(2*f*x + 2*e))*cos(6*f*x + 6*e) + 36*cos(6*f*x + 6*e)^2 +
16*cos(4*f*x + 4*e)^2 + 8*cos(4*f*x + 4*e)*cos(2*f*x + 2*e) + cos(2*f*x + 2*e)^2 + 2*(4*sin(8*f*x + 8*e) + 6*s
in(6*f*x + 6*e) + 4*sin(4*f*x + 4*e) + sin(2*f*x + 2*e))*sin(10*f*x + 10*e) + sin(10*f*x + 10*e)^2 + 8*(6*sin(
6*f*x + 6*e) + 4*sin(4*f*x + 4*e) + sin(2*f*x + 2*e))*sin(8*f*x + 8*e) + 16*sin(8*f*x + 8*e)^2 + 12*(4*sin(4*f
*x + 4*e) + sin(2*f*x + 2*e))*sin(6*f*x + 6*e) + 36*sin(6*f*x + 6*e)^2 + 16*sin(4*f*x + 4*e)^2 + 8*sin(4*f*x +
4*e)*sin(2*f*x + 2*e) + sin(2*f*x + 2*e)^2)*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e) + 1))^2 + (2*(
4*cos(8*f*x + 8*e) + 6*cos(6*f*x + 6*e) + 4*cos(4*f*x + 4*e) + cos(2*f*x + 2*e))*cos(10*f*x + 10*e) + cos(10*f
*x + 10*e)^2 + 8*(6*cos(6*f*x + 6*e) + 4*cos(4*f*x + 4*e) + cos(2*f*x + 2*e))*cos(8*f*x + 8*e) + 16*cos(8*f*x
+ 8*e)^2 + 12*(4*cos(4*f*x + 4*e) + cos(2*f*x + 2*e))*cos(6*f*x + 6*e) + 36*cos(6*f*x + 6*e)^2 + 16*cos(4*f*x
+ 4*e)^2 + 8*cos(4*f*x + 4*e)*cos(2*f*x + 2*e) + cos(2*f*x + 2*e)^2 + 2*(4*sin(8*f*x + 8*e) + 6*sin(6*f*x + 6*
e) + 4*sin(4*f*x + 4*e) + sin(2*f*x + 2*e))*sin(10*f*x + 10*e) + sin(10*f*x + 10*e)^2 + 8*(6*sin(6*f*x + 6*e)
+ 4*sin(4*f*x + 4*e) + sin(2*f*x + 2*e))*sin(8*f*x + 8*e) + 16*sin(8*f*x + 8*e)^2 + 12*(4*sin(4*f*x + 4*e) + s
in(2*f*x + 2*e))*sin(6*f*x + 6*e) + 36*sin(6*f*...

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Fricas [A]
time = 3.05, size = 503, normalized size = 1.86 \begin {gather*} \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 ] \end {gather*}

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

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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 535 vs. \(2 (250) = 500\).
time = 1.53, size = 535, normalized size = 1.97 \begin {gather*} -\frac {\frac {105 \, \sqrt {-a} a c^{4} \log \left (\frac {{\left | 2 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} - 4 \, \sqrt {2} {\left | a \right |} - 6 \, a \right |}}{{\left | 2 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} + 4 \, \sqrt {2} {\left | a \right |} - 6 \, a \right |}}\right ) \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}{{\left | a \right |}} + \frac {2 \, {\left (420 \, \sqrt {2} a^{4} c^{3} d \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 630 \, \sqrt {2} a^{4} c^{2} d^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 420 \, \sqrt {2} a^{4} c d^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 105 \, \sqrt {2} a^{4} d^{4} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - {\left (1260 \, \sqrt {2} a^{4} c^{3} d \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 1470 \, \sqrt {2} a^{4} c^{2} d^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 700 \, \sqrt {2} a^{4} c d^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 105 \, \sqrt {2} a^{4} d^{4} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - {\left (1260 \, \sqrt {2} a^{4} c^{3} d \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 1050 \, \sqrt {2} a^{4} c^{2} d^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 476 \, \sqrt {2} a^{4} c d^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 147 \, \sqrt {2} a^{4} d^{4} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - {\left (420 \, \sqrt {2} a^{4} c^{3} d \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 210 \, \sqrt {2} a^{4} c^{2} d^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 196 \, \sqrt {2} a^{4} c d^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 27 \, \sqrt {2} a^{4} d^{4} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )\right )} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}\right )} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}\right )} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}\right )} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{{\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a\right )}^{3} \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}}}{105 \, f} \end {gather*}

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]

-1/105*(105*sqrt(-a)*a*c^4*log(abs(2*(sqrt(-a)*tan(1/2*f*x + 1/2*e) - sqrt(-a*tan(1/2*f*x + 1/2*e)^2 + a))^2 -
4*sqrt(2)*abs(a) - 6*a)/abs(2*(sqrt(-a)*tan(1/2*f*x + 1/2*e) - sqrt(-a*tan(1/2*f*x + 1/2*e)^2 + a))^2 + 4*sqr
t(2)*abs(a) - 6*a))*sgn(cos(f*x + e))/abs(a) + 2*(420*sqrt(2)*a^4*c^3*d*sgn(cos(f*x + e)) + 630*sqrt(2)*a^4*c^
2*d^2*sgn(cos(f*x + e)) + 420*sqrt(2)*a^4*c*d^3*sgn(cos(f*x + e)) + 105*sqrt(2)*a^4*d^4*sgn(cos(f*x + e)) - (1
260*sqrt(2)*a^4*c^3*d*sgn(cos(f*x + e)) + 1470*sqrt(2)*a^4*c^2*d^2*sgn(cos(f*x + e)) + 700*sqrt(2)*a^4*c*d^3*s
gn(cos(f*x + e)) + 105*sqrt(2)*a^4*d^4*sgn(cos(f*x + e)) - (1260*sqrt(2)*a^4*c^3*d*sgn(cos(f*x + e)) + 1050*sq
rt(2)*a^4*c^2*d^2*sgn(cos(f*x + e)) + 476*sqrt(2)*a^4*c*d^3*sgn(cos(f*x + e)) + 147*sqrt(2)*a^4*d^4*sgn(cos(f*
x + e)) - (420*sqrt(2)*a^4*c^3*d*sgn(cos(f*x + e)) + 210*sqrt(2)*a^4*c^2*d^2*sgn(cos(f*x + e)) + 196*sqrt(2)*a
^4*c*d^3*sgn(cos(f*x + e)) + 27*sqrt(2)*a^4*d^4*sgn(cos(f*x + e)))*tan(1/2*f*x + 1/2*e)^2)*tan(1/2*f*x + 1/2*e
)^2)*tan(1/2*f*x + 1/2*e)^2)*tan(1/2*f*x + 1/2*e)/((a*tan(1/2*f*x + 1/2*e)^2 - a)^3*sqrt(-a*tan(1/2*f*x + 1/2*
e)^2 + a)))/f

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}

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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