3.2.61 \(\int (a+a \sec (e+f x))^{5/2} (c+d \sec (e+f x))^2 \, dx\) [161]

Optimal. Leaf size=258 \[ \frac {2 a^3 (c+2 d) (3 c+2 d) \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)}}+\frac {2 a^{7/2} c^2 \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a}}\right ) \tan (e+f x)}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {2 a d (2 c+5 d) (a-a \sec (e+f x))^2 \tan (e+f x)}{5 f \sqrt {a+a \sec (e+f x)}}-\frac {2 d^2 (a-a \sec (e+f x))^3 \tan (e+f x)}{7 f \sqrt {a+a \sec (e+f x)}}-\frac {2 \left (c^2+8 c d+8 d^2\right ) \left (a^3-a^3 \sec (e+f x)\right ) \tan (e+f x)}{3 f \sqrt {a+a \sec (e+f x)}} \]

[Out]

2*a^3*(c+2*d)*(3*c+2*d)*tan(f*x+e)/f/(a+a*sec(f*x+e))^(1/2)+2/5*a*d*(2*c+5*d)*(a-a*sec(f*x+e))^2*tan(f*x+e)/f/
(a+a*sec(f*x+e))^(1/2)-2/7*d^2*(a-a*sec(f*x+e))^3*tan(f*x+e)/f/(a+a*sec(f*x+e))^(1/2)-2/3*(c^2+8*c*d+8*d^2)*(a
^3-a^3*sec(f*x+e))*tan(f*x+e)/f/(a+a*sec(f*x+e))^(1/2)+2*a^(7/2)*c^2*arctanh((a-a*sec(f*x+e))^(1/2)/a^(1/2))*t
an(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 = 258, 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, 186, 65, 212} \begin {gather*} \frac {2 a^{7/2} c^2 \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 \left (c^2+8 c d+8 d^2\right ) \tan (e+f x) \left (a^3-a^3 \sec (e+f x)\right )}{3 f \sqrt {a \sec (e+f x)+a}}+\frac {2 a^3 (c+2 d) (3 c+2 d) \tan (e+f x)}{f \sqrt {a \sec (e+f x)+a}}+\frac {2 a d (2 c+5 d) \tan (e+f x) (a-a \sec (e+f x))^2}{5 f \sqrt {a \sec (e+f x)+a}}-\frac {2 d^2 \tan (e+f x) (a-a \sec (e+f x))^3}{7 f \sqrt {a \sec (e+f x)+a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + a*Sec[e + f*x])^(5/2)*(c + d*Sec[e + f*x])^2,x]

[Out]

(2*a^3*(c + 2*d)*(3*c + 2*d)*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) + (2*a^(7/2)*c^2*ArcTanh[Sqrt[a - a*Se
c[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*a*d*(2*c + 5*d)*
(a - a*Sec[e + f*x])^2*Tan[e + f*x])/(5*f*Sqrt[a + a*Sec[e + f*x]]) - (2*d^2*(a - a*Sec[e + f*x])^3*Tan[e + f*
x])/(7*f*Sqrt[a + a*Sec[e + f*x]]) - (2*(c^2 + 8*c*d + 8*d^2)*(a^3 - a^3*Sec[e + f*x])*Tan[e + f*x])/(3*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 186

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_))^(q_), x
_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x)^q, x], x] /; FreeQ[{a, b, c, d,
e, f, g, h, m, n}, x] && IntegersQ[p, q]

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

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Mathematica [A]
time = 2.78, size = 191, normalized size = 0.74 \begin {gather*} \frac {a^2 \sec \left (\frac {1}{2} (e+f x)\right ) \sec ^3(e+f x) \sqrt {a (1+\sec (e+f x))} \left (420 \sqrt {2} c^2 \text {ArcSin}\left (\sqrt {2} \sin \left (\frac {1}{2} (e+f x)\right )\right ) \cos ^{\frac {7}{2}}(e+f x)+4 \left (35 c^2+196 c d+145 d^2+\left (420 c^2+987 c d+465 d^2\right ) \cos (e+f x)+\left (35 c^2+196 c d+115 d^2\right ) \cos (2 (e+f x))+140 c^2 \cos (3 (e+f x))+301 c d \cos (3 (e+f x))+115 d^2 \cos (3 (e+f x))\right ) \sin \left (\frac {1}{2} (e+f x)\right )\right )}{420 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sec[e + f*x])^(5/2)*(c + d*Sec[e + f*x])^2,x]

[Out]

(a^2*Sec[(e + f*x)/2]*Sec[e + f*x]^3*Sqrt[a*(1 + Sec[e + f*x])]*(420*Sqrt[2]*c^2*ArcSin[Sqrt[2]*Sin[(e + f*x)/
2]]*Cos[e + f*x]^(7/2) + 4*(35*c^2 + 196*c*d + 145*d^2 + (420*c^2 + 987*c*d + 465*d^2)*Cos[e + f*x] + (35*c^2
+ 196*c*d + 115*d^2)*Cos[2*(e + f*x)] + 140*c^2*Cos[3*(e + f*x)] + 301*c*d*Cos[3*(e + f*x)] + 115*d^2*Cos[3*(e
+ f*x)])*Sin[(e + f*x)/2]))/(420*f)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(503\) vs. \(2(234)=468\).
time = 1.48, size = 504, normalized size = 1.95

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 (-\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 ) \sin \left (f x +e \right ) \left (\cos ^{3}\left (f x +e \right )\right ) c^{2}+315 \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 ) \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right ) c^{2}+315 \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 ) \sin \left (f x +e \right ) \cos \left (f x +e \right ) c^{2}+105 \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 ) \left (-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}\right )^{\frac {7}{2}} c^{2} \sin \left (f x +e \right )-4480 \left (\cos ^{4}\left (f x +e \right )\right ) c^{2}-9632 \left (\cos ^{4}\left (f x +e \right )\right ) c d -3680 \left (\cos ^{4}\left (f x +e \right )\right ) d^{2}+3920 \left (\cos ^{3}\left (f x +e \right )\right ) c^{2}+6496 \left (\cos ^{3}\left (f x +e \right )\right ) c d +1840 \left (\cos ^{3}\left (f x +e \right )\right ) d^{2}+560 \left (\cos ^{2}\left (f x +e \right )\right ) c^{2}+2464 \left (\cos ^{2}\left (f x +e \right )\right ) c d +880 \left (\cos ^{2}\left (f x +e \right )\right ) d^{2}+672 \cos \left (f x +e \right ) c d +720 \cos \left (f x +e \right ) d^{2}+240 d^{2}\right ) a^{2}}{840 f \cos \left (f x +e \right )^{3} \sin \left (f x +e \right )}\)	\(504\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sec(f*x+e))^(5/2)*(c+d*sec(f*x+e))^2,x,method=_RETURNVERBOSE)

[Out]

1/840/f*(a*(cos(f*x+e)+1)/cos(f*x+e))^(1/2)*(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))*sin(f*x+e)*cos(f*x+e)^3*c^2+315*(-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))
*sin(f*x+e)*cos(f*x+e)^2*c^2+315*(-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))*sin(f*x+e)*cos(f*x+e)*c^2+105*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))*(-2*cos(f*x+e)/(cos(f*x+e)+1))^(7/2)*c^2*sin(f*x+e)-44
80*cos(f*x+e)^4*c^2-9632*cos(f*x+e)^4*c*d-3680*cos(f*x+e)^4*d^2+3920*cos(f*x+e)^3*c^2+6496*cos(f*x+e)^3*c*d+18
40*cos(f*x+e)^3*d^2+560*cos(f*x+e)^2*c^2+2464*cos(f*x+e)^2*c*d+880*cos(f*x+e)^2*d^2+672*cos(f*x+e)*c*d+720*cos
(f*x+e)*d^2+240*d^2)/cos(f*x+e)^3/sin(f*x+e)*a^2

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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((a+a*sec(f*x+e))^(5/2)*(c+d*sec(f*x+e))^2,x, algorithm="maxima")

[Out]

-1/210*(105*((a^2*c^2*cos(2*f*x + 2*e)^2 + a^2*c^2*sin(2*f*x + 2*e)^2 + 2*a^2*c^2*cos(2*f*x + 2*e) + a^2*c^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) - (a^2*c^2*cos(2*f*x + 2*e)^2 + a^2*c^2*sin(2*f*x + 2*e)
^2 + 2*a^2*c^2*cos(2*f*x + 2*e) + a^2*c^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*(a^2*c^2*
f*cos(2*f*x + 2*e)^2 + a^2*c^2*f*sin(2*f*x + 2*e)^2 + 2*a^2*c^2*f*cos(2*f*x + 2*e) + a^2*c^2*f)*integrate((((c
os(6*f*x + 6*e)*cos(2*f*x + 2*e) + 2*cos(4*f*x + 4*e)*cos(2*f*x + 2*e) + cos(2*f*x + 2*e)^2 + sin(6*f*x + 6*e)
*sin(2*f*x + 2*e) + 2*sin(4*f*x + 4*e)*sin(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(6*f*x + 6*e) + 2*cos(2*f*x + 2*e)*sin(4*f*x + 4*e) - cos(6*f*x +
6*e)*sin(2*f*x + 2*e) - 2*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(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e) + 1)) - ((cos(2*f*x + 2*e)*sin(6*f*x + 6*e) + 2*cos(2*
f*x + 2*e)*sin(4*f*x + 4*e) - cos(6*f*x + 6*e)*sin(2*f*x + 2*e) - 2*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(6*f*x + 6*e)*cos(2*f*x + 2*e) + 2*cos(4*f*x + 4*e)*cos(2*
f*x + 2*e) + cos(2*f*x + 2*e)^2 + sin(6*f*x + 6*e)*sin(2*f*x + 2*e) + 2*sin(4*f*x + 4*e)*sin(2*f*x + 2*e) + si
n(2*f*x + 2*e)^2)*sin(9/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(
2*f*x + 2*e) + 1)))/(((cos(2*f*x + 2*e)^4 + sin(2*f*x + 2*e)^4 + (cos(2*f*x + 2*e)^2 + sin(2*f*x + 2*e)^2 + 2*
cos(2*f*x + 2*e) + 1)*cos(6*f*x + 6*e)^2 + 4*(cos(2*f*x + 2*e)^2 + sin(2*f*x + 2*e)^2 + 2*cos(2*f*x + 2*e) + 1
)*cos(4*f*x + 4*e)^2 + 2*cos(2*f*x + 2*e)^3 + (cos(2*f*x + 2*e)^2 + sin(2*f*x + 2*e)^2 + 2*cos(2*f*x + 2*e) +
1)*sin(6*f*x + 6*e)^2 + 4*(cos(2*f*x + 2*e)^2 + sin(2*f*x + 2*e)^2 + 2*cos(2*f*x + 2*e) + 1)*sin(4*f*x + 4*e)^
2 + (2*cos(2*f*x + 2*e)^2 + 2*cos(2*f*x + 2*e) + 1)*sin(2*f*x + 2*e)^2 + 2*(cos(2*f*x + 2*e)^3 + cos(2*f*x + 2
*e)*sin(2*f*x + 2*e)^2 + 2*(cos(2*f*x + 2*e)^2 + sin(2*f*x + 2*e)^2 + 2*cos(2*f*x + 2*e) + 1)*cos(4*f*x + 4*e)
+ 2*cos(2*f*x + 2*e)^2 + cos(2*f*x + 2*e))*cos(6*f*x + 6*e) + 4*(cos(2*f*x + 2*e)^3 + cos(2*f*x + 2*e)*sin(2*
f*x + 2*e)^2 + 2*cos(2*f*x + 2*e)^2 + cos(2*f*x + 2*e))*cos(4*f*x + 4*e) + cos(2*f*x + 2*e)^2 + 2*(sin(2*f*x +
2*e)^3 + 2*(cos(2*f*x + 2*e)^2 + sin(2*f*x + 2*e)^2 + 2*cos(2*f*x + 2*e) + 1)*sin(4*f*x + 4*e) + (cos(2*f*x +
2*e)^2 + 2*cos(2*f*x + 2*e) + 1)*sin(2*f*x + 2*e))*sin(6*f*x + 6*e) + 4*(sin(2*f*x + 2*e)^3 + (cos(2*f*x + 2*
e)^2 + 2*cos(2*f*x + 2*e) + 1)*sin(2*f*x + 2*e))*sin(4*f*x + 4*e))*cos(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x
+ 2*e) + 1))^2 + (cos(2*f*x + 2*e)^4 + sin(2*f*x + 2*e)^4 + (cos(2*f*x + 2*e)^2 + sin(2*f*x + 2*e)^2 + 2*cos(
2*f*x + 2*e) + 1)*cos(6*f*x + 6*e)^2 + 4*(cos(2*f*x + 2*e)^2 + sin(2*f*x + 2*e)^2 + 2*cos(2*f*x + 2*e) + 1)*co
s(4*f*x + 4*e)^2 + 2*cos(2*f*x + 2*e)^3 + (cos(2*f*x + 2*e)^2 + sin(2*f*x + 2*e)^2 + 2*cos(2*f*x + 2*e) + 1)*s
in(6*f*x + 6*e)^2 + 4*(cos(2*f*x + 2*e)^2 + sin(2*f*x + 2*e)^2 + 2*cos(2*f*x + 2*e) + 1)*sin(4*f*x + 4*e)^2 +
(2*cos(2*f*x + 2*e)^2 + 2*cos(2*f*x + 2*e) + 1)*sin(2*f*x + 2*e)^2 + 2*(cos(2*f*x + 2*e)^3 + cos(2*f*x + 2*e)*
sin(2*f*x + 2*e)^2 + 2*(cos(2*f*x + 2*e)^2 + sin(2*f*x + 2*e)^2 + 2*cos(2*f*x + 2*e) + 1)*cos(4*f*x + 4*e) + 2
*cos(2*f*x + 2*e)^2 + cos(2*f*x + 2*e))*cos(6*f*x + 6*e) + 4*(cos(2*f*x + 2*e)^3 + cos(2*f*x + 2*e)*sin(2*f*x
+ 2*e)^2 + 2*cos(2*f*x + 2*e)^2 + cos(2*f*x + 2*e))*cos(4*f*x + 4*e) + cos(2*f*x + 2*e)^2 + 2*(sin(2*f*x + 2*e
)^3 + 2*(cos(2*f*x + 2*e)^2 + sin(2*f*x + 2*e)^2 + 2*cos(2*f*x + 2*e) + 1)*sin(4*f*x + 4*e) + (cos(2*f*x + 2*e
)^2 + 2*cos(2*f*x + 2*e) + 1)*sin(2*f*x + 2*e))*sin(6*f*x + 6*e) + 4*(sin(2*f*x + 2*e)^3 + (cos(2*f*x + 2*e)^2
+ 2*cos(2*f*x + 2*e) + 1)*sin(2*f*x + 2*e))*sin(4*f*x + 4*e))*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2
*e) + 1))^2)*(cos(2*f*x + 2*e)^2 + sin(2*f*x + 2*e)^2 + 2*cos(2*f*x + 2*e) + 1)^(1/4)), x) - 8*((3*a^2*c^2 + 5
*a^2*c*d + a^2*d^2)*f*cos(2*f*x + 2*e)^2 + (3*a^2*c^2 + 5*a^2*c*d + a^2*d^2)*f*sin(2*f*x + 2*e)^2 + 2*(3*a^2*c
^2 + 5*a^2*c*d + a^2*d^2)*f*cos(2*f*x + 2*e) + (3*a^2*c^2 + 5*a^2*c*d + a^2*d^2)*f)*integrate((((cos(6*f*x + 6
*e)*cos(2*f*x + 2*e) + 2*cos(4*f*x + 4*e)*cos(2*f*x + 2*e) + cos(2*f*x + 2*e)^2 + sin(6*f*x + 6*e)*sin(2*f*x +
2*e) + 2*sin(4*f*x + 4*e)*sin(2*f*x + 2*e) + sin(2*f*x + 2*e)^2)*cos(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x
+ 2*e))) + (cos(2*f*x + 2*e)*sin(6*f*x + 6*e) + 2*cos(2*f*x + 2*e)*sin(4*f*x + 4*e) - cos(6*f*x + 6*e)*sin(2*f
*x + 2*e) - 2*cos(4*f*x + 4*e)*sin(2*f*x + 2*e)...

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Fricas [A]
time = 3.92, size = 531, normalized size = 2.06 \begin {gather*} \left [\frac {105 \, {\left (a^{2} c^{2} \cos \left (f x + e\right )^{4} + a^{2} c^{2} \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 \, a^{2} d^{2} + 2 \, {\left (140 \, a^{2} c^{2} + 301 \, a^{2} c d + 115 \, a^{2} d^{2}\right )} \cos \left (f x + e\right )^{3} + {\left (35 \, a^{2} c^{2} + 196 \, a^{2} c d + 115 \, a^{2} d^{2}\right )} \cos \left (f x + e\right )^{2} + 6 \, {\left (7 \, a^{2} c d + 10 \, a^{2} d^{2}\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 (a^{2} c^{2} \cos \left (f x + e\right )^{4} + a^{2} c^{2} \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 \, a^{2} d^{2} + 2 \, {\left (140 \, a^{2} c^{2} + 301 \, a^{2} c d + 115 \, a^{2} d^{2}\right )} \cos \left (f x + e\right )^{3} + {\left (35 \, a^{2} c^{2} + 196 \, a^{2} c d + 115 \, a^{2} d^{2}\right )} \cos \left (f x + e\right )^{2} + 6 \, {\left (7 \, a^{2} c d + 10 \, a^{2} d^{2}\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((a+a*sec(f*x+e))^(5/2)*(c+d*sec(f*x+e))^2,x, algorithm="fricas")

[Out]

[1/105*(105*(a^2*c^2*cos(f*x + e)^4 + a^2*c^2*cos(f*x + e)^3)*sqrt(-a)*log((2*a*cos(f*x + e)^2 - 2*sqrt(-a)*sq
rt((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*
(15*a^2*d^2 + 2*(140*a^2*c^2 + 301*a^2*c*d + 115*a^2*d^2)*cos(f*x + e)^3 + (35*a^2*c^2 + 196*a^2*c*d + 115*a^2
*d^2)*cos(f*x + e)^2 + 6*(7*a^2*c*d + 10*a^2*d^2)*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*(a^2*c^2*cos(f*x + e)^4 + a^2*c^2*cos(f*x + e)^3)*s
qrt(a)*arctan(sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*cos(f*x + e)/(sqrt(a)*sin(f*x + e))) - (15*a^2*d^2 + 2*(
140*a^2*c^2 + 301*a^2*c*d + 115*a^2*d^2)*cos(f*x + e)^3 + (35*a^2*c^2 + 196*a^2*c*d + 115*a^2*d^2)*cos(f*x + e
)^2 + 6*(7*a^2*c*d + 10*a^2*d^2)*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 \left (a \left (\sec {\left (e + f x \right )} + 1\right )\right )^{\frac {5}{2}} \left (c + d \sec {\left (e + f x \right )}\right )^{2}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(f*x+e))**(5/2)*(c+d*sec(f*x+e))**2,x)

[Out]

Integral((a*(sec(e + f*x) + 1))**(5/2)*(c + d*sec(e + f*x))**2, x)

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Giac [A]
time = 3.45, size = 441, normalized size = 1.71 \begin {gather*} -\frac {\frac {105 \, \sqrt {-a} a^{3} c^{2} \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 (315 \, \sqrt {2} a^{6} c^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 840 \, \sqrt {2} a^{6} c d \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 420 \, \sqrt {2} a^{6} d^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - {\left (875 \, \sqrt {2} a^{6} c^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 1960 \, \sqrt {2} a^{6} c d \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 700 \, \sqrt {2} a^{6} d^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - {\left (805 \, \sqrt {2} a^{6} c^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 1568 \, \sqrt {2} a^{6} c d \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 560 \, \sqrt {2} a^{6} d^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - {\left (245 \, \sqrt {2} a^{6} c^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 448 \, \sqrt {2} a^{6} c d \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 160 \, \sqrt {2} a^{6} d^{2} \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((a+a*sec(f*x+e))^(5/2)*(c+d*sec(f*x+e))^2,x, algorithm="giac")

[Out]

-1/105*(105*sqrt(-a)*a^3*c^2*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*s
qrt(2)*abs(a) - 6*a))*sgn(cos(f*x + e))/abs(a) + 2*(315*sqrt(2)*a^6*c^2*sgn(cos(f*x + e)) + 840*sqrt(2)*a^6*c*
d*sgn(cos(f*x + e)) + 420*sqrt(2)*a^6*d^2*sgn(cos(f*x + e)) - (875*sqrt(2)*a^6*c^2*sgn(cos(f*x + e)) + 1960*sq
rt(2)*a^6*c*d*sgn(cos(f*x + e)) + 700*sqrt(2)*a^6*d^2*sgn(cos(f*x + e)) - (805*sqrt(2)*a^6*c^2*sgn(cos(f*x + e
)) + 1568*sqrt(2)*a^6*c*d*sgn(cos(f*x + e)) + 560*sqrt(2)*a^6*d^2*sgn(cos(f*x + e)) - (245*sqrt(2)*a^6*c^2*sgn
(cos(f*x + e)) + 448*sqrt(2)*a^6*c*d*sgn(cos(f*x + e)) + 160*sqrt(2)*a^6*d^2*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 {\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{5/2}\,{\left (c+\frac {d}{\cos \left (e+f\,x\right )}\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a/cos(e + f*x))^(5/2)*(c + d/cos(e + f*x))^2,x)

[Out]

int((a + a/cos(e + f*x))^(5/2)*(c + d/cos(e + f*x))^2, x)

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