3.2.54 \(\int (a+a \sec (e+f x))^{3/2} (c+d \sec (e+f x))^3 \, dx\) [154]

Optimal. Leaf size=241 \[ \frac {2 a^{5/2} c^3 \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a}}\right ) \tan (e+f x)}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {2 a^2 (6 c+13 d) (c+d \sec (e+f x))^2 \tan (e+f x)}{35 f \sqrt {a+a \sec (e+f x)}}+\frac {2 a^2 (c+d \sec (e+f x))^3 \tan (e+f x)}{7 f \sqrt {a+a \sec (e+f x)}}+\frac {2 a^2 \left (2 \left (36 c^3+243 c^2 d+189 c d^2+52 d^3\right )+d \left (24 c^2+111 c d+52 d^2\right ) \sec (e+f x)\right ) \tan (e+f x)}{105 f \sqrt {a+a \sec (e+f x)}} \]

[Out]

2/35*a^2*(6*c+13*d)*(c+d*sec(f*x+e))^2*tan(f*x+e)/f/(a+a*sec(f*x+e))^(1/2)+2/7*a^2*(c+d*sec(f*x+e))^3*tan(f*x+
e)/f/(a+a*sec(f*x+e))^(1/2)+2/105*a^2*(72*c^3+486*c^2*d+378*c*d^2+104*d^3+d*(24*c^2+111*c*d+52*d^2)*sec(f*x+e)
)*tan(f*x+e)/f/(a+a*sec(f*x+e))^(1/2)+2*a^(5/2)*c^3*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 = 0.14, antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {4025, 158, 152, 65, 212} \begin {gather*} \frac {2 a^{5/2} c^3 \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 a^2 \tan (e+f x) \left (d \left (24 c^2+111 c d+52 d^2\right ) \sec (e+f x)+2 \left (36 c^3+243 c^2 d+189 c d^2+52 d^3\right )\right )}{105 f \sqrt {a \sec (e+f x)+a}}+\frac {2 a^2 \tan (e+f x) (c+d \sec (e+f x))^3}{7 f \sqrt {a \sec (e+f x)+a}}+\frac {2 a^2 (6 c+13 d) \tan (e+f x) (c+d \sec (e+f x))^2}{35 f \sqrt {a \sec (e+f x)+a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + a*Sec[e + f*x])^(3/2)*(c + d*Sec[e + f*x])^3,x]

[Out]

(2*a^(5/2)*c^3*ArcTanh[Sqrt[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*a^2*(6*c + 13*d)*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(35*f*Sqrt[a + a*Sec[e + f*x]]) + (2
*a^2*(c + d*Sec[e + f*x])^3*Tan[e + f*x])/(7*f*Sqrt[a + a*Sec[e + f*x]]) + (2*a^2*(2*(36*c^3 + 243*c^2*d + 189
*c*d^2 + 52*d^3) + d*(24*c^2 + 111*c*d + 52*d^2)*Sec[e + f*x])*Tan[e + f*x])/(105*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 152

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)
^(m + 1)*((c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d
*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1
)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)
^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]

Rule 158

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegerQ[m]

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))^{3/2} (c+d \sec (e+f x))^3 \, dx &=-\frac {\left (a^2 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {(a+a x) (c+d x)^3}{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^2 (c+d \sec (e+f x))^3 \tan (e+f x)}{7 f \sqrt {a+a \sec (e+f x)}}+\frac {(2 a \tan (e+f x)) \text {Subst}\left (\int \frac {(c+d x)^2 \left (-\frac {7 a^2 c}{2}-\frac {1}{2} a^2 (6 c+13 d) x\right )}{x \sqrt {a-a x}} \, dx,x,\sec (e+f x)\right )}{7 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {2 a^2 (6 c+13 d) (c+d \sec (e+f x))^2 \tan (e+f x)}{35 f \sqrt {a+a \sec (e+f x)}}+\frac {2 a^2 (c+d \sec (e+f x))^3 \tan (e+f x)}{7 f \sqrt {a+a \sec (e+f x)}}-\frac {(4 \tan (e+f x)) \text {Subst}\left (\int \frac {(c+d x) \left (\frac {35 a^3 c^2}{4}+\frac {1}{4} a^3 \left (24 c^2+111 c d+52 d^2\right ) x\right )}{x \sqrt {a-a x}} \, dx,x,\sec (e+f x)\right )}{35 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {2 a^2 (6 c+13 d) (c+d \sec (e+f x))^2 \tan (e+f x)}{35 f \sqrt {a+a \sec (e+f x)}}+\frac {2 a^2 (c+d \sec (e+f x))^3 \tan (e+f x)}{7 f \sqrt {a+a \sec (e+f x)}}+\frac {2 a^2 \left (2 \left (36 c^3+243 c^2 d+189 c d^2+52 d^3\right )+d \left (24 c^2+111 c d+52 d^2\right ) \sec (e+f x)\right ) \tan (e+f x)}{105 f \sqrt {a+a \sec (e+f x)}}-\frac {\left (a^3 c^3 \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^2 (6 c+13 d) (c+d \sec (e+f x))^2 \tan (e+f x)}{35 f \sqrt {a+a \sec (e+f x)}}+\frac {2 a^2 (c+d \sec (e+f x))^3 \tan (e+f x)}{7 f \sqrt {a+a \sec (e+f x)}}+\frac {2 a^2 \left (2 \left (36 c^3+243 c^2 d+189 c d^2+52 d^3\right )+d \left (24 c^2+111 c d+52 d^2\right ) \sec (e+f x)\right ) \tan (e+f x)}{105 f \sqrt {a+a \sec (e+f x)}}+\frac {\left (2 a^2 c^3 \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^{5/2} c^3 \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^2 (6 c+13 d) (c+d \sec (e+f x))^2 \tan (e+f x)}{35 f \sqrt {a+a \sec (e+f x)}}+\frac {2 a^2 (c+d \sec (e+f x))^3 \tan (e+f x)}{7 f \sqrt {a+a \sec (e+f x)}}+\frac {2 a^2 \left (2 \left (36 c^3+243 c^2 d+189 c d^2+52 d^3\right )+d \left (24 c^2+111 c d+52 d^2\right ) \sec (e+f x)\right ) \tan (e+f x)}{105 f \sqrt {a+a \sec (e+f x)}}\\ \end {align*}

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Mathematica [A]
time = 3.91, size = 219, normalized size = 0.91 \begin {gather*} \frac {a \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^3 \text {ArcSin}\left (\sqrt {2} \sin \left (\frac {1}{2} (e+f x)\right )\right ) \cos ^{\frac {7}{2}}(e+f x)+2 \left (210 c^2 d+378 c d^2+164 d^3+9 \left (35 c^3+175 c^2 d+154 c d^2+52 d^3\right ) \cos (e+f x)+2 d \left (105 c^2+189 c d+52 d^2\right ) \cos (2 (e+f x))+105 c^3 \cos (3 (e+f x))+525 c^2 d \cos (3 (e+f x))+378 c d^2 \cos (3 (e+f x))+104 d^3 \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])^(3/2)*(c + d*Sec[e + f*x])^3,x]

[Out]

(a*Sec[(e + f*x)/2]*Sec[e + f*x]^3*Sqrt[a*(1 + Sec[e + f*x])]*(420*Sqrt[2]*c^3*ArcSin[Sqrt[2]*Sin[(e + f*x)/2]
]*Cos[e + f*x]^(7/2) + 2*(210*c^2*d + 378*c*d^2 + 164*d^3 + 9*(35*c^3 + 175*c^2*d + 154*c*d^2 + 52*d^3)*Cos[e
+ f*x] + 2*d*(105*c^2 + 189*c*d + 52*d^2)*Cos[2*(e + f*x)] + 105*c^3*Cos[3*(e + f*x)] + 525*c^2*d*Cos[3*(e + f
*x)] + 378*c*d^2*Cos[3*(e + f*x)] + 104*d^3*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. \(538\) vs. \(2(216)=432\).
time = 3.30, size = 539, normalized size = 2.24

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 \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 (\cos ^{3}\left (f x +e \right )\right ) \sin \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}} \sqrt {2}\, c^{3}+315 \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 (\cos ^{2}\left (f x +e \right )\right ) \sin \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}} \sqrt {2}\, c^{3}+315 \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 ) \cos \left (f x +e \right ) \sin \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}} \sqrt {2}\, c^{3}+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^{3} \sin \left (f x +e \right )-1680 \left (\cos ^{4}\left (f x +e \right )\right ) c^{3}-8400 \left (\cos ^{4}\left (f x +e \right )\right ) c^{2} d -6048 \left (\cos ^{4}\left (f x +e \right )\right ) c \,d^{2}-1664 \left (\cos ^{4}\left (f x +e \right )\right ) d^{3}+1680 \left (\cos ^{3}\left (f x +e \right )\right ) c^{3}+6720 \left (\cos ^{3}\left (f x +e \right )\right ) c^{2} d +3024 \left (\cos ^{3}\left (f x +e \right )\right ) c \,d^{2}+832 \left (\cos ^{3}\left (f x +e \right )\right ) d^{3}+1680 \left (\cos ^{2}\left (f x +e \right )\right ) c^{2} d +2016 \left (\cos ^{2}\left (f x +e \right )\right ) c \,d^{2}+208 \left (\cos ^{2}\left (f x +e \right )\right ) d^{3}+1008 \cos \left (f x +e \right ) c \,d^{2}+384 \cos \left (f x +e \right ) d^{3}+240 d^{3}\right ) a}{840 f \cos \left (f x +e \right )^{3} \sin \left (f x +e \right )}\)	\(539\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sec(f*x+e))^(3/2)*(c+d*sec(f*x+e))^3,x,method=_RETURNVERBOSE)

[Out]

1/840/f*(a*(cos(f*x+e)+1)/cos(f*x+e))^(1/2)*(105*arctanh(1/2*(-2*cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)/c
os(f*x+e)*2^(1/2))*cos(f*x+e)^3*sin(f*x+e)*(-2*cos(f*x+e)/(cos(f*x+e)+1))^(7/2)*2^(1/2)*c^3+315*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))*cos(f*x+e)^2*sin(f*x+e)*(-2*cos(f*x+e)/(cos(
f*x+e)+1))^(7/2)*2^(1/2)*c^3+315*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
))*cos(f*x+e)*sin(f*x+e)*(-2*cos(f*x+e)/(cos(f*x+e)+1))^(7/2)*2^(1/2)*c^3+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^3*sin(f*x+e)-16
80*cos(f*x+e)^4*c^3-8400*cos(f*x+e)^4*c^2*d-6048*cos(f*x+e)^4*c*d^2-1664*cos(f*x+e)^4*d^3+1680*cos(f*x+e)^3*c^
3+6720*cos(f*x+e)^3*c^2*d+3024*cos(f*x+e)^3*c*d^2+832*cos(f*x+e)^3*d^3+1680*cos(f*x+e)^2*c^2*d+2016*cos(f*x+e)
^2*c*d^2+208*cos(f*x+e)^2*d^3+1008*cos(f*x+e)*c*d^2+384*cos(f*x+e)*d^3+240*d^3)/cos(f*x+e)^3/sin(f*x+e)*a

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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))^(3/2)*(c+d*sec(f*x+e))^3,x, algorithm="maxima")

[Out]

-1/210*(4*(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*(a*c^3 + 3*a*c^2*d)*
sin(6*f*x + 6*e) + 5*(9*a*c^3 + 33*a*c^2*d + 18*a*c*d^2 + 4*a*d^3)*sin(4*f*x + 4*e) + (45*a*c^3 + 195*a*c^2*d
+ 144*a*c*d^2 + 52*a*d^3)*sin(2*f*x + 2*e))*cos(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e) + 1)) - (105*a*
c^3 + 525*a*c^2*d + 378*a*c*d^2 + 104*a*d^3 + 105*(a*c^3 + 3*a*c^2*d)*cos(6*f*x + 6*e) + 35*(9*a*c^3 + 33*a*c^
2*d + 18*a*c*d^2 + 4*a*d^3)*cos(4*f*x + 4*e) + 7*(45*a*c^3 + 195*a*c^2*d + 144*a*c*d^2 + 52*a*d^3)*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*((a*c^3*cos(2*f*x + 2*e)^4 + a*
c^3*sin(2*f*x + 2*e)^4 + 4*a*c^3*cos(2*f*x + 2*e)^3 + 6*a*c^3*cos(2*f*x + 2*e)^2 + 4*a*c^3*cos(2*f*x + 2*e) +
a*c^3 + 2*(a*c^3*cos(2*f*x + 2*e)^2 + 2*a*c^3*cos(2*f*x + 2*e) + a*c^3)*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) - (a*c^3*cos(2*f*x + 2*e)^4 + a*c^3*sin(2*f*x + 2*e)^4 + 4*a*c^3*cos(2*f*x
+ 2*e)^3 + 6*a*c^3*cos(2*f*x + 2*e)^2 + 4*a*c^3*cos(2*f*x + 2*e) + a*c^3 + 2*(a*c^3*cos(2*f*x + 2*e)^2 + 2*a*
c^3*cos(2*f*x + 2*e) + a*c^3)*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*(a
*c^3*f*cos(2*f*x + 2*e)^4 + a*c^3*f*sin(2*f*x + 2*e)^4 + 4*a*c^3*f*cos(2*f*x + 2*e)^3 + 6*a*c^3*f*cos(2*f*x +
2*e)^2 + 4*a*c^3*f*cos(2*f*x + 2*e) + a*c^3*f + 2*(a*c^3*f*cos(2*f*x + 2*e)^2 + 2*a*c^3*f*cos(2*f*x + 2*e) + a
*c^3*f)*sin(2*f*x + 2*e)^2)*integrate((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(8*f*x + 8*e)*cos(2*f*x + 2*e) + 3*cos(6*f*x + 6*e)*cos(2*f*x + 2*e) + 3*cos(4*f*x + 4*e)*cos(2*f*x + 2
*e) + cos(2*f*x + 2*e)^2 + sin(8*f*x + 8*e)*sin(2*f*x + 2*e) + 3*sin(6*f*x + 6*e)*sin(2*f*x + 2*e) + 3*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(8*f*x + 8*e) + 3*cos(2*f*x + 2*e)*sin(6*f*x + 6*e) + 3*cos(2*f*x + 2*e)*sin(4*f*x + 4*e) - cos
(8*f*x + 8*e)*sin(2*f*x + 2*e) - 3*cos(6*f*x + 6*e)*sin(2*f*x + 2*e) - 3*cos(4*f*x + 4*e)*sin(2*f*x + 2*e))*si
n(9/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e) + 1)) -
((cos(2*f*x + 2*e)*sin(8*f*x + 8*e) + 3*cos(2*f*x + 2*e)*sin(6*f*x + 6*e) + 3*cos(2*f*x + 2*e)*sin(4*f*x + 4*
e) - cos(8*f*x + 8*e)*sin(2*f*x + 2*e) - 3*cos(6*f*x + 6*e)*sin(2*f*x + 2*e) - 3*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(8*f*x + 8*e)*cos(2*f*x + 2*e) + 3*cos(6*f*x
+ 6*e)*cos(2*f*x + 2*e) + 3*cos(4*f*x + 4*e)*cos(2*f*x + 2*e) + cos(2*f*x + 2*e)^2 + sin(8*f*x + 8*e)*sin(2*f*
x + 2*e) + 3*sin(6*f*x + 6*e)*sin(2*f*x + 2*e) + 3*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(3/2*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(8*f*x + 8*e)^2 + 9*(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 +
9*(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(8*f*x + 8*e)^2 + 9*(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 + 9*(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 + 3*(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) + 3*(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(8*f*x + 8*e) + 6*(cos(2*f*x + 2*e)^3
+ cos(2*f*x + 2*e)*sin(2*f*x + 2*e)^2 + 3*(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) + 6*(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 + 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
) + 3*(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(8*f*x + 8*e) + 6*(sin(2*f*x + 2*e)^3 + 3*(cos(2*f*x + 2*e)^2
+ sin(2*f*x + 2*e)^2 + 2*cos(2*f*x + 2*e) + 1)...

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Fricas [A]
time = 3.41, size = 513, normalized size = 2.13 \begin {gather*} \left [\frac {105 \, {\left (a c^{3} \cos \left (f x + e\right )^{4} + a c^{3} \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 d^{3} + {\left (105 \, a c^{3} + 525 \, a c^{2} d + 378 \, a c d^{2} + 104 \, a d^{3}\right )} \cos \left (f x + e\right )^{3} + {\left (105 \, a c^{2} d + 189 \, a c d^{2} + 52 \, a d^{3}\right )} \cos \left (f x + e\right )^{2} + 3 \, {\left (21 \, a c d^{2} + 13 \, a d^{3}\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 c^{3} \cos \left (f x + e\right )^{4} + a c^{3} \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 d^{3} + {\left (105 \, a c^{3} + 525 \, a c^{2} d + 378 \, a c d^{2} + 104 \, a d^{3}\right )} \cos \left (f x + e\right )^{3} + {\left (105 \, a c^{2} d + 189 \, a c d^{2} + 52 \, a d^{3}\right )} \cos \left (f x + e\right )^{2} + 3 \, {\left (21 \, a c d^{2} + 13 \, a d^{3}\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))^(3/2)*(c+d*sec(f*x+e))^3,x, algorithm="fricas")

[Out]

[1/105*(105*(a*c^3*cos(f*x + e)^4 + a*c^3*cos(f*x + e)^3)*sqrt(-a)*log((2*a*cos(f*x + e)^2 - 2*sqrt(-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*(15*
a*d^3 + (105*a*c^3 + 525*a*c^2*d + 378*a*c*d^2 + 104*a*d^3)*cos(f*x + e)^3 + (105*a*c^2*d + 189*a*c*d^2 + 52*a
*d^3)*cos(f*x + e)^2 + 3*(21*a*c*d^2 + 13*a*d^3)*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*c^3*cos(f*x + e)^4 + a*c^3*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*a*d^3 + (105*a*c^3
+ 525*a*c^2*d + 378*a*c*d^2 + 104*a*d^3)*cos(f*x + e)^3 + (105*a*c^2*d + 189*a*c*d^2 + 52*a*d^3)*cos(f*x + e)
^2 + 3*(21*a*c*d^2 + 13*a*d^3)*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 {3}{2}} \left (c + d \sec {\left (e + f x \right )}\right )^{3}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(f*x+e))**(3/2)*(c+d*sec(f*x+e))**3,x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 525 vs. \(2 (216) = 432\).
time = 1.87, size = 525, normalized size = 2.18 \begin {gather*} -\frac {\frac {105 \, \sqrt {-a} a^{2} c^{3} \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 (105 \, \sqrt {2} a^{5} c^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 630 \, \sqrt {2} a^{5} c^{2} d \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 630 \, \sqrt {2} a^{5} c d^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 210 \, \sqrt {2} a^{5} d^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - {\left (315 \, \sqrt {2} a^{5} c^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 1680 \, \sqrt {2} a^{5} c^{2} d \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 1260 \, \sqrt {2} a^{5} c d^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 280 \, \sqrt {2} a^{5} d^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - {\left (315 \, \sqrt {2} a^{5} c^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 1470 \, \sqrt {2} a^{5} c^{2} d \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 882 \, \sqrt {2} a^{5} c d^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 266 \, \sqrt {2} a^{5} d^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - {\left (105 \, \sqrt {2} a^{5} c^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 420 \, \sqrt {2} a^{5} c^{2} d \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 252 \, \sqrt {2} a^{5} c d^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 76 \, \sqrt {2} a^{5} d^{3} \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))^(3/2)*(c+d*sec(f*x+e))^3,x, algorithm="giac")

[Out]

-1/105*(105*sqrt(-a)*a^2*c^3*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*(105*sqrt(2)*a^5*c^3*sgn(cos(f*x + e)) + 630*sqrt(2)*a^5*c^
2*d*sgn(cos(f*x + e)) + 630*sqrt(2)*a^5*c*d^2*sgn(cos(f*x + e)) + 210*sqrt(2)*a^5*d^3*sgn(cos(f*x + e)) - (315
*sqrt(2)*a^5*c^3*sgn(cos(f*x + e)) + 1680*sqrt(2)*a^5*c^2*d*sgn(cos(f*x + e)) + 1260*sqrt(2)*a^5*c*d^2*sgn(cos
(f*x + e)) + 280*sqrt(2)*a^5*d^3*sgn(cos(f*x + e)) - (315*sqrt(2)*a^5*c^3*sgn(cos(f*x + e)) + 1470*sqrt(2)*a^5
*c^2*d*sgn(cos(f*x + e)) + 882*sqrt(2)*a^5*c*d^2*sgn(cos(f*x + e)) + 266*sqrt(2)*a^5*d^3*sgn(cos(f*x + e)) - (
105*sqrt(2)*a^5*c^3*sgn(cos(f*x + e)) + 420*sqrt(2)*a^5*c^2*d*sgn(cos(f*x + e)) + 252*sqrt(2)*a^5*c*d^2*sgn(co
s(f*x + e)) + 76*sqrt(2)*a^5*d^3*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 )}^{3/2}\,{\left (c+\frac {d}{\cos \left (e+f\,x\right )}\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a/cos(e + f*x))^(3/2)*(c + d/cos(e + f*x))^3,x)

[Out]

int((a + a/cos(e + f*x))^(3/2)*(c + d/cos(e + f*x))^3, x)

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