3.2.60 \(\int (a+a \sec (e+f x))^{5/2} (c+d \sec (e+f x))^3 \, dx\) [160]

Optimal. Leaf size=336 \[ \frac {2 a^3 \left (3 c^3+12 c^2 d+12 c d^2+4 d^3\right ) \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)}}+\frac {2 a^{7/2} c^3 \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a}}\right ) \tan (e+f x)}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {2 a d \left (3 c^2+15 c d+13 d^2\right ) (a-a \sec (e+f x))^2 \tan (e+f x)}{5 f \sqrt {a+a \sec (e+f x)}}-\frac {6 d^2 (c+2 d) (a-a \sec (e+f x))^3 \tan (e+f x)}{7 f \sqrt {a+a \sec (e+f x)}}+\frac {2 d^3 (a-a \sec (e+f x))^4 \tan (e+f x)}{9 a f \sqrt {a+a \sec (e+f x)}}-\frac {2 \left (c^3+12 c^2 d+24 c d^2+12 d^3\right ) \left (a^3-a^3 \sec (e+f x)\right ) \tan (e+f x)}{3 f \sqrt {a+a \sec (e+f x)}} \]

[Out]

2*a^3*(3*c^3+12*c^2*d+12*c*d^2+4*d^3)*tan(f*x+e)/f/(a+a*sec(f*x+e))^(1/2)+2/5*a*d*(3*c^2+15*c*d+13*d^2)*(a-a*s
ec(f*x+e))^2*tan(f*x+e)/f/(a+a*sec(f*x+e))^(1/2)-6/7*d^2*(c+2*d)*(a-a*sec(f*x+e))^3*tan(f*x+e)/f/(a+a*sec(f*x+
e))^(1/2)+2/9*d^3*(a-a*sec(f*x+e))^4*tan(f*x+e)/a/f/(a+a*sec(f*x+e))^(1/2)-2/3*(c^3+12*c^2*d+24*c*d^2+12*d^3)*
(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^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 = 336, 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^3 \tan (e+f x) \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a}}\right )}{f \sqrt {a \sec (e+f x)+a} \sqrt {a-a \sec (e+f x)}}-\frac {2 \left (c^3+12 c^2 d+24 c d^2+12 d^3\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 \left (3 c^3+12 c^2 d+12 c d^2+4 d^3\right ) \tan (e+f x)}{f \sqrt {a \sec (e+f x)+a}}+\frac {2 a d \left (3 c^2+15 c d+13 d^2\right ) \tan (e+f x) (a-a \sec (e+f x))^2}{5 f \sqrt {a \sec (e+f x)+a}}-\frac {6 d^2 (c+2 d) \tan (e+f x) (a-a \sec (e+f x))^3}{7 f \sqrt {a \sec (e+f x)+a}}+\frac {2 d^3 \tan (e+f x) (a-a \sec (e+f x))^4}{9 a 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])^3,x]

[Out]

(2*a^3*(3*c^3 + 12*c^2*d + 12*c*d^2 + 4*d^3)*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) + (2*a^(7/2)*c^3*ArcTa
nh[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*d*(3*c^2 + 15*c*d + 13*d^2)*(a - a*Sec[e + f*x])^2*Tan[e + f*x])/(5*f*Sqrt[a + a*Sec[e + f*x]]) - (6*d^2*(c
+ 2*d)*(a - a*Sec[e + f*x])^3*Tan[e + f*x])/(7*f*Sqrt[a + a*Sec[e + f*x]]) + (2*d^3*(a - a*Sec[e + f*x])^4*Tan
[e + f*x])/(9*a*f*Sqrt[a + a*Sec[e + f*x]]) - (2*(c^3 + 12*c^2*d + 24*c*d^2 + 12*d^3)*(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))^3 \, dx &=-\frac {\left (a^2 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {(a+a x)^2 (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 {\left (a^2 \tan (e+f x)\right ) \text {Subst}\left (\int \left (\frac {a^2 \left (3 c^3+12 c^2 d+12 c d^2+4 d^3\right )}{\sqrt {a-a x}}+\frac {a^2 c^3}{x \sqrt {a-a x}}-a \left (c^3+12 c^2 d+24 c d^2+12 d^3\right ) \sqrt {a-a x}+d \left (3 c^2+15 c d+13 d^2\right ) (a-a x)^{3/2}-\frac {3 d^2 (c+2 d) (a-a x)^{5/2}}{a}+\frac {d^3 (a-a x)^{7/2}}{a^2}\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 \left (3 c^3+12 c^2 d+12 c d^2+4 d^3\right ) \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)}}+\frac {2 a d \left (3 c^2+15 c d+13 d^2\right ) (a-a \sec (e+f x))^2 \tan (e+f x)}{5 f \sqrt {a+a \sec (e+f x)}}-\frac {6 d^2 (c+2 d) (a-a \sec (e+f x))^3 \tan (e+f x)}{7 f \sqrt {a+a \sec (e+f x)}}+\frac {2 d^3 (a-a \sec (e+f x))^4 \tan (e+f x)}{9 a f \sqrt {a+a \sec (e+f x)}}-\frac {2 \left (c^3+12 c^2 d+24 c d^2+12 d^3\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^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^3 \left (3 c^3+12 c^2 d+12 c d^2+4 d^3\right ) \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)}}+\frac {2 a d \left (3 c^2+15 c d+13 d^2\right ) (a-a \sec (e+f x))^2 \tan (e+f x)}{5 f \sqrt {a+a \sec (e+f x)}}-\frac {6 d^2 (c+2 d) (a-a \sec (e+f x))^3 \tan (e+f x)}{7 f \sqrt {a+a \sec (e+f x)}}+\frac {2 d^3 (a-a \sec (e+f x))^4 \tan (e+f x)}{9 a f \sqrt {a+a \sec (e+f x)}}-\frac {2 \left (c^3+12 c^2 d+24 c d^2+12 d^3\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^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^3 \left (3 c^3+12 c^2 d+12 c d^2+4 d^3\right ) \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)}}+\frac {2 a^{7/2} 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 d \left (3 c^2+15 c d+13 d^2\right ) (a-a \sec (e+f x))^2 \tan (e+f x)}{5 f \sqrt {a+a \sec (e+f x)}}-\frac {6 d^2 (c+2 d) (a-a \sec (e+f x))^3 \tan (e+f x)}{7 f \sqrt {a+a \sec (e+f x)}}+\frac {2 d^3 (a-a \sec (e+f x))^4 \tan (e+f x)}{9 a f \sqrt {a+a \sec (e+f x)}}-\frac {2 \left (c^3+12 c^2 d+24 c d^2+12 d^3\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 = 6.38, size = 286, normalized size = 0.85 \begin {gather*} \frac {a^2 \sec \left (\frac {1}{2} (e+f x)\right ) \sec ^4(e+f x) \sqrt {a (1+\sec (e+f x))} \left (2520 \sqrt {2} c^3 \text {ArcSin}\left (\sqrt {2} \sin \left (\frac {1}{2} (e+f x)\right )\right ) \cos ^{\frac {9}{2}}(e+f x)+2 \left (2520 c^3+8883 c^2 d+8370 c d^2+2908 d^3+\left (630 c^3+5292 c^2 d+7290 c d^2+2792 d^3\right ) \cos (e+f x)+4 \left (840 c^3+2898 c^2 d+2610 c d^2+803 d^3\right ) \cos (2 (e+f x))+210 c^3 \cos (3 (e+f x))+1764 c^2 d \cos (3 (e+f x))+2070 c d^2 \cos (3 (e+f x))+584 d^3 \cos (3 (e+f x))+840 c^3 \cos (4 (e+f x))+2709 c^2 d \cos (4 (e+f x))+2070 c d^2 \cos (4 (e+f x))+584 d^3 \cos (4 (e+f x))\right ) \sin \left (\frac {1}{2} (e+f x)\right )\right )}{2520 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sec[e + f*x])^(5/2)*(c + d*Sec[e + f*x])^3,x]

[Out]

(a^2*Sec[(e + f*x)/2]*Sec[e + f*x]^4*Sqrt[a*(1 + Sec[e + f*x])]*(2520*Sqrt[2]*c^3*ArcSin[Sqrt[2]*Sin[(e + f*x)
/2]]*Cos[e + f*x]^(9/2) + 2*(2520*c^3 + 8883*c^2*d + 8370*c*d^2 + 2908*d^3 + (630*c^3 + 5292*c^2*d + 7290*c*d^
2 + 2792*d^3)*Cos[e + f*x] + 4*(840*c^3 + 2898*c^2*d + 2610*c*d^2 + 803*d^3)*Cos[2*(e + f*x)] + 210*c^3*Cos[3*
(e + f*x)] + 1764*c^2*d*Cos[3*(e + f*x)] + 2070*c*d^2*Cos[3*(e + f*x)] + 584*d^3*Cos[3*(e + f*x)] + 840*c^3*Co
s[4*(e + f*x)] + 2709*c^2*d*Cos[4*(e + f*x)] + 2070*c*d^2*Cos[4*(e + f*x)] + 584*d^3*Cos[4*(e + f*x)])*Sin[(e
+ f*x)/2]))/(2520*f)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(676\) vs. \(2(308)=616\).
time = 1.56, size = 677, 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 (315 \sin \left (f x +e \right ) \sqrt {2}\, \left (\cos ^{4}\left (f x +e \right )\right ) \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 {9}{2}} c^{3}+1260 \sin \left (f x +e \right ) \sqrt {2}\, \left (\cos ^{3}\left (f x +e \right )\right ) \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 {9}{2}} c^{3}+1890 \sin \left (f x +e \right ) \sqrt {2}\, \left (\cos ^{2}\left (f x +e \right )\right ) \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 {9}{2}} c^{3}+1260 \sin \left (f x +e \right ) \sqrt {2}\, \cos \left (f x +e \right ) \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 {9}{2}} c^{3}+315 \sqrt {2}\, \left (-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}\right )^{\frac {9}{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 )+26880 \left (\cos ^{5}\left (f x +e \right )\right ) c^{3}+86688 \left (\cos ^{5}\left (f x +e \right )\right ) c^{2} d +66240 \left (\cos ^{5}\left (f x +e \right )\right ) c \,d^{2}+18688 \left (\cos ^{5}\left (f x +e \right )\right ) d^{3}-23520 \left (\cos ^{4}\left (f x +e \right )\right ) c^{3}-58464 \left (\cos ^{4}\left (f x +e \right )\right ) c^{2} d -33120 \left (\cos ^{4}\left (f x +e \right )\right ) c \,d^{2}-9344 \left (\cos ^{4}\left (f x +e \right )\right ) d^{3}-3360 \left (\cos ^{3}\left (f x +e \right )\right ) c^{3}-22176 \left (\cos ^{3}\left (f x +e \right )\right ) c^{2} d -15840 \left (\cos ^{3}\left (f x +e \right )\right ) c \,d^{2}-2336 \left (\cos ^{3}\left (f x +e \right )\right ) d^{3}-6048 \left (\cos ^{2}\left (f x +e \right )\right ) c^{2} d -12960 \left (\cos ^{2}\left (f x +e \right )\right ) c \,d^{2}-2848 \left (\cos ^{2}\left (f x +e \right )\right ) d^{3}-4320 \cos \left (f x +e \right ) c \,d^{2}-3040 \cos \left (f x +e \right ) d^{3}-1120 d^{3}\right ) a^{2}}{5040 f \cos \left (f x +e \right )^{4} \sin \left (f x +e \right )}\)	\(677\)

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))^3,x,method=_RETURNVERBOSE)

[Out]

-1/5040/f*(a*(cos(f*x+e)+1)/cos(f*x+e))^(1/2)*(315*sin(f*x+e)*2^(1/2)*cos(f*x+e)^4*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))^(9/2)*c^3+1260*sin(f*x+e)*
2^(1/2)*cos(f*x+e)^3*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))^(9/2)*c^3+1890*sin(f*x+e)*2^(1/2)*cos(f*x+e)^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))^(9/2)*c^3+1260*sin(f*x+e)*2^(1/2)*cos(f*
x+e)*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))^(9/2)*c^3+315*2^(1/2)*(-2*cos(f*x+e)/(cos(f*x+e)+1))^(9/2)*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)+26880*cos(f*x+e)^5*c^3+86688*cos(f*x+e)^5*c^2*d+66240*cos(f*
x+e)^5*c*d^2+18688*cos(f*x+e)^5*d^3-23520*cos(f*x+e)^4*c^3-58464*cos(f*x+e)^4*c^2*d-33120*cos(f*x+e)^4*c*d^2-9
344*cos(f*x+e)^4*d^3-3360*cos(f*x+e)^3*c^3-22176*cos(f*x+e)^3*c^2*d-15840*cos(f*x+e)^3*c*d^2-2336*cos(f*x+e)^3
*d^3-6048*cos(f*x+e)^2*c^2*d-12960*cos(f*x+e)^2*c*d^2-2848*cos(f*x+e)^2*d^3-4320*cos(f*x+e)*c*d^2-3040*cos(f*x
+e)*d^3-1120*d^3)/cos(f*x+e)^4/sin(f*x+e)*a^2

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Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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))^3,x, algorithm="maxima")

[Out]

Timed out

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Fricas [A]
time = 3.17, size = 653, normalized size = 1.94 \begin {gather*} \left [\frac {315 \, {\left (a^{2} c^{3} \cos \left (f x + e\right )^{5} + a^{2} c^{3} \cos \left (f x + e\right )^{4}\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 (35 \, a^{2} d^{3} + {\left (840 \, a^{2} c^{3} + 2709 \, a^{2} c^{2} d + 2070 \, a^{2} c d^{2} + 584 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{4} + {\left (105 \, a^{2} c^{3} + 882 \, a^{2} c^{2} d + 1035 \, a^{2} c d^{2} + 292 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (63 \, a^{2} c^{2} d + 180 \, a^{2} c d^{2} + 73 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{2} + 5 \, {\left (27 \, a^{2} c d^{2} + 26 \, a^{2} 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 )}{315 \, {\left (f \cos \left (f x + e\right )^{5} + f \cos \left (f x + e\right )^{4}\right )}}, -\frac {2 \, {\left (315 \, {\left (a^{2} c^{3} \cos \left (f x + e\right )^{5} + a^{2} c^{3} \cos \left (f x + e\right )^{4}\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 (35 \, a^{2} d^{3} + {\left (840 \, a^{2} c^{3} + 2709 \, a^{2} c^{2} d + 2070 \, a^{2} c d^{2} + 584 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{4} + {\left (105 \, a^{2} c^{3} + 882 \, a^{2} c^{2} d + 1035 \, a^{2} c d^{2} + 292 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (63 \, a^{2} c^{2} d + 180 \, a^{2} c d^{2} + 73 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{2} + 5 \, {\left (27 \, a^{2} c d^{2} + 26 \, a^{2} 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 )}}{315 \, {\left (f \cos \left (f x + e\right )^{5} + f \cos \left (f x + e\right )^{4}\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))^3,x, algorithm="fricas")

[Out]

[1/315*(315*(a^2*c^3*cos(f*x + e)^5 + a^2*c^3*cos(f*x + e)^4)*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*
(35*a^2*d^3 + (840*a^2*c^3 + 2709*a^2*c^2*d + 2070*a^2*c*d^2 + 584*a^2*d^3)*cos(f*x + e)^4 + (105*a^2*c^3 + 88
2*a^2*c^2*d + 1035*a^2*c*d^2 + 292*a^2*d^3)*cos(f*x + e)^3 + 3*(63*a^2*c^2*d + 180*a^2*c*d^2 + 73*a^2*d^3)*cos
(f*x + e)^2 + 5*(27*a^2*c*d^2 + 26*a^2*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)^5 + f*cos(f*x + e)^4), -2/315*(315*(a^2*c^3*cos(f*x + e)^5 + a^2*c^3*cos(f*x + e)^4)*sqrt(a)
*arctan(sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*cos(f*x + e)/(sqrt(a)*sin(f*x + e))) - (35*a^2*d^3 + (840*a^2*
c^3 + 2709*a^2*c^2*d + 2070*a^2*c*d^2 + 584*a^2*d^3)*cos(f*x + e)^4 + (105*a^2*c^3 + 882*a^2*c^2*d + 1035*a^2*
c*d^2 + 292*a^2*d^3)*cos(f*x + e)^3 + 3*(63*a^2*c^2*d + 180*a^2*c*d^2 + 73*a^2*d^3)*cos(f*x + e)^2 + 5*(27*a^2
*c*d^2 + 26*a^2*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)^5 + f
*cos(f*x + e)^4)]

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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 )^{3}\, 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))**3,x)

[Out]

Integral((a*(sec(e + f*x) + 1))**(5/2)*(c + d*sec(e + f*x))**3, x)

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Giac [A]
time = 1.75, size = 613, normalized size = 1.82 \begin {gather*} -\frac {\frac {315 \, \sqrt {-a} a^{3} 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 (945 \, \sqrt {2} a^{7} c^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 3780 \, \sqrt {2} a^{7} c^{2} d \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 3780 \, \sqrt {2} a^{7} c d^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 1260 \, \sqrt {2} a^{7} d^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - {\left (3570 \, \sqrt {2} a^{7} c^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 12600 \, \sqrt {2} a^{7} c^{2} d \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 10080 \, \sqrt {2} a^{7} c d^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 2520 \, \sqrt {2} a^{7} d^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - {\left (5040 \, \sqrt {2} a^{7} c^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 15876 \, \sqrt {2} a^{7} c^{2} d \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 11340 \, \sqrt {2} a^{7} c d^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 3276 \, \sqrt {2} a^{7} d^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - {\left (3150 \, \sqrt {2} a^{7} c^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 9072 \, \sqrt {2} a^{7} c^{2} d \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 6480 \, \sqrt {2} a^{7} c d^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 1872 \, \sqrt {2} a^{7} d^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - {\left (735 \, \sqrt {2} a^{7} c^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 2016 \, \sqrt {2} a^{7} c^{2} d \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 1440 \, \sqrt {2} a^{7} c d^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 416 \, \sqrt {2} a^{7} 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 )^{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 )}^{4} \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}}}{315 \, 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))^3,x, algorithm="giac")

[Out]

-1/315*(315*sqrt(-a)*a^3*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*(945*sqrt(2)*a^7*c^3*sgn(cos(f*x + e)) + 3780*sqrt(2)*a^7*c
^2*d*sgn(cos(f*x + e)) + 3780*sqrt(2)*a^7*c*d^2*sgn(cos(f*x + e)) + 1260*sqrt(2)*a^7*d^3*sgn(cos(f*x + e)) - (
3570*sqrt(2)*a^7*c^3*sgn(cos(f*x + e)) + 12600*sqrt(2)*a^7*c^2*d*sgn(cos(f*x + e)) + 10080*sqrt(2)*a^7*c*d^2*s
gn(cos(f*x + e)) + 2520*sqrt(2)*a^7*d^3*sgn(cos(f*x + e)) - (5040*sqrt(2)*a^7*c^3*sgn(cos(f*x + e)) + 15876*sq
rt(2)*a^7*c^2*d*sgn(cos(f*x + e)) + 11340*sqrt(2)*a^7*c*d^2*sgn(cos(f*x + e)) + 3276*sqrt(2)*a^7*d^3*sgn(cos(f
*x + e)) - (3150*sqrt(2)*a^7*c^3*sgn(cos(f*x + e)) + 9072*sqrt(2)*a^7*c^2*d*sgn(cos(f*x + e)) + 6480*sqrt(2)*a
^7*c*d^2*sgn(cos(f*x + e)) + 1872*sqrt(2)*a^7*d^3*sgn(cos(f*x + e)) - (735*sqrt(2)*a^7*c^3*sgn(cos(f*x + e)) +
2016*sqrt(2)*a^7*c^2*d*sgn(cos(f*x + e)) + 1440*sqrt(2)*a^7*c*d^2*sgn(cos(f*x + e)) + 416*sqrt(2)*a^7*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)^2
)*tan(1/2*f*x + 1/2*e)/((a*tan(1/2*f*x + 1/2*e)^2 - a)^4*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 )}^3 \,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))^3,x)

[Out]

int((a + a/cos(e + f*x))^(5/2)*(c + d/cos(e + f*x))^3, x)

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