3.1.57 \(\int (a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^3 \, dx\) [57]

Optimal. Leaf size=212 \[ \frac {2 a^{5/2} c^3 \text {ArcTan}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{f}-\frac {2 a^3 c^3 \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)}}+\frac {2 a^4 c^3 \tan ^3(e+f x)}{3 f (a+a \sec (e+f x))^{3/2}}-\frac {2 a^5 c^3 \tan ^5(e+f x)}{5 f (a+a \sec (e+f x))^{5/2}}-\frac {6 a^6 c^3 \tan ^7(e+f x)}{7 f (a+a \sec (e+f x))^{7/2}}-\frac {2 a^7 c^3 \tan ^9(e+f x)}{9 f (a+a \sec (e+f x))^{9/2}} \]

[Out]

2*a^(5/2)*c^3*arctan(a^(1/2)*tan(f*x+e)/(a+a*sec(f*x+e))^(1/2))/f-2*a^3*c^3*tan(f*x+e)/f/(a+a*sec(f*x+e))^(1/2
)+2/3*a^4*c^3*tan(f*x+e)^3/f/(a+a*sec(f*x+e))^(3/2)-2/5*a^5*c^3*tan(f*x+e)^5/f/(a+a*sec(f*x+e))^(5/2)-6/7*a^6*
c^3*tan(f*x+e)^7/f/(a+a*sec(f*x+e))^(7/2)-2/9*a^7*c^3*tan(f*x+e)^9/f/(a+a*sec(f*x+e))^(9/2)

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Rubi [A]
time = 0.13, antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3989, 3972, 472, 209} \begin {gather*} \frac {2 a^{5/2} c^3 \text {ArcTan}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a}}\right )}{f}-\frac {2 a^7 c^3 \tan ^9(e+f x)}{9 f (a \sec (e+f x)+a)^{9/2}}-\frac {6 a^6 c^3 \tan ^7(e+f x)}{7 f (a \sec (e+f x)+a)^{7/2}}-\frac {2 a^5 c^3 \tan ^5(e+f x)}{5 f (a \sec (e+f x)+a)^{5/2}}+\frac {2 a^4 c^3 \tan ^3(e+f x)}{3 f (a \sec (e+f x)+a)^{3/2}}-\frac {2 a^3 c^3 \tan (e+f x)}{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 - c*Sec[e + f*x])^3,x]

[Out]

(2*a^(5/2)*c^3*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/f - (2*a^3*c^3*Tan[e + f*x])/(f*Sqrt[a
+ a*Sec[e + f*x]]) + (2*a^4*c^3*Tan[e + f*x]^3)/(3*f*(a + a*Sec[e + f*x])^(3/2)) - (2*a^5*c^3*Tan[e + f*x]^5)
/(5*f*(a + a*Sec[e + f*x])^(5/2)) - (6*a^6*c^3*Tan[e + f*x]^7)/(7*f*(a + a*Sec[e + f*x])^(7/2)) - (2*a^7*c^3*T
an[e + f*x]^9)/(9*f*(a + a*Sec[e + f*x])^(9/2))

Rule 209

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

Rule 472

Int[(((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegr
and[(e*x)^m*((a + b*x^n)^p/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n
, 0] && IGtQ[p, 0] && (IntegerQ[m] || IGtQ[2*(m + 1), 0] ||  !RationalQ[m])

Rule 3972

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

Rule 3989

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

Rubi steps

\begin {align*} \int (a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^3 \, dx &=-\left (\left (a^3 c^3\right ) \int \frac {\tan ^6(e+f x)}{\sqrt {a+a \sec (e+f x)}} \, dx\right )\\ &=\frac {\left (2 a^6 c^3\right ) \text {Subst}\left (\int \frac {x^6 \left (2+a x^2\right )^2}{1+a x^2} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{f}\\ &=\frac {\left (2 a^6 c^3\right ) \text {Subst}\left (\int \left (\frac {1}{a^3}-\frac {x^2}{a^2}+\frac {x^4}{a}+3 x^6+a x^8-\frac {1}{a^3 \left (1+a x^2\right )}\right ) \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{f}\\ &=-\frac {2 a^3 c^3 \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)}}+\frac {2 a^4 c^3 \tan ^3(e+f x)}{3 f (a+a \sec (e+f x))^{3/2}}-\frac {2 a^5 c^3 \tan ^5(e+f x)}{5 f (a+a \sec (e+f x))^{5/2}}-\frac {6 a^6 c^3 \tan ^7(e+f x)}{7 f (a+a \sec (e+f x))^{7/2}}-\frac {2 a^7 c^3 \tan ^9(e+f x)}{9 f (a+a \sec (e+f x))^{9/2}}-\frac {\left (2 a^3 c^3\right ) \text {Subst}\left (\int \frac {1}{1+a x^2} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{f}\\ &=\frac {2 a^{5/2} c^3 \tan ^{-1}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{f}-\frac {2 a^3 c^3 \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)}}+\frac {2 a^4 c^3 \tan ^3(e+f x)}{3 f (a+a \sec (e+f x))^{3/2}}-\frac {2 a^5 c^3 \tan ^5(e+f x)}{5 f (a+a \sec (e+f x))^{5/2}}-\frac {6 a^6 c^3 \tan ^7(e+f x)}{7 f (a+a \sec (e+f x))^{7/2}}-\frac {2 a^7 c^3 \tan ^9(e+f x)}{9 f (a+a \sec (e+f x))^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 1.28, size = 134, normalized size = 0.63 \begin {gather*} -\frac {a^2 c^3 \left (-2520 \text {ArcTan}\left (\sqrt {-1+\sec (e+f x)}\right ) \cos ^4(e+f x)+(901+164 \cos (e+f x)+1004 \cos (2 (e+f x))+68 \cos (3 (e+f x))+383 \cos (4 (e+f x))) \sqrt {-1+\sec (e+f x)}\right ) \sec ^4(e+f x) \sqrt {a (1+\sec (e+f x))} \tan \left (\frac {1}{2} (e+f x)\right )}{1260 f \sqrt {-1+\sec (e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/1260*(a^2*c^3*(-2520*ArcTan[Sqrt[-1 + Sec[e + f*x]]]*Cos[e + f*x]^4 + (901 + 164*Cos[e + f*x] + 1004*Cos[2*
(e + f*x)] + 68*Cos[3*(e + f*x)] + 383*Cos[4*(e + f*x)])*Sqrt[-1 + Sec[e + f*x]])*Sec[e + f*x]^4*Sqrt[a*(1 + S
ec[e + f*x])]*Tan[(e + f*x)/2])/(f*Sqrt[-1 + Sec[e + f*x]])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(482\) vs. \(2(188)=376\).
time = 0.30, size = 483, normalized size = 2.28

method	result	size
default	\(-\frac {c^{3} \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 ) \left (\cos ^{4}\left (f x +e \right )\right ) \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 {9}{2}}+1260 \sin \left (f x +e \right ) \left (\cos ^{3}\left (f x +e \right )\right ) \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 {9}{2}}+1890 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right ) \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 {9}{2}}+1260 \sin \left (f x +e \right ) \cos \left (f x +e \right ) \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 {9}{2}}+315 \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 {9}{2}} \sin \left (f x +e \right )-12256 \left (\cos ^{5}\left (f x +e \right )\right )+11168 \left (\cos ^{4}\left (f x +e \right )\right )+5312 \left (\cos ^{3}\left (f x +e \right )\right )-4064 \left (\cos ^{2}\left (f x +e \right )\right )-1280 \cos \left (f x +e \right )+1120\right ) a^{2}}{5040 f \cos \left (f x +e \right )^{4} \sin \left (f x +e \right )}\)	\(483\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sec(f*x+e))^(5/2)*(c-c*sec(f*x+e))^3,x,method=_RETURNVERBOSE)

[Out]

-1/5040*c^3/f*(a*(cos(f*x+e)+1)/cos(f*x+e))^(1/2)*(315*sin(f*x+e)*cos(f*x+e)^4*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))^(9/2)+1260*sin(f*x+e)*
cos(f*x+e)^3*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))^(9/2)+1890*sin(f*x+e)*cos(f*x+e)^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))*(-2*cos(f*x+e)/(cos(f*x+e)+1))^(9/2)+1260*sin(f*x+e)*cos(f*x+e)*2^(1/2)*arc
tanh(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)+315*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))^(9/2)*sin(f*x+e)-12256*cos(f*x+e)^5+11168*cos(f*x+e)^4+5312*cos(f*x+e)^3-4064*cos(f*x+e)^2-12
80*cos(f*x+e)+1120)/cos(f*x+e)^4/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-c*sec(f*x+e))^3,x, algorithm="maxima")

[Out]

-1/630*(315*((a^2*c^3*cos(2*f*x + 2*e)^4 + a^2*c^3*sin(2*f*x + 2*e)^4 + 4*a^2*c^3*cos(2*f*x + 2*e)^3 + 6*a^2*c
^3*cos(2*f*x + 2*e)^2 + 4*a^2*c^3*cos(2*f*x + 2*e) + a^2*c^3 + 2*(a^2*c^3*cos(2*f*x + 2*e)^2 + 2*a^2*c^3*cos(2
*f*x + 2*e) + a^2*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^2*c^3*co
s(2*f*x + 2*e)^4 + a^2*c^3*sin(2*f*x + 2*e)^4 + 4*a^2*c^3*cos(2*f*x + 2*e)^3 + 6*a^2*c^3*cos(2*f*x + 2*e)^2 +
4*a^2*c^3*cos(2*f*x + 2*e) + a^2*c^3 + 2*(a^2*c^3*cos(2*f*x + 2*e)^2 + 2*a^2*c^3*cos(2*f*x + 2*e) + a^2*c^3)*s
in(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*ar
ctan2(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^3*f*cos(2*f*x + 2*e)^4 + a
^2*c^3*f*sin(2*f*x + 2*e)^4 + 4*a^2*c^3*f*cos(2*f*x + 2*e)^3 + 6*a^2*c^3*f*cos(2*f*x + 2*e)^2 + 4*a^2*c^3*f*co
s(2*f*x + 2*e) + a^2*c^3*f + 2*(a^2*c^3*f*cos(2*f*x + 2*e)^2 + 2*a^2*c^3*f*cos(2*f*x + 2*e) + a^2*c^3*f)*sin(2
*f*x + 2*e)^2)*integrate((((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(11/2*arctan2(sin(2*f*x + 2*e), co
s(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))*sin(11/2*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(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(11/2*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(11/2*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(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)*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) + 6*(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(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*c
os(2*f*x + 2*e)^3 + (cos(2*f*x + 2*e)^2 + sin(2...

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Fricas [A]
time = 2.75, size = 474, normalized size = 2.24 \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 (383 \, a^{2} c^{3} \cos \left (f x + e\right )^{4} + 34 \, a^{2} c^{3} \cos \left (f x + e\right )^{3} - 132 \, a^{2} c^{3} \cos \left (f x + e\right )^{2} - 5 \, a^{2} c^{3} \cos \left (f x + e\right ) + 35 \, a^{2} c^{3}\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 (383 \, a^{2} c^{3} \cos \left (f x + e\right )^{4} + 34 \, a^{2} c^{3} \cos \left (f x + e\right )^{3} - 132 \, a^{2} c^{3} \cos \left (f x + e\right )^{2} - 5 \, a^{2} c^{3} \cos \left (f x + e\right ) + 35 \, a^{2} c^{3}\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-c*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*
(383*a^2*c^3*cos(f*x + e)^4 + 34*a^2*c^3*cos(f*x + e)^3 - 132*a^2*c^3*cos(f*x + e)^2 - 5*a^2*c^3*cos(f*x + e)
+ 35*a^2*c^3)*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))) + (383*a^2*c^3*cos(f*x + e)^4 + 34*a^2*c^3*cos(f*x + e)^3 - 132*a^2*c
^3*cos(f*x + e)^2 - 5*a^2*c^3*cos(f*x + e) + 35*a^2*c^3)*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*} - c^{3} \left (\int \left (- a^{2} \sqrt {a \sec {\left (e + f x \right )} + a}\right )\, dx + \int a^{2} \sqrt {a \sec {\left (e + f x \right )} + a} \sec {\left (e + f x \right )}\, dx + \int 2 a^{2} \sqrt {a \sec {\left (e + f x \right )} + a} \sec ^{2}{\left (e + f x \right )}\, dx + \int \left (- 2 a^{2} \sqrt {a \sec {\left (e + f x \right )} + a} \sec ^{3}{\left (e + f x \right )}\right )\, dx + \int \left (- a^{2} \sqrt {a \sec {\left (e + f x \right )} + a} \sec ^{4}{\left (e + f x \right )}\right )\, dx + \int a^{2} \sqrt {a \sec {\left (e + f x \right )} + a} \sec ^{5}{\left (e + f x \right )}\, dx\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(f*x+e))**(5/2)*(c-c*sec(f*x+e))**3,x)

[Out]

-c**3*(Integral(-a**2*sqrt(a*sec(e + f*x) + a), x) + Integral(a**2*sqrt(a*sec(e + f*x) + a)*sec(e + f*x), x) +
Integral(2*a**2*sqrt(a*sec(e + f*x) + a)*sec(e + f*x)**2, x) + Integral(-2*a**2*sqrt(a*sec(e + f*x) + a)*sec(
e + f*x)**3, x) + Integral(-a**2*sqrt(a*sec(e + f*x) + a)*sec(e + f*x)**4, x) + Integral(a**2*sqrt(a*sec(e + f
*x) + a)*sec(e + f*x)**5, x))

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Giac [A]
time = 1.77, size = 329, normalized size = 1.55 \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 (315 \, \sqrt {2} a^{7} c^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - {\left (1470 \, \sqrt {2} a^{7} c^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - {\left (2772 \, \sqrt {2} a^{7} c^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + {\left (257 \, \sqrt {2} a^{7} c^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1314 \, \sqrt {2} a^{7} c^{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 )}^{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-c*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*(315*sqrt(2)*a^7*c^3*sgn(cos(f*x + e)) - (1470*sqrt(2)*a^7*
c^3*sgn(cos(f*x + e)) - (2772*sqrt(2)*a^7*c^3*sgn(cos(f*x + e)) + (257*sqrt(2)*a^7*c^3*sgn(cos(f*x + e))*tan(1
/2*f*x + 1/2*e)^2 - 1314*sqrt(2)*a^7*c^3*sgn(cos(f*x + e)))*tan(1/2*f*x + 1/2*e)^2)*tan(1/2*f*x + 1/2*e)^2)*ta
n(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 {c}{\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 - c/cos(e + f*x))^3,x)

[Out]

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

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