3.7.24 \(\int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}} \, dx\) [624]

Optimal. Leaf size=313 \[ -\frac {(283 A-163 B+75 C) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {\sec (c+d x)} \sin (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{16 \sqrt {2} a^{5/2} d}-\frac {(A-B+C) \sin (c+d x)}{4 d \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}}-\frac {(21 A-13 B+5 C) \sin (c+d x)}{16 a d \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}}+\frac {(157 A-85 B+45 C) \sin (c+d x)}{80 a^2 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}-\frac {(787 A-475 B+195 C) \sin (c+d x)}{240 a^2 d \sqrt {\sec (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {(2671 A-1495 B+735 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{240 a^2 d \sqrt {a+a \sec (c+d x)}} \]

[Out]

-1/4*(A-B+C)*sin(d*x+c)/d/sec(d*x+c)^(3/2)/(a+a*sec(d*x+c))^(5/2)-1/16*(21*A-13*B+5*C)*sin(d*x+c)/a/d/sec(d*x+
c)^(3/2)/(a+a*sec(d*x+c))^(3/2)-1/32*(283*A-163*B+75*C)*arctanh(1/2*sin(d*x+c)*a^(1/2)*sec(d*x+c)^(1/2)*2^(1/2
)/(a+a*sec(d*x+c))^(1/2))/a^(5/2)/d*2^(1/2)+1/80*(157*A-85*B+45*C)*sin(d*x+c)/a^2/d/sec(d*x+c)^(3/2)/(a+a*sec(
d*x+c))^(1/2)-1/240*(787*A-475*B+195*C)*sin(d*x+c)/a^2/d/sec(d*x+c)^(1/2)/(a+a*sec(d*x+c))^(1/2)+1/240*(2671*A
-1495*B+735*C)*sin(d*x+c)*sec(d*x+c)^(1/2)/a^2/d/(a+a*sec(d*x+c))^(1/2)

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Rubi [A]
time = 0.66, antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {4169, 4105, 4107, 4098, 3893, 212} \begin {gather*} -\frac {(283 A-163 B+75 C) \tanh ^{-1}\left (\frac {\sqrt {a} \sin (c+d x) \sqrt {\sec (c+d x)}}{\sqrt {2} \sqrt {a \sec (c+d x)+a}}\right )}{16 \sqrt {2} a^{5/2} d}+\frac {(157 A-85 B+45 C) \sin (c+d x)}{80 a^2 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}+\frac {(2671 A-1495 B+735 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{240 a^2 d \sqrt {a \sec (c+d x)+a}}-\frac {(787 A-475 B+195 C) \sin (c+d x)}{240 a^2 d \sqrt {\sec (c+d x)} \sqrt {a \sec (c+d x)+a}}-\frac {(21 A-13 B+5 C) \sin (c+d x)}{16 a d \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}-\frac {(A-B+C) \sin (c+d x)}{4 d \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)),x]

[Out]

-1/16*((283*A - 163*B + 75*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*
x]])])/(Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sin[c + d*x])/(4*d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)) -
((21*A - 13*B + 5*C)*Sin[c + d*x])/(16*a*d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)) + ((157*A - 85*B + 4
5*C)*Sin[c + d*x])/(80*a^2*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) - ((787*A - 475*B + 195*C)*Sin[c + d
*x])/(240*a^2*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + ((2671*A - 1495*B + 735*C)*Sqrt[Sec[c + d*x]]*S
in[c + d*x])/(240*a^2*d*Sqrt[a + a*Sec[c + d*x]])

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 3893

Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[-2*b*(d/
(a*f)), Subst[Int[1/(2*b - d*x^2), x], x, b*(Cot[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*Sqrt[d*Csc[e + f*x]]))], x
] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0]

Rule 4098

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*
(B_.) + (A_)), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(f*n)), x] - Dist[(
a*A*m - b*B*n)/(b*d*n), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, A
, B, m, n}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && EqQ[m + n + 1, 0] &&  !LeQ[m, -1]

Rule 4105

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*
(B_.) + (A_)), x_Symbol] :> Simp[(-(A*b - a*B))*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(b*f*(
2*m + 1))), x] - Dist[1/(a^2*(2*m + 1)), Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n*Simp[b*B*n - a*A*
(2*m + n + 1) + (A*b - a*B)*(m + n + 1)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[
A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)] &&  !GtQ[n, 0]

Rule 4107

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*
(B_.) + (A_)), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(f*n)), x] - Dist[1
/(b*d*n), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)*Simp[a*A*m - b*B*n - A*b*(m + n + 1)*Csc[e + f*x
], x], x], x] /; FreeQ[{a, b, d, e, f, A, B, m}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && LtQ[n, 0]

Rule 4169

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^
(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(-(a*A - b*B + a*C))*Cot[e + f*x]*(a + b*C
sc[e + f*x])^m*((d*Csc[e + f*x])^n/(a*f*(2*m + 1))), x] - Dist[1/(a*b*(2*m + 1)), Int[(a + b*Csc[e + f*x])^(m
+ 1)*(d*Csc[e + f*x])^n*Simp[a*B*n - b*C*n - A*b*(2*m + n + 1) - (b*B*(m + n + 1) - a*(A*(m + n + 1) - C*(m -
n)))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)]

Rubi steps

\begin {align*} \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}} \, dx &=-\frac {(A-B+C) \sin (c+d x)}{4 d \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}}+\frac {\int \frac {\frac {1}{2} a (13 A-5 B+5 C)-4 a (A-B) \sec (c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}} \, dx}{4 a^2}\\ &=-\frac {(A-B+C) \sin (c+d x)}{4 d \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}}-\frac {(21 A-13 B+5 C) \sin (c+d x)}{16 a d \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}}+\frac {\int \frac {\frac {1}{4} a^2 (157 A-85 B+45 C)-\frac {3}{2} a^2 (21 A-13 B+5 C) \sec (c+d x)}{\sec ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}} \, dx}{8 a^4}\\ &=-\frac {(A-B+C) \sin (c+d x)}{4 d \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}}-\frac {(21 A-13 B+5 C) \sin (c+d x)}{16 a d \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}}+\frac {(157 A-85 B+45 C) \sin (c+d x)}{80 a^2 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {\int \frac {-\frac {1}{8} a^3 (787 A-475 B+195 C)+\frac {1}{2} a^3 (157 A-85 B+45 C) \sec (c+d x)}{\sec ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}} \, dx}{20 a^5}\\ &=-\frac {(A-B+C) \sin (c+d x)}{4 d \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}}-\frac {(21 A-13 B+5 C) \sin (c+d x)}{16 a d \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}}+\frac {(157 A-85 B+45 C) \sin (c+d x)}{80 a^2 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}-\frac {(787 A-475 B+195 C) \sin (c+d x)}{240 a^2 d \sqrt {\sec (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {\int \frac {\frac {1}{16} a^4 (2671 A-1495 B+735 C)-\frac {1}{8} a^4 (787 A-475 B+195 C) \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+a \sec (c+d x)}} \, dx}{30 a^6}\\ &=-\frac {(A-B+C) \sin (c+d x)}{4 d \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}}-\frac {(21 A-13 B+5 C) \sin (c+d x)}{16 a d \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}}+\frac {(157 A-85 B+45 C) \sin (c+d x)}{80 a^2 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}-\frac {(787 A-475 B+195 C) \sin (c+d x)}{240 a^2 d \sqrt {\sec (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {(2671 A-1495 B+735 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{240 a^2 d \sqrt {a+a \sec (c+d x)}}-\frac {(283 A-163 B+75 C) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+a \sec (c+d x)}} \, dx}{32 a^2}\\ &=-\frac {(A-B+C) \sin (c+d x)}{4 d \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}}-\frac {(21 A-13 B+5 C) \sin (c+d x)}{16 a d \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}}+\frac {(157 A-85 B+45 C) \sin (c+d x)}{80 a^2 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}-\frac {(787 A-475 B+195 C) \sin (c+d x)}{240 a^2 d \sqrt {\sec (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {(2671 A-1495 B+735 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{240 a^2 d \sqrt {a+a \sec (c+d x)}}+\frac {(283 A-163 B+75 C) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,-\frac {a \sqrt {\sec (c+d x)} \sin (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{16 a^2 d}\\ &=-\frac {(283 A-163 B+75 C) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {\sec (c+d x)} \sin (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{16 \sqrt {2} a^{5/2} d}-\frac {(A-B+C) \sin (c+d x)}{4 d \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}}-\frac {(21 A-13 B+5 C) \sin (c+d x)}{16 a d \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}}+\frac {(157 A-85 B+45 C) \sin (c+d x)}{80 a^2 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}-\frac {(787 A-475 B+195 C) \sin (c+d x)}{240 a^2 d \sqrt {\sec (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {(2671 A-1495 B+735 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{240 a^2 d \sqrt {a+a \sec (c+d x)}}\\ \end {align*}

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Mathematica [A]
time = 3.32, size = 221, normalized size = 0.71 \begin {gather*} -\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (15 (283 A-163 B+75 C) \tanh ^{-1}\left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \cos ^4\left (\frac {1}{2} (c+d x)\right )-\frac {1}{2} (3491 A-1895 B+975 C+5 (887 A-479 B+255 C) \cos (c+d x)+16 (52 A-25 B+15 C) \cos (2 (c+d x))-40 A \cos (3 (c+d x))+40 B \cos (3 (c+d x))+12 A \cos (4 (c+d x))) \sin \left (\frac {1}{2} (c+d x)\right )\right )}{60 a d (A+2 C+2 B \cos (c+d x)+A \cos (2 (c+d x))) \sqrt {\sec (c+d x)} (a (1+\sec (c+d x)))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)),x]

[Out]

-1/60*(Sec[(c + d*x)/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(15*(283*A - 163*B + 75*C)*ArcTanh[Sin[(c + d*
x)/2]]*Cos[(c + d*x)/2]^4 - ((3491*A - 1895*B + 975*C + 5*(887*A - 479*B + 255*C)*Cos[c + d*x] + 16*(52*A - 25
*B + 15*C)*Cos[2*(c + d*x)] - 40*A*Cos[3*(c + d*x)] + 40*B*Cos[3*(c + d*x)] + 12*A*Cos[4*(c + d*x)])*Sin[(c +
d*x)/2])/2))/(a*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*Sqrt[Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^
(3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(656\) vs. \(2(270)=540\).
time = 0.27, size = 657, normalized size = 2.10

method	result	size
default	\(\frac {\sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (-1+\cos \left (d x +c \right )\right )^{2} \left (4245 A \arctan \left (\frac {\sin \left (d x +c \right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}}{2}\right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )-192 A \left (\cos ^{5}\left (d x +c \right )\right )-2445 B \arctan \left (\frac {\sin \left (d x +c \right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}}{2}\right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )+1125 C \arctan \left (\frac {\sin \left (d x +c \right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}}{2}\right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )+8490 A \arctan \left (\frac {\sin \left (d x +c \right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}}{2}\right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \cos \left (d x +c \right )+512 A \left (\cos ^{4}\left (d x +c \right )\right )-4890 B \arctan \left (\frac {\sin \left (d x +c \right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}}{2}\right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \cos \left (d x +c \right )-320 B \left (\cos ^{4}\left (d x +c \right )\right )+2250 C \arctan \left (\frac {\sin \left (d x +c \right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}}{2}\right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \cos \left (d x +c \right )+4245 \arctan \left (\frac {\sin \left (d x +c \right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}}{2}\right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, A \sin \left (d x +c \right )-3456 A \left (\cos ^{3}\left (d x +c \right )\right )-2445 \arctan \left (\frac {\sin \left (d x +c \right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}}{2}\right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, B \sin \left (d x +c \right )+1920 B \left (\cos ^{3}\left (d x +c \right )\right )+1125 C \arctan \left (\frac {\sin \left (d x +c \right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}}{2}\right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )-960 C \left (\cos ^{3}\left (d x +c \right )\right )-5974 A \left (\cos ^{2}\left (d x +c \right )\right )+3430 B \left (\cos ^{2}\left (d x +c \right )\right )-1590 C \left (\cos ^{2}\left (d x +c \right )\right )+3768 A \cos \left (d x +c \right )-2040 B \cos \left (d x +c \right )+1080 C \cos \left (d x +c \right )+5342 A -2990 B +1470 C \right ) \left (\cos ^{3}\left (d x +c \right )\right ) \left (\frac {1}{\cos \left (d x +c \right )}\right )^{\frac {5}{2}}}{480 d \sin \left (d x +c \right )^{5} a^{3}}\)	\(657\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(5/2)/(a+a*sec(d*x+c))^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/480/d*(a*(1+cos(d*x+c))/cos(d*x+c))^(1/2)*(-1+cos(d*x+c))^2*(4245*A*arctan(1/2*sin(d*x+c)*(-2/(1+cos(d*x+c))
)^(1/2))*(-2/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)*cos(d*x+c)^2-192*A*cos(d*x+c)^5-2445*B*arctan(1/2*sin(d*x+c)*(-2
/(1+cos(d*x+c)))^(1/2))*(-2/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)*cos(d*x+c)^2+1125*C*arctan(1/2*sin(d*x+c)*(-2/(1+
cos(d*x+c)))^(1/2))*(-2/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)*cos(d*x+c)^2+8490*A*arctan(1/2*sin(d*x+c)*(-2/(1+cos(
d*x+c)))^(1/2))*(-2/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)*cos(d*x+c)+512*A*cos(d*x+c)^4-4890*B*arctan(1/2*sin(d*x+c
)*(-2/(1+cos(d*x+c)))^(1/2))*(-2/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)*cos(d*x+c)-320*B*cos(d*x+c)^4+2250*C*arctan(
1/2*sin(d*x+c)*(-2/(1+cos(d*x+c)))^(1/2))*(-2/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)*cos(d*x+c)+4245*arctan(1/2*sin(
d*x+c)*(-2/(1+cos(d*x+c)))^(1/2))*(-2/(1+cos(d*x+c)))^(1/2)*A*sin(d*x+c)-3456*A*cos(d*x+c)^3-2445*arctan(1/2*s
in(d*x+c)*(-2/(1+cos(d*x+c)))^(1/2))*(-2/(1+cos(d*x+c)))^(1/2)*B*sin(d*x+c)+1920*B*cos(d*x+c)^3+1125*C*arctan(
1/2*sin(d*x+c)*(-2/(1+cos(d*x+c)))^(1/2))*(-2/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)-960*C*cos(d*x+c)^3-5974*A*cos(d
*x+c)^2+3430*B*cos(d*x+c)^2-1590*C*cos(d*x+c)^2+3768*A*cos(d*x+c)-2040*B*cos(d*x+c)+1080*C*cos(d*x+c)+5342*A-2
990*B+1470*C)*cos(d*x+c)^3*(1/cos(d*x+c))^(5/2)/sin(d*x+c)^5/a^3

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 645572 vs. \(2 (270) = 540\).
time = 11.65, size = 645572, normalized size = 2062.53 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(5/2)/(a+a*sec(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

1/480*((16*(3*cos(5*d*x + 5*c)^2*sin(5/2*d*x + 5/2*c) + 3*sin(5*d*x + 5*c)^2*sin(5/2*d*x + 5/2*c) - 25*(cos(5*
d*x + 5*c)^2 + sin(5*d*x + 5*c)^2)*sin(3/5*arctan2(sin(5/2*d*x + 5/2*c), cos(5/2*d*x + 5/2*c))) + 300*(cos(5*d
*x + 5*c)^2 + sin(5*d*x + 5*c)^2)*sin(1/5*arctan2(sin(5/2*d*x + 5/2*c), cos(5/2*d*x + 5/2*c))))*cos(13/5*arcta
n2(sin(5/2*d*x + 5/2*c), cos(5/2*d*x + 5/2*c)))^4 + 4096*(3*cos(5*d*x + 5*c)^2*sin(5/2*d*x + 5/2*c) + 3*sin(5*
d*x + 5*c)^2*sin(5/2*d*x + 5/2*c) - 25*(cos(5*d*x + 5*c)^2 + sin(5*d*x + 5*c)^2)*sin(3/5*arctan2(sin(5/2*d*x +
5/2*c), cos(5/2*d*x + 5/2*c))) + 300*(cos(5*d*x + 5*c)^2 + sin(5*d*x + 5*c)^2)*sin(1/5*arctan2(sin(5/2*d*x +
5/2*c), cos(5/2*d*x + 5/2*c))))*cos(11/5*arctan2(sin(5/2*d*x + 5/2*c), cos(5/2*d*x + 5/2*c)))^4 + 20736*(3*cos
(5*d*x + 5*c)^2*sin(5/2*d*x + 5/2*c) + 3*sin(5*d*x + 5*c)^2*sin(5/2*d*x + 5/2*c) - 25*(cos(5*d*x + 5*c)^2 + si
n(5*d*x + 5*c)^2)*sin(3/5*arctan2(sin(5/2*d*x + 5/2*c), cos(5/2*d*x + 5/2*c))) + 300*(cos(5*d*x + 5*c)^2 + sin
(5*d*x + 5*c)^2)*sin(1/5*arctan2(sin(5/2*d*x + 5/2*c), cos(5/2*d*x + 5/2*c))))*cos(9/5*arctan2(sin(5/2*d*x + 5
/2*c), cos(5/2*d*x + 5/2*c)))^4 + 4096*(3*cos(5*d*x + 5*c)^2*sin(5/2*d*x + 5/2*c) + 3*sin(5*d*x + 5*c)^2*sin(5
/2*d*x + 5/2*c) - 25*(cos(5*d*x + 5*c)^2 + sin(5*d*x + 5*c)^2)*sin(3/5*arctan2(sin(5/2*d*x + 5/2*c), cos(5/2*d
*x + 5/2*c))) + 300*(cos(5*d*x + 5*c)^2 + sin(5*d*x + 5*c)^2)*sin(1/5*arctan2(sin(5/2*d*x + 5/2*c), cos(5/2*d*
x + 5/2*c))))*cos(7/5*arctan2(sin(5/2*d*x + 5/2*c), cos(5/2*d*x + 5/2*c)))^4 + 48*sin(5/2*d*x + 5/2*c)^5 + 16*
(3*cos(5*d*x + 5*c)^2*sin(5/2*d*x + 5/2*c) + 3*sin(5*d*x + 5*c)^2*sin(5/2*d*x + 5/2*c) - 25*(cos(5*d*x + 5*c)^
2 + sin(5*d*x + 5*c)^2)*sin(3/5*arctan2(sin(5/2*d*x + 5/2*c), cos(5/2*d*x + 5/2*c))) + 300*(cos(5*d*x + 5*c)^2
+ sin(5*d*x + 5*c)^2)*sin(1/5*arctan2(sin(5/2*d*x + 5/2*c), cos(5/2*d*x + 5/2*c))))*sin(13/5*arctan2(sin(5/2*
d*x + 5/2*c), cos(5/2*d*x + 5/2*c)))^4 + 4096*(3*cos(5*d*x + 5*c)^2*sin(5/2*d*x + 5/2*c) + 3*sin(5*d*x + 5*c)^
2*sin(5/2*d*x + 5/2*c) - 25*(cos(5*d*x + 5*c)^2 + sin(5*d*x + 5*c)^2)*sin(3/5*arctan2(sin(5/2*d*x + 5/2*c), co
s(5/2*d*x + 5/2*c))) + 300*(cos(5*d*x + 5*c)^2 + sin(5*d*x + 5*c)^2)*sin(1/5*arctan2(sin(5/2*d*x + 5/2*c), cos
(5/2*d*x + 5/2*c))))*sin(11/5*arctan2(sin(5/2*d*x + 5/2*c), cos(5/2*d*x + 5/2*c)))^4 + 20736*(3*cos(5*d*x + 5*
c)^2*sin(5/2*d*x + 5/2*c) + 3*sin(5*d*x + 5*c)^2*sin(5/2*d*x + 5/2*c) - 25*(cos(5*d*x + 5*c)^2 + sin(5*d*x + 5
*c)^2)*sin(3/5*arctan2(sin(5/2*d*x + 5/2*c), cos(5/2*d*x + 5/2*c))) + 300*(cos(5*d*x + 5*c)^2 + sin(5*d*x + 5*
c)^2)*sin(1/5*arctan2(sin(5/2*d*x + 5/2*c), cos(5/2*d*x + 5/2*c))))*sin(9/5*arctan2(sin(5/2*d*x + 5/2*c), cos(
5/2*d*x + 5/2*c)))^4 + 4096*(3*cos(5*d*x + 5*c)^2*sin(5/2*d*x + 5/2*c) + 3*sin(5*d*x + 5*c)^2*sin(5/2*d*x + 5/
2*c) - 25*(cos(5*d*x + 5*c)^2 + sin(5*d*x + 5*c)^2)*sin(3/5*arctan2(sin(5/2*d*x + 5/2*c), cos(5/2*d*x + 5/2*c)
)) + 300*(cos(5*d*x + 5*c)^2 + sin(5*d*x + 5*c)^2)*sin(1/5*arctan2(sin(5/2*d*x + 5/2*c), cos(5/2*d*x + 5/2*c))
))*sin(7/5*arctan2(sin(5/2*d*x + 5/2*c), cos(5/2*d*x + 5/2*c)))^4 + 4*(24*cos(5*d*x + 5*c)^2*cos(5/2*d*x + 5/2
*c)*sin(5/2*d*x + 5/2*c) + 24*cos(5/2*d*x + 5/2*c)*sin(5*d*x + 5*c)^2*sin(5/2*d*x + 5/2*c) + 24*cos(5*d*x + 5*
c)*cos(5/2*d*x + 5/2*c)*sin(5/2*d*x + 5/2*c) + 64*(3*cos(5*d*x + 5*c)^2*sin(5/2*d*x + 5/2*c) + 3*sin(5*d*x + 5
*c)^2*sin(5/2*d*x + 5/2*c) - 25*(cos(5*d*x + 5*c)^2 + sin(5*d*x + 5*c)^2)*sin(3/5*arctan2(sin(5/2*d*x + 5/2*c)
, cos(5/2*d*x + 5/2*c))) + 300*(cos(5*d*x + 5*c)^2 + sin(5*d*x + 5*c)^2)*sin(1/5*arctan2(sin(5/2*d*x + 5/2*c),
cos(5/2*d*x + 5/2*c))))*cos(11/5*arctan2(sin(5/2*d*x + 5/2*c), cos(5/2*d*x + 5/2*c))) + 96*(3*cos(5*d*x + 5*c
)^2*sin(5/2*d*x + 5/2*c) + 3*sin(5*d*x + 5*c)^2*sin(5/2*d*x + 5/2*c) - 25*(cos(5*d*x + 5*c)^2 + sin(5*d*x + 5*
c)^2)*sin(3/5*arctan2(sin(5/2*d*x + 5/2*c), cos(5/2*d*x + 5/2*c))) + 300*(cos(5*d*x + 5*c)^2 + sin(5*d*x + 5*c
)^2)*sin(1/5*arctan2(sin(5/2*d*x + 5/2*c), cos(5/2*d*x + 5/2*c))))*cos(9/5*arctan2(sin(5/2*d*x + 5/2*c), cos(5
/2*d*x + 5/2*c))) + 64*(3*cos(5*d*x + 5*c)^2*sin(5/2*d*x + 5/2*c) + 3*sin(5*d*x + 5*c)^2*sin(5/2*d*x + 5/2*c)
- 25*(cos(5*d*x + 5*c)^2 + sin(5*d*x + 5*c)^2)*sin(3/5*arctan2(sin(5/2*d*x + 5/2*c), cos(5/2*d*x + 5/2*c))) +
300*(cos(5*d*x + 5*c)^2 + sin(5*d*x + 5*c)^2)*sin(1/5*arctan2(sin(5/2*d*x + 5/2*c), cos(5/2*d*x + 5/2*c))))*co
s(7/5*arctan2(sin(5/2*d*x + 5/2*c), cos(5/2*d*x + 5/2*c))) + (24*sin(5/2*d*x + 5/2*c)^2 - 4265)*sin(5*d*x + 5*
c) - 645*(cos(5*d*x + 5*c)^2 + sin(5*d*x + 5*c)^2)*sin(12/5*arctan2(sin(5/2*d*x + 5/2*c), cos(5/2*d*x + 5/2*c)
)) - 7247*(cos(5*d*x + 5*c)^2 + sin(5*d*x + 5*c)^2)*sin(8/5*arctan2(sin(5/2*d*x + 5/2*c), cos(5/2*d*x + 5/2*c)
)) - 4803*(cos(5*d*x + 5*c)^2 + sin(5*d*x + 5*c)^2)*sin(6/5*arctan2(sin(5/2*d*x + 5/2*c), cos(5/2*d*x + 5/2*c)
)) - 872*(cos(5*d*x + 5*c)^2 + sin(5*d*x + 5*c)^2)*sin(4/5*arctan2(sin(5/2*d*x + 5/2*c), cos(5/2*d*x + 5/2*c))
) - 200*(cos(5*d*x + 5*c)^2*cos(5/2*d*x + 5/2*c) + cos(5/2*d*x + 5/2*c)*sin(5*d*x + 5*c)^2 + cos(5*d*x + 5*c)*
cos(5/2*d*x + 5/2*c) + sin(5*d*x + 5*c)*sin(5/2...

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Fricas [A]
time = 2.41, size = 634, normalized size = 2.03 \begin {gather*} \left [\frac {15 \, \sqrt {2} {\left ({\left (283 \, A - 163 \, B + 75 \, C\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (283 \, A - 163 \, B + 75 \, C\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (283 \, A - 163 \, B + 75 \, C\right )} \cos \left (d x + c\right ) + 283 \, A - 163 \, B + 75 \, C\right )} \sqrt {a} \log \left (-\frac {a \cos \left (d x + c\right )^{2} + 2 \, \sqrt {2} \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 2 \, a \cos \left (d x + c\right ) - 3 \, a}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right ) + \frac {4 \, {\left (96 \, A \cos \left (d x + c\right )^{5} - 160 \, {\left (A - B\right )} \cos \left (d x + c\right )^{4} + 32 \, {\left (49 \, A - 25 \, B + 15 \, C\right )} \cos \left (d x + c\right )^{3} + 5 \, {\left (911 \, A - 503 \, B + 255 \, C\right )} \cos \left (d x + c\right )^{2} + {\left (2671 \, A - 1495 \, B + 735 \, C\right )} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{960 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}}, \frac {15 \, \sqrt {2} {\left ({\left (283 \, A - 163 \, B + 75 \, C\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (283 \, A - 163 \, B + 75 \, C\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (283 \, A - 163 \, B + 75 \, C\right )} \cos \left (d x + c\right ) + 283 \, A - 163 \, B + 75 \, C\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {2} \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )}}{a \sin \left (d x + c\right )}\right ) + \frac {2 \, {\left (96 \, A \cos \left (d x + c\right )^{5} - 160 \, {\left (A - B\right )} \cos \left (d x + c\right )^{4} + 32 \, {\left (49 \, A - 25 \, B + 15 \, C\right )} \cos \left (d x + c\right )^{3} + 5 \, {\left (911 \, A - 503 \, B + 255 \, C\right )} \cos \left (d x + c\right )^{2} + {\left (2671 \, A - 1495 \, B + 735 \, C\right )} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{480 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(5/2)/(a+a*sec(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

[1/960*(15*sqrt(2)*((283*A - 163*B + 75*C)*cos(d*x + c)^3 + 3*(283*A - 163*B + 75*C)*cos(d*x + c)^2 + 3*(283*A
- 163*B + 75*C)*cos(d*x + c) + 283*A - 163*B + 75*C)*sqrt(a)*log(-(a*cos(d*x + c)^2 + 2*sqrt(2)*sqrt(a)*sqrt(
(a*cos(d*x + c) + a)/cos(d*x + c))*sqrt(cos(d*x + c))*sin(d*x + c) - 2*a*cos(d*x + c) - 3*a)/(cos(d*x + c)^2 +
2*cos(d*x + c) + 1)) + 4*(96*A*cos(d*x + c)^5 - 160*(A - B)*cos(d*x + c)^4 + 32*(49*A - 25*B + 15*C)*cos(d*x
+ c)^3 + 5*(911*A - 503*B + 255*C)*cos(d*x + c)^2 + (2671*A - 1495*B + 735*C)*cos(d*x + c))*sqrt((a*cos(d*x +
c) + a)/cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)))/(a^3*d*cos(d*x + c)^3 + 3*a^3*d*cos(d*x + c)^2 + 3*a^3*
d*cos(d*x + c) + a^3*d), 1/480*(15*sqrt(2)*((283*A - 163*B + 75*C)*cos(d*x + c)^3 + 3*(283*A - 163*B + 75*C)*c
os(d*x + c)^2 + 3*(283*A - 163*B + 75*C)*cos(d*x + c) + 283*A - 163*B + 75*C)*sqrt(-a)*arctan(sqrt(2)*sqrt(-a)
*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sqrt(cos(d*x + c))/(a*sin(d*x + c))) + 2*(96*A*cos(d*x + c)^5 - 160*(
A - B)*cos(d*x + c)^4 + 32*(49*A - 25*B + 15*C)*cos(d*x + c)^3 + 5*(911*A - 503*B + 255*C)*cos(d*x + c)^2 + (2
671*A - 1495*B + 735*C)*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)))
/(a^3*d*cos(d*x + c)^3 + 3*a^3*d*cos(d*x + c)^2 + 3*a^3*d*cos(d*x + c) + a^3*d)]

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*sec(d*x+c)+C*sec(d*x+c)**2)/sec(d*x+c)**(5/2)/(a+a*sec(d*x+c))**(5/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 6192 deep

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(5/2)/(a+a*sec(d*x+c))^(5/2),x, algorithm="giac")

[Out]

integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)/((a*sec(d*x + c) + a)^(5/2)*sec(d*x + c)^(5/2)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B/cos(c + d*x) + C/cos(c + d*x)^2)/((a + a/cos(c + d*x))^(5/2)*(1/cos(c + d*x))^(5/2)),x)

[Out]

int((A + B/cos(c + d*x) + C/cos(c + d*x)^2)/((a + a/cos(c + d*x))^(5/2)*(1/cos(c + d*x))^(5/2)), x)

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