3.417 \(\int \frac {(A+B \cos (c+d x)+C \cos ^2(c+d x)) \sec ^4(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx\)
Optimal. Leaf size=284 \[ -\frac {(47 A-38 B+24 C) \tanh ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{8 a^{3/2} d}+\frac {(17 A-13 B+9 C) \tanh ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {a \cos (c+d x)+a}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {(21 A-14 B+12 C) \tan (c+d x)}{8 a d \sqrt {a \cos (c+d x)+a}}+\frac {(5 A-3 B+3 C) \tan (c+d x) \sec ^2(c+d x)}{6 a d \sqrt {a \cos (c+d x)+a}}-\frac {(A-B+C) \tan (c+d x) \sec ^2(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac {(13 A-12 B+6 C) \tan (c+d x) \sec (c+d x)}{12 a d \sqrt {a \cos (c+d x)+a}} \]
[Out]
-1/8*(47*A-38*B+24*C)*arctanh(sin(d*x+c)*a^(1/2)/(a+a*cos(d*x+c))^(1/2))/a^(3/2)/d+1/4*(17*A-13*B+9*C)*arctanh
(1/2*sin(d*x+c)*a^(1/2)*2^(1/2)/(a+a*cos(d*x+c))^(1/2))/a^(3/2)/d*2^(1/2)-1/2*(A-B+C)*sec(d*x+c)^2*tan(d*x+c)/
d/(a+a*cos(d*x+c))^(3/2)+1/8*(21*A-14*B+12*C)*tan(d*x+c)/a/d/(a+a*cos(d*x+c))^(1/2)-1/12*(13*A-12*B+6*C)*sec(d
*x+c)*tan(d*x+c)/a/d/(a+a*cos(d*x+c))^(1/2)+1/6*(5*A-3*B+3*C)*sec(d*x+c)^2*tan(d*x+c)/a/d/(a+a*cos(d*x+c))^(1/
2)
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Rubi [A] time = 0.99, antiderivative size = 284, normalized size of antiderivative = 1.00,
number of steps used = 9, number of rules used = 6, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used =
{3041, 2984, 2985, 2649, 206, 2773} \[ -\frac {(47 A-38 B+24 C) \tanh ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{8 a^{3/2} d}+\frac {(17 A-13 B+9 C) \tanh ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {a \cos (c+d x)+a}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {(21 A-14 B+12 C) \tan (c+d x)}{8 a d \sqrt {a \cos (c+d x)+a}}+\frac {(5 A-3 B+3 C) \tan (c+d x) \sec ^2(c+d x)}{6 a d \sqrt {a \cos (c+d x)+a}}-\frac {(A-B+C) \tan (c+d x) \sec ^2(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac {(13 A-12 B+6 C) \tan (c+d x) \sec (c+d x)}{12 a d \sqrt {a \cos (c+d x)+a}} \]
Antiderivative was successfully verified.
[In]
Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + a*Cos[c + d*x])^(3/2),x]
[Out]
-((47*A - 38*B + 24*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*a^(3/2)*d) + ((17*A - 13*B
+ 9*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + ((21*A - 1
4*B + 12*C)*Tan[c + d*x])/(8*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((13*A - 12*B + 6*C)*Sec[c + d*x]*Tan[c + d*x])/(
12*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B + C)*Sec[c + d*x]^2*Tan[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2))
+ ((5*A - 3*B + 3*C)*Sec[c + d*x]^2*Tan[c + d*x])/(6*a*d*Sqrt[a + a*Cos[c + d*x]])
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 2649
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[-2/d, Subst[Int[1/(2*a - x^2), x], x, (b*C
os[c + d*x])/Sqrt[a + b*Sin[c + d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
Rule 2773
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(-2*
b)/f, Subst[Int[1/(b*c + a*d - d*x^2), x], x, (b*Cos[e + f*x])/Sqrt[a + b*Sin[e + f*x]]], x] /; FreeQ[{a, b, c
, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Rule 2984
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[((B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x]
)^(n + 1))/(f*(n + 1)*(c^2 - d^2)), x] + Dist[1/(b*(n + 1)*(c^2 - d^2)), Int[(a + b*Sin[e + f*x])^m*(c + d*Sin
[e + f*x])^(n + 1)*Simp[A*(a*d*m + b*c*(n + 1)) - B*(a*c*m + b*d*(n + 1)) + b*(B*c - A*d)*(m + n + 2)*Sin[e +
f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2
- d^2, 0] && LtQ[n, -1] && (IntegerQ[n] || EqQ[m + 1/2, 0])
Rule 2985
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])), x_Symbol] :> Dist[(A*b - a*B)/(b*c - a*d), Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] + Dist[(
B*c - A*d)/(b*c - a*d), Int[Sqrt[a + b*Sin[e + f*x]]/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f,
A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Rule 3041
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[((a*A - b*B + a*C)*Cos[e + f*x]*(
a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(f*(b*c - a*d)*(2*m + 1)), x] + Dist[1/(b*(b*c - a*d)*(2*m
+ 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[A*(a*c*(m + 1) - b*d*(2*m + n + 2)) + B*(
b*c*m + a*d*(n + 1)) - C*(a*c*m + b*d*(n + 1)) + (d*(a*A - b*B)*(m + n + 2) + C*(b*c*(2*m + 1) - a*d*(m - n -
1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^
2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)]
Rubi steps
\begin {align*} \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx &=-\frac {(A-B+C) \sec ^2(c+d x) \tan (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {\int \frac {\left (a (5 A-3 B+3 C)-\frac {1}{2} a (7 A-7 B+3 C) \cos (c+d x)\right ) \sec ^4(c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx}{2 a^2}\\ &=-\frac {(A-B+C) \sec ^2(c+d x) \tan (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {(5 A-3 B+3 C) \sec ^2(c+d x) \tan (c+d x)}{6 a d \sqrt {a+a \cos (c+d x)}}+\frac {\int \frac {\left (-a^2 (13 A-12 B+6 C)+\frac {5}{2} a^2 (5 A-3 B+3 C) \cos (c+d x)\right ) \sec ^3(c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx}{6 a^3}\\ &=-\frac {(13 A-12 B+6 C) \sec (c+d x) \tan (c+d x)}{12 a d \sqrt {a+a \cos (c+d x)}}-\frac {(A-B+C) \sec ^2(c+d x) \tan (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {(5 A-3 B+3 C) \sec ^2(c+d x) \tan (c+d x)}{6 a d \sqrt {a+a \cos (c+d x)}}+\frac {\int \frac {\left (\frac {3}{2} a^3 (21 A-14 B+12 C)-\frac {3}{2} a^3 (13 A-12 B+6 C) \cos (c+d x)\right ) \sec ^2(c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx}{12 a^4}\\ &=\frac {(21 A-14 B+12 C) \tan (c+d x)}{8 a d \sqrt {a+a \cos (c+d x)}}-\frac {(13 A-12 B+6 C) \sec (c+d x) \tan (c+d x)}{12 a d \sqrt {a+a \cos (c+d x)}}-\frac {(A-B+C) \sec ^2(c+d x) \tan (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {(5 A-3 B+3 C) \sec ^2(c+d x) \tan (c+d x)}{6 a d \sqrt {a+a \cos (c+d x)}}+\frac {\int \frac {\left (-\frac {3}{4} a^4 (47 A-38 B+24 C)+\frac {3}{4} a^4 (21 A-14 B+12 C) \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx}{12 a^5}\\ &=\frac {(21 A-14 B+12 C) \tan (c+d x)}{8 a d \sqrt {a+a \cos (c+d x)}}-\frac {(13 A-12 B+6 C) \sec (c+d x) \tan (c+d x)}{12 a d \sqrt {a+a \cos (c+d x)}}-\frac {(A-B+C) \sec ^2(c+d x) \tan (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {(5 A-3 B+3 C) \sec ^2(c+d x) \tan (c+d x)}{6 a d \sqrt {a+a \cos (c+d x)}}+\frac {(17 A-13 B+9 C) \int \frac {1}{\sqrt {a+a \cos (c+d x)}} \, dx}{4 a}-\frac {(47 A-38 B+24 C) \int \sqrt {a+a \cos (c+d x)} \sec (c+d x) \, dx}{16 a^2}\\ &=\frac {(21 A-14 B+12 C) \tan (c+d x)}{8 a d \sqrt {a+a \cos (c+d x)}}-\frac {(13 A-12 B+6 C) \sec (c+d x) \tan (c+d x)}{12 a d \sqrt {a+a \cos (c+d x)}}-\frac {(A-B+C) \sec ^2(c+d x) \tan (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {(5 A-3 B+3 C) \sec ^2(c+d x) \tan (c+d x)}{6 a d \sqrt {a+a \cos (c+d x)}}-\frac {(17 A-13 B+9 C) \operatorname {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,-\frac {a \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{2 a d}+\frac {(47 A-38 B+24 C) \operatorname {Subst}\left (\int \frac {1}{a-x^2} \, dx,x,-\frac {a \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{8 a d}\\ &=-\frac {(47 A-38 B+24 C) \tanh ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{8 a^{3/2} d}+\frac {(17 A-13 B+9 C) \tanh ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {a+a \cos (c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {(21 A-14 B+12 C) \tan (c+d x)}{8 a d \sqrt {a+a \cos (c+d x)}}-\frac {(13 A-12 B+6 C) \sec (c+d x) \tan (c+d x)}{12 a d \sqrt {a+a \cos (c+d x)}}-\frac {(A-B+C) \sec ^2(c+d x) \tan (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {(5 A-3 B+3 C) \sec ^2(c+d x) \tan (c+d x)}{6 a d \sqrt {a+a \cos (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 2.70, size = 223, normalized size = 0.79 \[ \frac {\cos ^3\left (\frac {1}{2} (c+d x)\right ) \left (12 (17 A-13 B+9 C) \tanh ^{-1}\left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {3 \sqrt {2} (47 A-38 B+24 C) \cos ^2\left (\frac {1}{2} (c+d x)\right ) \tanh ^{-1}\left (\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right )-\frac {1}{4} \sin \left (\frac {1}{2} (c+d x)\right ) \sec ^3(c+d x) (3 (55 A-26 B+36 C) \cos (c+d x)+(74 A-36 B+48 C) \cos (2 (c+d x))+63 A \cos (3 (c+d x))+106 A-42 B \cos (3 (c+d x))-36 B+36 C \cos (3 (c+d x))+48 C)}{\sin ^2\left (\frac {1}{2} (c+d x)\right )-1}\right )}{12 d (a (\cos (c+d x)+1))^{3/2}} \]
Antiderivative was successfully verified.
[In]
Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + a*Cos[c + d*x])^(3/2),x]
[Out]
(Cos[(c + d*x)/2]^3*(12*(17*A - 13*B + 9*C)*ArcTanh[Sin[(c + d*x)/2]] + (3*Sqrt[2]*(47*A - 38*B + 24*C)*ArcTan
h[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^2 - ((106*A - 36*B + 48*C + 3*(55*A - 26*B + 36*C)*Cos[c + d*x] +
(74*A - 36*B + 48*C)*Cos[2*(c + d*x)] + 63*A*Cos[3*(c + d*x)] - 42*B*Cos[3*(c + d*x)] + 36*C*Cos[3*(c + d*x)]
)*Sec[c + d*x]^3*Sin[(c + d*x)/2])/4)/(-1 + Sin[(c + d*x)/2]^2)))/(12*d*(a*(1 + Cos[c + d*x]))^(3/2))
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fricas [A] time = 0.88, size = 402, normalized size = 1.42 \[ \frac {12 \, \sqrt {2} {\left ({\left (17 \, A - 13 \, B + 9 \, C\right )} \cos \left (d x + c\right )^{5} + 2 \, {\left (17 \, A - 13 \, B + 9 \, C\right )} \cos \left (d x + c\right )^{4} + {\left (17 \, A - 13 \, B + 9 \, C\right )} \cos \left (d x + c\right )^{3}\right )} \sqrt {a} \log \left (-\frac {a \cos \left (d x + c\right )^{2} - 2 \, \sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {a} \sin \left (d x + c\right ) - 2 \, a \cos \left (d x + c\right ) - 3 \, a}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right ) + 3 \, {\left ({\left (47 \, A - 38 \, B + 24 \, C\right )} \cos \left (d x + c\right )^{5} + 2 \, {\left (47 \, A - 38 \, B + 24 \, C\right )} \cos \left (d x + c\right )^{4} + {\left (47 \, A - 38 \, B + 24 \, C\right )} \cos \left (d x + c\right )^{3}\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} + 4 \, \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {a} {\left (\cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right ) + 8 \, a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}}\right ) + 4 \, {\left (3 \, {\left (21 \, A - 14 \, B + 12 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (37 \, A - 18 \, B + 24 \, C\right )} \cos \left (d x + c\right )^{2} - 6 \, {\left (A - 2 \, B\right )} \cos \left (d x + c\right ) + 8 \, A\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{96 \, {\left (a^{2} d \cos \left (d x + c\right )^{5} + 2 \, a^{2} d \cos \left (d x + c\right )^{4} + a^{2} d \cos \left (d x + c\right )^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^4/(a+a*cos(d*x+c))^(3/2),x, algorithm="fricas")
[Out]
1/96*(12*sqrt(2)*((17*A - 13*B + 9*C)*cos(d*x + c)^5 + 2*(17*A - 13*B + 9*C)*cos(d*x + c)^4 + (17*A - 13*B + 9
*C)*cos(d*x + c)^3)*sqrt(a)*log(-(a*cos(d*x + c)^2 - 2*sqrt(2)*sqrt(a*cos(d*x + c) + a)*sqrt(a)*sin(d*x + c) -
2*a*cos(d*x + c) - 3*a)/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)) + 3*((47*A - 38*B + 24*C)*cos(d*x + c)^5 + 2*(
47*A - 38*B + 24*C)*cos(d*x + c)^4 + (47*A - 38*B + 24*C)*cos(d*x + c)^3)*sqrt(a)*log((a*cos(d*x + c)^3 - 7*a*
cos(d*x + c)^2 + 4*sqrt(a*cos(d*x + c) + a)*sqrt(a)*(cos(d*x + c) - 2)*sin(d*x + c) + 8*a)/(cos(d*x + c)^3 + c
os(d*x + c)^2)) + 4*(3*(21*A - 14*B + 12*C)*cos(d*x + c)^3 + (37*A - 18*B + 24*C)*cos(d*x + c)^2 - 6*(A - 2*B)
*cos(d*x + c) + 8*A)*sqrt(a*cos(d*x + c) + a)*sin(d*x + c))/(a^2*d*cos(d*x + c)^5 + 2*a^2*d*cos(d*x + c)^4 + a
^2*d*cos(d*x + c)^3)
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^4/(a+a*cos(d*x+c))^(3/2),x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unab
le to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*p
i/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unab
le to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*p
i/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unab
le to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*p
i/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unab
le to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*p
i/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unab
le to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*p
i/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Warning, integration of abs or sign assumes cons
tant sign by intervals (correct if the argument is real):Check [abs(cos((d*t_nostep+c)/2))]Discontinuities at
zeroes of cos((d*t_nostep+c)/2) were not checkedEvaluation time: 1.45Unable to divide, perhaps due to rounding
error%%%{%%{[%%%{%%{[7975367974709495237422842361682067456000,0]:[1,0,-2]%%},[30]%%%},0]:[1,0,%%%{-1,[1]%%%}]
%%},[0]%%%} / %%%{%%%{%%{[63802943797675961899382738893456539648,0]:[1,0,-2]%%},[30]%%%},[0]%%%} Error: Bad Ar
gument Value
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maple [B] time = 3.84, size = 2993, normalized size = 10.54 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^4/(a+a*cos(d*x+c))^(3/2),x)
[Out]
-1/6*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-141*A*ln(-4*(a*2^(1/2)*cos(1/2*d*x+1/2*c)-2^(1/2)*(a*sin(1/2*d*x+1/2*c)^
2)^(1/2)*a^(1/2)-2*a)/(2*cos(1/2*d*x+1/2*c)-2^(1/2)))*cos(1/2*d*x+1/2*c)^2*a-912*B*ln(4/(2*cos(1/2*d*x+1/2*c)+
2^(1/2))*(2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+a*2^(1/2)*cos(1/2*d*x+1/2*c)+2*a))*cos(1/2*d*x+1/2*c)
^8*a+1128*A*ln(4/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+a*2^(1/2)*cos(
1/2*d*x+1/2*c)+2*a))*cos(1/2*d*x+1/2*c)^8*a-912*B*ln(-4*(a*2^(1/2)*cos(1/2*d*x+1/2*c)-2^(1/2)*(a*sin(1/2*d*x+1
/2*c)^2)^(1/2)*a^(1/2)-2*a)/(2*cos(1/2*d*x+1/2*c)-2^(1/2)))*cos(1/2*d*x+1/2*c)^8*a+1128*A*ln(-4*(a*2^(1/2)*cos
(1/2*d*x+1/2*c)-2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)-2*a)/(2*cos(1/2*d*x+1/2*c)-2^(1/2)))*cos(1/2*d*
x+1/2*c)^8*a-1692*A*ln(-4*(a*2^(1/2)*cos(1/2*d*x+1/2*c)-2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)-2*a)/(2
*cos(1/2*d*x+1/2*c)-2^(1/2)))*cos(1/2*d*x+1/2*c)^6*a-1692*A*ln(4/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(2^(1/2)*(a*si
n(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+a*2^(1/2)*cos(1/2*d*x+1/2*c)+2*a))*cos(1/2*d*x+1/2*c)^6*a+1368*B*ln(-4*(a*2^
(1/2)*cos(1/2*d*x+1/2*c)-2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)-2*a)/(2*cos(1/2*d*x+1/2*c)-2^(1/2)))*c
os(1/2*d*x+1/2*c)^6*a+1368*B*ln(4/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/
2)+a*2^(1/2)*cos(1/2*d*x+1/2*c)+2*a))*cos(1/2*d*x+1/2*c)^6*a-864*C*ln(-4*(a*2^(1/2)*cos(1/2*d*x+1/2*c)-2^(1/2)
*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)-2*a)/(2*cos(1/2*d*x+1/2*c)-2^(1/2)))*cos(1/2*d*x+1/2*c)^6*a-72*C*ln(-4
*(a*2^(1/2)*cos(1/2*d*x+1/2*c)-2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)-2*a)/(2*cos(1/2*d*x+1/2*c)-2^(1/
2)))*cos(1/2*d*x+1/2*c)^2*a+432*C*ln(-4*(a*2^(1/2)*cos(1/2*d*x+1/2*c)-2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a
^(1/2)-2*a)/(2*cos(1/2*d*x+1/2*c)-2^(1/2)))*cos(1/2*d*x+1/2*c)^4*a-864*C*ln(4/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(
2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+a*2^(1/2)*cos(1/2*d*x+1/2*c)+2*a))*cos(1/2*d*x+1/2*c)^6*a-72*C*
ln(4/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+a*2^(1/2)*cos(1/2*d*x+1/2*
c)+2*a))*cos(1/2*d*x+1/2*c)^2*a+336*C*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)*cos(1/2*d*x+1/2*c)^4+204*
A*ln(2*(2*a^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)+2*a)/cos(1/2*d*x+1/2*c))*2^(1/2)*cos(1/2*d*x+1/2*c)^2*a-12*B*
2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)-141*A*ln(4/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(2^(1/2)*(a*sin(1/2*d
*x+1/2*c)^2)^(1/2)*a^(1/2)+a*2^(1/2)*cos(1/2*d*x+1/2*c)+2*a))*cos(1/2*d*x+1/2*c)^2*a+2448*A*ln(2*(2*a^(1/2)*(a
*sin(1/2*d*x+1/2*c)^2)^(1/2)+2*a)/cos(1/2*d*x+1/2*c))*2^(1/2)*cos(1/2*d*x+1/2*c)^6*a-1872*B*ln(2*(2*a^(1/2)*(a
*sin(1/2*d*x+1/2*c)^2)^(1/2)+2*a)/cos(1/2*d*x+1/2*c))*2^(1/2)*cos(1/2*d*x+1/2*c)^6*a+608*A*2^(1/2)*(a*sin(1/2*
d*x+1/2*c)^2)^(1/2)*a^(1/2)*cos(1/2*d*x+1/2*c)^4+576*C*ln(4/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(2^(1/2)*(a*sin(1/2
*d*x+1/2*c)^2)^(1/2)*a^(1/2)+a*2^(1/2)*cos(1/2*d*x+1/2*c)+2*a))*cos(1/2*d*x+1/2*c)^8*a+576*C*ln(-4*(a*2^(1/2)*
cos(1/2*d*x+1/2*c)-2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)-2*a)/(2*cos(1/2*d*x+1/2*c)-2^(1/2)))*cos(1/2
*d*x+1/2*c)^8*a+846*A*ln(4/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+a*2^
(1/2)*cos(1/2*d*x+1/2*c)+2*a))*cos(1/2*d*x+1/2*c)^4*a+846*A*ln(-4*(a*2^(1/2)*cos(1/2*d*x+1/2*c)-2^(1/2)*(a*sin
(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)-2*a)/(2*cos(1/2*d*x+1/2*c)-2^(1/2)))*cos(1/2*d*x+1/2*c)^4*a-684*B*ln(4/(2*cos
(1/2*d*x+1/2*c)+2^(1/2))*(2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+a*2^(1/2)*cos(1/2*d*x+1/2*c)+2*a))*co
s(1/2*d*x+1/2*c)^4*a-684*B*ln(-4*(a*2^(1/2)*cos(1/2*d*x+1/2*c)-2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)-
2*a)/(2*cos(1/2*d*x+1/2*c)-2^(1/2)))*cos(1/2*d*x+1/2*c)^4*a-648*C*2^(1/2)*ln(2*(2*a^(1/2)*(a*sin(1/2*d*x+1/2*c
)^2)^(1/2)+2*a)/cos(1/2*d*x+1/2*c))*cos(1/2*d*x+1/2*c)^4*a+12*C*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)
-156*B*ln(2*(2*a^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)+2*a)/cos(1/2*d*x+1/2*c))*2^(1/2)*cos(1/2*d*x+1/2*c)^2*a+
108*C*2^(1/2)*ln(2*(2*a^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)+2*a)/cos(1/2*d*x+1/2*c))*cos(1/2*d*x+1/2*c)^2*a-1
20*C*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)*cos(1/2*d*x+1/2*c)^2-218*A*a^(1/2)*2^(1/2)*(a*sin(1/2*d*x+
1/2*c)^2)^(1/2)*cos(1/2*d*x+1/2*c)^2+156*B*a^(1/2)*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*cos(1/2*d*x+1/2*c)^2
+1296*C*2^(1/2)*ln(2*(2*a^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)+2*a)/cos(1/2*d*x+1/2*c))*cos(1/2*d*x+1/2*c)^6*a
+114*B*ln(-4*(a*2^(1/2)*cos(1/2*d*x+1/2*c)-2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)-2*a)/(2*cos(1/2*d*x+
1/2*c)-2^(1/2)))*cos(1/2*d*x+1/2*c)^2*a+114*B*ln(4/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(2^(1/2)*(a*sin(1/2*d*x+1/2*
c)^2)^(1/2)*a^(1/2)+a*2^(1/2)*cos(1/2*d*x+1/2*c)+2*a))*cos(1/2*d*x+1/2*c)^2*a-288*C*2^(1/2)*(a*sin(1/2*d*x+1/2
*c)^2)^(1/2)*a^(1/2)*cos(1/2*d*x+1/2*c)^6-864*C*2^(1/2)*ln(2*(2*a^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)+2*a)/co
s(1/2*d*x+1/2*c))*cos(1/2*d*x+1/2*c)^8*a-432*B*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)*cos(1/2*d*x+1/2*
c)^4+432*C*ln(4/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+a*2^(1/2)*cos(1
/2*d*x+1/2*c)+2*a))*cos(1/2*d*x+1/2*c)^4*a+12*A*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)-1224*A*ln(2*(2*
a^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)+2*a)/cos(1/2*d*x+1/2*c))*2^(1/2)*cos(1/2*d*x+1/2*c)^4*a+936*B*ln(2*(2*a
^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)+2*a)/cos(1/2*d*x+1/2*c))*2^(1/2)*cos(1/2*d*x+1/2*c)^4*a-504*A*2^(1/2)*(a
*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)*cos(1/2*d*x+1/2*c)^6+336*B*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)
*cos(1/2*d*x+1/2*c)^6-1632*A*ln(2*(2*a^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)+2*a)/cos(1/2*d*x+1/2*c))*2^(1/2)*c
os(1/2*d*x+1/2*c)^8*a+1248*B*ln(2*(2*a^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)+2*a)/cos(1/2*d*x+1/2*c))*2^(1/2)*c
os(1/2*d*x+1/2*c)^8*a)/a^(5/2)/cos(1/2*d*x+1/2*c)/(2*cos(1/2*d*x+1/2*c)-2^(1/2))^3/(2*cos(1/2*d*x+1/2*c)+2^(1/
2))^3/sin(1/2*d*x+1/2*c)/(a*cos(1/2*d*x+1/2*c)^2)^(1/2)/d
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^4/(a+a*cos(d*x+c))^(3/2),x, algorithm="maxima")
[Out]
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A}{{\cos \left (c+d\,x\right )}^4\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(cos(c + d*x)^4*(a + a*cos(c + d*x))^(3/2)),x)
[Out]
int((A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(cos(c + d*x)^4*(a + a*cos(c + d*x))^(3/2)), x)
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)**2)*sec(d*x+c)**4/(a+a*cos(d*x+c))**(3/2),x)
[Out]
Timed out
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