3.2.13 \(\int \frac {(A+B \cos (c+d x)) \sec ^2(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx\) [113]
Optimal. Leaf size=170 \[ -\frac {(3 A-2 B) \tanh ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{a^{3/2} d}+\frac {(9 A-5 B) \tanh ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {a+a \cos (c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {(A-B) \tan (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {(3 A-B) \tan (c+d x)}{2 a d \sqrt {a+a \cos (c+d x)}} \]
[Out]
-(3*A-2*B)*arctanh(sin(d*x+c)*a^(1/2)/(a+a*cos(d*x+c))^(1/2))/a^(3/2)/d+1/4*(9*A-5*B)*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)*tan(d*x+c)/d/(a+a*cos(d*x+c))^(3/2)+1/2*(3*
A-B)*tan(d*x+c)/a/d/(a+a*cos(d*x+c))^(1/2)
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Rubi [A]
time = 0.33, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3057, 3063,
3064, 2728, 212, 2852} \begin {gather*} -\frac {(3 A-2 B) \tanh ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{a^{3/2} d}+\frac {(9 A-5 B) \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 {(3 A-B) \tan (c+d x)}{2 a d \sqrt {a \cos (c+d x)+a}}-\frac {(A-B) \tan (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^2)/(a + a*Cos[c + d*x])^(3/2),x]
[Out]
-(((3*A - 2*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(3/2)*d)) + ((9*A - 5*B)*ArcTanh[(
Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B)*Tan[c + d*x])/(2*d
*(a + a*Cos[c + d*x])^(3/2)) + ((3*A - B)*Tan[c + d*x])/(2*a*d*Sqrt[a + a*Cos[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 2728
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 2852
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 3057
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*(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*
x])^(n + 1)/(a*f*(2*m + 1)*(b*c - a*d))), x] + Dist[1/(a*(2*m + 1)*(b*c - a*d)), Int[(a + b*Sin[e + f*x])^(m +
1)*(c + d*Sin[e + f*x])^n*Simp[B*(a*c*m + b*d*(n + 1)) + A*(b*c*(m + 1) - a*d*(2*m + n + 2)) + d*(A*b - a*B)*
(m + n + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2
- b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)] && !GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c,
0])
Rule 3063
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 3064
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]
Rubi steps
\begin {align*} \int \frac {(A+B \cos (c+d x)) \sec ^2(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx &=-\frac {(A-B) \tan (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {\int \frac {\left (a (3 A-B)-\frac {3}{2} a (A-B) \cos (c+d x)\right ) \sec ^2(c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx}{2 a^2}\\ &=-\frac {(A-B) \tan (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {(3 A-B) \tan (c+d x)}{2 a d \sqrt {a+a \cos (c+d x)}}+\frac {\int \frac {\left (-a^2 (3 A-2 B)+\frac {1}{2} a^2 (3 A-B) \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx}{2 a^3}\\ &=-\frac {(A-B) \tan (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {(3 A-B) \tan (c+d x)}{2 a d \sqrt {a+a \cos (c+d x)}}+\frac {(9 A-5 B) \int \frac {1}{\sqrt {a+a \cos (c+d x)}} \, dx}{4 a}-\frac {(3 A-2 B) \int \sqrt {a+a \cos (c+d x)} \sec (c+d x) \, dx}{2 a^2}\\ &=-\frac {(A-B) \tan (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {(3 A-B) \tan (c+d x)}{2 a d \sqrt {a+a \cos (c+d x)}}-\frac {(9 A-5 B) \text {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 {(3 A-2 B) \text {Subst}\left (\int \frac {1}{a-x^2} \, dx,x,-\frac {a \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{a d}\\ &=-\frac {(3 A-2 B) \tanh ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{a^{3/2} d}+\frac {(9 A-5 B) \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 {(A-B) \tan (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {(3 A-B) \tan (c+d x)}{2 a d \sqrt {a+a \cos (c+d x)}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 6.39, size = 608, normalized size = 3.58 \begin {gather*} \frac {\cos ^3\left (\frac {1}{2} (c+d x)\right ) \left (2 (-9 A+5 B) \log \left (\cos \left (\frac {1}{4} (c+d x)\right )-\sin \left (\frac {1}{4} (c+d x)\right )\right )+2 (9 A-5 B) \log \left (\cos \left (\frac {1}{4} (c+d x)\right )+\sin \left (\frac {1}{4} (c+d x)\right )\right )-2 \sqrt {2} (3 A-2 B) \log \left (\sqrt {2}+2 \sin \left (\frac {1}{2} (c+d x)\right )\right )-\frac {2 i (3 A-2 B) \text {ArcTan}\left (\frac {\cos \left (\frac {1}{4} (c+d x)\right )-\left (-1+\sqrt {2}\right ) \sin \left (\frac {1}{4} (c+d x)\right )}{\left (1+\sqrt {2}\right ) \cos \left (\frac {1}{4} (c+d x)\right )-\sin \left (\frac {1}{4} (c+d x)\right )}\right ) \left (\sqrt {2}-2 \sin \left (\frac {c}{2}\right )\right )}{-1+\sqrt {2} \sin \left (\frac {c}{2}\right )}-\frac {2 i (3 A-2 B) \text {ArcTan}\left (\frac {\cos \left (\frac {1}{4} (c+d x)\right )-\left (1+\sqrt {2}\right ) \sin \left (\frac {1}{4} (c+d x)\right )}{\left (-1+\sqrt {2}\right ) \cos \left (\frac {1}{4} (c+d x)\right )-\sin \left (\frac {1}{4} (c+d x)\right )}\right ) \left (\sqrt {2}-2 \sin \left (\frac {c}{2}\right )\right )}{-1+\sqrt {2} \sin \left (\frac {c}{2}\right )}-\frac {(3 A-2 B) \log \left (2-\sqrt {2} \cos \left (\frac {1}{2} (c+d x)\right )-\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (\sqrt {2}-2 \sin \left (\frac {c}{2}\right )\right )}{-1+\sqrt {2} \sin \left (\frac {c}{2}\right )}-\frac {(3 A-2 B) \log \left (2+\sqrt {2} \cos \left (\frac {1}{2} (c+d x)\right )-\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (\sqrt {2}-2 \sin \left (\frac {c}{2}\right )\right )}{-1+\sqrt {2} \sin \left (\frac {c}{2}\right )}+\frac {A-B}{\left (\cos \left (\frac {1}{4} (c+d x)\right )-\sin \left (\frac {1}{4} (c+d x)\right )\right )^2}+\frac {-A+B}{\left (\cos \left (\frac {1}{4} (c+d x)\right )+\sin \left (\frac {1}{4} (c+d x)\right )\right )^2}+\frac {4 A}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}-\frac {4 A}{\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )}\right )}{2 d (a (1+\cos (c+d x)))^{3/2}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^2)/(a + a*Cos[c + d*x])^(3/2),x]
[Out]
(Cos[(c + d*x)/2]^3*(2*(-9*A + 5*B)*Log[Cos[(c + d*x)/4] - Sin[(c + d*x)/4]] + 2*(9*A - 5*B)*Log[Cos[(c + d*x)
/4] + Sin[(c + d*x)/4]] - 2*Sqrt[2]*(3*A - 2*B)*Log[Sqrt[2] + 2*Sin[(c + d*x)/2]] - ((2*I)*(3*A - 2*B)*ArcTan[
(Cos[(c + d*x)/4] - (-1 + Sqrt[2])*Sin[(c + d*x)/4])/((1 + Sqrt[2])*Cos[(c + d*x)/4] - Sin[(c + d*x)/4])]*(Sqr
t[2] - 2*Sin[c/2]))/(-1 + Sqrt[2]*Sin[c/2]) - ((2*I)*(3*A - 2*B)*ArcTan[(Cos[(c + d*x)/4] - (1 + Sqrt[2])*Sin[
(c + d*x)/4])/((-1 + Sqrt[2])*Cos[(c + d*x)/4] - Sin[(c + d*x)/4])]*(Sqrt[2] - 2*Sin[c/2]))/(-1 + Sqrt[2]*Sin[
c/2]) - ((3*A - 2*B)*Log[2 - Sqrt[2]*Cos[(c + d*x)/2] - Sqrt[2]*Sin[(c + d*x)/2]]*(Sqrt[2] - 2*Sin[c/2]))/(-1
+ Sqrt[2]*Sin[c/2]) - ((3*A - 2*B)*Log[2 + Sqrt[2]*Cos[(c + d*x)/2] - Sqrt[2]*Sin[(c + d*x)/2]]*(Sqrt[2] - 2*S
in[c/2]))/(-1 + Sqrt[2]*Sin[c/2]) + (A - B)/(Cos[(c + d*x)/4] - Sin[(c + d*x)/4])^2 + (-A + B)/(Cos[(c + d*x)/
4] + Sin[(c + d*x)/4])^2 + (4*A)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) - (4*A)/(Cos[(c + d*x)/2] + Sin[(c + d*
x)/2])))/(2*d*(a*(1 + Cos[c + d*x]))^(3/2))
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1050\) vs.
\(2(145)=290\).
time = 0.44, size = 1051, normalized size = 6.18
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result |
size |
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\(\text {Expression too large to display}\) |
\(1051\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int((A+B*cos(d*x+c))*sec(d*x+c)^2/(a+a*cos(d*x+c))^(3/2),x,method=_RETURNVERBOSE)
[Out]
1/2*(sin(1/2*d*x+1/2*c)^2*a)^(1/2)*(18*A*2^(1/2)*ln(2*(2*a^(1/2)*(sin(1/2*d*x+1/2*c)^2*a)^(1/2)+2*a)/cos(1/2*d
*x+1/2*c))*cos(1/2*d*x+1/2*c)^4*a-10*B*2^(1/2)*ln(2*(2*a^(1/2)*(sin(1/2*d*x+1/2*c)^2*a)^(1/2)+2*a)/cos(1/2*d*x
+1/2*c))*cos(1/2*d*x+1/2*c)^4*a-12*A*ln(-4/(2*cos(1/2*d*x+1/2*c)-2^(1/2))*(a*2^(1/2)*cos(1/2*d*x+1/2*c)-a^(1/2
)*2^(1/2)*(sin(1/2*d*x+1/2*c)^2*a)^(1/2)-2*a))*cos(1/2*d*x+1/2*c)^4*a-12*A*ln(4/(2*cos(1/2*d*x+1/2*c)+2^(1/2))
*(a*2^(1/2)*cos(1/2*d*x+1/2*c)+a^(1/2)*2^(1/2)*(sin(1/2*d*x+1/2*c)^2*a)^(1/2)+2*a))*cos(1/2*d*x+1/2*c)^4*a+8*B
*ln(-4/(2*cos(1/2*d*x+1/2*c)-2^(1/2))*(a*2^(1/2)*cos(1/2*d*x+1/2*c)-a^(1/2)*2^(1/2)*(sin(1/2*d*x+1/2*c)^2*a)^(
1/2)-2*a))*cos(1/2*d*x+1/2*c)^4*a+8*B*ln(4/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(a*2^(1/2)*cos(1/2*d*x+1/2*c)+a^(1/2
)*2^(1/2)*(sin(1/2*d*x+1/2*c)^2*a)^(1/2)+2*a))*cos(1/2*d*x+1/2*c)^4*a-9*A*ln(2*(2*a^(1/2)*(sin(1/2*d*x+1/2*c)^
2*a)^(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+5*B*ln(2*(2*a^(1/2)*(sin(1/2*d*x+1/2*c)^2*a
)^(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+6*A*a^(1/2)*2^(1/2)*(sin(1/2*d*x+1/2*c)^2*a)^(
1/2)*cos(1/2*d*x+1/2*c)^2+6*A*ln(-4/(2*cos(1/2*d*x+1/2*c)-2^(1/2))*(a*2^(1/2)*cos(1/2*d*x+1/2*c)-a^(1/2)*2^(1/
2)*(sin(1/2*d*x+1/2*c)^2*a)^(1/2)-2*a))*cos(1/2*d*x+1/2*c)^2*a+6*A*ln(4/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(a*2^(1
/2)*cos(1/2*d*x+1/2*c)+a^(1/2)*2^(1/2)*(sin(1/2*d*x+1/2*c)^2*a)^(1/2)+2*a))*cos(1/2*d*x+1/2*c)^2*a-2*B*a^(1/2)
*2^(1/2)*(sin(1/2*d*x+1/2*c)^2*a)^(1/2)*cos(1/2*d*x+1/2*c)^2-4*B*ln(-4/(2*cos(1/2*d*x+1/2*c)-2^(1/2))*(a*2^(1/
2)*cos(1/2*d*x+1/2*c)-a^(1/2)*2^(1/2)*(sin(1/2*d*x+1/2*c)^2*a)^(1/2)-2*a))*cos(1/2*d*x+1/2*c)^2*a-4*B*ln(4/(2*
cos(1/2*d*x+1/2*c)+2^(1/2))*(a*2^(1/2)*cos(1/2*d*x+1/2*c)+a^(1/2)*2^(1/2)*(sin(1/2*d*x+1/2*c)^2*a)^(1/2)+2*a))
*cos(1/2*d*x+1/2*c)^2*a-A*a^(1/2)*2^(1/2)*(sin(1/2*d*x+1/2*c)^2*a)^(1/2)+B*2^(1/2)*(sin(1/2*d*x+1/2*c)^2*a)^(1
/2)*a^(1/2))/a^(5/2)/cos(1/2*d*x+1/2*c)/(2*cos(1/2*d*x+1/2*c)-2^(1/2))/(2*cos(1/2*d*x+1/2*c)+2^(1/2))/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 [B] Leaf count of result is larger than twice the leaf count of optimal. 47933 vs.
\(2 (145) = 290\).
time = 1.99, size = 47933, normalized size = 281.96 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((A+B*cos(d*x+c))*sec(d*x+c)^2/(a+a*cos(d*x+c))^(3/2),x, algorithm="maxima")
[Out]
1/4*(128*cos(3/2*d*x + 3/2*c)*sin(3*d*x + 3*c)^3 + 1152*cos(3/2*d*x + 3/2*c)*sin(2*d*x + 2*c)^3 - 128*cos(3*d*
x + 3*c)^3*sin(3/2*d*x + 3/2*c) - 1152*cos(2*d*x + 2*c)^3*sin(3/2*d*x + 3/2*c) + 32*(4*cos(3/2*d*x + 3/2*c)*si
n(2*d*x + 2*c) - 9*(3*cos(d*x + c) + 1)*sin(3/2*d*x + 3/2*c) - 28*cos(2*d*x + 2*c)*sin(3/2*d*x + 3/2*c) + 3*co
s(3/2*d*x + 3/2*c)*sin(d*x + c))*cos(3*d*x + 3*c)^2 - 96*(11*(3*cos(d*x + c) + 1)*sin(3/2*d*x + 3/2*c) - 9*cos
(3/2*d*x + 3/2*c)*sin(d*x + c))*cos(2*d*x + 2*c)^2 + 32*(28*cos(3/2*d*x + 3/2*c)*sin(2*d*x + 2*c) - (3*cos(d*x
+ c) + 1)*sin(3/2*d*x + 3/2*c) - 4*cos(3*d*x + 3*c)*sin(3/2*d*x + 3/2*c) - 4*cos(2*d*x + 2*c)*sin(3/2*d*x + 3
/2*c) + 27*cos(3/2*d*x + 3/2*c)*sin(d*x + c))*sin(3*d*x + 3*c)^2 - 288*((3*cos(d*x + c) + 1)*sin(3/2*d*x + 3/2
*c) + 4*cos(2*d*x + 2*c)*sin(3/2*d*x + 3/2*c) - 11*cos(3/2*d*x + 3/2*c)*sin(d*x + c))*sin(2*d*x + 2*c)^2 - 32*
(cos(3*d*x + 3*c)^2*sin(3/2*d*x + 3/2*c) + 6*(3*cos(d*x + c) + 1)*cos(2*d*x + 2*c)*sin(3/2*d*x + 3/2*c) + 9*co
s(2*d*x + 2*c)^2*sin(3/2*d*x + 3/2*c) + sin(3*d*x + 3*c)^2*sin(3/2*d*x + 3/2*c) + 9*sin(2*d*x + 2*c)^2*sin(3/2
*d*x + 3/2*c) + 18*sin(2*d*x + 2*c)*sin(3/2*d*x + 3/2*c)*sin(d*x + c) + 2*((3*cos(d*x + c) + 1)*sin(3/2*d*x +
3/2*c) + 3*cos(2*d*x + 2*c)*sin(3/2*d*x + 3/2*c))*cos(3*d*x + 3*c) + 6*(sin(2*d*x + 2*c)*sin(3/2*d*x + 3/2*c)
+ sin(3/2*d*x + 3/2*c)*sin(d*x + c))*sin(3*d*x + 3*c) + (9*cos(d*x + c)^2 + 9*sin(d*x + c)^2 + 6*cos(d*x + c)
+ 1)*sin(3/2*d*x + 3/2*c))*cos(5*d*x + 5*c) - 96*(cos(3*d*x + 3*c)^2*sin(3/2*d*x + 3/2*c) + 6*(3*cos(d*x + c)
+ 1)*cos(2*d*x + 2*c)*sin(3/2*d*x + 3/2*c) + 9*cos(2*d*x + 2*c)^2*sin(3/2*d*x + 3/2*c) + sin(3*d*x + 3*c)^2*si
n(3/2*d*x + 3/2*c) + 9*sin(2*d*x + 2*c)^2*sin(3/2*d*x + 3/2*c) + 18*sin(2*d*x + 2*c)*sin(3/2*d*x + 3/2*c)*sin(
d*x + c) + 2*((3*cos(d*x + c) + 1)*sin(3/2*d*x + 3/2*c) + 3*cos(2*d*x + 2*c)*sin(3/2*d*x + 3/2*c))*cos(3*d*x +
3*c) + 6*(sin(2*d*x + 2*c)*sin(3/2*d*x + 3/2*c) + sin(3/2*d*x + 3/2*c)*sin(d*x + c))*sin(3*d*x + 3*c) + (9*co
s(d*x + c)^2 + 9*sin(d*x + c)^2 + 6*cos(d*x + c) + 1)*sin(3/2*d*x + 3/2*c))*cos(4*d*x + 4*c) - 64*(30*cos(2*d*
x + 2*c)^2*sin(3/2*d*x + 3/2*c) + 18*sin(2*d*x + 2*c)^2*sin(3/2*d*x + 3/2*c) - 3*(3*cos(d*x + c) + 1)*cos(3/2*
d*x + 3/2*c)*sin(d*x + c) + (19*(3*cos(d*x + c) + 1)*sin(3/2*d*x + 3/2*c) - 9*cos(3/2*d*x + 3/2*c)*sin(d*x + c
))*cos(2*d*x + 2*c) - 4*((3*cos(d*x + c) + 1)*cos(3/2*d*x + 3/2*c) + 3*cos(2*d*x + 2*c)*cos(3/2*d*x + 3/2*c) -
9*sin(3/2*d*x + 3/2*c)*sin(d*x + c))*sin(2*d*x + 2*c) + 3*(9*cos(d*x + c)^2 + 6*sin(d*x + c)^2 + 6*cos(d*x +
c) + 1)*sin(3/2*d*x + 3/2*c))*cos(3*d*x + 3*c) + 64*(9*(3*cos(d*x + c) + 1)*cos(3/2*d*x + 3/2*c)*sin(d*x + c)
- (45*cos(d*x + c)^2 + 18*sin(d*x + c)^2 + 30*cos(d*x + c) + 5)*sin(3/2*d*x + 3/2*c))*cos(2*d*x + 2*c) + 96*(9
*sin(d*x + c)^3 + (9*cos(d*x + c)^2 + 6*cos(d*x + c) + 1)*sin(d*x + c))*cos(3/2*d*x + 3/2*c) - 12*((sin(3*d*x
+ 3*c) + 3*sin(2*d*x + 2*c) + 3*sin(d*x + c))*cos(5*d*x + 5*c)^2 + 9*(sin(3*d*x + 3*c) + 3*sin(2*d*x + 2*c) +
3*sin(d*x + c))*cos(4*d*x + 4*c)^2 + 48*(sin(2*d*x + 2*c) + sin(d*x + c))*cos(3*d*x + 3*c)^2 + (sin(3*d*x + 3*
c) + 3*sin(2*d*x + 2*c) + 3*sin(d*x + c))*sin(5*d*x + 5*c)^2 + 9*(sin(3*d*x + 3*c) + 3*sin(2*d*x + 2*c) + 3*si
n(d*x + c))*sin(4*d*x + 4*c)^2 + 8*(10*sin(2*d*x + 2*c) + 9*sin(d*x + c))*sin(3*d*x + 3*c)^2 + 16*sin(3*d*x +
3*c)^3 + 48*sin(2*d*x + 2*c)^3 + 24*(3*cos(d*x + c) + 1)*cos(2*d*x + 2*c)*sin(d*x + c) + 48*cos(2*d*x + 2*c)^2
*sin(d*x + c) + 120*sin(2*d*x + 2*c)^2*sin(d*x + c) + 27*sin(d*x + c)^3 + 2*(3*(sin(3*d*x + 3*c) + 3*sin(2*d*x
+ 2*c) + 3*sin(d*x + c))*cos(4*d*x + 4*c) + 12*(sin(2*d*x + 2*c) + sin(d*x + c))*cos(3*d*x + 3*c) + (4*cos(3*
d*x + 3*c) + 4*cos(2*d*x + 2*c) + 3*cos(d*x + c) + 1)*sin(3*d*x + 3*c) + 3*(4*cos(2*d*x + 2*c) + 3*cos(d*x + c
) + 1)*sin(2*d*x + 2*c) + 3*(3*cos(d*x + c) + 1)*sin(d*x + c) + 12*cos(2*d*x + 2*c)*sin(d*x + c))*cos(5*d*x +
5*c) + 6*(12*(sin(2*d*x + 2*c) + sin(d*x + c))*cos(3*d*x + 3*c) + (4*cos(3*d*x + 3*c) + 4*cos(2*d*x + 2*c) + 3
*cos(d*x + c) + 1)*sin(3*d*x + 3*c) + 3*(4*cos(2*d*x + 2*c) + 3*cos(d*x + c) + 1)*sin(2*d*x + 2*c) + 3*(3*cos(
d*x + c) + 1)*sin(d*x + c) + 12*cos(2*d*x + 2*c)*sin(d*x + c))*cos(4*d*x + 4*c) + 24*((4*cos(2*d*x + 2*c) + 3*
cos(d*x + c) + 1)*sin(2*d*x + 2*c) + (3*cos(d*x + c) + 1)*sin(d*x + c) + 4*cos(2*d*x + 2*c)*sin(d*x + c))*cos(
3*d*x + 3*c) + 2*(3*(sin(3*d*x + 3*c) + 3*sin(2*d*x + 2*c) + 3*sin(d*x + c))*sin(4*d*x + 4*c) + (16*sin(2*d*x
+ 2*c) + 15*sin(d*x + c))*sin(3*d*x + 3*c) + 4*sin(3*d*x + 3*c)^2 + 12*sin(2*d*x + 2*c)^2 + 21*sin(2*d*x + 2*c
)*sin(d*x + c) + 9*sin(d*x + c)^2)*sin(5*d*x + 5*c) + 6*((16*sin(2*d*x + 2*c) + 15*sin(d*x + c))*sin(3*d*x + 3
*c) + 4*sin(3*d*x + 3*c)^2 + 12*sin(2*d*x + 2*c)^2 + 21*sin(2*d*x + 2*c)*sin(d*x + c) + 9*sin(d*x + c)^2)*sin(
4*d*x + 4*c) + (8*(4*cos(2*d*x + 2*c) + 3*cos(d*x + c) + 1)*cos(3*d*x + 3*c) + 16*cos(3*d*x + 3*c)^2 + 8*(3*co
s(d*x + c) + 1)*cos(2*d*x + 2*c) + 16*cos(2*d*x + 2*c)^2 + 9*cos(d*x + c)^2 + 112*sin(2*d*x + 2*c)^2 + 192*sin
(2*d*x + 2*c)*sin(d*x + c) + 81*sin(d*x + c)^2 ...
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 339 vs.
\(2 (145) = 290\).
time = 0.40, size = 339, normalized size = 1.99 \begin {gather*} -\frac {\sqrt {2} {\left ({\left (9 \, A - 5 \, B\right )} \cos \left (d x + c\right )^{3} + 2 \, {\left (9 \, A - 5 \, B\right )} \cos \left (d x + c\right )^{2} + {\left (9 \, A - 5 \, B\right )} \cos \left (d x + c\right )\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 ) + 2 \, {\left ({\left (3 \, A - 2 \, B\right )} \cos \left (d x + c\right )^{3} + 2 \, {\left (3 \, A - 2 \, B\right )} \cos \left (d x + c\right )^{2} + {\left (3 \, A - 2 \, B\right )} \cos \left (d x + c\right )\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 ({\left (3 \, A - B\right )} \cos \left (d x + c\right ) + 2 \, A\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{8 \, {\left (a^{2} d \cos \left (d x + c\right )^{3} + 2 \, a^{2} d \cos \left (d x + c\right )^{2} + a^{2} d \cos \left (d x + c\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((A+B*cos(d*x+c))*sec(d*x+c)^2/(a+a*cos(d*x+c))^(3/2),x, algorithm="fricas")
[Out]
-1/8*(sqrt(2)*((9*A - 5*B)*cos(d*x + c)^3 + 2*(9*A - 5*B)*cos(d*x + c)^2 + (9*A - 5*B)*cos(d*x + c))*sqrt(a)*l
og(-(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)) + 2*((3*A - 2*B)*cos(d*x + c)^3 + 2*(3*A - 2*B)*cos(d*x + c)^2 + (3*A - 2*B
)*cos(d*x + c))*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 + cos(d*x + c)^2)) - 4*((3*A - B)*cos(d*x + c) + 2*A)*sqrt(a*
cos(d*x + c) + a)*sin(d*x + c))/(a^2*d*cos(d*x + c)^3 + 2*a^2*d*cos(d*x + c)^2 + a^2*d*cos(d*x + c))
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A + B \cos {\left (c + d x \right )}\right ) \sec ^{2}{\left (c + d x \right )}}{\left (a \left (\cos {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((A+B*cos(d*x+c))*sec(d*x+c)**2/(a+a*cos(d*x+c))**(3/2),x)
[Out]
Integral((A + B*cos(c + d*x))*sec(c + d*x)**2/(a*(cos(c + d*x) + 1))**(3/2), x)
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((A+B*cos(d*x+c))*sec(d*x+c)^2/(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,sageVARx):;OUTP
UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch
eck [abs(co
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {A+B\,\cos \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^2\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((A + B*cos(c + d*x))/(cos(c + d*x)^2*(a + a*cos(c + d*x))^(3/2)),x)
[Out]
int((A + B*cos(c + d*x))/(cos(c + d*x)^2*(a + a*cos(c + d*x))^(3/2)), x)
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