3.116 \(\int (a+a \cos (c+d x))^{5/2} \sec (c+d x) \, dx\)
Optimal. Leaf size=98 \[ \frac {2 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{d}+\frac {14 a^3 \sin (c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}+\frac {2 a^2 \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3 d} \]
[Out]
2*a^(5/2)*arctanh(sin(d*x+c)*a^(1/2)/(a+a*cos(d*x+c))^(1/2))/d+14/3*a^3*sin(d*x+c)/d/(a+a*cos(d*x+c))^(1/2)+2/
3*a^2*sin(d*x+c)*(a+a*cos(d*x+c))^(1/2)/d
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Rubi [A] time = 0.20, antiderivative size = 98, normalized size of antiderivative = 1.00,
number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used =
{2763, 2981, 2773, 206} \[ \frac {14 a^3 \sin (c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}+\frac {2 a^2 \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3 d}+\frac {2 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{d} \]
Antiderivative was successfully verified.
[In]
Int[(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x],x]
[Out]
(2*a^(5/2)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (14*a^3*Sin[c + d*x])/(3*d*Sqrt[a + a
*Cos[c + d*x]]) + (2*a^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d)
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 2763
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Si
mp[(b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(m + n)), x] + Dist[1/(d*
(m + n)), Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^n*Simp[a*b*c*(m - 2) + b^2*d*(n + 1) + a^2*d*(
m + n) - b*(b*c*(m - 1) - a*d*(3*m + 2*n - 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&
NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && !LtQ[n, -1] && (IntegersQ[2*m, 2*
n] || IntegerQ[m + 1/2] || (IntegerQ[m] && EqQ[c, 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 2981
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.
) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-2*b*B*Cos[e + f*x]*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(2*n + 3)*Sqr
t[a + b*Sin[e + f*x]]), x] + Dist[(A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)))/(b*d*(2*n + 3)), Int[Sqrt[a + b*
Sin[e + f*x]]*(c + d*Sin[e + f*x])^n, 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[n, -1]
Rubi steps
\begin {align*} \int (a+a \cos (c+d x))^{5/2} \sec (c+d x) \, dx &=\frac {2 a^2 \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{3 d}+\frac {2}{3} \int \sqrt {a+a \cos (c+d x)} \left (\frac {3 a^2}{2}+\frac {7}{2} a^2 \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac {14 a^3 \sin (c+d x)}{3 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{3 d}+a^2 \int \sqrt {a+a \cos (c+d x)} \sec (c+d x) \, dx\\ &=\frac {14 a^3 \sin (c+d x)}{3 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{3 d}-\frac {\left (2 a^3\right ) \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 )}{d}\\ &=\frac {2 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{d}+\frac {14 a^3 \sin (c+d x)}{3 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.51, size = 89, normalized size = 0.91 \[ \frac {2 a^2 \tan \left (\frac {1}{2} (c+d x)\right ) \sqrt {a (\cos (c+d x)+1)} \left (\sqrt {1-\cos (c+d x)} (\cos (c+d x)+8)+3 \tanh ^{-1}\left (\sqrt {1-\cos (c+d x)}\right )\right )}{3 d \sqrt {1-\cos (c+d x)}} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x],x]
[Out]
(2*a^2*Sqrt[a*(1 + Cos[c + d*x])]*(3*ArcTanh[Sqrt[1 - Cos[c + d*x]]] + Sqrt[1 - Cos[c + d*x]]*(8 + Cos[c + d*x
]))*Tan[(c + d*x)/2])/(3*d*Sqrt[1 - Cos[c + d*x]])
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fricas [A] time = 1.22, size = 147, normalized size = 1.50 \[ \frac {3 \, {\left (a^{2} \cos \left (d x + c\right ) + a^{2}\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - 4 \, \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {a} {\left (\cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right ) + 8 \, a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}}\right ) + 4 \, {\left (a^{2} \cos \left (d x + c\right ) + 8 \, a^{2}\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{6 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*cos(d*x+c))^(5/2)*sec(d*x+c),x, algorithm="fricas")
[Out]
1/6*(3*(a^2*cos(d*x + c) + a^2)*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*(a^2*cos(d*x + c) + 8*
a^2)*sqrt(a*cos(d*x + c) + a)*sin(d*x + c))/(d*cos(d*x + c) + d)
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giac [B] time = 56.70, size = 5671, normalized size = 57.87 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*cos(d*x+c))^(5/2)*sec(d*x+c),x, algorithm="giac")
[Out]
1/6*sqrt(2)*sqrt(a)*(3*sqrt(2)*(a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/2*c)^3*tan(1/4*c)^6 - 6*a^2*sgn(cos(1/2*d*
x + 1/2*c))*tan(1/2*c)^3*tan(1/4*c)^5 + 3*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/2*c)^2*tan(1/4*c)^6 - 15*a^2*sgn
(cos(1/2*d*x + 1/2*c))*tan(1/2*c)^3*tan(1/4*c)^4 + 18*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/2*c)^2*tan(1/4*c)^5
- 3*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/2*c)*tan(1/4*c)^6 + 20*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/2*c)^3*tan(
1/4*c)^3 - 45*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/2*c)^2*tan(1/4*c)^4 + 18*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1
/2*c)*tan(1/4*c)^5 - a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*c)^6 + 15*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/2*c)^
3*tan(1/4*c)^2 - 60*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/2*c)^2*tan(1/4*c)^3 + 45*a^2*sgn(cos(1/2*d*x + 1/2*c))
*tan(1/2*c)*tan(1/4*c)^4 - 6*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*c)^5 - 6*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(
1/2*c)^3*tan(1/4*c) + 45*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/2*c)^2*tan(1/4*c)^2 - 60*a^2*sgn(cos(1/2*d*x + 1/
2*c))*tan(1/2*c)*tan(1/4*c)^3 + 15*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*c)^4 - a^2*sgn(cos(1/2*d*x + 1/2*c))*
tan(1/2*c)^3 + 18*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/2*c)^2*tan(1/4*c) - 45*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan
(1/2*c)*tan(1/4*c)^2 + 20*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*c)^3 - 3*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/2
*c)^2 + 18*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/2*c)*tan(1/4*c) - 15*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*c)^2
+ 3*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/2*c) - 6*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*c) + a^2*sgn(cos(1/2*d
*x + 1/2*c)))*log(abs(-2*tan(1/4*d*x + c)*tan(1/2*c)^3 + 6*tan(1/4*d*x + c)*tan(1/2*c)^2 - 2*tan(1/2*c)^3 - 2*
sqrt(2)*(tan(1/2*c)^2 + 1)^(3/2) + 6*tan(1/4*d*x + c)*tan(1/2*c) - 6*tan(1/2*c)^2 - 2*tan(1/4*d*x + c) + 6*tan
(1/2*c) + 2)/abs(-2*tan(1/4*d*x + c)*tan(1/2*c)^3 + 6*tan(1/4*d*x + c)*tan(1/2*c)^2 - 2*tan(1/2*c)^3 + 2*sqrt(
2)*(tan(1/2*c)^2 + 1)^(3/2) + 6*tan(1/4*d*x + c)*tan(1/2*c) - 6*tan(1/2*c)^2 - 2*tan(1/4*d*x + c) + 6*tan(1/2*
c) + 2))/((tan(1/4*c)^6 + 3*tan(1/4*c)^4 + 3*tan(1/4*c)^2 + 1)*(tan(1/2*c)^2 + 1)^(3/2)) + 3*sqrt(2)*(a^2*sgn(
cos(1/2*d*x + 1/2*c))*tan(1/2*c)^3*tan(1/4*c)^6 + 6*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/2*c)^3*tan(1/4*c)^5 -
3*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/2*c)^2*tan(1/4*c)^6 - 15*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/2*c)^3*tan(
1/4*c)^4 + 18*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/2*c)^2*tan(1/4*c)^5 - 3*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/
2*c)*tan(1/4*c)^6 - 20*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/2*c)^3*tan(1/4*c)^3 + 45*a^2*sgn(cos(1/2*d*x + 1/2*
c))*tan(1/2*c)^2*tan(1/4*c)^4 - 18*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/2*c)*tan(1/4*c)^5 + a^2*sgn(cos(1/2*d*x
+ 1/2*c))*tan(1/4*c)^6 + 15*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/2*c)^3*tan(1/4*c)^2 - 60*a^2*sgn(cos(1/2*d*x
+ 1/2*c))*tan(1/2*c)^2*tan(1/4*c)^3 + 45*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/2*c)*tan(1/4*c)^4 - 6*a^2*sgn(cos
(1/2*d*x + 1/2*c))*tan(1/4*c)^5 + 6*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/2*c)^3*tan(1/4*c) - 45*a^2*sgn(cos(1/2
*d*x + 1/2*c))*tan(1/2*c)^2*tan(1/4*c)^2 + 60*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/2*c)*tan(1/4*c)^3 - 15*a^2*s
gn(cos(1/2*d*x + 1/2*c))*tan(1/4*c)^4 - a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/2*c)^3 + 18*a^2*sgn(cos(1/2*d*x +
1/2*c))*tan(1/2*c)^2*tan(1/4*c) - 45*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/2*c)*tan(1/4*c)^2 + 20*a^2*sgn(cos(1/
2*d*x + 1/2*c))*tan(1/4*c)^3 + 3*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/2*c)^2 - 18*a^2*sgn(cos(1/2*d*x + 1/2*c))
*tan(1/2*c)*tan(1/4*c) + 15*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*c)^2 + 3*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1
/2*c) - 6*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*c) - a^2*sgn(cos(1/2*d*x + 1/2*c)))*log(abs(-2*tan(1/4*d*x + c
)*tan(1/2*c)^3 - 6*tan(1/4*d*x + c)*tan(1/2*c)^2 + 2*tan(1/2*c)^3 - 2*sqrt(2)*(tan(1/2*c)^2 + 1)^(3/2) + 6*tan
(1/4*d*x + c)*tan(1/2*c) - 6*tan(1/2*c)^2 + 2*tan(1/4*d*x + c) - 6*tan(1/2*c) + 2)/abs(-2*tan(1/4*d*x + c)*tan
(1/2*c)^3 - 6*tan(1/4*d*x + c)*tan(1/2*c)^2 + 2*tan(1/2*c)^3 + 2*sqrt(2)*(tan(1/2*c)^2 + 1)^(3/2) + 6*tan(1/4*
d*x + c)*tan(1/2*c) - 6*tan(1/2*c)^2 + 2*tan(1/4*d*x + c) - 6*tan(1/2*c) + 2))/((tan(1/4*c)^6 + 3*tan(1/4*c)^4
+ 3*tan(1/4*c)^2 + 1)*(tan(1/2*c)^2 + 1)^(3/2)) - 8*(3*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^5*tan(1
/2*c)^6*tan(1/4*c)^6 - 45*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^5*tan(1/2*c)^6*tan(1/4*c)^4 + 18*a^2*
sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^4*tan(1/2*c)^6*tan(1/4*c)^5 + 63*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(
1/4*d*x + c)^5*tan(1/2*c)^4*tan(1/4*c)^6 - 36*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^4*tan(1/2*c)^5*ta
n(1/4*c)^6 + 14*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)^6*tan(1/4*c)^6 + 45*a^2*sgn(cos(1/
2*d*x + 1/2*c))*tan(1/4*d*x + c)^5*tan(1/2*c)^6*tan(1/4*c)^2 - 60*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x +
c)^4*tan(1/2*c)^6*tan(1/4*c)^3 - 945*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^5*tan(1/2*c)^4*tan(1/4*c)^
4 + 540*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^4*tan(1/2*c)^5*tan(1/4*c)^4 - 210*a^2*sgn(cos(1/2*d*x +
1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)^6*tan(1/4*c)^4 + 378*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^4*t
an(1/2*c)^4*tan(1/4*c)^5 - 288*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)^5*tan(1/4*c)^5 + 10
8*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^2*tan(1/2*c)^6*tan(1/4*c)^5 - 27*a^2*sgn(cos(1/2*d*x + 1/2*c)
)*tan(1/4*d*x + c)^5*tan(1/2*c)^2*tan(1/4*c)^6 + 120*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^4*tan(1/2*
c)^3*tan(1/4*c)^6 + 6*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)^4*tan(1/4*c)^6 + 3*a^2*sgn(c
os(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/2*c)^6*tan(1/4*c)^6 - 3*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x
+ c)^5*tan(1/2*c)^6 + 18*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^4*tan(1/2*c)^6*tan(1/4*c) + 945*a^2*sg
n(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^5*tan(1/2*c)^4*tan(1/4*c)^2 - 540*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1
/4*d*x + c)^4*tan(1/2*c)^5*tan(1/4*c)^2 + 210*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)^6*ta
n(1/4*c)^2 - 1260*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^4*tan(1/2*c)^4*tan(1/4*c)^3 + 960*a^2*sgn(cos
(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)^5*tan(1/4*c)^3 - 360*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*
x + c)^2*tan(1/2*c)^6*tan(1/4*c)^3 + 405*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^5*tan(1/2*c)^2*tan(1/4
*c)^4 - 1800*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^4*tan(1/2*c)^3*tan(1/4*c)^4 - 90*a^2*sgn(cos(1/2*d
*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)^4*tan(1/4*c)^4 - 45*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*
tan(1/2*c)^6*tan(1/4*c)^4 - 162*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^4*tan(1/2*c)^2*tan(1/4*c)^5 + 9
60*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)^3*tan(1/4*c)^5 - 324*a^2*sgn(cos(1/2*d*x + 1/2*
c))*tan(1/4*d*x + c)^2*tan(1/2*c)^4*tan(1/4*c)^5 + 42*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/2*c)^6*tan(1/4*c)^5
+ 9*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^5*tan(1/4*c)^6 - 36*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d
*x + c)^4*tan(1/2*c)*tan(1/4*c)^6 + 66*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)^2*tan(1/4*c
)^6 + 63*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/2*c)^4*tan(1/4*c)^6 - 12*a^2*sgn(cos(1/2*d*x + 1
/2*c))*tan(1/2*c)^5*tan(1/4*c)^6 - 63*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^5*tan(1/2*c)^4 + 36*a^2*s
gn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^4*tan(1/2*c)^5 - 14*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3
*tan(1/2*c)^6 + 378*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^4*tan(1/2*c)^4*tan(1/4*c) - 288*a^2*sgn(cos
(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)^5*tan(1/4*c) + 108*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x
+ c)^2*tan(1/2*c)^6*tan(1/4*c) - 405*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^5*tan(1/2*c)^2*tan(1/4*c)^
2 + 1800*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^4*tan(1/2*c)^3*tan(1/4*c)^2 + 90*a^2*sgn(cos(1/2*d*x +
1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)^4*tan(1/4*c)^2 + 45*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*tan(
1/2*c)^6*tan(1/4*c)^2 + 540*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^4*tan(1/2*c)^2*tan(1/4*c)^3 - 3200*
a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)^3*tan(1/4*c)^3 + 1080*a^2*sgn(cos(1/2*d*x + 1/2*c)
)*tan(1/4*d*x + c)^2*tan(1/2*c)^4*tan(1/4*c)^3 - 140*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/2*c)^6*tan(1/4*c)^3 -
135*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^5*tan(1/4*c)^4 + 540*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4
*d*x + c)^4*tan(1/2*c)*tan(1/4*c)^4 - 990*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)^2*tan(1/
4*c)^4 - 945*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/2*c)^4*tan(1/4*c)^4 + 180*a^2*sgn(cos(1/2*d*
x + 1/2*c))*tan(1/2*c)^5*tan(1/4*c)^4 + 54*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^4*tan(1/4*c)^5 - 288
*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)*tan(1/4*c)^5 + 756*a^2*sgn(cos(1/2*d*x + 1/2*c))*
tan(1/4*d*x + c)^2*tan(1/2*c)^2*tan(1/4*c)^5 + 18*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/2*c)^4*tan(1/4*c)^5 + 10
*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/4*c)^6 - 27*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x
+ c)*tan(1/2*c)^2*tan(1/4*c)^6 + 40*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/2*c)^3*tan(1/4*c)^6 + 27*a^2*sgn(cos(1
/2*d*x + 1/2*c))*tan(1/4*d*x + c)^5*tan(1/2*c)^2 - 120*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^4*tan(1/
2*c)^3 - 6*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)^4 - 3*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan
(1/4*d*x + c)*tan(1/2*c)^6 - 162*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^4*tan(1/2*c)^2*tan(1/4*c) + 96
0*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)^3*tan(1/4*c) - 324*a^2*sgn(cos(1/2*d*x + 1/2*c))
*tan(1/4*d*x + c)^2*tan(1/2*c)^4*tan(1/4*c) + 42*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/2*c)^6*tan(1/4*c) + 135*a
^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^5*tan(1/4*c)^2 - 540*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x +
c)^4*tan(1/2*c)*tan(1/4*c)^2 + 990*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)^2*tan(1/4*c)^2
+ 945*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/2*c)^4*tan(1/4*c)^2 - 180*a^2*sgn(cos(1/2*d*x + 1/
2*c))*tan(1/2*c)^5*tan(1/4*c)^2 - 180*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^4*tan(1/4*c)^3 + 960*a^2*
sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)*tan(1/4*c)^3 - 2520*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(
1/4*d*x + c)^2*tan(1/2*c)^2*tan(1/4*c)^3 - 60*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/2*c)^4*tan(1/4*c)^3 - 150*a^
2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/4*c)^4 + 405*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x +
c)*tan(1/2*c)^2*tan(1/4*c)^4 - 600*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/2*c)^3*tan(1/4*c)^4 + 36*a^2*sgn(cos(1/
2*d*x + 1/2*c))*tan(1/4*d*x + c)^2*tan(1/4*c)^5 + 198*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/2*c)^2*tan(1/4*c)^5
+ 9*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/4*c)^6 - 12*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/2*c)*
tan(1/4*c)^6 - 9*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^5 + 36*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d
*x + c)^4*tan(1/2*c) - 66*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)^2 - 63*a^2*sgn(cos(1/2*d
*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/2*c)^4 + 12*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/2*c)^5 + 54*a^2*sgn(cos(1/
2*d*x + 1/2*c))*tan(1/4*d*x + c)^4*tan(1/4*c) - 288*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c
)*tan(1/4*c) + 756*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^2*tan(1/2*c)^2*tan(1/4*c) + 18*a^2*sgn(cos(1
/2*d*x + 1/2*c))*tan(1/2*c)^4*tan(1/4*c) + 150*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/4*c)^2 -
405*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/2*c)^2*tan(1/4*c)^2 + 600*a^2*sgn(cos(1/2*d*x + 1/2*
c))*tan(1/2*c)^3*tan(1/4*c)^2 - 120*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^2*tan(1/4*c)^3 - 660*a^2*sg
n(cos(1/2*d*x + 1/2*c))*tan(1/2*c)^2*tan(1/4*c)^3 - 135*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/4
*c)^4 + 180*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/2*c)*tan(1/4*c)^4 + 30*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*c
)^5 - 10*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3 + 27*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*
tan(1/2*c)^2 - 40*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/2*c)^3 + 36*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x +
c)^2*tan(1/4*c) + 198*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/2*c)^2*tan(1/4*c) + 135*a^2*sgn(cos(1/2*d*x + 1/2*c)
)*tan(1/4*d*x + c)*tan(1/4*c)^2 - 180*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/2*c)*tan(1/4*c)^2 - 100*a^2*sgn(cos(
1/2*d*x + 1/2*c))*tan(1/4*c)^3 - 9*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*d*x + c) + 12*a^2*sgn(cos(1/2*d*x + 1
/2*c))*tan(1/2*c) + 30*a^2*sgn(cos(1/2*d*x + 1/2*c))*tan(1/4*c))/((tan(1/2*c)^6*tan(1/4*c)^6 + 3*tan(1/2*c)^6*
tan(1/4*c)^4 + 3*tan(1/2*c)^4*tan(1/4*c)^6 + 3*tan(1/2*c)^6*tan(1/4*c)^2 + 9*tan(1/2*c)^4*tan(1/4*c)^4 + 3*tan
(1/2*c)^2*tan(1/4*c)^6 + tan(1/2*c)^6 + 9*tan(1/2*c)^4*tan(1/4*c)^2 + 9*tan(1/2*c)^2*tan(1/4*c)^4 + tan(1/4*c)
^6 + 3*tan(1/2*c)^4 + 9*tan(1/2*c)^2*tan(1/4*c)^2 + 3*tan(1/4*c)^4 + 3*tan(1/2*c)^2 + 3*tan(1/4*c)^2 + 1)*(tan
(1/4*d*x + c)^2 + 1)^3))/d
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maple [B] time = 0.56, size = 244, normalized size = 2.49 \[ \frac {a^{\frac {3}{2}} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (-4 \sqrt {a}\, \sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+18 \sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {a}+3 \ln \left (\frac {4 \sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {a}+4 a \sqrt {2}\, \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+8 a}{2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\sqrt {2}}\right ) a +3 \ln \left (-\frac {4 \left (\sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {a}-a \sqrt {2}\, \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+2 a \right )}{-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\sqrt {2}}\right ) a \right )}{3 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+a*cos(d*x+c))^(5/2)*sec(d*x+c),x)
[Out]
1/3*a^(3/2)*cos(1/2*d*x+1/2*c)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-4*a^(1/2)*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/
2)*sin(1/2*d*x+1/2*c)^2+18*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+3*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))*a+3*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))*a)/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] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*cos(d*x+c))^(5/2)*sec(d*x+c),x, algorithm="maxima")
[Out]
integrate((a*cos(d*x + c) + a)^(5/2)*sec(d*x + c), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{5/2}}{\cos \left (c+d\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + a*cos(c + d*x))^(5/2)/cos(c + d*x),x)
[Out]
int((a + a*cos(c + d*x))^(5/2)/cos(c + d*x), 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+a*cos(d*x+c))**(5/2)*sec(d*x+c),x)
[Out]
Timed out
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