3.181 \(\int \cos ^5(c+d x) (a+a \sec (c+d x))^{5/2} (A+C \sec ^2(c+d x)) \, dx\)
Optimal. Leaf size=245 \[ \frac{a^3 (283 A+400 C) \sin (c+d x)}{128 d \sqrt{a \sec (c+d x)+a}}+\frac{a^{5/2} (283 A+400 C) \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{128 d}+\frac{a^2 (79 A+80 C) \sin (c+d x) \cos ^2(c+d x) \sqrt{a \sec (c+d x)+a}}{240 d}+\frac{a^3 (787 A+1040 C) \sin (c+d x) \cos (c+d x)}{960 d \sqrt{a \sec (c+d x)+a}}+\frac{A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{5/2}}{5 d}+\frac{a A \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{8 d} \]
[Out]
(a^(5/2)*(283*A + 400*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(128*d) + (a^3*(283*A + 400*
C)*Sin[c + d*x])/(128*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(787*A + 1040*C)*Cos[c + d*x]*Sin[c + d*x])/(960*d*Sq
rt[a + a*Sec[c + d*x]]) + (a^2*(79*A + 80*C)*Cos[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(240*d) + (
a*A*Cos[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(8*d) + (A*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(5/
2)*Sin[c + d*x])/(5*d)
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Rubi [A] time = 0.761214, antiderivative size = 245, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used =
{4087, 4017, 4015, 3805, 3774, 203} \[ \frac{a^3 (283 A+400 C) \sin (c+d x)}{128 d \sqrt{a \sec (c+d x)+a}}+\frac{a^{5/2} (283 A+400 C) \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{128 d}+\frac{a^2 (79 A+80 C) \sin (c+d x) \cos ^2(c+d x) \sqrt{a \sec (c+d x)+a}}{240 d}+\frac{a^3 (787 A+1040 C) \sin (c+d x) \cos (c+d x)}{960 d \sqrt{a \sec (c+d x)+a}}+\frac{A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{5/2}}{5 d}+\frac{a A \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{8 d} \]
Antiderivative was successfully verified.
[In]
Int[Cos[c + d*x]^5*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]
[Out]
(a^(5/2)*(283*A + 400*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(128*d) + (a^3*(283*A + 400*
C)*Sin[c + d*x])/(128*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(787*A + 1040*C)*Cos[c + d*x]*Sin[c + d*x])/(960*d*Sq
rt[a + a*Sec[c + d*x]]) + (a^2*(79*A + 80*C)*Cos[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(240*d) + (
a*A*Cos[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(8*d) + (A*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(5/
2)*Sin[c + d*x])/(5*d)
Rule 4087
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b
_.) + (a_))^(m_), x_Symbol] :> Simp[(A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n)/(f*n), x] - Dis
t[1/(b*d*n), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)*Simp[a*A*m - b*(A*(m + n + 1) + C*n)*Csc[e +
f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, C, m}, x] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)] && (LtQ[n, -2
^(-1)] || EqQ[m + n + 1, 0])
Rule 4017
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*
(B_.) + (A_)), x_Symbol] :> Simp[(a*A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^n)/(f*n), x]
- Dist[b/(a*d*n), Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n + 1)*Simp[a*A*(m - n - 1) - b*B*n - (a*
B*n + A*b*(m + n))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2
- b^2, 0] && GtQ[m, 1/2] && LtQ[n, -1]
Rule 4015
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]*(csc[(e_.) + (f_.)*(x_)]*(
B_.) + (A_)), x_Symbol] :> Simp[(A*b^2*Cot[e + f*x]*(d*Csc[e + f*x])^n)/(a*f*n*Sqrt[a + b*Csc[e + f*x]]), x] +
Dist[(A*b*(2*n + 1) + 2*a*B*n)/(2*a*d*n), Int[Sqrt[a + b*Csc[e + f*x]]*(d*Csc[e + f*x])^(n + 1), x], x] /; Fr
eeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && NeQ[A*b*(2*n + 1) + 2*a*B*n, 0] &&
LtQ[n, 0]
Rule 3805
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[(a*Cot[
e + f*x]*(d*Csc[e + f*x])^n)/(f*n*Sqrt[a + b*Csc[e + f*x]]), x] + Dist[(a*(2*n + 1))/(2*b*d*n), Int[Sqrt[a + b
*Csc[e + f*x]]*(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[n, -2
^(-1)] && IntegerQ[2*n]
Rule 3774
Int[Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[(-2*b)/d, Subst[Int[1/(a + x^2), x], x, (b*C
ot[c + d*x])/Sqrt[a + b*Csc[c + d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
Rule 203
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])
Rubi steps
\begin{align*} \int \cos ^5(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac{A \cos ^4(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{5 d}+\frac{\int \cos ^4(c+d x) (a+a \sec (c+d x))^{5/2} \left (\frac{5 a A}{2}+\frac{1}{2} a (3 A+10 C) \sec (c+d x)\right ) \, dx}{5 a}\\ &=\frac{a A \cos ^3(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{8 d}+\frac{A \cos ^4(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{5 d}+\frac{\int \cos ^3(c+d x) (a+a \sec (c+d x))^{3/2} \left (\frac{1}{4} a^2 (79 A+80 C)+\frac{1}{4} a^2 (39 A+80 C) \sec (c+d x)\right ) \, dx}{20 a}\\ &=\frac{a^2 (79 A+80 C) \cos ^2(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{240 d}+\frac{a A \cos ^3(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{8 d}+\frac{A \cos ^4(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{5 d}+\frac{\int \cos ^2(c+d x) \sqrt{a+a \sec (c+d x)} \left (\frac{1}{8} a^3 (787 A+1040 C)+\frac{3}{8} a^3 (157 A+240 C) \sec (c+d x)\right ) \, dx}{60 a}\\ &=\frac{a^3 (787 A+1040 C) \cos (c+d x) \sin (c+d x)}{960 d \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (79 A+80 C) \cos ^2(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{240 d}+\frac{a A \cos ^3(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{8 d}+\frac{A \cos ^4(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{5 d}+\frac{1}{128} \left (a^2 (283 A+400 C)\right ) \int \cos (c+d x) \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{a^3 (283 A+400 C) \sin (c+d x)}{128 d \sqrt{a+a \sec (c+d x)}}+\frac{a^3 (787 A+1040 C) \cos (c+d x) \sin (c+d x)}{960 d \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (79 A+80 C) \cos ^2(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{240 d}+\frac{a A \cos ^3(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{8 d}+\frac{A \cos ^4(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{5 d}+\frac{1}{256} \left (a^2 (283 A+400 C)\right ) \int \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{a^3 (283 A+400 C) \sin (c+d x)}{128 d \sqrt{a+a \sec (c+d x)}}+\frac{a^3 (787 A+1040 C) \cos (c+d x) \sin (c+d x)}{960 d \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (79 A+80 C) \cos ^2(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{240 d}+\frac{a A \cos ^3(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{8 d}+\frac{A \cos ^4(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{5 d}-\frac{\left (a^3 (283 A+400 C)\right ) \operatorname{Subst}\left (\int \frac{1}{a+x^2} \, dx,x,-\frac{a \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{128 d}\\ &=\frac{a^{5/2} (283 A+400 C) \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{128 d}+\frac{a^3 (283 A+400 C) \sin (c+d x)}{128 d \sqrt{a+a \sec (c+d x)}}+\frac{a^3 (787 A+1040 C) \cos (c+d x) \sin (c+d x)}{960 d \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (79 A+80 C) \cos ^2(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{240 d}+\frac{a A \cos ^3(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{8 d}+\frac{A \cos ^4(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 2.47798, size = 160, normalized size = 0.65 \[ \frac{a^2 \sec \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\sec (c+d x)+1)} \left (15 \sqrt{2} (283 A+400 C) \sin ^{-1}\left (\sqrt{2} \sin \left (\frac{1}{2} (c+d x)\right )\right ) \sqrt{\cos (c+d x)}+\left (\sin \left (\frac{3}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right ) ((3874 A+2720 C) \cos (c+d x)+4 (331 A+80 C) \cos (2 (c+d x))+348 A \cos (3 (c+d x))+48 A \cos (4 (c+d x))+5521 A+6320 C)\right )}{3840 d} \]
Antiderivative was successfully verified.
[In]
Integrate[Cos[c + d*x]^5*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]
[Out]
(a^2*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(15*Sqrt[2]*(283*A + 400*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*
Sqrt[Cos[c + d*x]] + (5521*A + 6320*C + (3874*A + 2720*C)*Cos[c + d*x] + 4*(331*A + 80*C)*Cos[2*(c + d*x)] + 3
48*A*Cos[3*(c + d*x)] + 48*A*Cos[4*(c + d*x)])*(-Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2])))/(3840*d)
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Maple [B] time = 0.329, size = 936, normalized size = 3.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(d*x+c)^5*(a+a*sec(d*x+c))^(5/2)*(A+C*sec(d*x+c)^2),x)
[Out]
-1/61440/d*a^2*(4245*A*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*(-2*cos
(d*x+c)/(cos(d*x+c)+1))^(9/2)*2^(1/2)*cos(d*x+c)^4*sin(d*x+c)+6000*C*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d
*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(9/2)*2^(1/2)*cos(d*x+c)^4*sin(d*x+c)+16
980*A*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*(-2*cos(d*x+c)/(cos(d*x+
c)+1))^(9/2)*2^(1/2)*cos(d*x+c)^3*sin(d*x+c)+24000*C*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*
sin(d*x+c)/cos(d*x+c))*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(9/2)*2^(1/2)*cos(d*x+c)^3*sin(d*x+c)+25470*A*arctanh(1/
2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(9/2)*2^(
1/2)*cos(d*x+c)^2*sin(d*x+c)+36000*C*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d
*x+c))*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(9/2)*2^(1/2)*cos(d*x+c)^2*sin(d*x+c)+16980*A*arctanh(1/2*2^(1/2)*(-2*co
s(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(9/2)*2^(1/2)*cos(d*x+c)*
sin(d*x+c)+24000*C*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*(-2*cos(d*x
+c)/(cos(d*x+c)+1))^(9/2)*2^(1/2)*cos(d*x+c)*sin(d*x+c)+4245*A*2^(1/2)*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos
(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(9/2)*sin(d*x+c)+6000*C*2^(1/2)*arctan
h(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(9/2)
*sin(d*x+c)+12288*A*cos(d*x+c)^10+32256*A*cos(d*x+c)^9+27904*A*cos(d*x+c)^8+20480*C*cos(d*x+c)^8+18112*A*cos(d
*x+c)^7+66560*C*cos(d*x+c)^7+45280*A*cos(d*x+c)^6+104960*C*cos(d*x+c)^6-135840*A*cos(d*x+c)^5-192000*C*cos(d*x
+c)^5)*(a*(cos(d*x+c)+1)/cos(d*x+c))^(1/2)/cos(d*x+c)^4/sin(d*x+c)
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^5*(a+a*sec(d*x+c))^(5/2)*(A+C*sec(d*x+c)^2),x, algorithm="maxima")
[Out]
Timed out
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Fricas [A] time = 0.766085, size = 1197, normalized size = 4.89 \begin{align*} \left [\frac{15 \,{\left ({\left (283 \, A + 400 \, C\right )} a^{2} \cos \left (d x + c\right ) +{\left (283 \, A + 400 \, C\right )} a^{2}\right )} \sqrt{-a} \log \left (\frac{2 \, a \cos \left (d x + c\right )^{2} - 2 \, \sqrt{-a} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right ) + 1}\right ) + 2 \,{\left (384 \, A a^{2} \cos \left (d x + c\right )^{5} + 1392 \, A a^{2} \cos \left (d x + c\right )^{4} + 8 \,{\left (283 \, A + 80 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 10 \,{\left (283 \, A + 272 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 15 \,{\left (283 \, A + 400 \, C\right )} a^{2} \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 )}{3840 \,{\left (d \cos \left (d x + c\right ) + d\right )}}, -\frac{15 \,{\left ({\left (283 \, A + 400 \, C\right )} a^{2} \cos \left (d x + c\right ) +{\left (283 \, A + 400 \, C\right )} a^{2}\right )} \sqrt{a} \arctan \left (\frac{\sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt{a} \sin \left (d x + c\right )}\right ) -{\left (384 \, A a^{2} \cos \left (d x + c\right )^{5} + 1392 \, A a^{2} \cos \left (d x + c\right )^{4} + 8 \,{\left (283 \, A + 80 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 10 \,{\left (283 \, A + 272 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 15 \,{\left (283 \, A + 400 \, C\right )} a^{2} \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 )}{1920 \,{\left (d \cos \left (d x + c\right ) + d\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^5*(a+a*sec(d*x+c))^(5/2)*(A+C*sec(d*x+c)^2),x, algorithm="fricas")
[Out]
[1/3840*(15*((283*A + 400*C)*a^2*cos(d*x + c) + (283*A + 400*C)*a^2)*sqrt(-a)*log((2*a*cos(d*x + c)^2 - 2*sqrt
(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) + a*cos(d*x + c) - a)/(cos(d*x + c) + 1
)) + 2*(384*A*a^2*cos(d*x + c)^5 + 1392*A*a^2*cos(d*x + c)^4 + 8*(283*A + 80*C)*a^2*cos(d*x + c)^3 + 10*(283*A
+ 272*C)*a^2*cos(d*x + c)^2 + 15*(283*A + 400*C)*a^2*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*si
n(d*x + c))/(d*cos(d*x + c) + d), -1/1920*(15*((283*A + 400*C)*a^2*cos(d*x + c) + (283*A + 400*C)*a^2)*sqrt(a)
*arctan(sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c))) - (384*A*a^2*cos(d*x + c)
^5 + 1392*A*a^2*cos(d*x + c)^4 + 8*(283*A + 80*C)*a^2*cos(d*x + c)^3 + 10*(283*A + 272*C)*a^2*cos(d*x + c)^2 +
15*(283*A + 400*C)*a^2*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c) +
d)]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)**5*(a+a*sec(d*x+c))**(5/2)*(A+C*sec(d*x+c)**2),x)
[Out]
Timed out
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Giac [B] time = 8.16618, size = 1782, normalized size = 7.27 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^5*(a+a*sec(d*x+c))^(5/2)*(A+C*sec(d*x+c)^2),x, algorithm="giac")
[Out]
-1/3840*(15*(283*A*sqrt(-a)*a^2*sgn(cos(d*x + c)) + 400*C*sqrt(-a)*a^2*sgn(cos(d*x + c)))*log(abs((sqrt(-a)*ta
n(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^2 - a*(2*sqrt(2) + 3))) - 15*(283*A*sqrt(-a)*a^2*sgn
(cos(d*x + c)) + 400*C*sqrt(-a)*a^2*sgn(cos(d*x + c)))*log(abs((sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/
2*d*x + 1/2*c)^2 + a))^2 + a*(2*sqrt(2) - 3))) + 4*sqrt(2)*(4245*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(
1/2*d*x + 1/2*c)^2 + a))^18*A*sqrt(-a)*a^3*sgn(cos(d*x + c)) + 6000*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*t
an(1/2*d*x + 1/2*c)^2 + a))^18*C*sqrt(-a)*a^3*sgn(cos(d*x + c)) - 114615*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt
(-a*tan(1/2*d*x + 1/2*c)^2 + a))^16*A*sqrt(-a)*a^4*sgn(cos(d*x + c)) - 162000*(sqrt(-a)*tan(1/2*d*x + 1/2*c) -
sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^16*C*sqrt(-a)*a^4*sgn(cos(d*x + c)) + 1298820*(sqrt(-a)*tan(1/2*d*x + 1/
2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^14*A*sqrt(-a)*a^5*sgn(cos(d*x + c)) + 1801920*(sqrt(-a)*tan(1/2*d*
x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^14*C*sqrt(-a)*a^5*sgn(cos(d*x + c)) - 6176700*(sqrt(-a)*tan(
1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^12*A*sqrt(-a)*a^6*sgn(cos(d*x + c)) - 9764160*(sqrt(-a
)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^12*C*sqrt(-a)*a^6*sgn(cos(d*x + c)) + 16394598*(
sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^10*A*sqrt(-a)*a^7*sgn(cos(d*x + c)) + 240
60960*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^10*C*sqrt(-a)*a^7*sgn(cos(d*x + c)
) - 14042770*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^8*A*sqrt(-a)*a^8*sgn(cos(d*
x + c)) - 19910240*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^8*C*sqrt(-a)*a^8*sgn(
cos(d*x + c)) + 4791060*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^6*A*sqrt(-a)*a^9
*sgn(cos(d*x + c)) + 7135680*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^6*C*sqrt(-a
)*a^9*sgn(cos(d*x + c)) - 860300*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^4*A*sqr
t(-a)*a^10*sgn(cos(d*x + c)) - 1268800*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^4
*C*sqrt(-a)*a^10*sgn(cos(d*x + c)) + 75885*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a
))^2*A*sqrt(-a)*a^11*sgn(cos(d*x + c)) + 111600*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^
2 + a))^2*C*sqrt(-a)*a^11*sgn(cos(d*x + c)) - 2671*A*sqrt(-a)*a^12*sgn(cos(d*x + c)) - 3920*C*sqrt(-a)*a^12*sg
n(cos(d*x + c)))/((sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^4 - 6*(sqrt(-a)*tan(1/
2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^2*a + a^2)^5)/d