3.1150 \(\int \sqrt{\cos (c+d x)} (a+a \sec (c+d x))^{5/2} (A+C \sec ^2(c+d x)) \, dx\)

Optimal. Leaf size=238 \[ \frac{a^3 (24 A-49 C) \sin (c+d x)}{24 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (24 A+31 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{24 d \sqrt{\cos (c+d x)}}+\frac{5 a^{5/2} (8 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{8 d}+\frac{5 a C \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{12 d \sqrt{\cos (c+d x)}}+\frac{C \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{3 d \sqrt{\cos (c+d x)}} \]

[Out]

(5*a^(5/2)*(8*A + 5*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c
+ d*x]])/(8*d) + (a^3*(24*A - 49*C)*Sin[c + d*x])/(24*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(2
4*A + 31*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(24*d*Sqrt[Cos[c + d*x]]) + (5*a*C*(a + a*Sec[c + d*x])^(3/
2)*Sin[c + d*x])/(12*d*Sqrt[Cos[c + d*x]]) + (C*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x
]])

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Rubi [A]  time = 0.792775, antiderivative size = 238, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.162, Rules used = {4265, 4089, 4018, 4015, 3801, 215} \[ \frac{a^3 (24 A-49 C) \sin (c+d x)}{24 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (24 A+31 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{24 d \sqrt{\cos (c+d x)}}+\frac{5 a^{5/2} (8 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{8 d}+\frac{5 a C \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{12 d \sqrt{\cos (c+d x)}}+\frac{C \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{3 d \sqrt{\cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*a^(5/2)*(8*A + 5*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c
+ d*x]])/(8*d) + (a^3*(24*A - 49*C)*Sin[c + d*x])/(24*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(2
4*A + 31*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(24*d*Sqrt[Cos[c + d*x]]) + (5*a*C*(a + a*Sec[c + d*x])^(3/
2)*Sin[c + d*x])/(12*d*Sqrt[Cos[c + d*x]]) + (C*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x
]])

Rule 4265

Int[(cos[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Dist[(c*Cos[a + b*x])^m*(c*Sec[a + b*x])^m, Int[A
ctivateTrig[u]/(c*Sec[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[m] && KnownSecantIntegrandQ[
u, x]

Rule 4089

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

Rule 4018

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*
(B_.) + (A_)), x_Symbol] :> -Simp[(b*B*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^n)/(f*(m + n
)), x] + Dist[1/(d*(m + n)), Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^n*Simp[a*A*d*(m + n) + B*(b*d*n
) + (A*b*d*(m + n) + a*B*d*(2*m + n - 1))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] && Ne
Q[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 3801

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

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

\begin{align*} \int \sqrt{\cos (c+d x)} (a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right )}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \sec (c+d x))^{5/2} \left (\frac{1}{2} a (6 A-C)+\frac{5}{2} a C \sec (c+d x)\right )}{\sqrt{\sec (c+d x)}} \, dx}{3 a}\\ &=\frac{5 a C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \sqrt{\cos (c+d x)}}+\frac{C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \sec (c+d x))^{3/2} \left (\frac{3}{4} a^2 (8 A-3 C)+\frac{1}{4} a^2 (24 A+31 C) \sec (c+d x)\right )}{\sqrt{\sec (c+d x)}} \, dx}{6 a}\\ &=\frac{a^2 (24 A+31 C) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{24 d \sqrt{\cos (c+d x)}}+\frac{5 a C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \sqrt{\cos (c+d x)}}+\frac{C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+a \sec (c+d x)} \left (\frac{1}{8} a^3 (24 A-49 C)+\frac{15}{8} a^3 (8 A+5 C) \sec (c+d x)\right )}{\sqrt{\sec (c+d x)}} \, dx}{6 a}\\ &=\frac{a^3 (24 A-49 C) \sin (c+d x)}{24 d \sqrt{\cos (c+d x)} \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (24 A+31 C) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{24 d \sqrt{\cos (c+d x)}}+\frac{5 a C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \sqrt{\cos (c+d x)}}+\frac{C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}}+\frac{1}{16} \left (5 a^2 (8 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{a^3 (24 A-49 C) \sin (c+d x)}{24 d \sqrt{\cos (c+d x)} \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (24 A+31 C) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{24 d \sqrt{\cos (c+d x)}}+\frac{5 a C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \sqrt{\cos (c+d x)}}+\frac{C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}}-\frac{\left (5 a^2 (8 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{a}}} \, dx,x,-\frac{a \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{8 d}\\ &=\frac{5 a^{5/2} (8 A+5 C) \sinh ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}}{8 d}+\frac{a^3 (24 A-49 C) \sin (c+d x)}{24 d \sqrt{\cos (c+d x)} \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (24 A+31 C) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{24 d \sqrt{\cos (c+d x)}}+\frac{5 a C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \sqrt{\cos (c+d x)}}+\frac{C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 2.38492, size = 144, normalized size = 0.61 \[ \frac{a^2 \sec \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\sec (c+d x)+1)} \left (\sin \left (\frac{1}{2} (c+d x)\right ) ((72 A+68 C) \cos (c+d x)+3 (8 A+25 C) \cos (2 (c+d x))+24 A \cos (3 (c+d x))+24 A+91 C)+15 \sqrt{2} (8 A+5 C) \cos ^3(c+d x) \tanh ^{-1}\left (\sqrt{2} \sin \left (\frac{1}{2} (c+d x)\right )\right )\right )}{48 d \cos ^{\frac{5}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[Cos[c + d*x]]*(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]*(8*A + 5*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos
[c + d*x]^3 + (24*A + 91*C + (72*A + 68*C)*Cos[c + d*x] + 3*(8*A + 25*C)*Cos[2*(c + d*x)] + 24*A*Cos[3*(c + d*
x)])*Sin[(c + d*x)/2]))/(48*d*Cos[c + d*x]^(5/2))

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Maple [B]  time = 0.309, size = 409, normalized size = 1.7 \begin{align*} -{\frac{{a}^{2} \left ( -1+\cos \left ( dx+c \right ) \right ) }{48\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}\sqrt{{\frac{a \left ( \cos \left ( dx+c \right ) +1 \right ) }{\cos \left ( dx+c \right ) }}} \left ( 120\,A \left ( \cos \left ( dx+c \right ) \right ) ^{3}\arctan \left ( 1/4\,\sqrt{2}\sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}} \left ( \cos \left ( dx+c \right ) +1+\sin \left ( dx+c \right ) \right ) \right ) \sqrt{2}-120\,A \left ( \cos \left ( dx+c \right ) \right ) ^{3}\arctan \left ( 1/4\,\sqrt{2}\sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}} \left ( \cos \left ( dx+c \right ) +1-\sin \left ( dx+c \right ) \right ) \right ) \sqrt{2}+96\,A\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}+75\,C \left ( \cos \left ( dx+c \right ) \right ) ^{3}\arctan \left ( 1/4\,\sqrt{2}\sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}} \left ( \cos \left ( dx+c \right ) +1+\sin \left ( dx+c \right ) \right ) \right ) \sqrt{2}-75\,C \left ( \cos \left ( dx+c \right ) \right ) ^{3}\arctan \left ( 1/4\,\sqrt{2}\sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}} \left ( \cos \left ( dx+c \right ) +1-\sin \left ( dx+c \right ) \right ) \right ) \sqrt{2}+48\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) \sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}+150\,C \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sin \left ( dx+c \right ) +68\,C\sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +16\,C\sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sin \left ( dx+c \right ) \right ){\frac{1}{\sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}}} \left ( \cos \left ( dx+c \right ) \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/48/d*a^2*(a*(cos(d*x+c)+1)/cos(d*x+c))^(1/2)*(-1+cos(d*x+c))*(120*A*cos(d*x+c)^3*arctan(1/4*2^(1/2)*(-2/(co
s(d*x+c)+1))^(1/2)*(cos(d*x+c)+1+sin(d*x+c)))*2^(1/2)-120*A*cos(d*x+c)^3*arctan(1/4*2^(1/2)*(-2/(cos(d*x+c)+1)
)^(1/2)*(cos(d*x+c)+1-sin(d*x+c)))*2^(1/2)+96*A*sin(d*x+c)*cos(d*x+c)^3*(-2/(cos(d*x+c)+1))^(1/2)+75*C*cos(d*x
+c)^3*arctan(1/4*2^(1/2)*(-2/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)+1+sin(d*x+c)))*2^(1/2)-75*C*cos(d*x+c)^3*arctan
(1/4*2^(1/2)*(-2/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)+1-sin(d*x+c)))*2^(1/2)+48*A*cos(d*x+c)^2*sin(d*x+c)*(-2/(co
s(d*x+c)+1))^(1/2)+150*C*cos(d*x+c)^2*(-2/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)+68*C*(-2/(cos(d*x+c)+1))^(1/2)*cos(
d*x+c)*sin(d*x+c)+16*C*(-2/(cos(d*x+c)+1))^(1/2)*sin(d*x+c))/(-2/(cos(d*x+c)+1))^(1/2)/cos(d*x+c)^(5/2)/sin(d*
x+c)^2

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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((a+a*sec(d*x+c))^(5/2)*(A+C*sec(d*x+c)^2)*cos(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [A]  time = 0.716152, size = 1237, normalized size = 5.2 \begin{align*} \left [\frac{4 \,{\left (48 \, A a^{2} \cos \left (d x + c\right )^{3} + 3 \,{\left (8 \, A + 25 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 34 \, C a^{2} \cos \left (d x + c\right ) + 8 \, C a^{2}\right )} \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 ) + 15 \,{\left ({\left (8 \, A + 5 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} +{\left (8 \, A + 5 \, C\right )} a^{2} \cos \left (d x + c\right )^{3}\right )} \sqrt{a} \log \left (\frac{a \cos \left (d x + c\right )^{3} - 4 \, \sqrt{a} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}{\left (\cos \left (d x + c\right ) - 2\right )} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 7 \, a \cos \left (d x + c\right )^{2} + 8 \, a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}}\right )}{96 \,{\left (d \cos \left (d x + c\right )^{4} + d \cos \left (d x + c\right )^{3}\right )}}, \frac{2 \,{\left (48 \, A a^{2} \cos \left (d x + c\right )^{3} + 3 \,{\left (8 \, A + 25 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 34 \, C a^{2} \cos \left (d x + c\right ) + 8 \, C a^{2}\right )} \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 ) + 15 \,{\left ({\left (8 \, A + 5 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} +{\left (8 \, A + 5 \, C\right )} a^{2} \cos \left (d x + c\right )^{3}\right )} \sqrt{-a} \arctan \left (\frac{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 )}{a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) - 2 \, a}\right )}{48 \,{\left (d \cos \left (d x + c\right )^{4} + d \cos \left (d x + c\right )^{3}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^(5/2)*(A+C*sec(d*x+c)^2)*cos(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

[1/96*(4*(48*A*a^2*cos(d*x + c)^3 + 3*(8*A + 25*C)*a^2*cos(d*x + c)^2 + 34*C*a^2*cos(d*x + c) + 8*C*a^2)*sqrt(
(a*cos(d*x + c) + a)/cos(d*x + c))*sqrt(cos(d*x + c))*sin(d*x + c) + 15*((8*A + 5*C)*a^2*cos(d*x + c)^4 + (8*A
+ 5*C)*a^2*cos(d*x + c)^3)*sqrt(a)*log((a*cos(d*x + c)^3 - 4*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*
(cos(d*x + c) - 2)*sqrt(cos(d*x + c))*sin(d*x + c) - 7*a*cos(d*x + c)^2 + 8*a)/(cos(d*x + c)^3 + cos(d*x + c)^
2)))/(d*cos(d*x + c)^4 + d*cos(d*x + c)^3), 1/48*(2*(48*A*a^2*cos(d*x + c)^3 + 3*(8*A + 25*C)*a^2*cos(d*x + c)
^2 + 34*C*a^2*cos(d*x + c) + 8*C*a^2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sqrt(cos(d*x + c))*sin(d*x + c)
+ 15*((8*A + 5*C)*a^2*cos(d*x + c)^4 + (8*A + 5*C)*a^2*cos(d*x + c)^3)*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(
d*x + c) + a)/cos(d*x + c))*sqrt(cos(d*x + c))*sin(d*x + c)/(a*cos(d*x + c)^2 - a*cos(d*x + c) - 2*a)))/(d*cos
(d*x + c)^4 + d*cos(d*x + c)^3)]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \sec \left (d x + c\right )^{2} + A\right )}{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \sqrt{\cos \left (d x + c\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^(5/2)*(A+C*sec(d*x+c)^2)*cos(d*x+c)^(1/2),x, algorithm="giac")

[Out]

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