3.265 \(\int \frac{\sec ^{\frac{7}{2}}(c+d x)}{\sqrt{1+\sec (c+d x)}} \, dx\)

Optimal. Leaf size=126 \[ \frac{\sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{2 d \sqrt{\sec (c+d x)+1}}-\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 d \sqrt{\sec (c+d x)+1}}-\frac{\sqrt{2} \sinh ^{-1}\left (\frac{\tan (c+d x)}{\sec (c+d x)+1}\right )}{d}+\frac{7 \sinh ^{-1}\left (\frac{\tan (c+d x)}{\sqrt{\sec (c+d x)+1}}\right )}{4 d} \]

[Out]

-((Sqrt[2]*ArcSinh[Tan[c + d*x]/(1 + Sec[c + d*x])])/d) + (7*ArcSinh[Tan[c + d*x]/Sqrt[1 + Sec[c + d*x]]])/(4*
d) - (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*Sqrt[1 + Sec[c + d*x]]) + (Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*d*S
qrt[1 + Sec[c + d*x]])

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Rubi [A]  time = 0.285309, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {3822, 4021, 4023, 3807, 215, 3801} \[ \frac{\sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{2 d \sqrt{\sec (c+d x)+1}}-\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 d \sqrt{\sec (c+d x)+1}}-\frac{\sqrt{2} \sinh ^{-1}\left (\frac{\tan (c+d x)}{\sec (c+d x)+1}\right )}{d}+\frac{7 \sinh ^{-1}\left (\frac{\tan (c+d x)}{\sqrt{\sec (c+d x)+1}}\right )}{4 d} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^(7/2)/Sqrt[1 + Sec[c + d*x]],x]

[Out]

-((Sqrt[2]*ArcSinh[Tan[c + d*x]/(1 + Sec[c + d*x])])/d) + (7*ArcSinh[Tan[c + d*x]/Sqrt[1 + Sec[c + d*x]]])/(4*
d) - (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*Sqrt[1 + Sec[c + d*x]]) + (Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*d*S
qrt[1 + Sec[c + d*x]])

Rule 3822

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[(-2*d^2
*Cot[e + f*x]*(d*Csc[e + f*x])^(n - 2))/(f*(2*n - 3)*Sqrt[a + b*Csc[e + f*x]]), x] + Dist[d^2/(b*(2*n - 3)), I
nt[((d*Csc[e + f*x])^(n - 2)*(2*b*(n - 2) - a*Csc[e + f*x]))/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ[{a, b,
d, e, f}, x] && EqQ[a^2 - b^2, 0] && GtQ[n, 2] && IntegerQ[2*n]

Rule 4021

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

Rule 4023

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

Rule 3807

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

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]

Rubi steps

\begin{align*} \int \frac{\sec ^{\frac{7}{2}}(c+d x)}{\sqrt{1+\sec (c+d x)}} \, dx &=\frac{\sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{2 d \sqrt{1+\sec (c+d x)}}+\frac{1}{4} \int \frac{(3-\sec (c+d x)) \sec ^{\frac{3}{2}}(c+d x)}{\sqrt{1+\sec (c+d x)}} \, dx\\ &=-\frac{\sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{4 d \sqrt{1+\sec (c+d x)}}+\frac{\sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{2 d \sqrt{1+\sec (c+d x)}}+\frac{1}{4} \int \frac{\sqrt{\sec (c+d x)} \left (-\frac{1}{2}+\frac{7}{2} \sec (c+d x)\right )}{\sqrt{1+\sec (c+d x)}} \, dx\\ &=-\frac{\sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{4 d \sqrt{1+\sec (c+d x)}}+\frac{\sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{2 d \sqrt{1+\sec (c+d x)}}+\frac{7}{8} \int \sqrt{\sec (c+d x)} \sqrt{1+\sec (c+d x)} \, dx-\int \frac{\sqrt{\sec (c+d x)}}{\sqrt{1+\sec (c+d x)}} \, dx\\ &=-\frac{\sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{4 d \sqrt{1+\sec (c+d x)}}+\frac{\sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{2 d \sqrt{1+\sec (c+d x)}}-\frac{7 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^2}} \, dx,x,-\frac{\tan (c+d x)}{\sqrt{1+\sec (c+d x)}}\right )}{4 d}+\frac{\sqrt{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^2}} \, dx,x,-\frac{\tan (c+d x)}{1+\sec (c+d x)}\right )}{d}\\ &=-\frac{\sqrt{2} \sinh ^{-1}\left (\frac{\tan (c+d x)}{1+\sec (c+d x)}\right )}{d}+\frac{7 \sinh ^{-1}\left (\frac{\tan (c+d x)}{\sqrt{1+\sec (c+d x)}}\right )}{4 d}-\frac{\sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{4 d \sqrt{1+\sec (c+d x)}}+\frac{\sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{2 d \sqrt{1+\sec (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 0.400099, size = 140, normalized size = 1.11 \[ \frac{\sqrt{-\tan ^2(c+d x)} \cot (c+d x) \left (-2 \sqrt{1-\sec (c+d x)} \sec ^{\frac{3}{2}}(c+d x)+\sqrt{-(\sec (c+d x)-1) \sec (c+d x)}+\sin ^{-1}\left (\sqrt{1-\sec (c+d x)}\right )+8 \sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )-4 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right )\right )}{4 d} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sec[c + d*x]^(7/2)/Sqrt[1 + Sec[c + d*x]],x]

[Out]

(Cot[c + d*x]*(ArcSin[Sqrt[1 - Sec[c + d*x]]] + 8*ArcSin[Sqrt[Sec[c + d*x]]] - 4*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[
Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]] - 2*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(3/2) + Sqrt[-((-1 + Sec[c + d*
x])*Sec[c + d*x])])*Sqrt[-Tan[c + d*x]^2])/(4*d)

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Maple [B]  time = 0.239, size = 255, normalized size = 2. \begin{align*}{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{2}-1 \right ) }{16\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}} \left ( 7\,\sqrt{2}\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 ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}-7\,\sqrt{2}\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 ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}-2\,\sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) -16\,\arctan \left ( 1/2\,\sin \left ( dx+c \right ) \sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}} \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}+4\,\sin \left ( dx+c \right ) \sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}} \right ) \sqrt{{\frac{\cos \left ( dx+c \right ) +1}{\cos \left ( dx+c \right ) }}} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{-1} \right ) ^{{\frac{7}{2}}}\sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^(7/2)/(1+sec(d*x+c))^(1/2),x)

[Out]

1/16/d*(7*2^(1/2)*arctan(1/4*2^(1/2)*(-2/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)+1+sin(d*x+c)))*cos(d*x+c)^2-7*2^(1/
2)*arctan(1/4*2^(1/2)*(-2/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)+1-sin(d*x+c)))*cos(d*x+c)^2-2*(-2/(cos(d*x+c)+1))^
(1/2)*cos(d*x+c)*sin(d*x+c)-16*arctan(1/2*sin(d*x+c)*(-2/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)^2+4*sin(d*x+c)*(-2/
(cos(d*x+c)+1))^(1/2))*((cos(d*x+c)+1)/cos(d*x+c))^(1/2)*(1/cos(d*x+c))^(7/2)*cos(d*x+c)^2*(-2/(cos(d*x+c)+1))
^(1/2)/sin(d*x+c)^2*(cos(d*x+c)^2-1)

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Maxima [B]  time = 2.34053, size = 2218, normalized size = 17.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^(7/2)/(1+sec(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

1/16*(4*(sqrt(2)*sin(4*d*x + 4*c) + 2*sqrt(2)*sin(2*d*x + 2*c))*cos(7/2*arctan2(sin(d*x + c), cos(d*x + c))) -
20*(sqrt(2)*sin(4*d*x + 4*c) + 2*sqrt(2)*sin(2*d*x + 2*c))*cos(5/2*arctan2(sin(d*x + c), cos(d*x + c))) + 20*
(sqrt(2)*sin(4*d*x + 4*c) + 2*sqrt(2)*sin(2*d*x + 2*c))*cos(3/2*arctan2(sin(d*x + c), cos(d*x + c))) - 4*(sqrt
(2)*sin(4*d*x + 4*c) + 2*sqrt(2)*sin(2*d*x + 2*c))*cos(1/2*arctan2(sin(d*x + c), cos(d*x + c))) + 7*(2*(2*cos(
2*d*x + 2*c) + 1)*cos(4*d*x + 4*c) + cos(4*d*x + 4*c)^2 + 4*cos(2*d*x + 2*c)^2 + sin(4*d*x + 4*c)^2 + 4*sin(4*
d*x + 4*c)*sin(2*d*x + 2*c) + 4*sin(2*d*x + 2*c)^2 + 4*cos(2*d*x + 2*c) + 1)*log(2*cos(1/2*arctan2(sin(d*x + c
), cos(d*x + c)))^2 + 2*sin(1/2*arctan2(sin(d*x + c), cos(d*x + c)))^2 + 2*sqrt(2)*cos(1/2*arctan2(sin(d*x + c
), cos(d*x + c))) + 2*sqrt(2)*sin(1/2*arctan2(sin(d*x + c), cos(d*x + c))) + 2) - 7*(2*(2*cos(2*d*x + 2*c) + 1
)*cos(4*d*x + 4*c) + cos(4*d*x + 4*c)^2 + 4*cos(2*d*x + 2*c)^2 + sin(4*d*x + 4*c)^2 + 4*sin(4*d*x + 4*c)*sin(2
*d*x + 2*c) + 4*sin(2*d*x + 2*c)^2 + 4*cos(2*d*x + 2*c) + 1)*log(2*cos(1/2*arctan2(sin(d*x + c), cos(d*x + c))
)^2 + 2*sin(1/2*arctan2(sin(d*x + c), cos(d*x + c)))^2 + 2*sqrt(2)*cos(1/2*arctan2(sin(d*x + c), cos(d*x + c))
) - 2*sqrt(2)*sin(1/2*arctan2(sin(d*x + c), cos(d*x + c))) + 2) + 7*(2*(2*cos(2*d*x + 2*c) + 1)*cos(4*d*x + 4*
c) + cos(4*d*x + 4*c)^2 + 4*cos(2*d*x + 2*c)^2 + sin(4*d*x + 4*c)^2 + 4*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 4*
sin(2*d*x + 2*c)^2 + 4*cos(2*d*x + 2*c) + 1)*log(2*cos(1/2*arctan2(sin(d*x + c), cos(d*x + c)))^2 + 2*sin(1/2*
arctan2(sin(d*x + c), cos(d*x + c)))^2 - 2*sqrt(2)*cos(1/2*arctan2(sin(d*x + c), cos(d*x + c))) + 2*sqrt(2)*si
n(1/2*arctan2(sin(d*x + c), cos(d*x + c))) + 2) - 7*(2*(2*cos(2*d*x + 2*c) + 1)*cos(4*d*x + 4*c) + cos(4*d*x +
4*c)^2 + 4*cos(2*d*x + 2*c)^2 + sin(4*d*x + 4*c)^2 + 4*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 4*sin(2*d*x + 2*c)
^2 + 4*cos(2*d*x + 2*c) + 1)*log(2*cos(1/2*arctan2(sin(d*x + c), cos(d*x + c)))^2 + 2*sin(1/2*arctan2(sin(d*x
+ c), cos(d*x + c)))^2 - 2*sqrt(2)*cos(1/2*arctan2(sin(d*x + c), cos(d*x + c))) - 2*sqrt(2)*sin(1/2*arctan2(si
n(d*x + c), cos(d*x + c))) + 2) - 8*(sqrt(2)*cos(4*d*x + 4*c)^2 + 4*sqrt(2)*cos(2*d*x + 2*c)^2 + sqrt(2)*sin(4
*d*x + 4*c)^2 + 4*sqrt(2)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 4*sqrt(2)*sin(2*d*x + 2*c)^2 + 2*(2*sqrt(2)*cos(
2*d*x + 2*c) + sqrt(2))*cos(4*d*x + 4*c) + 4*sqrt(2)*cos(2*d*x + 2*c) + sqrt(2))*log(cos(1/2*arctan2(sin(d*x +
c), cos(d*x + c)))^2 + sin(1/2*arctan2(sin(d*x + c), cos(d*x + c)))^2 + 2*sin(1/2*arctan2(sin(d*x + c), cos(d
*x + c))) + 1) + 8*(sqrt(2)*cos(4*d*x + 4*c)^2 + 4*sqrt(2)*cos(2*d*x + 2*c)^2 + sqrt(2)*sin(4*d*x + 4*c)^2 + 4
*sqrt(2)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 4*sqrt(2)*sin(2*d*x + 2*c)^2 + 2*(2*sqrt(2)*cos(2*d*x + 2*c) + sq
rt(2))*cos(4*d*x + 4*c) + 4*sqrt(2)*cos(2*d*x + 2*c) + sqrt(2))*log(cos(1/2*arctan2(sin(d*x + c), cos(d*x + c)
))^2 + sin(1/2*arctan2(sin(d*x + c), cos(d*x + c)))^2 - 2*sin(1/2*arctan2(sin(d*x + c), cos(d*x + c))) + 1) -
4*(sqrt(2)*cos(4*d*x + 4*c) + 2*sqrt(2)*cos(2*d*x + 2*c) + sqrt(2))*sin(7/2*arctan2(sin(d*x + c), cos(d*x + c)
)) + 20*(sqrt(2)*cos(4*d*x + 4*c) + 2*sqrt(2)*cos(2*d*x + 2*c) + sqrt(2))*sin(5/2*arctan2(sin(d*x + c), cos(d*
x + c))) - 20*(sqrt(2)*cos(4*d*x + 4*c) + 2*sqrt(2)*cos(2*d*x + 2*c) + sqrt(2))*sin(3/2*arctan2(sin(d*x + c),
cos(d*x + c))) + 4*(sqrt(2)*cos(4*d*x + 4*c) + 2*sqrt(2)*cos(2*d*x + 2*c) + sqrt(2))*sin(1/2*arctan2(sin(d*x +
c), cos(d*x + c))))/((2*(2*cos(2*d*x + 2*c) + 1)*cos(4*d*x + 4*c) + cos(4*d*x + 4*c)^2 + 4*cos(2*d*x + 2*c)^2
+ sin(4*d*x + 4*c)^2 + 4*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 4*sin(2*d*x + 2*c)^2 + 4*cos(2*d*x + 2*c) + 1)*d
)

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Fricas [B]  time = 2.08105, size = 936, normalized size = 7.43 \begin{align*} \frac{8 \,{\left (\sqrt{2} \cos \left (d x + c\right )^{2} + \sqrt{2} \cos \left (d x + c\right )\right )} \log \left (-\frac{2 \, \sqrt{2} \sqrt{\frac{\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right )}} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right ) - 3}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right ) - 7 \,{\left (\cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \log \left (-\frac{\cos \left (d x + c\right )^{2} + 2 \, \sqrt{\frac{\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right )}} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}{\cos \left (d x + c\right ) + 1}\right ) + 7 \,{\left (\cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \log \left (-\frac{\cos \left (d x + c\right )^{2} - 2 \, \sqrt{\frac{\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right )}} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}{\cos \left (d x + c\right ) + 1}\right ) - \frac{4 \, \sqrt{\frac{\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right )}}{\left (\cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right )}{\sqrt{\cos \left (d x + c\right )}}}{16 \,{\left (d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^(7/2)/(1+sec(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

1/16*(8*(sqrt(2)*cos(d*x + c)^2 + sqrt(2)*cos(d*x + c))*log(-(2*sqrt(2)*sqrt((cos(d*x + c) + 1)/cos(d*x + c))*
sqrt(cos(d*x + c))*sin(d*x + c) + cos(d*x + c)^2 - 2*cos(d*x + c) - 3)/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1))
- 7*(cos(d*x + c)^2 + cos(d*x + c))*log(-(cos(d*x + c)^2 + 2*sqrt((cos(d*x + c) + 1)/cos(d*x + c))*sqrt(cos(d*
x + c))*sin(d*x + c) - cos(d*x + c) - 2)/(cos(d*x + c) + 1)) + 7*(cos(d*x + c)^2 + cos(d*x + c))*log(-(cos(d*x
+ c)^2 - 2*sqrt((cos(d*x + c) + 1)/cos(d*x + c))*sqrt(cos(d*x + c))*sin(d*x + c) - cos(d*x + c) - 2)/(cos(d*x
+ c) + 1)) - 4*sqrt((cos(d*x + c) + 1)/cos(d*x + c))*(cos(d*x + c) - 2)*sin(d*x + c)/sqrt(cos(d*x + c)))/(d*c
os(d*x + c)^2 + d*cos(d*x + c))

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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(sec(d*x+c)**(7/2)/(1+sec(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 \frac{\sec \left (d x + c\right )^{\frac{7}{2}}}{\sqrt{\sec \left (d x + c\right ) + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^(7/2)/(1+sec(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate(sec(d*x + c)^(7/2)/sqrt(sec(d*x + c) + 1), x)