3.358 \(\int \frac{a+a \sec (c+d x)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx\)

Optimal. Leaf size=135 \[ \frac{10 a \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{21 d}-\frac{6 a E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{10 a \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{6 a \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}} \]

[Out]

(-6*a*EllipticE[(c + d*x)/2, 2])/(5*d) + (10*a*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a*Sin[c + d*x])/(7*d*Cos
[c + d*x]^(7/2)) + (2*a*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (10*a*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2))
+ (6*a*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])

________________________________________________________________________________________

Rubi [A]  time = 0.101516, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {4225, 2748, 2636, 2641, 2639} \[ \frac{10 a F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}-\frac{6 a E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{10 a \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{6 a \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*a*EllipticE[(c + d*x)/2, 2])/(5*d) + (10*a*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a*Sin[c + d*x])/(7*d*Cos
[c + d*x]^(7/2)) + (2*a*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (10*a*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2))
+ (6*a*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])

Rule 4225

Int[(csc[(a_.) + (b_.)*(x_)]*(B_.) + (A_))*(u_), x_Symbol] :> Int[(ActivateTrig[u]*(B + A*Sin[a + b*x]))/Sin[a
+ b*x], x] /; FreeQ[{a, b, A, B}, x] && KnownSineIntegrandQ[u, x]

Rule 2748

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*S
in[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]

Rule 2636

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1))/(b*d*(n +
1)), x] + Dist[(n + 2)/(b^2*(n + 1)), Int[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1
] && IntegerQ[2*n]

Rule 2641

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{c, d}, x]

Rule 2639

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{
c, d}, x]

Rubi steps

\begin{align*} \int \frac{a+a \sec (c+d x)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx &=\int \frac{a+a \cos (c+d x)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx\\ &=a \int \frac{1}{\cos ^{\frac{9}{2}}(c+d x)} \, dx+a \int \frac{1}{\cos ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{2 a \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 a \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{1}{5} (3 a) \int \frac{1}{\cos ^{\frac{3}{2}}(c+d x)} \, dx+\frac{1}{7} (5 a) \int \frac{1}{\cos ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{2 a \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 a \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{10 a \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{6 a \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{1}{21} (5 a) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx-\frac{1}{5} (3 a) \int \sqrt{\cos (c+d x)} \, dx\\ &=-\frac{6 a E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{10 a F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 a \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 a \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{10 a \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{6 a \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}\\ \end{align*}

Mathematica [C]  time = 4.68017, size = 294, normalized size = 2.18 \[ \frac{a (\cos (c+d x)+1) \sec ^2\left (\frac{1}{2} (c+d x)\right ) \left (-\frac{126 \sec (c) \cos ^3(c+d x) \left (\csc (c) \sqrt{\sin ^2\left (\tan ^{-1}(\tan (c))+d x\right )} \left (3 \cos \left (c-\tan ^{-1}(\tan (c))-d x\right )+\cos \left (c+\tan ^{-1}(\tan (c))+d x\right )\right )-2 \sin \left (\tan ^{-1}(\tan (c))+d x\right ) \text{HypergeometricPFQ}\left (\left \{-\frac{1}{2},-\frac{1}{4}\right \},\left \{\frac{3}{4}\right \},\cos ^2\left (\tan ^{-1}(\tan (c))+d x\right )\right )\right )}{\sqrt{\sec ^2(c)} \sqrt{\sin ^2\left (\tan ^{-1}(\tan (c))+d x\right )}}-200 \sin (c) \sqrt{\csc ^2(c)} \cos ^4(c+d x) \sqrt{\cos ^2\left (d x-\tan ^{-1}(\cot (c))\right )} \sec \left (d x-\tan ^{-1}(\cot (c))\right ) \text{HypergeometricPFQ}\left (\left \{\frac{1}{4},\frac{1}{2}\right \},\left \{\frac{5}{4}\right \},\sin ^2\left (d x-\tan ^{-1}(\cot (c))\right )\right )+\csc (c) (-85 \cos (2 c+d x)+231 \cos (c+2 d x)+21 \cos (3 c+2 d x)+25 \cos (2 c+3 d x)-25 \cos (4 c+3 d x)+63 \cos (3 c+4 d x)+189 \cos (c)+85 \cos (d x))\right )}{840 d \cos ^{\frac{7}{2}}(c+d x)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a*(1 + Cos[c + d*x])*Sec[(c + d*x)/2]^2*((189*Cos[c] + 85*Cos[d*x] - 85*Cos[2*c + d*x] + 231*Cos[c + 2*d*x] +
21*Cos[3*c + 2*d*x] + 25*Cos[2*c + 3*d*x] - 25*Cos[4*c + 3*d*x] + 63*Cos[3*c + 4*d*x])*Csc[c] - 200*Cos[c + d
*x]^4*Sqrt[Cos[d*x - ArcTan[Cot[c]]]^2]*Sqrt[Csc[c]^2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[C
ot[c]]]^2]*Sec[d*x - ArcTan[Cot[c]]]*Sin[c] - (126*Cos[c + d*x]^3*Sec[c]*(-2*HypergeometricPFQ[{-1/2, -1/4}, {
3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]] + (3*Cos[c - d*x - ArcTan[Tan[c]]] + Cos[c + d*x
+ ArcTan[Tan[c]]])*Csc[c]*Sqrt[Sin[d*x + ArcTan[Tan[c]]]^2]))/(Sqrt[Sec[c]^2]*Sqrt[Sin[d*x + ArcTan[Tan[c]]]^2
])))/(840*d*Cos[c + d*x]^(7/2))

________________________________________________________________________________________

Maple [B]  time = 2.555, size = 437, normalized size = 3.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*a*(-1/40*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c
)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^3-3/5*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)/(-(-2
*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)+44/105*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c
)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-3/10*(
sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(
1/2)*(EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-EllipticE(cos(1/2*d*x+1/2*c),2^(1/2)))-1/112*cos(1/2*d*x+1/2*c)*(-
2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^4-5/84*cos(1/2*d*x+1/2*c)*(-2*si
n(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^2)/sin(1/2*d*x+1/2*c)/(2*cos(1/2*d*x
+1/2*c)^2-1)^(1/2)/d

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a \sec \left (d x + c\right ) + a}{\cos \left (d x + c\right )^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*sec(d*x + c) + a)/cos(d*x + c)^(7/2), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{a \sec \left (d x + c\right ) + a}{\cos \left (d x + c\right )^{\frac{7}{2}}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a*sec(d*x + c) + a)/cos(d*x + c)^(7/2), x)

________________________________________________________________________________________

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))/cos(d*x+c)**(7/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a \sec \left (d x + c\right ) + a}{\cos \left (d x + c\right )^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*sec(d*x + c) + a)/cos(d*x + c)^(7/2), x)