3.173 \(\int \frac{(a+a \cos (c+d x))^4}{\cos ^{\frac{9}{2}}(c+d x)} \, dx\)

Optimal. Leaf size=147 \[ \frac{136 a^4 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}-\frac{64 a^4 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{94 a^4 \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{8 a^4 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a^4 \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{64 a^4 \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}} \]

[Out]

(-64*a^4*EllipticE[(c + d*x)/2, 2])/(5*d) + (136*a^4*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a^4*Sin[c + d*x])/
(7*d*Cos[c + d*x]^(7/2)) + (8*a^4*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (94*a^4*Sin[c + d*x])/(21*d*Cos[c +
d*x]^(3/2)) + (64*a^4*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])

________________________________________________________________________________________

Rubi [A]  time = 0.166529, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {2757, 2636, 2641, 2639} \[ \frac{136 a^4 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}-\frac{64 a^4 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{94 a^4 \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{8 a^4 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a^4 \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{64 a^4 \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Cos[c + d*x])^4/Cos[c + d*x]^(9/2),x]

[Out]

(-64*a^4*EllipticE[(c + d*x)/2, 2])/(5*d) + (136*a^4*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a^4*Sin[c + d*x])/
(7*d*Cos[c + d*x]^(7/2)) + (8*a^4*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (94*a^4*Sin[c + d*x])/(21*d*Cos[c +
d*x]^(3/2)) + (64*a^4*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])

Rule 2757

Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Int[Expan
dTrig[(a + b*sin[e + f*x])^m*(d*sin[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] &
& IGtQ[m, 0] && RationalQ[n]

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 \cos (c+d x))^4}{\cos ^{\frac{9}{2}}(c+d x)} \, dx &=\int \left (\frac{a^4}{\cos ^{\frac{9}{2}}(c+d x)}+\frac{4 a^4}{\cos ^{\frac{7}{2}}(c+d x)}+\frac{6 a^4}{\cos ^{\frac{5}{2}}(c+d x)}+\frac{4 a^4}{\cos ^{\frac{3}{2}}(c+d x)}+\frac{a^4}{\sqrt{\cos (c+d x)}}\right ) \, dx\\ &=a^4 \int \frac{1}{\cos ^{\frac{9}{2}}(c+d x)} \, dx+a^4 \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx+\left (4 a^4\right ) \int \frac{1}{\cos ^{\frac{7}{2}}(c+d x)} \, dx+\left (4 a^4\right ) \int \frac{1}{\cos ^{\frac{3}{2}}(c+d x)} \, dx+\left (6 a^4\right ) \int \frac{1}{\cos ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{2 a^4 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 a^4 \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{8 a^4 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{4 a^4 \sin (c+d x)}{d \cos ^{\frac{3}{2}}(c+d x)}+\frac{8 a^4 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{1}{7} \left (5 a^4\right ) \int \frac{1}{\cos ^{\frac{5}{2}}(c+d x)} \, dx+\left (2 a^4\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx+\frac{1}{5} \left (12 a^4\right ) \int \frac{1}{\cos ^{\frac{3}{2}}(c+d x)} \, dx-\left (4 a^4\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=-\frac{8 a^4 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{6 a^4 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 a^4 \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{8 a^4 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{94 a^4 \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{64 a^4 \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{1}{21} \left (5 a^4\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx-\frac{1}{5} \left (12 a^4\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=-\frac{64 a^4 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{136 a^4 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 a^4 \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{8 a^4 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{94 a^4 \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{64 a^4 \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}\\ \end{align*}

Mathematica [C]  time = 5.12433, size = 298, normalized size = 2.03 \[ \frac{a^4 (\cos (c+d x)+1)^4 \sec ^8\left (\frac{1}{2} (c+d x)\right ) \left (-\frac{1344 \sec (c) \cos ^3(c+d x) \left (\csc (c) \sqrt{\sin ^2\left (\tan ^{-1}(\tan (c))+d x\right )} \left (3 \cos \left (c-\tan ^{-1}(\tan (c))-d x\right )+\cos \left (c+\tan ^{-1}(\tan (c))+d x\right )\right )-2 \sin \left (\tan ^{-1}(\tan (c))+d x\right ) \text{HypergeometricPFQ}\left (\left \{-\frac{1}{2},-\frac{1}{4}\right \},\left \{\frac{3}{4}\right \},\cos ^2\left (\tan ^{-1}(\tan (c))+d x\right )\right )\right )}{\sqrt{\sec ^2(c)} \sqrt{\sin ^2\left (\tan ^{-1}(\tan (c))+d x\right )}}-2720 \sin (c) \sqrt{\csc ^2(c)} \cos ^4(c+d x) \sqrt{\cos ^2\left (d x-\tan ^{-1}(\cot (c))\right )} \sec \left (d x-\tan ^{-1}(\cot (c))\right ) \text{HypergeometricPFQ}\left (\left \{\frac{1}{4},\frac{1}{2}\right \},\left \{\frac{5}{4}\right \},\sin ^2\left (d x-\tan ^{-1}(\cot (c))\right )\right )+\csc (c) (-295 \cos (2 c+d x)+2184 \cos (c+2 d x)+504 \cos (3 c+2 d x)+235 \cos (2 c+3 d x)-235 \cos (4 c+3 d x)+672 \cos (3 c+4 d x)+2016 \cos (c)+295 \cos (d x))\right )}{6720 d \cos ^{\frac{7}{2}}(c+d x)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + a*Cos[c + d*x])^4/Cos[c + d*x]^(9/2),x]

[Out]

(a^4*(1 + Cos[c + d*x])^4*Sec[(c + d*x)/2]^8*((2016*Cos[c] + 295*Cos[d*x] - 295*Cos[2*c + d*x] + 2184*Cos[c +
2*d*x] + 504*Cos[3*c + 2*d*x] + 235*Cos[2*c + 3*d*x] - 235*Cos[4*c + 3*d*x] + 672*Cos[3*c + 4*d*x])*Csc[c] - 2
720*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[Cot[c]]]^2]*Sec[d*x - ArcTan[Cot[c]]]*Sin[c] - (1344*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 + Arc
Tan[Tan[c]]]^2])))/(6720*d*Cos[c + d*x]^(7/2))

________________________________________________________________________________________

Maple [B]  time = 4.355, size = 439, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+cos(d*x+c)*a)^4/cos(d*x+c)^(9/2),x)

[Out]

-32*(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^4*(253/420*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*co
s(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))-47/672*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)^2-4/5*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)-2
/5*(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)))-1/80*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)^3-1/896*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)/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{{\left (a \cos \left (d x + c\right ) + a\right )}^{4}}{\cos \left (d x + c\right )^{\frac{9}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(d*x+c))^4/cos(d*x+c)^(9/2),x, algorithm="maxima")

[Out]

integrate((a*cos(d*x + c) + a)^4/cos(d*x + c)^(9/2), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{a^{4} \cos \left (d x + c\right )^{4} + 4 \, a^{4} \cos \left (d x + c\right )^{3} + 6 \, a^{4} \cos \left (d x + c\right )^{2} + 4 \, a^{4} \cos \left (d x + c\right ) + a^{4}}{\cos \left (d x + c\right )^{\frac{9}{2}}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(d*x+c))^4/cos(d*x+c)^(9/2),x, algorithm="fricas")

[Out]

integral((a^4*cos(d*x + c)^4 + 4*a^4*cos(d*x + c)^3 + 6*a^4*cos(d*x + c)^2 + 4*a^4*cos(d*x + c) + a^4)/cos(d*x
+ c)^(9/2), 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*cos(d*x+c))**4/cos(d*x+c)**(9/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a \cos \left (d x + c\right ) + a\right )}^{4}}{\cos \left (d x + c\right )^{\frac{9}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(d*x+c))^4/cos(d*x+c)^(9/2),x, algorithm="giac")

[Out]

integrate((a*cos(d*x + c) + a)^4/cos(d*x + c)^(9/2), x)