3.170 \(\int \frac{(a+a \cos (c+d x))^4}{\cos ^{\frac{3}{2}}(c+d x)} \, dx\)

Optimal. Leaf size=119 \[ \frac{32 a^4 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{56 a^4 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 a^4 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{8 a^4 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 a^4 \sin (c+d x)}{d \sqrt{\cos (c+d x)}} \]

[Out]

(56*a^4*EllipticE[(c + d*x)/2, 2])/(5*d) + (32*a^4*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^4*Sin[c + d*x])/(d*
Sqrt[Cos[c + d*x]]) + (8*a^4*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a^4*Cos[c + d*x]^(3/2)*Sin[c + d*x])/
(5*d)

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Rubi [A]  time = 0.12211, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {2757, 2636, 2639, 2641, 2635} \[ \frac{32 a^4 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{56 a^4 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 a^4 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{8 a^4 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 a^4 \sin (c+d x)}{d \sqrt{\cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Cos[c + d*x])^4/Cos[c + d*x]^(3/2),x]

[Out]

(56*a^4*EllipticE[(c + d*x)/2, 2])/(5*d) + (32*a^4*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^4*Sin[c + d*x])/(d*
Sqrt[Cos[c + d*x]]) + (8*a^4*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a^4*Cos[c + d*x]^(3/2)*Sin[c + d*x])/
(5*d)

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 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]

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 2635

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

Rubi steps

\begin{align*} \int \frac{(a+a \cos (c+d x))^4}{\cos ^{\frac{3}{2}}(c+d x)} \, dx &=\int \left (\frac{a^4}{\cos ^{\frac{3}{2}}(c+d x)}+\frac{4 a^4}{\sqrt{\cos (c+d x)}}+6 a^4 \sqrt{\cos (c+d x)}+4 a^4 \cos ^{\frac{3}{2}}(c+d x)+a^4 \cos ^{\frac{5}{2}}(c+d x)\right ) \, dx\\ &=a^4 \int \frac{1}{\cos ^{\frac{3}{2}}(c+d x)} \, dx+a^4 \int \cos ^{\frac{5}{2}}(c+d x) \, dx+\left (4 a^4\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx+\left (4 a^4\right ) \int \cos ^{\frac{3}{2}}(c+d x) \, dx+\left (6 a^4\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{12 a^4 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{8 a^4 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 a^4 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{8 a^4 \sqrt{\cos (c+d x)} \sin (c+d x)}{3 d}+\frac{2 a^4 \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac{1}{5} \left (3 a^4\right ) \int \sqrt{\cos (c+d x)} \, dx-a^4 \int \sqrt{\cos (c+d x)} \, dx+\frac{1}{3} \left (4 a^4\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{56 a^4 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{32 a^4 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{2 a^4 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{8 a^4 \sqrt{\cos (c+d x)} \sin (c+d x)}{3 d}+\frac{2 a^4 \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{5 d}\\ \end{align*}

Mathematica [C]  time = 6.14523, size = 245, normalized size = 2.06 \[ \frac{a^4 (\cos (c+d x)+1)^4 \sec ^8\left (\frac{1}{2} (c+d x)\right ) \left (-336 \cos (c) \sqrt{\sec ^2(c)} \sqrt{\sin ^2\left (\tan ^{-1}(\tan (c))+d x\right )} \csc \left (\tan ^{-1}(\tan (c))+d x\right ) \text{HypergeometricPFQ}\left (\left \{-\frac{1}{2},-\frac{1}{4}\right \},\left \{\frac{3}{4}\right \},\cos ^2\left (\tan ^{-1}(\tan (c))+d x\right )\right )-640 \sin (c) \sqrt{\csc ^2(c)} \cos (c+d x) \sqrt{\cos ^2\left (d x-\tan ^{-1}(\cot (c))\right )} \sec \left (d x-\tan ^{-1}(\cot (c))\right ) \text{HypergeometricPFQ}\left (\left \{\frac{1}{4},\frac{1}{2}\right \},\left \{\frac{5}{4}\right \},\sin ^2\left (d x-\tan ^{-1}(\cot (c))\right )\right )+80 \sin (2 (c+d x))+6 \sin (3 (c+d x))-273 \csc (c) \cos (d x)-399 \csc (c) \cos (2 c+d x)+\frac{168 \csc (c) \sec (c) \left (3 \cos \left (c-\tan ^{-1}(\tan (c))-d x\right )+\cos \left (c+\tan ^{-1}(\tan (c))+d x\right )\right )}{\sqrt{\sec ^2(c)}}\right )}{960 d \sqrt{\cos (c+d x)}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + a*Cos[c + d*x])^4/Cos[c + d*x]^(3/2),x]

[Out]

(a^4*(1 + Cos[c + d*x])^4*Sec[(c + d*x)/2]^8*(-273*Cos[d*x]*Csc[c] - 399*Cos[2*c + d*x]*Csc[c] + (168*(3*Cos[c
- d*x - ArcTan[Tan[c]]] + Cos[c + d*x + ArcTan[Tan[c]]])*Csc[c]*Sec[c])/Sqrt[Sec[c]^2] - 640*Cos[c + d*x]*Sqr
t[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] + 80*Sin[2*(c + d*x)] + 6*Sin[3*(c + d*x)] - 336*Cos[c]*Csc[d*x + ArcTan[Ta
n[c]]]*HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sqrt[Sec[c]^2]*Sqrt[Sin[d*x + ArcTa
n[Tan[c]]]^2]))/(960*d*Sqrt[Cos[c + d*x]])

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Maple [A]  time = 2.33, size = 194, normalized size = 1.6 \begin{align*} -{\frac{8\,{a}^{4}}{15\,d} \left ( -6\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}\cos \left ( 1/2\,dx+c/2 \right ) +26\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}\cos \left ( 1/2\,dx+c/2 \right ) +20\,\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) -21\,\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) -19\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}\cos \left ( 1/2\,dx+c/2 \right ) \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}}}} \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)^(3/2),x)

[Out]

-8/15*a^4*(-6*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+26*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+20*(2*sin(1/2
*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-21*(sin(1/2*d*x+1/2*
c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))-19*sin(1/2*d*x+1/2*c)^2*cos
(1/2*d*x+1/2*c))/sin(1/2*d*x+1/2*c)/(2*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d

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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{3}{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)^(3/2),x, algorithm="maxima")

[Out]

integrate((a*cos(d*x + c) + a)^4/cos(d*x + c)^(3/2), x)

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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{3}{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)^(3/2),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)^(3/2), x)

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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*cos(d*x+c))**4/cos(d*x+c)**(3/2),x)

[Out]

Timed out

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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{3}{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)^(3/2),x, algorithm="giac")

[Out]

integrate((a*cos(d*x + c) + a)^4/cos(d*x + c)^(3/2), x)