\(\int \frac {(a+a \cos (c+d x))^4}{\cos ^{\frac {3}{2}}(c+d x)} \, dx\) [170]

Optimal result

Integrand size = 23, antiderivative size = 119 \[ \int \frac {(a+a \cos (c+d x))^4}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\frac {56 a^4 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {32 a^4 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}+\frac {2 a^4 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+\frac {8 a^4 \sqrt {\cos (c+d x)} \sin (c+d x)}{3 d}+\frac {2 a^4 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 d} \]

[Out]

56/5*a^4*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/d+32/3*a^4*(cos
(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2))/d+2/5*a^4*cos(d*x+c)^(3/2)*s
in(d*x+c)/d+2*a^4*sin(d*x+c)/d/cos(d*x+c)^(1/2)+8/3*a^4*sin(d*x+c)*cos(d*x+c)^(1/2)/d

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2836, 2716, 2719, 2720, 2715} \[ \int \frac {(a+a \cos (c+d x))^4}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\frac {32 a^4 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}+\frac {56 a^4 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 a^4 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}+\frac {8 a^4 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}+\frac {2 a^4 \sin (c+d x)}{d \sqrt {\cos (c+d x)}} \]

[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 2715

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] && Integ
erQ[2*n]

Rule 2716

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 2719

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

Rule 2720

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

Rule 2836

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]

Rubi steps \begin{align*} \text {integral}& = \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 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),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 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),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] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.94 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.32 \[ \int \frac {(a+a \cos (c+d x))^4}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {2 a^4 \csc (c+d x) \left (-15 \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},\cos ^2(c+d x)\right ) \sqrt {\sin ^2(c+d x)}+\cos (c+d x) \left (-\left ((20+3 \cos (c+d x)) \sin ^2(c+d x)\right )+80 \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\cos ^2(c+d x)\right ) \sqrt {\sin ^2(c+d x)}+33 \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},\cos ^2(c+d x)\right ) \sqrt {\sin ^2(c+d x)}\right )\right )}{15 d \sqrt {\cos (c+d x)}} \]

[In]

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

[Out]

(-2*a^4*Csc[c + d*x]*(-15*Hypergeometric2F1[-1/4, 1/2, 3/4, Cos[c + d*x]^2]*Sqrt[Sin[c + d*x]^2] + Cos[c + d*x
]*(-((20 + 3*Cos[c + d*x])*Sin[c + d*x]^2) + 80*Hypergeometric2F1[1/4, 1/2, 5/4, Cos[c + d*x]^2]*Sqrt[Sin[c +
d*x]^2] + 33*Cos[c + d*x]*Hypergeometric2F1[1/2, 3/4, 7/4, Cos[c + d*x]^2]*Sqrt[Sin[c + d*x]^2])))/(15*d*Sqrt[
Cos[c + d*x]])

Maple [A] (verified)

Time = 8.78 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.63

method	result	size
default	\(\frac {8 a^{4} \left (6 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-26 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+19 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-20 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+21 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{15 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d}\)	\(194\)
parts	\(\text {Expression too large to display}\)	\(726\)

[In]

int((a+cos(d*x+c)*a)^4/cos(d*x+c)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.11 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.63 \[ \int \frac {(a+a \cos (c+d x))^4}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {2 \, {\left (40 i \, \sqrt {2} a^{4} \cos \left (d x + c\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 40 i \, \sqrt {2} a^{4} \cos \left (d x + c\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 42 i \, \sqrt {2} a^{4} \cos \left (d x + c\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 42 i \, \sqrt {2} a^{4} \cos \left (d x + c\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - {\left (3 \, a^{4} \cos \left (d x + c\right )^{2} + 20 \, a^{4} \cos \left (d x + c\right ) + 15 \, a^{4}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )\right )}}{15 \, d \cos \left (d x + c\right )} \]

[In]

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

[Out]

-2/15*(40*I*sqrt(2)*a^4*cos(d*x + c)*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) - 40*I*sqrt(2)*
a^4*cos(d*x + c)*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 42*I*sqrt(2)*a^4*cos(d*x + c)*wei
erstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 42*I*sqrt(2)*a^4*cos(d*x + c)
*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - (3*a^4*cos(d*x + c)^2 + 2
0*a^4*cos(d*x + c) + 15*a^4)*sqrt(cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c))

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+a \cos (c+d x))^4}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {(a+a \cos (c+d x))^4}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (a \cos \left (d x + c\right ) + a\right )}^{4}}{\cos \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {(a+a \cos (c+d x))^4}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (a \cos \left (d x + c\right ) + a\right )}^{4}}{\cos \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]

[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)

Mupad [B] (verification not implemented)

Time = 15.02 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.25 \[ \int \frac {(a+a \cos (c+d x))^4}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\frac {12\,a^4\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {32\,a^4\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{3\,d}+\frac {8\,a^4\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{3\,d}+\frac {2\,a^4\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{d\,\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,a^4\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]

[In]

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

[Out]

(12*a^4*ellipticE(c/2 + (d*x)/2, 2))/d + (32*a^4*ellipticF(c/2 + (d*x)/2, 2))/(3*d) + (8*a^4*cos(c + d*x)^(1/2
)*sin(c + d*x))/(3*d) + (2*a^4*sin(c + d*x)*hypergeom([-1/4, 1/2], 3/4, cos(c + d*x)^2))/(d*cos(c + d*x)^(1/2)
*(sin(c + d*x)^2)^(1/2)) - (2*a^4*cos(c + d*x)^(7/2)*sin(c + d*x)*hypergeom([1/2, 7/4], 11/4, cos(c + d*x)^2))
/(7*d*(sin(c + d*x)^2)^(1/2))