Integrand size = 23, antiderivative size = 147 \[ \int \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^4 \, dx=\frac {152 a^4 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {32 a^4 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{7 d}+\frac {32 a^4 \sqrt {\cos (c+d x)} \sin (c+d x)}{7 d}+\frac {122 a^4 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {8 a^4 \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {2 a^4 \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9 d} \] Output:
152/15*a^4*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/d+32/7*a^4*InverseJacobiA M(1/2*d*x+1/2*c,2^(1/2))/d+32/7*a^4*cos(d*x+c)^(1/2)*sin(d*x+c)/d+122/45*a ^4*cos(d*x+c)^(3/2)*sin(d*x+c)/d+8/7*a^4*cos(d*x+c)^(5/2)*sin(d*x+c)/d+2/9 *a^4*cos(d*x+c)^(7/2)*sin(d*x+c)/d
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 7.09 (sec) , antiderivative size = 532, normalized size of antiderivative = 3.62 \[ \int \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^4 \, dx=\sqrt {\cos (c+d x)} (a+a \cos (c+d x))^4 \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (-\frac {19 \cot (c)}{30 d}+\frac {17 \cos (d x) \sin (c)}{56 d}+\frac {127 \cos (2 d x) \sin (2 c)}{1440 d}+\frac {\cos (3 d x) \sin (3 c)}{56 d}+\frac {\cos (4 d x) \sin (4 c)}{576 d}+\frac {17 \cos (c) \sin (d x)}{56 d}+\frac {127 \cos (2 c) \sin (2 d x)}{1440 d}+\frac {\cos (3 c) \sin (3 d x)}{56 d}+\frac {\cos (4 c) \sin (4 d x)}{576 d}\right )-\frac {2 (a+a \cos (c+d x))^4 \csc (c) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\sin ^2(d x-\arctan (\cot (c)))\right ) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec (d x-\arctan (\cot (c))) \sqrt {1-\sin (d x-\arctan (\cot (c)))} \sqrt {-\sqrt {1+\cot ^2(c)} \sin (c) \sin (d x-\arctan (\cot (c)))} \sqrt {1+\sin (d x-\arctan (\cot (c)))}}{7 d \sqrt {1+\cot ^2(c)}}-\frac {19 (a+a \cos (c+d x))^4 \csc (c) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (\frac {\, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};\cos ^2(d x+\arctan (\tan (c)))\right ) \sin (d x+\arctan (\tan (c))) \tan (c)}{\sqrt {1-\cos (d x+\arctan (\tan (c)))} \sqrt {1+\cos (d x+\arctan (\tan (c)))} \sqrt {\cos (c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}} \sqrt {1+\tan ^2(c)}}-\frac {\frac {\sin (d x+\arctan (\tan (c))) \tan (c)}{\sqrt {1+\tan ^2(c)}}+\frac {2 \cos ^2(c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}}{\cos ^2(c)+\sin ^2(c)}}{\sqrt {\cos (c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}}}\right )}{60 d} \] Input:
Integrate[Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^4,x]
Output:
Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8*((-19*Cot[c ])/(30*d) + (17*Cos[d*x]*Sin[c])/(56*d) + (127*Cos[2*d*x]*Sin[2*c])/(1440* d) + (Cos[3*d*x]*Sin[3*c])/(56*d) + (Cos[4*d*x]*Sin[4*c])/(576*d) + (17*Co s[c]*Sin[d*x])/(56*d) + (127*Cos[2*c]*Sin[2*d*x])/(1440*d) + (Cos[3*c]*Sin [3*d*x])/(56*d) + (Cos[4*c]*Sin[4*d*x])/(576*d)) - (2*(a + a*Cos[c + d*x]) ^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2 ]*Sec[c/2 + (d*x)/2]^8*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan [Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sq rt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*Sqrt[1 + Cot[c]^2]) - (19*(a + a*C os[c + d*x])^4*Csc[c]*Sec[c/2 + (d*x)/2]^8*((HypergeometricPFQ[{-1/2, -1/4 }, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/( Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sq rt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2] ) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*C os[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[C os[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(60*d)
Time = 0.40 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3042, 3236, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^4 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^4dx\) |
| \(\Big \downarrow \) 3236 |
| \(\displaystyle \int \left (a^4 \cos ^{\frac {9}{2}}(c+d x)+4 a^4 \cos ^{\frac {7}{2}}(c+d x)+6 a^4 \cos ^{\frac {5}{2}}(c+d x)+4 a^4 \cos ^{\frac {3}{2}}(c+d x)+a^4 \sqrt {\cos (c+d x)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {32 a^4 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{7 d}+\frac {152 a^4 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 a^4 \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}+\frac {8 a^4 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}+\frac {122 a^4 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{45 d}+\frac {32 a^4 \sin (c+d x) \sqrt {\cos (c+d x)}}{7 d}\) |
Input:
Int[Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^4,x]
Output:
(152*a^4*EllipticE[(c + d*x)/2, 2])/(15*d) + (32*a^4*EllipticF[(c + d*x)/2 , 2])/(7*d) + (32*a^4*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(7*d) + (122*a^4*Co s[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (8*a^4*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*a^4*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)
Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*( x_)])^(m_.), x_Symbol] :> Int[ExpandTrig[(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] && IGt Q[m, 0] && RationalQ[n]
Time = 17.73 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.77
| method | result | size |
| default | \(-\frac {8 \sqrt {\left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, a^{4} \left (280 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}-120 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}+34 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+72 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-485 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+180 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-399 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+219 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{315 \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\) | \(260\) |
| parts | \(\text {Expression too large to display}\) | \(937\) |
Input:
int(cos(d*x+c)^(1/2)*(a+a*cos(d*x+c))^4,x,method=_RETURNVERBOSE)
Output:
-8/315*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^4*(280*co s(1/2*d*x+1/2*c)^11-120*cos(1/2*d*x+1/2*c)^9+34*cos(1/2*d*x+1/2*c)^7+72*co s(1/2*d*x+1/2*c)^5-485*cos(1/2*d*x+1/2*c)^3+180*(sin(1/2*d*x+1/2*c)^2)^(1/ 2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)) -399*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)*Ellipt icE(cos(1/2*d*x+1/2*c),2^(1/2))+219*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)/sin(1/2*d*x+1/2*c)/(2*cos(1/2*d*x+1/2*c )^2-1)^(1/2)/d
Result contains complex when optimal does not.
Time = 0.13 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.19 \[ \int \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^4 \, dx=-\frac {2 \, {\left (360 i \, \sqrt {2} a^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 360 i \, \sqrt {2} a^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 798 i \, \sqrt {2} a^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 798 i \, \sqrt {2} a^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - {\left (35 \, a^{4} \cos \left (d x + c\right )^{3} + 180 \, a^{4} \cos \left (d x + c\right )^{2} + 427 \, a^{4} \cos \left (d x + c\right ) + 720 \, a^{4}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )\right )}}{315 \, d} \] Input:
integrate(cos(d*x+c)^(1/2)*(a+a*cos(d*x+c))^4,x, algorithm="fricas")
Output:
-2/315*(360*I*sqrt(2)*a^4*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin( d*x + c)) - 360*I*sqrt(2)*a^4*weierstrassPInverse(-4, 0, cos(d*x + c) - I* sin(d*x + c)) - 798*I*sqrt(2)*a^4*weierstrassZeta(-4, 0, weierstrassPInver se(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 798*I*sqrt(2)*a^4*weierstrassZ eta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - (3 5*a^4*cos(d*x + c)^3 + 180*a^4*cos(d*x + c)^2 + 427*a^4*cos(d*x + c) + 720 *a^4)*sqrt(cos(d*x + c))*sin(d*x + c))/d
Timed out. \[ \int \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^4 \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**(1/2)*(a+a*cos(d*x+c))**4,x)
Output:
Timed out
\[ \int \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^4 \, dx=\int { {\left (a \cos \left (d x + c\right ) + a\right )}^{4} \sqrt {\cos \left (d x + c\right )} \,d x } \] Input:
integrate(cos(d*x+c)^(1/2)*(a+a*cos(d*x+c))^4,x, algorithm="maxima")
Output:
integrate((a*cos(d*x + c) + a)^4*sqrt(cos(d*x + c)), x)
\[ \int \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^4 \, dx=\int { {\left (a \cos \left (d x + c\right ) + a\right )}^{4} \sqrt {\cos \left (d x + c\right )} \,d x } \] Input:
integrate(cos(d*x+c)^(1/2)*(a+a*cos(d*x+c))^4,x, algorithm="giac")
Output:
integrate((a*cos(d*x + c) + a)^4*sqrt(cos(d*x + c)), x)
Time = 40.95 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.52 \[ \int \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^4 \, dx=\frac {2\,\left (3\,a^4\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+4\,a^4\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+4\,a^4\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )\right )}{3\,d}-\frac {2\,\left (\frac {66\,a^4\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )}{\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {17\,a^4\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )}{\sqrt {{\sin \left (c+d\,x\right )}^2}}\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {15}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{77\,d}-\frac {8\,a^4\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {208\,a^4\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {19}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{385\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \] Input:
int(cos(c + d*x)^(1/2)*(a + a*cos(c + d*x))^4,x)
Output:
(2*(3*a^4*ellipticE(c/2 + (d*x)/2, 2) + 4*a^4*ellipticF(c/2 + (d*x)/2, 2) + 4*a^4*cos(c + d*x)^(1/2)*sin(c + d*x)))/(3*d) - (2*((66*a^4*cos(c + d*x) ^(7/2)*sin(c + d*x))/(sin(c + d*x)^2)^(1/2) - (17*a^4*cos(c + d*x)^(11/2)* sin(c + d*x))/(sin(c + d*x)^2)^(1/2))*hypergeom([1/2, 11/4], 15/4, cos(c + d*x)^2))/(77*d) - (8*a^4*cos(c + d*x)^(9/2)*sin(c + d*x)*hypergeom([1/2, 9/4], 13/4, cos(c + d*x)^2))/(9*d*(sin(c + d*x)^2)^(1/2)) - (208*a^4*cos(c + d*x)^(11/2)*sin(c + d*x)*hypergeom([1/2, 11/4], 19/4, cos(c + d*x)^2))/ (385*d*(sin(c + d*x)^2)^(1/2))
\[ \int \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^4 \, dx=a^{4} \left (\int \sqrt {\cos \left (d x +c \right )}d x +4 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )d x \right )+\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4}d x +4 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}d x \right )+6 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}d x \right )\right ) \] Input:
int(cos(d*x+c)^(1/2)*(a+a*cos(d*x+c))^4,x)
Output:
a**4*(int(sqrt(cos(c + d*x)),x) + 4*int(sqrt(cos(c + d*x))*cos(c + d*x),x) + int(sqrt(cos(c + d*x))*cos(c + d*x)**4,x) + 4*int(sqrt(cos(c + d*x))*co s(c + d*x)**3,x) + 6*int(sqrt(cos(c + d*x))*cos(c + d*x)**2,x))