Integrand size = 23, antiderivative size = 298 \[ \int \cos ^4(c+d x) \sqrt {a+b \sin (c+d x)} \, dx=-\frac {4 a \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{21 b d}+\frac {2 \cos ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{9 b d}-\frac {8 \left (4 a^4-15 a^2 b^2-21 b^4\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{315 b^4 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {32 a \left (a^4-4 a^2 b^2+3 b^4\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{315 b^4 d \sqrt {a+b \sin (c+d x)}}-\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (a^2-3 b^2\right )-3 b \left (a^2+7 b^2\right ) \sin (c+d x)\right )}{315 b^3 d} \]
2/9*cos(d*x+c)^3*(a+b*sin(d*x+c))^(3/2)/b/d-4/21*a*cos(d*x+c)^3*(a+b*sin(d *x+c))^(1/2)/b/d-4/315*cos(d*x+c)*(4*a*(a^2-3*b^2)-3*b*(a^2+7*b^2)*sin(d*x +c))*(a+b*sin(d*x+c))^(1/2)/b^3/d+8/315*(4*a^4-15*a^2*b^2-21*b^4)*(sin(1/2 *c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/4*Pi+1/2*d*x)*EllipticE(cos(1/2*c+ 1/4*Pi+1/2*d*x),2^(1/2)*(b/(a+b))^(1/2))*(a+b*sin(d*x+c))^(1/2)/b^4/d/((a+ b*sin(d*x+c))/(a+b))^(1/2)-32/315*a*(a^4-4*a^2*b^2+3*b^4)*(sin(1/2*c+1/4*P i+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/4*Pi+1/2*d*x)*EllipticF(cos(1/2*c+1/4*Pi+1 /2*d*x),2^(1/2)*(b/(a+b))^(1/2))*((a+b*sin(d*x+c))/(a+b))^(1/2)/b^4/d/(a+b *sin(d*x+c))^(1/2)
Time = 0.62 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.78 \[ \int \cos ^4(c+d x) \sqrt {a+b \sin (c+d x)} \, dx=\frac {32 \left (a b^2 \left (a^2-33 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right )+\left (4 a^4-15 a^2 b^2-21 b^4\right ) \left ((a+b) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right )\right )\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}+2 b \cos (c+d x) (a+b \sin (c+d x)) \left (-32 a^3+106 a b^2+10 a b^2 \cos (2 (c+d x))+b \left (24 a^2+203 b^2\right ) \sin (c+d x)+35 b^3 \sin (3 (c+d x))\right )}{1260 b^4 d \sqrt {a+b \sin (c+d x)}} \]
(32*(a*b^2*(a^2 - 33*b^2)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)] + (4*a^4 - 15*a^2*b^2 - 21*b^4)*((a + b)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)] - a*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]))*Sqrt[ (a + b*Sin[c + d*x])/(a + b)] + 2*b*Cos[c + d*x]*(a + b*Sin[c + d*x])*(-32 *a^3 + 106*a*b^2 + 10*a*b^2*Cos[2*(c + d*x)] + b*(24*a^2 + 203*b^2)*Sin[c + d*x] + 35*b^3*Sin[3*(c + d*x)]))/(1260*b^4*d*Sqrt[a + b*Sin[c + d*x]])
Time = 1.58 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.05, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.739, Rules used = {3042, 3174, 3042, 3341, 27, 3042, 3344, 27, 3042, 3231, 3042, 3134, 3042, 3132, 3142, 3042, 3140}
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.
\(\displaystyle \int \cos ^4(c+d x) \sqrt {a+b \sin (c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \cos (c+d x)^4 \sqrt {a+b \sin (c+d x)}dx\) |
\(\Big \downarrow \) 3174 |
\(\displaystyle \frac {2 \int \cos ^2(c+d x) (b+a \sin (c+d x)) \sqrt {a+b \sin (c+d x)}dx}{3 b}+\frac {2 \cos ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{9 b d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 \int \cos (c+d x)^2 (b+a \sin (c+d x)) \sqrt {a+b \sin (c+d x)}dx}{3 b}+\frac {2 \cos ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{9 b d}\) |
\(\Big \downarrow \) 3341 |
\(\displaystyle \frac {2 \left (\frac {2}{7} \int \frac {\cos ^2(c+d x) \left (8 a b+\left (a^2+7 b^2\right ) \sin (c+d x)\right )}{2 \sqrt {a+b \sin (c+d x)}}dx-\frac {2 a \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{7 d}\right )}{3 b}+\frac {2 \cos ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{9 b d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2 \left (\frac {1}{7} \int \frac {\cos ^2(c+d x) \left (8 a b+\left (a^2+7 b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}}dx-\frac {2 a \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{7 d}\right )}{3 b}+\frac {2 \cos ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{9 b d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 \left (\frac {1}{7} \int \frac {\cos (c+d x)^2 \left (8 a b+\left (a^2+7 b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}}dx-\frac {2 a \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{7 d}\right )}{3 b}+\frac {2 \cos ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{9 b d}\) |
\(\Big \downarrow \) 3344 |
\(\displaystyle \frac {2 \left (\frac {1}{7} \left (\frac {4 \int -\frac {a b \left (a^2-33 b^2\right )+\left (4 a^4-15 b^2 a^2-21 b^4\right ) \sin (c+d x)}{2 \sqrt {a+b \sin (c+d x)}}dx}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (a^2-3 b^2\right )-3 b \left (a^2+7 b^2\right ) \sin (c+d x)\right )}{15 b^2 d}\right )-\frac {2 a \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{7 d}\right )}{3 b}+\frac {2 \cos ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{9 b d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2 \left (\frac {1}{7} \left (-\frac {2 \int \frac {a b \left (a^2-33 b^2\right )+\left (4 a^4-15 b^2 a^2-21 b^4\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}}dx}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (a^2-3 b^2\right )-3 b \left (a^2+7 b^2\right ) \sin (c+d x)\right )}{15 b^2 d}\right )-\frac {2 a \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{7 d}\right )}{3 b}+\frac {2 \cos ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{9 b d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 \left (\frac {1}{7} \left (-\frac {2 \int \frac {a b \left (a^2-33 b^2\right )+\left (4 a^4-15 b^2 a^2-21 b^4\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}}dx}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (a^2-3 b^2\right )-3 b \left (a^2+7 b^2\right ) \sin (c+d x)\right )}{15 b^2 d}\right )-\frac {2 a \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{7 d}\right )}{3 b}+\frac {2 \cos ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{9 b d}\) |
\(\Big \downarrow \) 3231 |
\(\displaystyle \frac {2 \left (\frac {1}{7} \left (-\frac {2 \left (\frac {\left (4 a^4-15 a^2 b^2-21 b^4\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}-\frac {4 a \left (a^4-4 a^2 b^2+3 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (a^2-3 b^2\right )-3 b \left (a^2+7 b^2\right ) \sin (c+d x)\right )}{15 b^2 d}\right )-\frac {2 a \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{7 d}\right )}{3 b}+\frac {2 \cos ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{9 b d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 \left (\frac {1}{7} \left (-\frac {2 \left (\frac {\left (4 a^4-15 a^2 b^2-21 b^4\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}-\frac {4 a \left (a^4-4 a^2 b^2+3 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (a^2-3 b^2\right )-3 b \left (a^2+7 b^2\right ) \sin (c+d x)\right )}{15 b^2 d}\right )-\frac {2 a \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{7 d}\right )}{3 b}+\frac {2 \cos ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{9 b d}\) |
\(\Big \downarrow \) 3134 |
\(\displaystyle \frac {2 \left (\frac {1}{7} \left (-\frac {2 \left (\frac {\left (4 a^4-15 a^2 b^2-21 b^4\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {4 a \left (a^4-4 a^2 b^2+3 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (a^2-3 b^2\right )-3 b \left (a^2+7 b^2\right ) \sin (c+d x)\right )}{15 b^2 d}\right )-\frac {2 a \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{7 d}\right )}{3 b}+\frac {2 \cos ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{9 b d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 \left (\frac {1}{7} \left (-\frac {2 \left (\frac {\left (4 a^4-15 a^2 b^2-21 b^4\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {4 a \left (a^4-4 a^2 b^2+3 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (a^2-3 b^2\right )-3 b \left (a^2+7 b^2\right ) \sin (c+d x)\right )}{15 b^2 d}\right )-\frac {2 a \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{7 d}\right )}{3 b}+\frac {2 \cos ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{9 b d}\) |
\(\Big \downarrow \) 3132 |
\(\displaystyle \frac {2 \left (\frac {1}{7} \left (-\frac {2 \left (\frac {2 \left (4 a^4-15 a^2 b^2-21 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {4 a \left (a^4-4 a^2 b^2+3 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (a^2-3 b^2\right )-3 b \left (a^2+7 b^2\right ) \sin (c+d x)\right )}{15 b^2 d}\right )-\frac {2 a \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{7 d}\right )}{3 b}+\frac {2 \cos ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{9 b d}\) |
\(\Big \downarrow \) 3142 |
\(\displaystyle \frac {2 \left (\frac {1}{7} \left (-\frac {2 \left (\frac {2 \left (4 a^4-15 a^2 b^2-21 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {4 a \left (a^4-4 a^2 b^2+3 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{b \sqrt {a+b \sin (c+d x)}}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (a^2-3 b^2\right )-3 b \left (a^2+7 b^2\right ) \sin (c+d x)\right )}{15 b^2 d}\right )-\frac {2 a \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{7 d}\right )}{3 b}+\frac {2 \cos ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{9 b d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 \left (\frac {1}{7} \left (-\frac {2 \left (\frac {2 \left (4 a^4-15 a^2 b^2-21 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {4 a \left (a^4-4 a^2 b^2+3 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{b \sqrt {a+b \sin (c+d x)}}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (a^2-3 b^2\right )-3 b \left (a^2+7 b^2\right ) \sin (c+d x)\right )}{15 b^2 d}\right )-\frac {2 a \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{7 d}\right )}{3 b}+\frac {2 \cos ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{9 b d}\) |
\(\Big \downarrow \) 3140 |
\(\displaystyle \frac {2 \left (\frac {1}{7} \left (-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (a^2-3 b^2\right )-3 b \left (a^2+7 b^2\right ) \sin (c+d x)\right )}{15 b^2 d}-\frac {2 \left (\frac {2 \left (4 a^4-15 a^2 b^2-21 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {8 a \left (a^4-4 a^2 b^2+3 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \sin (c+d x)}}\right )}{15 b^2}\right )-\frac {2 a \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{7 d}\right )}{3 b}+\frac {2 \cos ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{9 b d}\) |
(2*Cos[c + d*x]^3*(a + b*Sin[c + d*x])^(3/2))/(9*b*d) + (2*((-2*a*Cos[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]])/(7*d) + ((-2*Cos[c + d*x]*Sqrt[a + b*Sin[ c + d*x]]*(4*a*(a^2 - 3*b^2) - 3*b*(a^2 + 7*b^2)*Sin[c + d*x]))/(15*b^2*d) - (2*((2*(4*a^4 - 15*a^2*b^2 - 21*b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b )/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(b*d*Sqrt[(a + b*Sin[c + d*x])/(a + b )]) - (8*a*(a^4 - 4*a^2*b^2 + 3*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/( a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(b*d*Sqrt[a + b*Sin[c + d*x]]) ))/(15*b^2))/7))/(3*b)
3.5.79.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)] Int[Sqrt[a/(a + b) + ( b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 , 0] && !GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ {a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]] Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && !GtQ[a + b, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[g*(g*Cos[e + f*x])^(p - 1)*((a + b*Sin[e + f*x ])^(m + 1)/(b*f*(m + p))), x] + Simp[g^2*((p - 1)/(b*(m + p))) Int[(g*Cos [e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^m*(b + a*Sin[e + f*x]), x], x] /; F reeQ[{a, b, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && GtQ[p, 1] && NeQ[m + p, 0] && IntegersQ[2*m, 2*p]
Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + ( f_.)*(x_)]], x_Symbol] :> Simp[(b*c - a*d)/b Int[1/Sqrt[a + b*Sin[e + f*x ]], x], x] + Simp[d/b Int[Sqrt[a + b*Sin[e + f*x]], x], x] /; FreeQ[{a, b , c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d)* (g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(f*g*(m + p + 1))), x] + S imp[1/(m + p + 1) Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1)*Sim p[a*c*(m + p + 1) + b*d*m + (a*d*m + b*c*(m + p + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && !LtQ[p, -1] && IntegerQ[2*m] && !(EqQ[m, 1] && NeQ[c^2 - d^2, 0] && S implerQ[c + d*x, a + b*x])
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[g*(g* Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1)*((b*c*(m + p + 1) - a*d* p + b*d*(m + p)*Sin[e + f*x])/(b^2*f*(m + p)*(m + p + 1))), x] + Simp[g^2*( (p - 1)/(b^2*(m + p)*(m + p + 1))) Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Si n[e + f*x])^m*Simp[b*(a*d*m + b*c*(m + p + 1)) + (a*b*c*(m + p + 1) - d*(a^ 2*p - b^2*(m + p)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && GtQ[p, 1] && NeQ[m + p, 0] && NeQ[m + p + 1 , 0] && IntegerQ[2*m]
Leaf count of result is larger than twice the leaf count of optimal. \(1188\) vs. \(2(340)=680\).
Time = 3.07 (sec) , antiderivative size = 1189, normalized size of antiderivative = 3.99
-2/315*(-35*b^6*sin(d*x+c)^6-40*a*b^5*sin(d*x+c)^5+16*((a+b*sin(d*x+c))/(a -b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2) *EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^5*b-12*(( a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+ c))*b/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^ (1/2))*a^4*b^2-64*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b)) ^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^ (1/2),((a-b)/(a+b))^(1/2))*a^3*b^3-72*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(si n(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticF(((a+b *sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^2*b^4+48*((a+b*sin(d*x+c) )/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^( 1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a*b^5+8 4*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin( d*x+c))*b/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+ b))^(1/2))*b^6-16*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b)) ^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticE(((a+b*sin(d*x+c))/(a-b))^ (1/2),((a-b)/(a+b))^(1/2))*a^6+76*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d* x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticE(((a+b*sin (d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^4*b^2+24*((a+b*sin(d*x+c))/(a -b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 534, normalized size of antiderivative = 1.79 \[ \int \cos ^4(c+d x) \sqrt {a+b \sin (c+d x)} \, dx=\frac {2 \, {\left (2 \, \sqrt {2} {\left (8 \, a^{5} - 33 \, a^{3} b^{2} + 57 \, a b^{4}\right )} \sqrt {i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) - 2 i \, a}{3 \, b}\right ) + 2 \, \sqrt {2} {\left (8 \, a^{5} - 33 \, a^{3} b^{2} + 57 \, a b^{4}\right )} \sqrt {-i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 i \, a}{3 \, b}\right ) - 6 \, \sqrt {2} {\left (-4 i \, a^{4} b + 15 i \, a^{2} b^{3} + 21 i \, b^{5}\right )} \sqrt {i \, b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) - 2 i \, a}{3 \, b}\right )\right ) - 6 \, \sqrt {2} {\left (4 i \, a^{4} b - 15 i \, a^{2} b^{3} - 21 i \, b^{5}\right )} \sqrt {-i \, b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 i \, a}{3 \, b}\right )\right ) + 3 \, {\left (5 \, a b^{4} \cos \left (d x + c\right )^{3} - 8 \, {\left (a^{3} b^{2} - 3 \, a b^{4}\right )} \cos \left (d x + c\right ) + {\left (35 \, b^{5} \cos \left (d x + c\right )^{3} + 6 \, {\left (a^{2} b^{3} + 7 \, b^{5}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )} \sqrt {b \sin \left (d x + c\right ) + a}\right )}}{945 \, b^{5} d} \]
2/945*(2*sqrt(2)*(8*a^5 - 33*a^3*b^2 + 57*a*b^4)*sqrt(I*b)*weierstrassPInv erse(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*I*a^3 - 9*I*a*b^2)/b^3, 1/3*(3*b*c os(d*x + c) - 3*I*b*sin(d*x + c) - 2*I*a)/b) + 2*sqrt(2)*(8*a^5 - 33*a^3*b ^2 + 57*a*b^4)*sqrt(-I*b)*weierstrassPInverse(-4/3*(4*a^2 - 3*b^2)/b^2, -8 /27*(-8*I*a^3 + 9*I*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) + 3*I*b*sin(d*x + c) + 2*I*a)/b) - 6*sqrt(2)*(-4*I*a^4*b + 15*I*a^2*b^3 + 21*I*b^5)*sqrt(I*b)* weierstrassZeta(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*I*a^3 - 9*I*a*b^2)/b^3, weierstrassPInverse(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*I*a^3 - 9*I*a*b^2) /b^3, 1/3*(3*b*cos(d*x + c) - 3*I*b*sin(d*x + c) - 2*I*a)/b)) - 6*sqrt(2)* (4*I*a^4*b - 15*I*a^2*b^3 - 21*I*b^5)*sqrt(-I*b)*weierstrassZeta(-4/3*(4*a ^2 - 3*b^2)/b^2, -8/27*(-8*I*a^3 + 9*I*a*b^2)/b^3, weierstrassPInverse(-4/ 3*(4*a^2 - 3*b^2)/b^2, -8/27*(-8*I*a^3 + 9*I*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) + 3*I*b*sin(d*x + c) + 2*I*a)/b)) + 3*(5*a*b^4*cos(d*x + c)^3 - 8*(a^ 3*b^2 - 3*a*b^4)*cos(d*x + c) + (35*b^5*cos(d*x + c)^3 + 6*(a^2*b^3 + 7*b^ 5)*cos(d*x + c))*sin(d*x + c))*sqrt(b*sin(d*x + c) + a))/(b^5*d)
\[ \int \cos ^4(c+d x) \sqrt {a+b \sin (c+d x)} \, dx=\int \sqrt {a + b \sin {\left (c + d x \right )}} \cos ^{4}{\left (c + d x \right )}\, dx \]
\[ \int \cos ^4(c+d x) \sqrt {a+b \sin (c+d x)} \, dx=\int { \sqrt {b \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{4} \,d x } \]
\[ \int \cos ^4(c+d x) \sqrt {a+b \sin (c+d x)} \, dx=\int { \sqrt {b \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{4} \,d x } \]
Timed out. \[ \int \cos ^4(c+d x) \sqrt {a+b \sin (c+d x)} \, dx=\int {\cos \left (c+d\,x\right )}^4\,\sqrt {a+b\,\sin \left (c+d\,x\right )} \,d x \]