Integrand size = 25, antiderivative size = 486 \[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+b \sin (c+d x))} \, dx=\frac {b^{7/2} \arctan \left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{\left (-a^2+b^2\right )^{9/4} d e^{7/2}}-\frac {b^{7/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{\left (-a^2+b^2\right )^{9/4} d e^{7/2}}-\frac {2 a \left (3 a^2-8 b^2\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 \left (a^2-b^2\right )^2 d e^4 \sqrt {\cos (c+d x)}}+\frac {a b^3 \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} (c+d x),2\right )}{\left (a^2-b^2\right )^2 \left (b-\sqrt {-a^2+b^2}\right ) d e^3 \sqrt {e \cos (c+d x)}}+\frac {a b^3 \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} (c+d x),2\right )}{\left (a^2-b^2\right )^2 \left (b+\sqrt {-a^2+b^2}\right ) d e^3 \sqrt {e \cos (c+d x)}}-\frac {2 (b-a \sin (c+d x))}{5 \left (a^2-b^2\right ) d e (e \cos (c+d x))^{5/2}}+\frac {2 \left (5 b^3+a \left (3 a^2-8 b^2\right ) \sin (c+d x)\right )}{5 \left (a^2-b^2\right )^2 d e^3 \sqrt {e \cos (c+d x)}} \]
b^(7/2)*arctan(b^(1/2)*(e*cos(d*x+c))^(1/2)/(-a^2+b^2)^(1/4)/e^(1/2))/(-a^ 2+b^2)^(9/4)/d/e^(7/2)-b^(7/2)*arctanh(b^(1/2)*(e*cos(d*x+c))^(1/2)/(-a^2+ b^2)^(1/4)/e^(1/2))/(-a^2+b^2)^(9/4)/d/e^(7/2)-2/5*(b-a*sin(d*x+c))/(a^2-b ^2)/d/e/(e*cos(d*x+c))^(5/2)+2/5*(5*b^3+a*(3*a^2-8*b^2)*sin(d*x+c))/(a^2-b ^2)^2/d/e^3/(e*cos(d*x+c))^(1/2)+a*b^3*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/ 2*d*x+1/2*c)*EllipticPi(sin(1/2*d*x+1/2*c),2*b/(b-(-a^2+b^2)^(1/2)),2^(1/2 ))*cos(d*x+c)^(1/2)/(a^2-b^2)^2/d/e^3/(b-(-a^2+b^2)^(1/2))/(e*cos(d*x+c))^ (1/2)+a*b^3*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticPi(sin (1/2*d*x+1/2*c),2*b/(b+(-a^2+b^2)^(1/2)),2^(1/2))*cos(d*x+c)^(1/2)/(a^2-b^ 2)^2/d/e^3/(b+(-a^2+b^2)^(1/2))/(e*cos(d*x+c))^(1/2)-2/5*a*(3*a^2-8*b^2)*( 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))*(e*cos(d*x+c))^(1/2)/(a^2-b^2)^2/d/e^4/cos(d*x+c)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 6.55 (sec) , antiderivative size = 881, normalized size of antiderivative = 1.81 \[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+b \sin (c+d x))} \, dx=-\frac {\cos ^{\frac {7}{2}}(c+d x) \left (-\frac {2 \left (3 a^4-8 a^2 b^2-5 b^4\right ) \left (a+b \sqrt {1-\cos ^2(c+d x)}\right ) \left (\frac {a \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\cos ^2(c+d x),\frac {b^2 \cos ^2(c+d x)}{-a^2+b^2}\right ) \cos ^{\frac {3}{2}}(c+d x)}{3 \left (a^2-b^2\right )}+\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) \left (2 \arctan \left (1-\frac {(1+i) \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )-2 \arctan \left (1+\frac {(1+i) \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )-\log \left (\sqrt {-a^2+b^2}-(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\cos (c+d x)}+i b \cos (c+d x)\right )+\log \left (\sqrt {-a^2+b^2}+(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\cos (c+d x)}+i b \cos (c+d x)\right )\right )}{\sqrt {b} \sqrt [4]{-a^2+b^2}}\right ) \sin (c+d x)}{\sqrt {1-\cos ^2(c+d x)} (a+b \sin (c+d x))}-\frac {\left (3 a^3 b-8 a b^3\right ) \left (a+b \sqrt {1-\cos ^2(c+d x)}\right ) \left (8 b^{5/2} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{2},1,\frac {7}{4},\cos ^2(c+d x),\frac {b^2 \cos ^2(c+d x)}{-a^2+b^2}\right ) \cos ^{\frac {3}{2}}(c+d x)+3 \sqrt {2} a \left (a^2-b^2\right )^{3/4} \left (2 \arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-2 \arctan \left (1+\frac {\sqrt {2} \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-\log \left (\sqrt {a^2-b^2}-\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\cos (c+d x)}+b \cos (c+d x)\right )+\log \left (\sqrt {a^2-b^2}+\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\cos (c+d x)}+b \cos (c+d x)\right )\right )\right ) \sin ^2(c+d x)}{12 b^{3/2} \left (-a^2+b^2\right ) \left (1-\cos ^2(c+d x)\right ) (a+b \sin (c+d x))}\right )}{5 (a-b)^2 (a+b)^2 d (e \cos (c+d x))^{7/2}}+\frac {\cos ^4(c+d x) \left (\frac {2 \sec ^3(c+d x) (-b+a \sin (c+d x))}{5 \left (a^2-b^2\right )}+\frac {2 \sec (c+d x) \left (5 b^3+3 a^3 \sin (c+d x)-8 a b^2 \sin (c+d x)\right )}{5 \left (a^2-b^2\right )^2}\right )}{d (e \cos (c+d x))^{7/2}} \]
-1/5*(Cos[c + d*x]^(7/2)*((-2*(3*a^4 - 8*a^2*b^2 - 5*b^4)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((a*AppellF1[3/4, 1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2))/(3*(a^2 - b^2)) + ((1/8 + I/8 )*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - L og[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x] ] + I*b*Cos[c + d*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2 )^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/ 4)))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(a + b*Sin[c + d*x])) - ((3*a ^3*b - 8*a*b^3)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*(8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d *x]^(3/2) + 3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*S qrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt [Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b] *(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]] + Log[Sqrt[a^2 - b ^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x ]]))*Sin[c + d*x]^2)/(12*b^(3/2)*(-a^2 + b^2)*(1 - Cos[c + d*x]^2)*(a + b* Sin[c + d*x]))))/((a - b)^2*(a + b)^2*d*(e*Cos[c + d*x])^(7/2)) + (Cos[c + d*x]^4*((2*Sec[c + d*x]^3*(-b + a*Sin[c + d*x]))/(5*(a^2 - b^2)) + (2*Sec [c + d*x]*(5*b^3 + 3*a^3*Sin[c + d*x] - 8*a*b^2*Sin[c + d*x]))/(5*(a^2 ...
Time = 2.48 (sec) , antiderivative size = 479, normalized size of antiderivative = 0.99, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.840, Rules used = {3042, 3175, 27, 3042, 3345, 27, 3042, 3346, 3042, 3121, 3042, 3119, 3180, 266, 827, 218, 221, 3042, 3286, 3042, 3284}
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.
| \(\displaystyle \int \frac {1}{(e \cos (c+d x))^{7/2} (a+b \sin (c+d x))} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}dx\) |
| \(\Big \downarrow \) 3175 |
| \(\displaystyle -\frac {2 \int -\frac {3 a^2+3 b \sin (c+d x) a-5 b^2}{2 (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}dx}{5 e^2 \left (a^2-b^2\right )}-\frac {2 (b-a \sin (c+d x))}{5 d e \left (a^2-b^2\right ) (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {3 a^2+3 b \sin (c+d x) a-5 b^2}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}dx}{5 e^2 \left (a^2-b^2\right )}-\frac {2 (b-a \sin (c+d x))}{5 d e \left (a^2-b^2\right ) (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {3 a^2+3 b \sin (c+d x) a-5 b^2}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}dx}{5 e^2 \left (a^2-b^2\right )}-\frac {2 (b-a \sin (c+d x))}{5 d e \left (a^2-b^2\right ) (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3345 |
| \(\displaystyle \frac {\frac {2 \left (a \left (3 a^2-8 b^2\right ) \sin (c+d x)+5 b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}-\frac {2 \int \frac {\sqrt {e \cos (c+d x)} \left (3 a^4-8 b^2 a^2+b \left (3 a^2-8 b^2\right ) \sin (c+d x) a-5 b^4\right )}{2 (a+b \sin (c+d x))}dx}{e^2 \left (a^2-b^2\right )}}{5 e^2 \left (a^2-b^2\right )}-\frac {2 (b-a \sin (c+d x))}{5 d e \left (a^2-b^2\right ) (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 \left (a \left (3 a^2-8 b^2\right ) \sin (c+d x)+5 b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}-\frac {\int \frac {\sqrt {e \cos (c+d x)} \left (3 a^4-8 b^2 a^2+b \left (3 a^2-8 b^2\right ) \sin (c+d x) a-5 b^4\right )}{a+b \sin (c+d x)}dx}{e^2 \left (a^2-b^2\right )}}{5 e^2 \left (a^2-b^2\right )}-\frac {2 (b-a \sin (c+d x))}{5 d e \left (a^2-b^2\right ) (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 \left (a \left (3 a^2-8 b^2\right ) \sin (c+d x)+5 b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}-\frac {\int \frac {\sqrt {e \cos (c+d x)} \left (3 a^4-8 b^2 a^2+b \left (3 a^2-8 b^2\right ) \sin (c+d x) a-5 b^4\right )}{a+b \sin (c+d x)}dx}{e^2 \left (a^2-b^2\right )}}{5 e^2 \left (a^2-b^2\right )}-\frac {2 (b-a \sin (c+d x))}{5 d e \left (a^2-b^2\right ) (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3346 |
| \(\displaystyle \frac {\frac {2 \left (a \left (3 a^2-8 b^2\right ) \sin (c+d x)+5 b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}-\frac {a \left (3 a^2-8 b^2\right ) \int \sqrt {e \cos (c+d x)}dx-5 b^4 \int \frac {\sqrt {e \cos (c+d x)}}{a+b \sin (c+d x)}dx}{e^2 \left (a^2-b^2\right )}}{5 e^2 \left (a^2-b^2\right )}-\frac {2 (b-a \sin (c+d x))}{5 d e \left (a^2-b^2\right ) (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 \left (a \left (3 a^2-8 b^2\right ) \sin (c+d x)+5 b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}-\frac {a \left (3 a^2-8 b^2\right ) \int \sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}dx-5 b^4 \int \frac {\sqrt {e \cos (c+d x)}}{a+b \sin (c+d x)}dx}{e^2 \left (a^2-b^2\right )}}{5 e^2 \left (a^2-b^2\right )}-\frac {2 (b-a \sin (c+d x))}{5 d e \left (a^2-b^2\right ) (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {\frac {2 \left (a \left (3 a^2-8 b^2\right ) \sin (c+d x)+5 b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}-\frac {\frac {a \left (3 a^2-8 b^2\right ) \sqrt {e \cos (c+d x)} \int \sqrt {\cos (c+d x)}dx}{\sqrt {\cos (c+d x)}}-5 b^4 \int \frac {\sqrt {e \cos (c+d x)}}{a+b \sin (c+d x)}dx}{e^2 \left (a^2-b^2\right )}}{5 e^2 \left (a^2-b^2\right )}-\frac {2 (b-a \sin (c+d x))}{5 d e \left (a^2-b^2\right ) (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 \left (a \left (3 a^2-8 b^2\right ) \sin (c+d x)+5 b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}-\frac {\frac {a \left (3 a^2-8 b^2\right ) \sqrt {e \cos (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{\sqrt {\cos (c+d x)}}-5 b^4 \int \frac {\sqrt {e \cos (c+d x)}}{a+b \sin (c+d x)}dx}{e^2 \left (a^2-b^2\right )}}{5 e^2 \left (a^2-b^2\right )}-\frac {2 (b-a \sin (c+d x))}{5 d e \left (a^2-b^2\right ) (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {\frac {2 \left (a \left (3 a^2-8 b^2\right ) \sin (c+d x)+5 b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}-\frac {\frac {2 a \left (3 a^2-8 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}-5 b^4 \int \frac {\sqrt {e \cos (c+d x)}}{a+b \sin (c+d x)}dx}{e^2 \left (a^2-b^2\right )}}{5 e^2 \left (a^2-b^2\right )}-\frac {2 (b-a \sin (c+d x))}{5 d e \left (a^2-b^2\right ) (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3180 |
| \(\displaystyle \frac {\frac {2 \left (a \left (3 a^2-8 b^2\right ) \sin (c+d x)+5 b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}-\frac {\frac {2 a \left (3 a^2-8 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}-5 b^4 \left (\frac {b e \int \frac {\sqrt {e \cos (c+d x)}}{b^2 \cos ^2(c+d x) e^2+\left (a^2-b^2\right ) e^2}d(e \cos (c+d x))}{d}-\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {b^2-a^2}-b \cos (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (b \cos (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}\right )}{e^2 \left (a^2-b^2\right )}}{5 e^2 \left (a^2-b^2\right )}-\frac {2 (b-a \sin (c+d x))}{5 d e \left (a^2-b^2\right ) (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {\frac {2 \left (a \left (3 a^2-8 b^2\right ) \sin (c+d x)+5 b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}-\frac {\frac {2 a \left (3 a^2-8 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}-5 b^4 \left (\frac {2 b e \int \frac {e^2 \cos ^2(c+d x)}{b^2 e^4 \cos ^4(c+d x)+\left (a^2-b^2\right ) e^2}d\sqrt {e \cos (c+d x)}}{d}-\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {b^2-a^2}-b \cos (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (b \cos (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}\right )}{e^2 \left (a^2-b^2\right )}}{5 e^2 \left (a^2-b^2\right )}-\frac {2 (b-a \sin (c+d x))}{5 d e \left (a^2-b^2\right ) (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle \frac {\frac {2 \left (a \left (3 a^2-8 b^2\right ) \sin (c+d x)+5 b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}-\frac {\frac {2 a \left (3 a^2-8 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}-5 b^4 \left (\frac {2 b e \left (\frac {\int \frac {1}{b e^2 \cos ^2(c+d x)+\sqrt {b^2-a^2} e}d\sqrt {e \cos (c+d x)}}{2 b}-\frac {\int \frac {1}{\sqrt {b^2-a^2} e-b e^2 \cos ^2(c+d x)}d\sqrt {e \cos (c+d x)}}{2 b}\right )}{d}-\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {b^2-a^2}-b \cos (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (b \cos (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}\right )}{e^2 \left (a^2-b^2\right )}}{5 e^2 \left (a^2-b^2\right )}-\frac {2 (b-a \sin (c+d x))}{5 d e \left (a^2-b^2\right ) (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {2 \left (a \left (3 a^2-8 b^2\right ) \sin (c+d x)+5 b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}-\frac {\frac {2 a \left (3 a^2-8 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}-5 b^4 \left (\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\int \frac {1}{\sqrt {b^2-a^2} e-b e^2 \cos ^2(c+d x)}d\sqrt {e \cos (c+d x)}}{2 b}\right )}{d}-\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {b^2-a^2}-b \cos (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (b \cos (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}\right )}{e^2 \left (a^2-b^2\right )}}{5 e^2 \left (a^2-b^2\right )}-\frac {2 (b-a \sin (c+d x))}{5 d e \left (a^2-b^2\right ) (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {2 \left (a \left (3 a^2-8 b^2\right ) \sin (c+d x)+5 b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}-\frac {\frac {2 a \left (3 a^2-8 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}-5 b^4 \left (-\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {b^2-a^2}-b \cos (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (b \cos (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}+\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d}\right )}{e^2 \left (a^2-b^2\right )}}{5 e^2 \left (a^2-b^2\right )}-\frac {2 (b-a \sin (c+d x))}{5 d e \left (a^2-b^2\right ) (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 \left (a \left (3 a^2-8 b^2\right ) \sin (c+d x)+5 b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}-\frac {\frac {2 a \left (3 a^2-8 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}-5 b^4 \left (-\frac {a e \int \frac {1}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )} \left (\sqrt {b^2-a^2}-b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )} \left (b \sin \left (c+d x+\frac {\pi }{2}\right )+\sqrt {b^2-a^2}\right )}dx}{2 b}+\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d}\right )}{e^2 \left (a^2-b^2\right )}}{5 e^2 \left (a^2-b^2\right )}-\frac {2 (b-a \sin (c+d x))}{5 d e \left (a^2-b^2\right ) (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle \frac {\frac {2 \left (a \left (3 a^2-8 b^2\right ) \sin (c+d x)+5 b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}-\frac {\frac {2 a \left (3 a^2-8 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}-5 b^4 \left (-\frac {a e \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)} \left (\sqrt {b^2-a^2}-b \cos (c+d x)\right )}dx}{2 b \sqrt {e \cos (c+d x)}}+\frac {a e \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)} \left (b \cos (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b \sqrt {e \cos (c+d x)}}+\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d}\right )}{e^2 \left (a^2-b^2\right )}}{5 e^2 \left (a^2-b^2\right )}-\frac {2 (b-a \sin (c+d x))}{5 d e \left (a^2-b^2\right ) (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 \left (a \left (3 a^2-8 b^2\right ) \sin (c+d x)+5 b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}-\frac {\frac {2 a \left (3 a^2-8 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}-5 b^4 \left (-\frac {a e \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (\sqrt {b^2-a^2}-b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{2 b \sqrt {e \cos (c+d x)}}+\frac {a e \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (b \sin \left (c+d x+\frac {\pi }{2}\right )+\sqrt {b^2-a^2}\right )}dx}{2 b \sqrt {e \cos (c+d x)}}+\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d}\right )}{e^2 \left (a^2-b^2\right )}}{5 e^2 \left (a^2-b^2\right )}-\frac {2 (b-a \sin (c+d x))}{5 d e \left (a^2-b^2\right ) (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {\frac {2 \left (a \left (3 a^2-8 b^2\right ) \sin (c+d x)+5 b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}-\frac {\frac {2 a \left (3 a^2-8 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}-5 b^4 \left (\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d}+\frac {a e \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {b^2-a^2}},\frac {1}{2} (c+d x),2\right )}{b d \left (b-\sqrt {b^2-a^2}\right ) \sqrt {e \cos (c+d x)}}+\frac {a e \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {b^2-a^2}},\frac {1}{2} (c+d x),2\right )}{b d \left (\sqrt {b^2-a^2}+b\right ) \sqrt {e \cos (c+d x)}}\right )}{e^2 \left (a^2-b^2\right )}}{5 e^2 \left (a^2-b^2\right )}-\frac {2 (b-a \sin (c+d x))}{5 d e \left (a^2-b^2\right ) (e \cos (c+d x))^{5/2}}\) |
(-2*(b - a*Sin[c + d*x]))/(5*(a^2 - b^2)*d*e*(e*Cos[c + d*x])^(5/2)) + (-( ((2*a*(3*a^2 - 8*b^2)*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(d*S qrt[Cos[c + d*x]]) - 5*b^4*((2*b*e*(ArcTan[(Sqrt[b]*Sqrt[e]*Cos[c + d*x])/ (-a^2 + b^2)^(1/4)]/(2*b^(3/2)*(-a^2 + b^2)^(1/4)*Sqrt[e]) - ArcTanh[(Sqrt [b]*Sqrt[e]*Cos[c + d*x])/(-a^2 + b^2)^(1/4)]/(2*b^(3/2)*(-a^2 + b^2)^(1/4 )*Sqrt[e])))/d + (a*e*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(b*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) + (a*e*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d *x)/2, 2])/(b*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]])))/((a^2 - b^2 )*e^2)) + (2*(5*b^3 + a*(3*a^2 - 8*b^2)*Sin[c + d*x]))/((a^2 - b^2)*d*e*Sq rt[e*Cos[c + d*x]]))/(5*(a^2 - b^2)*e^2)
3.6.83.3.1 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
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]
Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Sin[c + d*x]) ^n/Sin[c + d*x]^n Int[Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && Lt Q[-1, n, 1] && IntegerQ[2*n]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^ (m + 1)*((b - a*Sin[e + f*x])/(f*g*(a^2 - b^2)*(p + 1))), x] + Simp[1/(g^2* (a^2 - b^2)*(p + 1)) Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^m* (a^2*(p + 2) - b^2*(m + p + 2) + a*b*(m + p + 3)*Sin[e + f*x]), x], x] /; F reeQ[{a, b, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && LtQ[p, -1] && IntegersQ [2*m, 2*p]
Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.)]/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_ )]), x_Symbol] :> With[{q = Rt[-a^2 + b^2, 2]}, Simp[a*(g/(2*b)) Int[1/(S qrt[g*Cos[e + f*x]]*(q + b*Cos[e + f*x])), x], x] + (-Simp[a*(g/(2*b)) In t[1/(Sqrt[g*Cos[e + f*x]]*(q - b*Cos[e + f*x])), x], x] + Simp[b*(g/f) Su bst[Int[Sqrt[x]/(g^2*(a^2 - b^2) + b^2*x^2), x], x, g*Cos[e + f*x]], x])] / ; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0]
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[ 2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(c + d))], x] /; FreeQ[{a, b, c , d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 0]
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt [c + d*Sin[e + f*x]] Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d/(c + d))*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] && NeQ[c^2 - d^2, 0] && !GtQ[c + d, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(g*Co s[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1)*((b*c - a*d - (a*c - b*d)* Sin[e + f*x])/(f*g*(a^2 - b^2)*(p + 1))), x] + Simp[1/(g^2*(a^2 - b^2)*(p + 1)) Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^m*Simp[c*(a^2*(p + 2) - b^2*(m + p + 2)) + a*b*d*m + b*(a*c - b*d)*(m + p + 3)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && Lt Q[p, -1] && IntegerQ[2*m]
Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((c_.) + (d_.)*sin[(e_.) + (f_.)* (x_)]))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d/b Int [(g*Cos[e + f*x])^p, x], x] + Simp[(b*c - a*d)/b Int[(g*Cos[e + f*x])^p/( a + b*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[a^2 - b^2, 0]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 9.45 (sec) , antiderivative size = 1723, normalized size of antiderivative = 3.55
(4/e^3*b*(-1/60/(4*a^2-4*b^2)/e/(4*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c) +6*2^(1/2)*sin(1/2*d*x+1/2*c)^2-10*cos(1/2*d*x+1/2*c)-7*2^(1/2))*(-2*sin(1 /2*d*x+1/2*c)^2*e+e)^(1/2)*(-4*2^(1/2)*sin(1/2*d*x+1/2*c)^2+12*cos(1/2*d*x +1/2*c)+11*2^(1/2))-1/60/(4*a^2-4*b^2)/e/(4*sin(1/2*d*x+1/2*c)^2*cos(1/2*d *x+1/2*c)-6*2^(1/2)*sin(1/2*d*x+1/2*c)^2-10*cos(1/2*d*x+1/2*c)+7*2^(1/2))* (-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*(4*2^(1/2)*sin(1/2*d*x+1/2*c)^2+12*cos (1/2*d*x+1/2*c)-11*2^(1/2))+1/6*2^(1/2)/(16*a^2-16*b^2)*(-2*sin(1/2*d*x+1/ 2*c)^2*e+e)^(1/2)*(2^(1/2)+cos(1/2*d*x+1/2*c))/e/(2*cos(1/2*d*x+1/2*c)*2^( 1/2)-2*sin(1/2*d*x+1/2*c)^2+3)+1/8/(a^2-b^2)^2*b^2/e*2^(1/2)/(cos(1/2*d*x+ 1/2*c)-1/2*2^(1/2))*(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)-1/8/(a^2-b^2)^2*b^ 2/e*2^(1/2)/(cos(1/2*d*x+1/2*c)+1/2*2^(1/2))*(-2*sin(1/2*d*x+1/2*c)^2*e+e) ^(1/2)+1/6*2^(1/2)/(16*a^2-16*b^2)*(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*(-2 ^(1/2)+cos(1/2*d*x+1/2*c))/e/(2*cos(1/2*d*x+1/2*c)*2^(1/2)+2*sin(1/2*d*x+1 /2*c)^2-3)+1/16*b^2/(a-b)^2/(a+b)^2/(e^2*(a^2-b^2)/b^2)^(1/4)*2^(1/2)*(ln( (2*e*cos(1/2*d*x+1/2*c)^2-e-(e^2*(a^2-b^2)/b^2)^(1/4)*(2*e*cos(1/2*d*x+1/2 *c)^2-e)^(1/2)*2^(1/2)+(e^2*(a^2-b^2)/b^2)^(1/2))/(2*e*cos(1/2*d*x+1/2*c)^ 2-e+(e^2*(a^2-b^2)/b^2)^(1/4)*(2*e*cos(1/2*d*x+1/2*c)^2-e)^(1/2)*2^(1/2)+( e^2*(a^2-b^2)/b^2)^(1/2)))+2*arctan((2^(1/2)*(2*e*cos(1/2*d*x+1/2*c)^2-e)^ (1/2)+(e^2*(a^2-b^2)/b^2)^(1/4))/(e^2*(a^2-b^2)/b^2)^(1/4))+2*arctan((2^(1 /2)*(2*e*cos(1/2*d*x+1/2*c)^2-e)^(1/2)-(e^2*(a^2-b^2)/b^2)^(1/4))/(e^2*...
Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+b \sin (c+d x))} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+b \sin (c+d x))} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+b \sin (c+d x))} \, dx=\int { \frac {1}{\left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}} \,d x } \]
\[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+b \sin (c+d x))} \, dx=\int { \frac {1}{\left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}} \,d x } \]
Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+b \sin (c+d x))} \, dx=\int \frac {1}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{7/2}\,\left (a+b\,\sin \left (c+d\,x\right )\right )} \,d x \]