Integrand size = 27, antiderivative size = 390 \[ \int (a+b \cos (x)+c \sin (x))^{5/2} (d+b e \cos (x)+c e \sin (x)) \, dx=\frac {2 \left (161 a^2 d+63 \left (b^2+c^2\right ) d+15 a^3 e+145 a \left (b^2+c^2\right ) e\right ) E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right ) \sqrt {a+b \cos (x)+c \sin (x)}}{105 \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}-\frac {2 \left (a^2-b^2-c^2\right ) \left (56 a d+15 a^2 e+25 \left (b^2+c^2\right ) e\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right ),\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right ) \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}{105 \sqrt {a+b \cos (x)+c \sin (x)}}-\frac {2}{7} (a+b \cos (x)+c \sin (x))^{5/2} (c e \cos (x)-b e \sin (x))-\frac {2}{35} (a+b \cos (x)+c \sin (x))^{3/2} (c (7 d+5 a e) \cos (x)-b (7 d+5 a e) \sin (x))-\frac {2}{105} \sqrt {a+b \cos (x)+c \sin (x)} \left (c \left (56 a d+15 a^2 e+25 \left (b^2+c^2\right ) e\right ) \cos (x)-b \left (56 a d+15 a^2 e+25 \left (b^2+c^2\right ) e\right ) \sin (x)\right ) \] Output:
2/105*(161*a^2*d+63*(b^2+c^2)*d+15*a^3*e+145*a*(b^2+c^2)*e)*EllipticE(sin( 1/2*x-1/2*arctan(c,b)),2^(1/2)*((b^2+c^2)^(1/2)/(a+(b^2+c^2)^(1/2)))^(1/2) )*(a+b*cos(x)+c*sin(x))^(1/2)/((a+b*cos(x)+c*sin(x))/(a+(b^2+c^2)^(1/2)))^ (1/2)-2/105*(a^2-b^2-c^2)*(56*a*d+15*a^2*e+25*(b^2+c^2)*e)*InverseJacobiAM (1/2*x-1/2*arctan(c,b),2^(1/2)*((b^2+c^2)^(1/2)/(a+(b^2+c^2)^(1/2)))^(1/2) )*((a+b*cos(x)+c*sin(x))/(a+(b^2+c^2)^(1/2)))^(1/2)/(a+b*cos(x)+c*sin(x))^ (1/2)-2/7*(a+b*cos(x)+c*sin(x))^(5/2)*(c*e*cos(x)-b*e*sin(x))-2/35*(a+b*co s(x)+c*sin(x))^(3/2)*(c*(5*a*e+7*d)*cos(x)-b*(5*a*e+7*d)*sin(x))-2/105*(a+ b*cos(x)+c*sin(x))^(1/2)*(c*(56*a*d+15*a^2*e+25*(b^2+c^2)*e)*cos(x)-b*(56* a*d+15*a^2*e+25*(b^2+c^2)*e)*sin(x))
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 6.63 (sec) , antiderivative size = 7823, normalized size of antiderivative = 20.06 \[ \int (a+b \cos (x)+c \sin (x))^{5/2} (d+b e \cos (x)+c e \sin (x)) \, dx=\text {Result too large to show} \] Input:
Integrate[(a + b*Cos[x] + c*Sin[x])^(5/2)*(d + b*e*Cos[x] + c*e*Sin[x]),x]
Output:
Result too large to show
Time = 2.31 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.09, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 3625, 27, 3042, 3625, 27, 3042, 3625, 27, 3042, 3628, 3042, 3598, 3042, 3132, 3606, 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.
| \(\displaystyle \int (a+b \cos (x)+c \sin (x))^{5/2} (b e \cos (x)+c e \sin (x)+d) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a+b \cos (x)+c \sin (x))^{5/2} (b e \cos (x)+c e \sin (x)+d)dx\) |
| \(\Big \downarrow \) 3625 |
| \(\displaystyle \frac {2 \int \frac {1}{2} (a+b \cos (x)+c \sin (x))^{3/2} \left (a \left (7 a d+5 \left (b^2+c^2\right ) e\right )+a b (7 d+5 a e) \cos (x)+a c (7 d+5 a e) \sin (x)\right )dx}{7 a}-\frac {2}{7} (a+b \cos (x)+c \sin (x))^{5/2} (c e \cos (x)-b e \sin (x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (a+b \cos (x)+c \sin (x))^{3/2} \left (a \left (7 a d+5 \left (b^2+c^2\right ) e\right )+a b (7 d+5 a e) \cos (x)+a c (7 d+5 a e) \sin (x)\right )dx}{7 a}-\frac {2}{7} (a+b \cos (x)+c \sin (x))^{5/2} (c e \cos (x)-b e \sin (x))\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int (a+b \cos (x)+c \sin (x))^{3/2} \left (a \left (7 a d+5 \left (b^2+c^2\right ) e\right )+a b (7 d+5 a e) \cos (x)+a c (7 d+5 a e) \sin (x)\right )dx}{7 a}-\frac {2}{7} (a+b \cos (x)+c \sin (x))^{5/2} (c e \cos (x)-b e \sin (x))\) |
| \(\Big \downarrow \) 3625 |
| \(\displaystyle \frac {\frac {2 \int \frac {1}{2} \sqrt {a+b \cos (x)+c \sin (x)} \left (\left (35 d a^2+40 \left (b^2+c^2\right ) e a+21 \left (b^2+c^2\right ) d\right ) a^2+b \left (15 e a^2+56 d a+25 \left (b^2+c^2\right ) e\right ) \cos (x) a^2+c \left (15 e a^2+56 d a+25 \left (b^2+c^2\right ) e\right ) \sin (x) a^2\right )dx}{5 a}-\frac {2}{5} (a+b \cos (x)+c \sin (x))^{3/2} (a c \cos (x) (5 a e+7 d)-a b \sin (x) (5 a e+7 d))}{7 a}-\frac {2}{7} (a+b \cos (x)+c \sin (x))^{5/2} (c e \cos (x)-b e \sin (x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \sqrt {a+b \cos (x)+c \sin (x)} \left (\left (35 d a^2+40 \left (b^2+c^2\right ) e a+21 \left (b^2+c^2\right ) d\right ) a^2+b \left (15 e a^2+56 d a+25 \left (b^2+c^2\right ) e\right ) \cos (x) a^2+c \left (15 e a^2+56 d a+25 \left (b^2+c^2\right ) e\right ) \sin (x) a^2\right )dx}{5 a}-\frac {2}{5} (a+b \cos (x)+c \sin (x))^{3/2} (a c \cos (x) (5 a e+7 d)-a b \sin (x) (5 a e+7 d))}{7 a}-\frac {2}{7} (a+b \cos (x)+c \sin (x))^{5/2} (c e \cos (x)-b e \sin (x))\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \sqrt {a+b \cos (x)+c \sin (x)} \left (\left (35 d a^2+40 \left (b^2+c^2\right ) e a+21 \left (b^2+c^2\right ) d\right ) a^2+b \left (15 e a^2+56 d a+25 \left (b^2+c^2\right ) e\right ) \cos (x) a^2+c \left (15 e a^2+56 d a+25 \left (b^2+c^2\right ) e\right ) \sin (x) a^2\right )dx}{5 a}-\frac {2}{5} (a+b \cos (x)+c \sin (x))^{3/2} (a c \cos (x) (5 a e+7 d)-a b \sin (x) (5 a e+7 d))}{7 a}-\frac {2}{7} (a+b \cos (x)+c \sin (x))^{5/2} (c e \cos (x)-b e \sin (x))\) |
| \(\Big \downarrow \) 3625 |
| \(\displaystyle \frac {\frac {\frac {2 \int \frac {\left (105 d a^3+135 \left (b^2+c^2\right ) e a^2+119 \left (b^2+c^2\right ) d a+25 \left (b^2+c^2\right )^2 e\right ) a^3+b \left (15 e a^3+161 d a^2+145 \left (b^2+c^2\right ) e a+63 \left (b^2+c^2\right ) d\right ) \cos (x) a^3+c \left (15 e a^3+161 d a^2+145 \left (b^2+c^2\right ) e a+63 \left (b^2+c^2\right ) d\right ) \sin (x) a^3}{2 \sqrt {a+b \cos (x)+c \sin (x)}}dx}{3 a}-\frac {2}{3} \sqrt {a+b \cos (x)+c \sin (x)} \left (a^2 c \cos (x) \left (15 a^2 e+56 a d+25 e \left (b^2+c^2\right )\right )-a^2 b \sin (x) \left (15 a^2 e+56 a d+25 e \left (b^2+c^2\right )\right )\right )}{5 a}-\frac {2}{5} (a+b \cos (x)+c \sin (x))^{3/2} (a c \cos (x) (5 a e+7 d)-a b \sin (x) (5 a e+7 d))}{7 a}-\frac {2}{7} (a+b \cos (x)+c \sin (x))^{5/2} (c e \cos (x)-b e \sin (x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\left (105 d a^3+135 \left (b^2+c^2\right ) e a^2+119 \left (b^2+c^2\right ) d a+25 \left (b^2+c^2\right )^2 e\right ) a^3+b \left (15 e a^3+161 d a^2+145 \left (b^2+c^2\right ) e a+63 \left (b^2+c^2\right ) d\right ) \cos (x) a^3+c \left (15 e a^3+161 d a^2+145 \left (b^2+c^2\right ) e a+63 \left (b^2+c^2\right ) d\right ) \sin (x) a^3}{\sqrt {a+b \cos (x)+c \sin (x)}}dx}{3 a}-\frac {2}{3} \sqrt {a+b \cos (x)+c \sin (x)} \left (a^2 c \cos (x) \left (15 a^2 e+56 a d+25 e \left (b^2+c^2\right )\right )-a^2 b \sin (x) \left (15 a^2 e+56 a d+25 e \left (b^2+c^2\right )\right )\right )}{5 a}-\frac {2}{5} (a+b \cos (x)+c \sin (x))^{3/2} (a c \cos (x) (5 a e+7 d)-a b \sin (x) (5 a e+7 d))}{7 a}-\frac {2}{7} (a+b \cos (x)+c \sin (x))^{5/2} (c e \cos (x)-b e \sin (x))\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\left (105 d a^3+135 \left (b^2+c^2\right ) e a^2+119 \left (b^2+c^2\right ) d a+25 \left (b^2+c^2\right )^2 e\right ) a^3+b \left (15 e a^3+161 d a^2+145 \left (b^2+c^2\right ) e a+63 \left (b^2+c^2\right ) d\right ) \cos (x) a^3+c \left (15 e a^3+161 d a^2+145 \left (b^2+c^2\right ) e a+63 \left (b^2+c^2\right ) d\right ) \sin (x) a^3}{\sqrt {a+b \cos (x)+c \sin (x)}}dx}{3 a}-\frac {2}{3} \sqrt {a+b \cos (x)+c \sin (x)} \left (a^2 c \cos (x) \left (15 a^2 e+56 a d+25 e \left (b^2+c^2\right )\right )-a^2 b \sin (x) \left (15 a^2 e+56 a d+25 e \left (b^2+c^2\right )\right )\right )}{5 a}-\frac {2}{5} (a+b \cos (x)+c \sin (x))^{3/2} (a c \cos (x) (5 a e+7 d)-a b \sin (x) (5 a e+7 d))}{7 a}-\frac {2}{7} (a+b \cos (x)+c \sin (x))^{5/2} (c e \cos (x)-b e \sin (x))\) |
| \(\Big \downarrow \) 3628 |
| \(\displaystyle \frac {\frac {\frac {a^3 \left (15 a^3 e+161 a^2 d+145 a e \left (b^2+c^2\right )+63 d \left (b^2+c^2\right )\right ) \int \sqrt {a+b \cos (x)+c \sin (x)}dx-a^3 \left (a^2-b^2-c^2\right ) \left (15 a^2 e+56 a d+25 e \left (b^2+c^2\right )\right ) \int \frac {1}{\sqrt {a+b \cos (x)+c \sin (x)}}dx}{3 a}-\frac {2}{3} \sqrt {a+b \cos (x)+c \sin (x)} \left (a^2 c \cos (x) \left (15 a^2 e+56 a d+25 e \left (b^2+c^2\right )\right )-a^2 b \sin (x) \left (15 a^2 e+56 a d+25 e \left (b^2+c^2\right )\right )\right )}{5 a}-\frac {2}{5} (a+b \cos (x)+c \sin (x))^{3/2} (a c \cos (x) (5 a e+7 d)-a b \sin (x) (5 a e+7 d))}{7 a}-\frac {2}{7} (a+b \cos (x)+c \sin (x))^{5/2} (c e \cos (x)-b e \sin (x))\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {a^3 \left (15 a^3 e+161 a^2 d+145 a e \left (b^2+c^2\right )+63 d \left (b^2+c^2\right )\right ) \int \sqrt {a+b \cos (x)+c \sin (x)}dx-a^3 \left (a^2-b^2-c^2\right ) \left (15 a^2 e+56 a d+25 e \left (b^2+c^2\right )\right ) \int \frac {1}{\sqrt {a+b \cos (x)+c \sin (x)}}dx}{3 a}-\frac {2}{3} \sqrt {a+b \cos (x)+c \sin (x)} \left (a^2 c \cos (x) \left (15 a^2 e+56 a d+25 e \left (b^2+c^2\right )\right )-a^2 b \sin (x) \left (15 a^2 e+56 a d+25 e \left (b^2+c^2\right )\right )\right )}{5 a}-\frac {2}{5} (a+b \cos (x)+c \sin (x))^{3/2} (a c \cos (x) (5 a e+7 d)-a b \sin (x) (5 a e+7 d))}{7 a}-\frac {2}{7} (a+b \cos (x)+c \sin (x))^{5/2} (c e \cos (x)-b e \sin (x))\) |
| \(\Big \downarrow \) 3598 |
| \(\displaystyle \frac {\frac {\frac {\frac {a^3 \left (15 a^3 e+161 a^2 d+145 a e \left (b^2+c^2\right )+63 d \left (b^2+c^2\right )\right ) \sqrt {a+b \cos (x)+c \sin (x)} \int \sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \cos \left (x-\tan ^{-1}(b,c)\right )}{a+\sqrt {b^2+c^2}}}dx}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}-a^3 \left (a^2-b^2-c^2\right ) \left (15 a^2 e+56 a d+25 e \left (b^2+c^2\right )\right ) \int \frac {1}{\sqrt {a+b \cos (x)+c \sin (x)}}dx}{3 a}-\frac {2}{3} \sqrt {a+b \cos (x)+c \sin (x)} \left (a^2 c \cos (x) \left (15 a^2 e+56 a d+25 e \left (b^2+c^2\right )\right )-a^2 b \sin (x) \left (15 a^2 e+56 a d+25 e \left (b^2+c^2\right )\right )\right )}{5 a}-\frac {2}{5} (a+b \cos (x)+c \sin (x))^{3/2} (a c \cos (x) (5 a e+7 d)-a b \sin (x) (5 a e+7 d))}{7 a}-\frac {2}{7} (a+b \cos (x)+c \sin (x))^{5/2} (c e \cos (x)-b e \sin (x))\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {a^3 \left (15 a^3 e+161 a^2 d+145 a e \left (b^2+c^2\right )+63 d \left (b^2+c^2\right )\right ) \sqrt {a+b \cos (x)+c \sin (x)} \int \sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \sin \left (x-\tan ^{-1}(b,c)+\frac {\pi }{2}\right )}{a+\sqrt {b^2+c^2}}}dx}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}-a^3 \left (a^2-b^2-c^2\right ) \left (15 a^2 e+56 a d+25 e \left (b^2+c^2\right )\right ) \int \frac {1}{\sqrt {a+b \cos (x)+c \sin (x)}}dx}{3 a}-\frac {2}{3} \sqrt {a+b \cos (x)+c \sin (x)} \left (a^2 c \cos (x) \left (15 a^2 e+56 a d+25 e \left (b^2+c^2\right )\right )-a^2 b \sin (x) \left (15 a^2 e+56 a d+25 e \left (b^2+c^2\right )\right )\right )}{5 a}-\frac {2}{5} (a+b \cos (x)+c \sin (x))^{3/2} (a c \cos (x) (5 a e+7 d)-a b \sin (x) (5 a e+7 d))}{7 a}-\frac {2}{7} (a+b \cos (x)+c \sin (x))^{5/2} (c e \cos (x)-b e \sin (x))\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 a^3 \left (15 a^3 e+161 a^2 d+145 a e \left (b^2+c^2\right )+63 d \left (b^2+c^2\right )\right ) \sqrt {a+b \cos (x)+c \sin (x)} E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}-a^3 \left (a^2-b^2-c^2\right ) \left (15 a^2 e+56 a d+25 e \left (b^2+c^2\right )\right ) \int \frac {1}{\sqrt {a+b \cos (x)+c \sin (x)}}dx}{3 a}-\frac {2}{3} \sqrt {a+b \cos (x)+c \sin (x)} \left (a^2 c \cos (x) \left (15 a^2 e+56 a d+25 e \left (b^2+c^2\right )\right )-a^2 b \sin (x) \left (15 a^2 e+56 a d+25 e \left (b^2+c^2\right )\right )\right )}{5 a}-\frac {2}{5} (a+b \cos (x)+c \sin (x))^{3/2} (a c \cos (x) (5 a e+7 d)-a b \sin (x) (5 a e+7 d))}{7 a}-\frac {2}{7} (a+b \cos (x)+c \sin (x))^{5/2} (c e \cos (x)-b e \sin (x))\) |
| \(\Big \downarrow \) 3606 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 a^3 \left (15 a^3 e+161 a^2 d+145 a e \left (b^2+c^2\right )+63 d \left (b^2+c^2\right )\right ) \sqrt {a+b \cos (x)+c \sin (x)} E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}-\frac {a^3 \left (a^2-b^2-c^2\right ) \left (15 a^2 e+56 a d+25 e \left (b^2+c^2\right )\right ) \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}} \int \frac {1}{\sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \cos \left (x-\tan ^{-1}(b,c)\right )}{a+\sqrt {b^2+c^2}}}}dx}{\sqrt {a+b \cos (x)+c \sin (x)}}}{3 a}-\frac {2}{3} \sqrt {a+b \cos (x)+c \sin (x)} \left (a^2 c \cos (x) \left (15 a^2 e+56 a d+25 e \left (b^2+c^2\right )\right )-a^2 b \sin (x) \left (15 a^2 e+56 a d+25 e \left (b^2+c^2\right )\right )\right )}{5 a}-\frac {2}{5} (a+b \cos (x)+c \sin (x))^{3/2} (a c \cos (x) (5 a e+7 d)-a b \sin (x) (5 a e+7 d))}{7 a}-\frac {2}{7} (a+b \cos (x)+c \sin (x))^{5/2} (c e \cos (x)-b e \sin (x))\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 a^3 \left (15 a^3 e+161 a^2 d+145 a e \left (b^2+c^2\right )+63 d \left (b^2+c^2\right )\right ) \sqrt {a+b \cos (x)+c \sin (x)} E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}-\frac {a^3 \left (a^2-b^2-c^2\right ) \left (15 a^2 e+56 a d+25 e \left (b^2+c^2\right )\right ) \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}} \int \frac {1}{\sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \sin \left (x-\tan ^{-1}(b,c)+\frac {\pi }{2}\right )}{a+\sqrt {b^2+c^2}}}}dx}{\sqrt {a+b \cos (x)+c \sin (x)}}}{3 a}-\frac {2}{3} \sqrt {a+b \cos (x)+c \sin (x)} \left (a^2 c \cos (x) \left (15 a^2 e+56 a d+25 e \left (b^2+c^2\right )\right )-a^2 b \sin (x) \left (15 a^2 e+56 a d+25 e \left (b^2+c^2\right )\right )\right )}{5 a}-\frac {2}{5} (a+b \cos (x)+c \sin (x))^{3/2} (a c \cos (x) (5 a e+7 d)-a b \sin (x) (5 a e+7 d))}{7 a}-\frac {2}{7} (a+b \cos (x)+c \sin (x))^{5/2} (c e \cos (x)-b e \sin (x))\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 a^3 \left (15 a^3 e+161 a^2 d+145 a e \left (b^2+c^2\right )+63 d \left (b^2+c^2\right )\right ) \sqrt {a+b \cos (x)+c \sin (x)} E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}-\frac {2 a^3 \left (a^2-b^2-c^2\right ) \left (15 a^2 e+56 a d+25 e \left (b^2+c^2\right )\right ) \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}} \operatorname {EllipticF}\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right ),\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{\sqrt {a+b \cos (x)+c \sin (x)}}}{3 a}-\frac {2}{3} \sqrt {a+b \cos (x)+c \sin (x)} \left (a^2 c \cos (x) \left (15 a^2 e+56 a d+25 e \left (b^2+c^2\right )\right )-a^2 b \sin (x) \left (15 a^2 e+56 a d+25 e \left (b^2+c^2\right )\right )\right )}{5 a}-\frac {2}{5} (a+b \cos (x)+c \sin (x))^{3/2} (a c \cos (x) (5 a e+7 d)-a b \sin (x) (5 a e+7 d))}{7 a}-\frac {2}{7} (a+b \cos (x)+c \sin (x))^{5/2} (c e \cos (x)-b e \sin (x))\) |
Input:
Int[(a + b*Cos[x] + c*Sin[x])^(5/2)*(d + b*e*Cos[x] + c*e*Sin[x]),x]
Output:
(-2*(a + b*Cos[x] + c*Sin[x])^(5/2)*(c*e*Cos[x] - b*e*Sin[x]))/7 + ((-2*(a + b*Cos[x] + c*Sin[x])^(3/2)*(a*c*(7*d + 5*a*e)*Cos[x] - a*b*(7*d + 5*a*e )*Sin[x]))/5 + ((-2*Sqrt[a + b*Cos[x] + c*Sin[x]]*(a^2*c*(56*a*d + 15*a^2* e + 25*(b^2 + c^2)*e)*Cos[x] - a^2*b*(56*a*d + 15*a^2*e + 25*(b^2 + c^2)*e )*Sin[x]))/3 + ((2*a^3*(161*a^2*d + 63*(b^2 + c^2)*d + 15*a^3*e + 145*a*(b ^2 + c^2)*e)*EllipticE[(x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt [b^2 + c^2])]*Sqrt[a + b*Cos[x] + c*Sin[x]])/Sqrt[(a + b*Cos[x] + c*Sin[x] )/(a + Sqrt[b^2 + c^2])] - (2*a^3*(a^2 - b^2 - c^2)*(56*a*d + 15*a^2*e + 2 5*(b^2 + c^2)*e)*EllipticF[(x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[(a + b*Cos[x] + c*Sin[x])/(a + Sqrt[b^2 + c^2])])/S qrt[a + b*Cos[x] + c*Sin[x]])/(3*a))/(5*a))/(7*a)
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[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[Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_ )]], x_Symbol] :> Simp[Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]/Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])] Int[Sqrt[a/(a + S qrt[b^2 + c^2]) + (Sqrt[b^2 + c^2]/(a + Sqrt[b^2 + c^2]))*Cos[d + e*x - Arc Tan[b, c]]], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[b^2 + c^2, 0] && !GtQ[a + Sqrt[b^2 + c^2], 0]
Int[1/Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*( x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sq rt[b^2 + c^2])]/Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]] Int[1/Sqrt[a/(a + Sqrt[b^2 + c^2]) + (Sqrt[b^2 + c^2]/(a + Sqrt[b^2 + c^2]))*Cos[d + e*x - ArcTan[b, c]]], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2 , 0] && NeQ[b^2 + c^2, 0] && !GtQ[a + Sqrt[b^2 + c^2], 0]
Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^ (n_.)*((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_) ]), x_Symbol] :> Simp[(B*c - b*C - a*C*Cos[d + e*x] + a*B*Sin[d + e*x])*((a + b*Cos[d + e*x] + c*Sin[d + e*x])^n/(a*e*(n + 1))), x] + Simp[1/(a*(n + 1 )) Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 1)*Simp[a*(b*B + c*C)*n + a^2*A*(n + 1) + (n*(a^2*B - B*c^2 + b*c*C) + a*b*A*(n + 1))*Cos[d + e*x] + (n*(b*B*c + a^2*C - b^2*C) + a*c*A*(n + 1))*Sin[d + e*x], x], x], x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && GtQ[n, 0] && NeQ[a^2 - b^2 - c^2, 0]
Int[((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_)]) /Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]] , x_Symbol] :> Simp[B/b Int[Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]], x] , x] + Simp[(A*b - a*B)/b Int[1/Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]] , x], x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && EqQ[B*c - b*C, 0] && NeQ[ A*b - a*B, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(3333\) vs. \(2(361)=722\).
Time = 3.31 (sec) , antiderivative size = 3334, normalized size of antiderivative = 8.55
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(3334\) |
| parts | \(\text {Expression too large to display}\) | \(40376\) |
Input:
int((a+b*cos(x)+c*sin(x))^(5/2)*(d+b*e*cos(x)+c*e*sin(x)),x,method=_RETURN VERBOSE)
Output:
(-(-b^2*sin(x-arctan(-b,c))-c^2*sin(x-arctan(-b,c))-a*(b^2+c^2)^(1/2))*cos (x-arctan(-b,c))^2/(b^2+c^2)^(1/2))^(1/2)/(b^2+c^2)*(2*a^2*(b^2+c^2)^(1/2) *(a*b^2*e+a*c^2*e+3*b^2*d+3*c^2*d)*(1/(b^2+c^2)^(1/2)*a+1)*(((b^2+c^2)^(1/ 2)*sin(x-arctan(-b,c))+a)/(a+(b^2+c^2)^(1/2)))^(1/2)*((sin(x-arctan(-b,c)) +1)*(b^2+c^2)^(1/2)/(-a+(b^2+c^2)^(1/2)))^(1/2)*(-(b^2+c^2)^(1/2)*(sin(x-a rctan(-b,c))-1)/(a+(b^2+c^2)^(1/2)))^(1/2)/(cos(x-arctan(-b,c))^2*((b^2+c^ 2)^(1/2)*sin(x-arctan(-b,c))+a))^(1/2)*((-1/(b^2+c^2)^(1/2)*a+1)*EllipticE ((((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)/(a+(b^2+c^2)^(1/2)))^(1/2),((-a- (b^2+c^2)^(1/2))/(-a+(b^2+c^2)^(1/2)))^(1/2))-EllipticF((((b^2+c^2)^(1/2)* sin(x-arctan(-b,c))+a)/(a+(b^2+c^2)^(1/2)))^(1/2),((-a-(b^2+c^2)^(1/2))/(- a+(b^2+c^2)^(1/2)))^(1/2)))+(b^2+c^2)^(1/2)*(3*a*b^4*e+6*a*b^2*c^2*e+3*a*c ^4*e+b^4*d+2*b^2*c^2*d+c^4*d)*(-2/5/(b^2+c^2)^(1/2)*sin(x-arctan(-b,c))*(c os(x-arctan(-b,c))^2*((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a))^(1/2)+8/15/( b^2+c^2)*a*(cos(x-arctan(-b,c))^2*((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)) ^(1/2)+4/15/(b^2+c^2)^(1/2)*a*(1/(b^2+c^2)^(1/2)*a+1)*(((b^2+c^2)^(1/2)*si n(x-arctan(-b,c))+a)/(a+(b^2+c^2)^(1/2)))^(1/2)*((sin(x-arctan(-b,c))+1)*( b^2+c^2)^(1/2)/(-a+(b^2+c^2)^(1/2)))^(1/2)*(-(b^2+c^2)^(1/2)*(sin(x-arctan (-b,c))-1)/(a+(b^2+c^2)^(1/2)))^(1/2)/(cos(x-arctan(-b,c))^2*((b^2+c^2)^(1 /2)*sin(x-arctan(-b,c))+a))^(1/2)*EllipticF((((b^2+c^2)^(1/2)*sin(x-arctan (-b,c))+a)/(a+(b^2+c^2)^(1/2)))^(1/2),((-a-(b^2+c^2)^(1/2))/(-a+(b^2+c^...
Result contains complex when optimal does not.
Time = 0.19 (sec) , antiderivative size = 2010, normalized size of antiderivative = 5.15 \[ \int (a+b \cos (x)+c \sin (x))^{5/2} (d+b e \cos (x)+c e \sin (x)) \, dx=\text {Too large to display} \] Input:
integrate((a+b*cos(x)+c*sin(x))^(5/2)*(d+b*e*cos(x)+c*e*sin(x)),x, algorit hm="fricas")
Output:
-2/315*((7*I*(a^3*b - 33*a*b^3 - 33*a*b*c^2)*d - 7*(33*a*c^3 - (a^3 - 33*a *b^2)*c)*d + 5*I*(6*a^4*b - 23*a^2*b^3 - 15*b^5 - 15*b*c^4 - (23*a^2*b + 3 0*b^3)*c^2)*e - 5*(15*c^5 + (23*a^2 + 30*b^2)*c^3 - (6*a^4 - 23*a^2*b^2 - 15*b^4)*c)*e)*sqrt(1/2*b + 1/2*I*c)*weierstrassPInverse(4/3*(4*a^2*b^2 - 3 *b^4 - 4*a^2*c^2 + 6*I*b*c^3 + 3*c^4 - 2*I*(4*a^2*b - 3*b^3)*c)/(b^4 + 2*b ^2*c^2 + c^4), -8/27*(8*a^3*b^3 - 9*a*b^5 + 27*a*b*c^4 - 9*I*a*c^5 + 2*I*( 4*a^3 + 9*a*b^2)*c^3 - 6*(4*a^3*b - 3*a*b^3)*c^2 - 3*I*(8*a^3*b^2 - 9*a*b^ 4)*c)/(b^6 + 3*b^4*c^2 + 3*b^2*c^4 + c^6), 1/3*(2*a*b - 2*I*a*c + 3*(b^2 + c^2)*cos(x) - 3*(I*b^2 + I*c^2)*sin(x))/(b^2 + c^2)) + (-7*I*(a^3*b - 33* a*b^3 - 33*a*b*c^2)*d - 7*(33*a*c^3 - (a^3 - 33*a*b^2)*c)*d - 5*I*(6*a^4*b - 23*a^2*b^3 - 15*b^5 - 15*b*c^4 - (23*a^2*b + 30*b^3)*c^2)*e - 5*(15*c^5 + (23*a^2 + 30*b^2)*c^3 - (6*a^4 - 23*a^2*b^2 - 15*b^4)*c)*e)*sqrt(1/2*b - 1/2*I*c)*weierstrassPInverse(4/3*(4*a^2*b^2 - 3*b^4 - 4*a^2*c^2 - 6*I*b* c^3 + 3*c^4 + 2*I*(4*a^2*b - 3*b^3)*c)/(b^4 + 2*b^2*c^2 + c^4), -8/27*(8*a ^3*b^3 - 9*a*b^5 + 27*a*b*c^4 + 9*I*a*c^5 - 2*I*(4*a^3 + 9*a*b^2)*c^3 - 6* (4*a^3*b - 3*a*b^3)*c^2 + 3*I*(8*a^3*b^2 - 9*a*b^4)*c)/(b^6 + 3*b^4*c^2 + 3*b^2*c^4 + c^6), 1/3*(2*a*b + 2*I*a*c + 3*(b^2 + c^2)*cos(x) - 3*(-I*b^2 - I*c^2)*sin(x))/(b^2 + c^2)) + 3*(7*I*(23*a^2*b^2 + 9*b^4 + 9*c^4 + (23*a ^2 + 18*b^2)*c^2)*d + 5*I*(3*a^3*b^2 + 29*a*b^4 + 29*a*c^4 + (3*a^3 + 58*a *b^2)*c^2)*e)*sqrt(1/2*b + 1/2*I*c)*weierstrassZeta(4/3*(4*a^2*b^2 - 3*...
Timed out. \[ \int (a+b \cos (x)+c \sin (x))^{5/2} (d+b e \cos (x)+c e \sin (x)) \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(x)+c*sin(x))**(5/2)*(d+b*e*cos(x)+c*e*sin(x)),x)
Output:
Timed out
\[ \int (a+b \cos (x)+c \sin (x))^{5/2} (d+b e \cos (x)+c e \sin (x)) \, dx=\int { {\left (b e \cos \left (x\right ) + c e \sin \left (x\right ) + d\right )} {\left (b \cos \left (x\right ) + c \sin \left (x\right ) + a\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((a+b*cos(x)+c*sin(x))^(5/2)*(d+b*e*cos(x)+c*e*sin(x)),x, algorit hm="maxima")
Output:
integrate((b*e*cos(x) + c*e*sin(x) + d)*(b*cos(x) + c*sin(x) + a)^(5/2), x )
\[ \int (a+b \cos (x)+c \sin (x))^{5/2} (d+b e \cos (x)+c e \sin (x)) \, dx=\int { {\left (b e \cos \left (x\right ) + c e \sin \left (x\right ) + d\right )} {\left (b \cos \left (x\right ) + c \sin \left (x\right ) + a\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((a+b*cos(x)+c*sin(x))^(5/2)*(d+b*e*cos(x)+c*e*sin(x)),x, algorit hm="giac")
Output:
integrate((b*e*cos(x) + c*e*sin(x) + d)*(b*cos(x) + c*sin(x) + a)^(5/2), x )
Timed out. \[ \int (a+b \cos (x)+c \sin (x))^{5/2} (d+b e \cos (x)+c e \sin (x)) \, dx=\int {\left (a+b\,\cos \left (x\right )+c\,\sin \left (x\right )\right )}^{5/2}\,\left (d+b\,e\,\cos \left (x\right )+c\,e\,\sin \left (x\right )\right ) \,d x \] Input:
int((a + b*cos(x) + c*sin(x))^(5/2)*(d + b*e*cos(x) + c*e*sin(x)),x)
Output:
int((a + b*cos(x) + c*sin(x))^(5/2)*(d + b*e*cos(x) + c*e*sin(x)), x)
\[ \int (a+b \cos (x)+c \sin (x))^{5/2} (d+b e \cos (x)+c e \sin (x)) \, dx=\int \left (\cos \left (x \right ) b +\sin \left (x \right ) c +a \right )^{\frac {5}{2}} \left (d +b e \cos \left (x \right )+c e \sin \left (x \right )\right )d x \] Input:
int((a+b*cos(x)+c*sin(x))^(5/2)*(d+b*e*cos(x)+c*e*sin(x)),x)
Output:
int((a+b*cos(x)+c*sin(x))^(5/2)*(d+b*e*cos(x)+c*e*sin(x)),x)