Integrand size = 33, antiderivative size = 550 \[ \int \frac {\cos ^4(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{5/2}} \, dx=-\frac {2 \left (80 a^5 A b-140 a^3 A b^3+40 a A b^5-128 a^6 B+212 a^4 b^2 B-55 a^2 b^4 B-9 b^6 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{15 b^5 \left (a^2-b^2\right )^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 \left (80 a^4 A b-80 a^2 A b^3-5 A b^5-128 a^5 B+116 a^3 b^2 B+17 a b^4 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{15 b^5 \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}+\frac {2 a (A b-a B) \cos ^3(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}+\frac {2 a \left (5 a^2 A b-9 A b^3-8 a^3 B+12 a b^2 B\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (40 a^4 A b-65 a^2 A b^3+5 A b^5-64 a^5 B+98 a^3 b^2 B-14 a b^4 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{15 b^4 \left (a^2-b^2\right )^2 d}-\frac {2 \left (30 a^3 A b-50 a A b^3-48 a^4 B+71 a^2 b^2 B-3 b^4 B\right ) \cos (c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{15 b^3 \left (a^2-b^2\right )^2 d} \] Output:
-2/15*(80*A*a^5*b-140*A*a^3*b^3+40*A*a*b^5-128*B*a^6+212*B*a^4*b^2-55*B*a^ 2*b^4-9*B*b^6)*(a+b*cos(d*x+c))^(1/2)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2) *(b/(a+b))^(1/2))/b^5/(a^2-b^2)^2/d/((a+b*cos(d*x+c))/(a+b))^(1/2)+2/15*(8 0*A*a^4*b-80*A*a^2*b^3-5*A*b^5-128*B*a^5+116*B*a^3*b^2+17*B*a*b^4)*((a+b*c os(d*x+c))/(a+b))^(1/2)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2)*(b/(a+b))^(1 /2))/b^5/(a^2-b^2)/d/(a+b*cos(d*x+c))^(1/2)+2/3*a*(A*b-B*a)*cos(d*x+c)^3*s in(d*x+c)/b/(a^2-b^2)/d/(a+b*cos(d*x+c))^(3/2)+2/3*a*(5*A*a^2*b-9*A*b^3-8* B*a^3+12*B*a*b^2)*cos(d*x+c)^2*sin(d*x+c)/b^2/(a^2-b^2)^2/d/(a+b*cos(d*x+c ))^(1/2)+2/15*(40*A*a^4*b-65*A*a^2*b^3+5*A*b^5-64*B*a^5+98*B*a^3*b^2-14*B* a*b^4)*(a+b*cos(d*x+c))^(1/2)*sin(d*x+c)/b^4/(a^2-b^2)^2/d-2/15*(30*A*a^3* b-50*A*a*b^3-48*B*a^4+71*B*a^2*b^2-3*B*b^4)*cos(d*x+c)*(a+b*cos(d*x+c))^(1 /2)*sin(d*x+c)/b^3/(a^2-b^2)^2/d
Time = 5.49 (sec) , antiderivative size = 372, normalized size of antiderivative = 0.68 \[ \int \frac {\cos ^4(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{5/2}} \, dx=\frac {-\frac {2 \left (\frac {a+b \cos (c+d x)}{a+b}\right )^{3/2} \left (b^2 \left (20 a^4 A b-35 a^2 A b^3-5 A b^5-32 a^5 B+44 a^3 b^2 B+8 a b^4 B\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )-\left (-80 a^5 A b+140 a^3 A b^3-40 a A b^5+128 a^6 B-212 a^4 b^2 B+55 a^2 b^4 B+9 b^6 B\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )\right )\right )}{(a-b)^2 (a+b)}+b \left (\frac {10 a^4 (-A b+a B) \sin (c+d x)}{a^2-b^2}-\frac {10 a^3 \left (-8 a^2 A b+12 A b^3+11 a^3 B-15 a b^2 B\right ) (a+b \cos (c+d x)) \sin (c+d x)}{\left (a^2-b^2\right )^2}+2 (5 A b-14 a B) (a+b \cos (c+d x))^2 \sin (c+d x)+3 b B (a+b \cos (c+d x))^2 \sin (2 (c+d x))\right )}{15 b^5 d (a+b \cos (c+d x))^{3/2}} \] Input:
Integrate[(Cos[c + d*x]^4*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^(5/2) ,x]
Output:
((-2*((a + b*Cos[c + d*x])/(a + b))^(3/2)*(b^2*(20*a^4*A*b - 35*a^2*A*b^3 - 5*A*b^5 - 32*a^5*B + 44*a^3*b^2*B + 8*a*b^4*B)*EllipticF[(c + d*x)/2, (2 *b)/(a + b)] - (-80*a^5*A*b + 140*a^3*A*b^3 - 40*a*A*b^5 + 128*a^6*B - 212 *a^4*b^2*B + 55*a^2*b^4*B + 9*b^6*B)*((a + b)*EllipticE[(c + d*x)/2, (2*b) /(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])))/((a - b)^2*(a + b)) + b*((10*a^4*(-(A*b) + a*B)*Sin[c + d*x])/(a^2 - b^2) - (10*a^3*(-8*a^2*A *b + 12*A*b^3 + 11*a^3*B - 15*a*b^2*B)*(a + b*Cos[c + d*x])*Sin[c + d*x])/ (a^2 - b^2)^2 + 2*(5*A*b - 14*a*B)*(a + b*Cos[c + d*x])^2*Sin[c + d*x] + 3 *b*B*(a + b*Cos[c + d*x])^2*Sin[2*(c + d*x)]))/(15*b^5*d*(a + b*Cos[c + d* x])^(3/2))
Time = 3.10 (sec) , antiderivative size = 555, normalized size of antiderivative = 1.01, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {3042, 3468, 27, 3042, 3526, 27, 3042, 3528, 27, 3042, 3502, 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.
| \(\displaystyle \int \frac {\cos ^4(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^4 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 3468 |
| \(\displaystyle \frac {2 a (A b-a B) \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}-\frac {2 \int -\frac {\cos ^2(c+d x) \left (-\left (\left (-8 B a^2+5 A b a+3 b^2 B\right ) \cos ^2(c+d x)\right )-3 b (A b-a B) \cos (c+d x)+6 a (A b-a B)\right )}{2 (a+b \cos (c+d x))^{3/2}}dx}{3 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\cos ^2(c+d x) \left (-\left (\left (-8 B a^2+5 A b a+3 b^2 B\right ) \cos ^2(c+d x)\right )-3 b (A b-a B) \cos (c+d x)+6 a (A b-a B)\right )}{(a+b \cos (c+d x))^{3/2}}dx}{3 b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (\left (8 B a^2-5 A b a-3 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-3 b (A b-a B) \sin \left (c+d x+\frac {\pi }{2}\right )+6 a (A b-a B)\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{3 b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {\frac {2 a \left (-8 a^3 B+5 a^2 A b+12 a b^2 B-9 A b^3\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}-\frac {2 \int -\frac {\cos (c+d x) \left (-\left (\left (-48 B a^4+30 A b a^3+71 b^2 B a^2-50 A b^3 a-3 b^4 B\right ) \cos ^2(c+d x)\right )+b \left (2 B a^3+A b a^2-6 b^2 B a+3 A b^3\right ) \cos (c+d x)+4 a \left (-8 B a^3+5 A b a^2+12 b^2 B a-9 A b^3\right )\right )}{2 \sqrt {a+b \cos (c+d x)}}dx}{b \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\cos (c+d x) \left (-\left (\left (-48 B a^4+30 A b a^3+71 b^2 B a^2-50 A b^3 a-3 b^4 B\right ) \cos ^2(c+d x)\right )+b \left (2 B a^3+A b a^2-6 b^2 B a+3 A b^3\right ) \cos (c+d x)+4 a \left (-8 B a^3+5 A b a^2+12 b^2 B a-9 A b^3\right )\right )}{\sqrt {a+b \cos (c+d x)}}dx}{b \left (a^2-b^2\right )}+\frac {2 a \left (-8 a^3 B+5 a^2 A b+12 a b^2 B-9 A b^3\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \left (\left (48 B a^4-30 A b a^3-71 b^2 B a^2+50 A b^3 a+3 b^4 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+b \left (2 B a^3+A b a^2-6 b^2 B a+3 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+4 a \left (-8 B a^3+5 A b a^2+12 b^2 B a-9 A b^3\right )\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b \left (a^2-b^2\right )}+\frac {2 a \left (-8 a^3 B+5 a^2 A b+12 a b^2 B-9 A b^3\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {\frac {2 \int -\frac {-3 \left (-64 B a^5+40 A b a^4+98 b^2 B a^3-65 A b^3 a^2-14 b^4 B a+5 A b^5\right ) \cos ^2(c+d x)-b \left (-16 B a^4+10 A b a^3+27 b^2 B a^2-30 A b^3 a+9 b^4 B\right ) \cos (c+d x)+2 a \left (-48 B a^4+30 A b a^3+71 b^2 B a^2-50 A b^3 a-3 b^4 B\right )}{2 \sqrt {a+b \cos (c+d x)}}dx}{5 b}-\frac {2 \left (-48 a^4 B+30 a^3 A b+71 a^2 b^2 B-50 a A b^3-3 b^4 B\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}+\frac {2 a \left (-8 a^3 B+5 a^2 A b+12 a b^2 B-9 A b^3\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {-3 \left (-64 B a^5+40 A b a^4+98 b^2 B a^3-65 A b^3 a^2-14 b^4 B a+5 A b^5\right ) \cos ^2(c+d x)-b \left (-16 B a^4+10 A b a^3+27 b^2 B a^2-30 A b^3 a+9 b^4 B\right ) \cos (c+d x)+2 a \left (-48 B a^4+30 A b a^3+71 b^2 B a^2-50 A b^3 a-3 b^4 B\right )}{\sqrt {a+b \cos (c+d x)}}dx}{5 b}-\frac {2 \left (-48 a^4 B+30 a^3 A b+71 a^2 b^2 B-50 a A b^3-3 b^4 B\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}+\frac {2 a \left (-8 a^3 B+5 a^2 A b+12 a b^2 B-9 A b^3\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {-3 \left (-64 B a^5+40 A b a^4+98 b^2 B a^3-65 A b^3 a^2-14 b^4 B a+5 A b^5\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-b \left (-16 B a^4+10 A b a^3+27 b^2 B a^2-30 A b^3 a+9 b^4 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+2 a \left (-48 B a^4+30 A b a^3+71 b^2 B a^2-50 A b^3 a-3 b^4 B\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{5 b}-\frac {2 \left (-48 a^4 B+30 a^3 A b+71 a^2 b^2 B-50 a A b^3-3 b^4 B\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}+\frac {2 a \left (-8 a^3 B+5 a^2 A b+12 a b^2 B-9 A b^3\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {-\frac {\frac {2 \int \frac {3 \left (b \left (-32 B a^5+20 A b a^4+44 b^2 B a^3-35 A b^3 a^2+8 b^4 B a-5 A b^5\right )+\left (-128 B a^6+80 A b a^5+212 b^2 B a^4-140 A b^3 a^3-55 b^4 B a^2+40 A b^5 a-9 b^6 B\right ) \cos (c+d x)\right )}{2 \sqrt {a+b \cos (c+d x)}}dx}{3 b}-\frac {2 \left (-64 a^5 B+40 a^4 A b+98 a^3 b^2 B-65 a^2 A b^3-14 a b^4 B+5 A b^5\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d}}{5 b}-\frac {2 \left (-48 a^4 B+30 a^3 A b+71 a^2 b^2 B-50 a A b^3-3 b^4 B\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}+\frac {2 a \left (-8 a^3 B+5 a^2 A b+12 a b^2 B-9 A b^3\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {\frac {\int \frac {b \left (-32 B a^5+20 A b a^4+44 b^2 B a^3-35 A b^3 a^2+8 b^4 B a-5 A b^5\right )+\left (-128 B a^6+80 A b a^5+212 b^2 B a^4-140 A b^3 a^3-55 b^4 B a^2+40 A b^5 a-9 b^6 B\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}-\frac {2 \left (-64 a^5 B+40 a^4 A b+98 a^3 b^2 B-65 a^2 A b^3-14 a b^4 B+5 A b^5\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d}}{5 b}-\frac {2 \left (-48 a^4 B+30 a^3 A b+71 a^2 b^2 B-50 a A b^3-3 b^4 B\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}+\frac {2 a \left (-8 a^3 B+5 a^2 A b+12 a b^2 B-9 A b^3\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {\frac {\int \frac {b \left (-32 B a^5+20 A b a^4+44 b^2 B a^3-35 A b^3 a^2+8 b^4 B a-5 A b^5\right )+\left (-128 B a^6+80 A b a^5+212 b^2 B a^4-140 A b^3 a^3-55 b^4 B a^2+40 A b^5 a-9 b^6 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 \left (-64 a^5 B+40 a^4 A b+98 a^3 b^2 B-65 a^2 A b^3-14 a b^4 B+5 A b^5\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d}}{5 b}-\frac {2 \left (-48 a^4 B+30 a^3 A b+71 a^2 b^2 B-50 a A b^3-3 b^4 B\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}+\frac {2 a \left (-8 a^3 B+5 a^2 A b+12 a b^2 B-9 A b^3\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {\frac {-\frac {\frac {\frac {\left (-128 a^6 B+80 a^5 A b+212 a^4 b^2 B-140 a^3 A b^3-55 a^2 b^4 B+40 a A b^5-9 b^6 B\right ) \int \sqrt {a+b \cos (c+d x)}dx}{b}-\frac {\left (a^2-b^2\right ) \left (-128 a^5 B+80 a^4 A b+116 a^3 b^2 B-80 a^2 A b^3+17 a b^4 B-5 A b^5\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx}{b}}{b}-\frac {2 \left (-64 a^5 B+40 a^4 A b+98 a^3 b^2 B-65 a^2 A b^3-14 a b^4 B+5 A b^5\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d}}{5 b}-\frac {2 \left (-48 a^4 B+30 a^3 A b+71 a^2 b^2 B-50 a A b^3-3 b^4 B\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}+\frac {2 a \left (-8 a^3 B+5 a^2 A b+12 a b^2 B-9 A b^3\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {\frac {\frac {\left (-128 a^6 B+80 a^5 A b+212 a^4 b^2 B-140 a^3 A b^3-55 a^2 b^4 B+40 a A b^5-9 b^6 B\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {\left (a^2-b^2\right ) \left (-128 a^5 B+80 a^4 A b+116 a^3 b^2 B-80 a^2 A b^3+17 a b^4 B-5 A b^5\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}}{b}-\frac {2 \left (-64 a^5 B+40 a^4 A b+98 a^3 b^2 B-65 a^2 A b^3-14 a b^4 B+5 A b^5\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d}}{5 b}-\frac {2 \left (-48 a^4 B+30 a^3 A b+71 a^2 b^2 B-50 a A b^3-3 b^4 B\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}+\frac {2 a \left (-8 a^3 B+5 a^2 A b+12 a b^2 B-9 A b^3\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {\frac {-\frac {\frac {\frac {\left (-128 a^6 B+80 a^5 A b+212 a^4 b^2 B-140 a^3 A b^3-55 a^2 b^4 B+40 a A b^5-9 b^6 B\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (-128 a^5 B+80 a^4 A b+116 a^3 b^2 B-80 a^2 A b^3+17 a b^4 B-5 A b^5\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}}{b}-\frac {2 \left (-64 a^5 B+40 a^4 A b+98 a^3 b^2 B-65 a^2 A b^3-14 a b^4 B+5 A b^5\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d}}{5 b}-\frac {2 \left (-48 a^4 B+30 a^3 A b+71 a^2 b^2 B-50 a A b^3-3 b^4 B\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}+\frac {2 a \left (-8 a^3 B+5 a^2 A b+12 a b^2 B-9 A b^3\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {\frac {\frac {\left (-128 a^6 B+80 a^5 A b+212 a^4 b^2 B-140 a^3 A b^3-55 a^2 b^4 B+40 a A b^5-9 b^6 B\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (-128 a^5 B+80 a^4 A b+116 a^3 b^2 B-80 a^2 A b^3+17 a b^4 B-5 A b^5\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}}{b}-\frac {2 \left (-64 a^5 B+40 a^4 A b+98 a^3 b^2 B-65 a^2 A b^3-14 a b^4 B+5 A b^5\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d}}{5 b}-\frac {2 \left (-48 a^4 B+30 a^3 A b+71 a^2 b^2 B-50 a A b^3-3 b^4 B\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}+\frac {2 a \left (-8 a^3 B+5 a^2 A b+12 a b^2 B-9 A b^3\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\frac {-\frac {\frac {\frac {2 \left (-128 a^6 B+80 a^5 A b+212 a^4 b^2 B-140 a^3 A b^3-55 a^2 b^4 B+40 a A b^5-9 b^6 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (-128 a^5 B+80 a^4 A b+116 a^3 b^2 B-80 a^2 A b^3+17 a b^4 B-5 A b^5\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}}{b}-\frac {2 \left (-64 a^5 B+40 a^4 A b+98 a^3 b^2 B-65 a^2 A b^3-14 a b^4 B+5 A b^5\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d}}{5 b}-\frac {2 \left (-48 a^4 B+30 a^3 A b+71 a^2 b^2 B-50 a A b^3-3 b^4 B\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}+\frac {2 a \left (-8 a^3 B+5 a^2 A b+12 a b^2 B-9 A b^3\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {\frac {-\frac {\frac {\frac {2 \left (-128 a^6 B+80 a^5 A b+212 a^4 b^2 B-140 a^3 A b^3-55 a^2 b^4 B+40 a A b^5-9 b^6 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (-128 a^5 B+80 a^4 A b+116 a^3 b^2 B-80 a^2 A b^3+17 a b^4 B-5 A b^5\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 \left (-64 a^5 B+40 a^4 A b+98 a^3 b^2 B-65 a^2 A b^3-14 a b^4 B+5 A b^5\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d}}{5 b}-\frac {2 \left (-48 a^4 B+30 a^3 A b+71 a^2 b^2 B-50 a A b^3-3 b^4 B\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}+\frac {2 a \left (-8 a^3 B+5 a^2 A b+12 a b^2 B-9 A b^3\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {\frac {\frac {2 \left (-128 a^6 B+80 a^5 A b+212 a^4 b^2 B-140 a^3 A b^3-55 a^2 b^4 B+40 a A b^5-9 b^6 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (-128 a^5 B+80 a^4 A b+116 a^3 b^2 B-80 a^2 A b^3+17 a b^4 B-5 A b^5\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 \left (-64 a^5 B+40 a^4 A b+98 a^3 b^2 B-65 a^2 A b^3-14 a b^4 B+5 A b^5\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d}}{5 b}-\frac {2 \left (-48 a^4 B+30 a^3 A b+71 a^2 b^2 B-50 a A b^3-3 b^4 B\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}+\frac {2 a \left (-8 a^3 B+5 a^2 A b+12 a b^2 B-9 A b^3\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {2 a (A b-a B) \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}+\frac {\frac {2 a \left (-8 a^3 B+5 a^2 A b+12 a b^2 B-9 A b^3\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}+\frac {-\frac {2 \left (-48 a^4 B+30 a^3 A b+71 a^2 b^2 B-50 a A b^3-3 b^4 B\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}-\frac {\frac {\frac {2 \left (-128 a^6 B+80 a^5 A b+212 a^4 b^2 B-140 a^3 A b^3-55 a^2 b^4 B+40 a A b^5-9 b^6 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (a^2-b^2\right ) \left (-128 a^5 B+80 a^4 A b+116 a^3 b^2 B-80 a^2 A b^3+17 a b^4 B-5 A b^5\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 \left (-64 a^5 B+40 a^4 A b+98 a^3 b^2 B-65 a^2 A b^3-14 a b^4 B+5 A b^5\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d}}{5 b}}{b \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}\) |
Input:
Int[(Cos[c + d*x]^4*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^(5/2),x]
Output:
(2*a*(A*b - a*B)*Cos[c + d*x]^3*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Co s[c + d*x])^(3/2)) + ((2*a*(5*a^2*A*b - 9*A*b^3 - 8*a^3*B + 12*a*b^2*B)*Co s[c + d*x]^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (( -2*(30*a^3*A*b - 50*a*A*b^3 - 48*a^4*B + 71*a^2*b^2*B - 3*b^4*B)*Cos[c + d *x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*b*d) - (((2*(80*a^5*A*b - 14 0*a^3*A*b^3 + 40*a*A*b^5 - 128*a^6*B + 212*a^4*b^2*B - 55*a^2*b^4*B - 9*b^ 6*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(b*d* Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(80*a^4*A*b - 80*a^2* A*b^3 - 5*A*b^5 - 128*a^5*B + 116*a^3*b^2*B + 17*a*b^4*B)*Sqrt[(a + b*Cos[ c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(b*d*Sqrt[a + b* Cos[c + d*x]]))/b - (2*(40*a^4*A*b - 65*a^2*A*b^3 + 5*A*b^5 - 64*a^5*B + 9 8*a^3*b^2*B - 14*a*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*d))/(5 *b))/(b*(a^2 - b^2)))/(3*b*(a^2 - b^2))
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[((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[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si mp[(-(b*c - a*d))*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1)*Simp[b*(b*c - a*d)*(B*c - A*d)*(m - 1) + a*d*(a*A*c + b*B*c - (A*b + a *B)*d)*(n + 1) + (b*(b*d*(B*c - A*d) + a*(A*c*d + B*(c^2 - 2*d^2)))*(n + 1) - a*(b*c - a*d)*(B*c - A*d)*(n + 2))*Sin[e + f*x] + b*(d*(A*b*c + a*B*c - a*A*d)*(m + n + 1) - b*B*(c^2*m + d^2*(n + 1)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2 , 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && LtQ[n, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C - B*c*d + A*d^2))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c*C - B* d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x ] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f *x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d , 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_ .) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x ])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + n + 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A* d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a *c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n} , x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[ m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
Leaf count of result is larger than twice the leaf count of optimal. \(1749\) vs. \(2(531)=1062\).
Time = 20.35 (sec) , antiderivative size = 1750, normalized size of antiderivative = 3.18
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1750\) |
| parts | \(\text {Expression too large to display}\) | \(2985\) |
Input:
int(cos(d*x+c)^4*(A+B*cos(d*x+c))/(a+cos(d*x+c)*b)^(5/2),x,method=_RETURNV ERBOSE)
Output:
-(-(-2*b*cos(1/2*d*x+1/2*c)^2-a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*(3*A*a^2 *b+2*A*a*b^2+A*b^3-4*B*a^3-3*B*a^2*b-2*B*a*b^2-B*b^3)/b^5*(sin(1/2*d*x+1/2 *c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)/(a-b))^(1/2)/(-2*b*sin(1/2*d* x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c), (-2*b/(a-b))^(1/2))+2*a^4*(A*b-B*a)/b^5*(1/6/b/(a-b)/(a+b)*cos(1/2*d*x+1/2 *c)*(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2* d*x+1/2*c)^2+1/2/b*(a-b))^2+8/3*sin(1/2*d*x+1/2*c)^2*b/(a-b)^2/(a+b)^2*cos (1/2*d*x+1/2*c)*a/(-(-2*b*cos(1/2*d*x+1/2*c)^2-a+b)*sin(1/2*d*x+1/2*c)^2)^ (1/2)+(3*a-b)/(3*a^3+3*a^2*b-3*a*b^2-3*b^3)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*( (2*b*cos(1/2*d*x+1/2*c)^2+a-b)/(a-b))^(1/2)/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+ b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^( 1/2))-4/3*a/(a-b)/(a+b)^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1 /2*c)^2+a-b)/(a-b))^(1/2)/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2 *c)^2)^(1/2)*(EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))-EllipticE(c os(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))))+2/b^5*(2*A*a*b+2*A*b^2-3*B*a^2-4*B *a*b-3*B*b^2)*(a-b)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c)^ 2+a-b)/(a-b))^(1/2)/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2) ^(1/2)*(EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))-EllipticE(cos(1/2 *d*x+1/2*c),(-2*b/(a-b))^(1/2)))+8/b^3*(A*b-2*B*a-3*B*b)*(-1/6/b*cos(1/2*d *x+1/2*c)*(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)+...
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 1504, normalized size of antiderivative = 2.73 \[ \int \frac {\cos ^4(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{5/2}} \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)^4*(A+B*cos(d*x+c))/(a+b*cos(d*x+c))^(5/2),x, algorith m="fricas")
Output:
-2/45*(sqrt(1/2)*(-256*I*B*a^9 + 160*I*A*a^8*b + 520*I*B*a^7*b^2 - 340*I*A *a^6*b^3 - 242*I*B*a^5*b^4 + 185*I*A*a^4*b^5 - 42*I*B*a^3*b^6 + 15*I*A*a^2 *b^7 + (-256*I*B*a^7*b^2 + 160*I*A*a^6*b^3 + 520*I*B*a^5*b^4 - 340*I*A*a^4 *b^5 - 242*I*B*a^3*b^6 + 185*I*A*a^2*b^7 - 42*I*B*a*b^8 + 15*I*A*b^9)*cos( d*x + c)^2 + 2*(-256*I*B*a^8*b + 160*I*A*a^7*b^2 + 520*I*B*a^6*b^3 - 340*I *A*a^5*b^4 - 242*I*B*a^4*b^5 + 185*I*A*a^3*b^6 - 42*I*B*a^2*b^7 + 15*I*A*a *b^8)*cos(d*x + c))*sqrt(b)*weierstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^2, - 8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) + 3*I*b*sin(d*x + c) + 2 *a)/b) + sqrt(1/2)*(256*I*B*a^9 - 160*I*A*a^8*b - 520*I*B*a^7*b^2 + 340*I* A*a^6*b^3 + 242*I*B*a^5*b^4 - 185*I*A*a^4*b^5 + 42*I*B*a^3*b^6 - 15*I*A*a^ 2*b^7 + (256*I*B*a^7*b^2 - 160*I*A*a^6*b^3 - 520*I*B*a^5*b^4 + 340*I*A*a^4 *b^5 + 242*I*B*a^3*b^6 - 185*I*A*a^2*b^7 + 42*I*B*a*b^8 - 15*I*A*b^9)*cos( d*x + c)^2 + 2*(256*I*B*a^8*b - 160*I*A*a^7*b^2 - 520*I*B*a^6*b^3 + 340*I* A*a^5*b^4 + 242*I*B*a^4*b^5 - 185*I*A*a^3*b^6 + 42*I*B*a^2*b^7 - 15*I*A*a* b^8)*cos(d*x + c))*sqrt(b)*weierstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8 /27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) - 3*I*b*sin(d*x + c) + 2* a)/b) + 3*sqrt(1/2)*(-128*I*B*a^8*b + 80*I*A*a^7*b^2 + 212*I*B*a^6*b^3 - 1 40*I*A*a^5*b^4 - 55*I*B*a^4*b^5 + 40*I*A*a^3*b^6 - 9*I*B*a^2*b^7 + (-128*I *B*a^6*b^3 + 80*I*A*a^5*b^4 + 212*I*B*a^4*b^5 - 140*I*A*a^3*b^6 - 55*I*B*a ^2*b^7 + 40*I*A*a*b^8 - 9*I*B*b^9)*cos(d*x + c)^2 + 2*(-128*I*B*a^7*b^2...
Timed out. \[ \int \frac {\cos ^4(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{5/2}} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**4*(A+B*cos(d*x+c))/(a+b*cos(d*x+c))**(5/2),x)
Output:
Timed out
\[ \int \frac {\cos ^4(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{5/2}} \, dx=\int { \frac {{\left (B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{4}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(cos(d*x+c)^4*(A+B*cos(d*x+c))/(a+b*cos(d*x+c))^(5/2),x, algorith m="maxima")
Output:
integrate((B*cos(d*x + c) + A)*cos(d*x + c)^4/(b*cos(d*x + c) + a)^(5/2), x)
\[ \int \frac {\cos ^4(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{5/2}} \, dx=\int { \frac {{\left (B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{4}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(cos(d*x+c)^4*(A+B*cos(d*x+c))/(a+b*cos(d*x+c))^(5/2),x, algorith m="giac")
Output:
integrate((B*cos(d*x + c) + A)*cos(d*x + c)^4/(b*cos(d*x + c) + a)^(5/2), x)
Timed out. \[ \int \frac {\cos ^4(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{5/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^4\,\left (A+B\,\cos \left (c+d\,x\right )\right )}{{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}} \,d x \] Input:
int((cos(c + d*x)^4*(A + B*cos(c + d*x)))/(a + b*cos(c + d*x))^(5/2),x)
Output:
int((cos(c + d*x)^4*(A + B*cos(c + d*x)))/(a + b*cos(c + d*x))^(5/2), x)
\[ \int \frac {\cos ^4(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{5/2}} \, dx=\int \frac {\sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{4}}{\cos \left (d x +c \right )^{2} b^{2}+2 \cos \left (d x +c \right ) a b +a^{2}}d x \] Input:
int(cos(d*x+c)^4*(A+B*cos(d*x+c))/(a+b*cos(d*x+c))^(5/2),x)
Output:
int((sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**4)/(cos(c + d*x)**2*b**2 + 2*c os(c + d*x)*a*b + a**2),x)