Integrand size = 35, antiderivative size = 522 \[ \int \frac {(a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\frac {2 (a-b) \sqrt {a+b} \left (147 a^4 A+279 a^2 A b^2-10 A b^4+435 a^3 b B+45 a b^3 B\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{315 a^3 d}-\frac {2 (a-b) \sqrt {a+b} \left (10 A b^3-6 a^2 b (19 A-60 B)+3 a^3 (49 A-25 B)+15 a b^2 (11 A-3 B)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{315 a^2 d}+\frac {2 a (4 A b+3 a B) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{21 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (49 a^2 A+75 A b^2+135 a b B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (163 a^2 A b+5 A b^3+75 a^3 B+135 a b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 a d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a A (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)} \] Output:
2/315*(a-b)*(a+b)^(1/2)*(147*A*a^4+279*A*a^2*b^2-10*A*b^4+435*B*a^3*b+45*B *a*b^3)*cot(d*x+c)*EllipticE((a+b*cos(d*x+c))^(1/2)/(a+b)^(1/2)/cos(d*x+c) ^(1/2),(-(a+b)/(a-b))^(1/2))*(a*(1-sec(d*x+c))/(a+b))^(1/2)*(a*(1+sec(d*x+ c))/(a-b))^(1/2)/a^3/d-2/315*(a-b)*(a+b)^(1/2)*(10*A*b^3-6*a^2*b*(19*A-60* B)+3*a^3*(49*A-25*B)+15*a*b^2*(11*A-3*B))*cot(d*x+c)*EllipticF((a+b*cos(d* x+c))^(1/2)/(a+b)^(1/2)/cos(d*x+c)^(1/2),(-(a+b)/(a-b))^(1/2))*(a*(1-sec(d *x+c))/(a+b))^(1/2)*(a*(1+sec(d*x+c))/(a-b))^(1/2)/a^2/d+2/21*a*(4*A*b+3*B *a)*(a+b*cos(d*x+c))^(1/2)*sin(d*x+c)/d/cos(d*x+c)^(7/2)+2/315*(49*A*a^2+7 5*A*b^2+135*B*a*b)*(a+b*cos(d*x+c))^(1/2)*sin(d*x+c)/d/cos(d*x+c)^(5/2)+2/ 315*(163*A*a^2*b+5*A*b^3+75*B*a^3+135*B*a*b^2)*(a+b*cos(d*x+c))^(1/2)*sin( d*x+c)/a/d/cos(d*x+c)^(3/2)+2/9*a*A*(a+b*cos(d*x+c))^(3/2)*sin(d*x+c)/d/co s(d*x+c)^(9/2)
Result contains complex when optimal does not.
Time = 7.65 (sec) , antiderivative size = 1517, normalized size of antiderivative = 2.91 \[ \int \frac {(a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac {11}{2}}(c+d x)} \, dx =\text {Too large to display} \] Input:
Integrate[((a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^( 11/2),x]
Output:
-1/315*((-4*a*(-114*a^4*A*b + 124*a^2*A*b^3 - 10*A*b^5 - 75*a^5*B + 30*a^3 *b^2*B + 45*a*b^4*B)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-((( a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc [(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x ])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/( (a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(147*a^5*A + 27 9*a^3*A*b^2 - 10*a*A*b^4 + 435*a^4*b*B + 45*a^2*b^3*B)*((Sqrt[((a + b)*Cot [(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2 )/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*Ellip ticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (- 2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b* Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2 ]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(147*a^4*A*b + 2 79*a^2*A*b^3 - 10*A*b^5 + 435*a^3*b^2*B + 45*a*b^4*B)*((I*Cos[(c + d*x)/2] *Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*S...
Time = 2.67 (sec) , antiderivative size = 533, normalized size of antiderivative = 1.02, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.486, Rules used = {3042, 3468, 27, 3042, 3526, 27, 3042, 3534, 27, 3042, 3534, 27, 3042, 3477, 3042, 3295, 3473}
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.
| \(\displaystyle \int \frac {(a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac {11}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{11/2}}dx\) |
| \(\Big \downarrow \) 3468 |
| \(\displaystyle \frac {2}{9} \int \frac {\sqrt {a+b \cos (c+d x)} \left (b (4 a A+9 b B) \cos ^2(c+d x)+\left (7 A a^2+18 b B a+9 A b^2\right ) \cos (c+d x)+3 a (4 A b+3 a B)\right )}{2 \cos ^{\frac {9}{2}}(c+d x)}dx+\frac {2 a A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \int \frac {\sqrt {a+b \cos (c+d x)} \left (b (4 a A+9 b B) \cos ^2(c+d x)+\left (7 A a^2+18 b B a+9 A b^2\right ) \cos (c+d x)+3 a (4 A b+3 a B)\right )}{\cos ^{\frac {9}{2}}(c+d x)}dx+\frac {2 a A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \int \frac {\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (b (4 a A+9 b B) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (7 A a^2+18 b B a+9 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+3 a (4 A b+3 a B)\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx+\frac {2 a A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{9} \left (\frac {2}{7} \int \frac {b \left (36 B a^2+76 A b a+63 b^2 B\right ) \cos ^2(c+d x)+\left (45 B a^3+137 A b a^2+189 b^2 B a+63 A b^3\right ) \cos (c+d x)+a \left (49 A a^2+135 b B a+75 A b^2\right )}{2 \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx+\frac {6 a (3 a B+4 A b) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int \frac {b \left (36 B a^2+76 A b a+63 b^2 B\right ) \cos ^2(c+d x)+\left (45 B a^3+137 A b a^2+189 b^2 B a+63 A b^3\right ) \cos (c+d x)+a \left (49 A a^2+135 b B a+75 A b^2\right )}{\cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx+\frac {6 a (3 a B+4 A b) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int \frac {b \left (36 B a^2+76 A b a+63 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (45 B a^3+137 A b a^2+189 b^2 B a+63 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a \left (49 A a^2+135 b B a+75 A b^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {6 a (3 a B+4 A b) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \int \frac {2 a b \left (49 A a^2+135 b B a+75 A b^2\right ) \cos ^2(c+d x)+a \left (147 A a^3+585 b B a^2+605 A b^2 a+315 b^3 B\right ) \cos (c+d x)+3 a \left (75 B a^3+163 A b a^2+135 b^2 B a+5 A b^3\right )}{2 \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{5 a}+\frac {2 \left (49 a^2 A+135 a b B+75 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 a (3 a B+4 A b) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {\int \frac {2 a b \left (49 A a^2+135 b B a+75 A b^2\right ) \cos ^2(c+d x)+a \left (147 A a^3+585 b B a^2+605 A b^2 a+315 b^3 B\right ) \cos (c+d x)+3 a \left (75 B a^3+163 A b a^2+135 b^2 B a+5 A b^3\right )}{\cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{5 a}+\frac {2 \left (49 a^2 A+135 a b B+75 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 a (3 a B+4 A b) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {\int \frac {2 a b \left (49 A a^2+135 b B a+75 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+a \left (147 A a^3+585 b B a^2+605 A b^2 a+315 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+3 a \left (75 B a^3+163 A b a^2+135 b^2 B a+5 A b^3\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{5 a}+\frac {2 \left (49 a^2 A+135 a b B+75 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 a (3 a B+4 A b) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {\frac {2 \int \frac {3 \left (\left (75 B a^3+261 A b a^2+405 b^2 B a+155 A b^3\right ) \cos (c+d x) a^2+\left (147 A a^4+435 b B a^3+279 A b^2 a^2+45 b^3 B a-10 A b^4\right ) a\right )}{2 \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{3 a}+\frac {2 \left (75 a^3 B+163 a^2 A b+135 a b^2 B+5 A b^3\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d \cos ^{\frac {3}{2}}(c+d x)}}{5 a}+\frac {2 \left (49 a^2 A+135 a b B+75 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 a (3 a B+4 A b) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {\frac {\int \frac {\left (75 B a^3+261 A b a^2+405 b^2 B a+155 A b^3\right ) \cos (c+d x) a^2+\left (147 A a^4+435 b B a^3+279 A b^2 a^2+45 b^3 B a-10 A b^4\right ) a}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{a}+\frac {2 \left (75 a^3 B+163 a^2 A b+135 a b^2 B+5 A b^3\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d \cos ^{\frac {3}{2}}(c+d x)}}{5 a}+\frac {2 \left (49 a^2 A+135 a b B+75 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 a (3 a B+4 A b) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {\frac {\int \frac {\left (75 B a^3+261 A b a^2+405 b^2 B a+155 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2+\left (147 A a^4+435 b B a^3+279 A b^2 a^2+45 b^3 B a-10 A b^4\right ) a}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {2 \left (75 a^3 B+163 a^2 A b+135 a b^2 B+5 A b^3\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d \cos ^{\frac {3}{2}}(c+d x)}}{5 a}+\frac {2 \left (49 a^2 A+135 a b B+75 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 a (3 a B+4 A b) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3477 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {\frac {a \left (147 a^4 A+435 a^3 b B+279 a^2 A b^2+45 a b^3 B-10 A b^4\right ) \int \frac {\cos (c+d x)+1}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx-a (a-b) \left (3 a^3 (49 A-25 B)-6 a^2 b (19 A-60 B)+15 a b^2 (11 A-3 B)+10 A b^3\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}dx}{a}+\frac {2 \left (75 a^3 B+163 a^2 A b+135 a b^2 B+5 A b^3\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d \cos ^{\frac {3}{2}}(c+d x)}}{5 a}+\frac {2 \left (49 a^2 A+135 a b B+75 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 a (3 a B+4 A b) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {\frac {a \left (147 a^4 A+435 a^3 b B+279 a^2 A b^2+45 a b^3 B-10 A b^4\right ) \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-a (a-b) \left (3 a^3 (49 A-25 B)-6 a^2 b (19 A-60 B)+15 a b^2 (11 A-3 B)+10 A b^3\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {2 \left (75 a^3 B+163 a^2 A b+135 a b^2 B+5 A b^3\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d \cos ^{\frac {3}{2}}(c+d x)}}{5 a}+\frac {2 \left (49 a^2 A+135 a b B+75 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 a (3 a B+4 A b) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3295 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {\frac {a \left (147 a^4 A+435 a^3 b B+279 a^2 A b^2+45 a b^3 B-10 A b^4\right ) \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 (a-b) \sqrt {a+b} \left (3 a^3 (49 A-25 B)-6 a^2 b (19 A-60 B)+15 a b^2 (11 A-3 B)+10 A b^3\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{d}}{a}+\frac {2 \left (75 a^3 B+163 a^2 A b+135 a b^2 B+5 A b^3\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d \cos ^{\frac {3}{2}}(c+d x)}}{5 a}+\frac {2 \left (49 a^2 A+135 a b B+75 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 a (3 a B+4 A b) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3473 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \left (49 a^2 A+135 a b B+75 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {\frac {2 \left (75 a^3 B+163 a^2 A b+135 a b^2 B+5 A b^3\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\frac {2 (a-b) \sqrt {a+b} \left (147 a^4 A+435 a^3 b B+279 a^2 A b^2+45 a b^3 B-10 A b^4\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{a d}-\frac {2 (a-b) \sqrt {a+b} \left (3 a^3 (49 A-25 B)-6 a^2 b (19 A-60 B)+15 a b^2 (11 A-3 B)+10 A b^3\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{d}}{a}}{5 a}\right )+\frac {6 a (3 a B+4 A b) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
Input:
Int[((a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(11/2), x]
Output:
(2*a*A*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2)) + ((6*a*(4*A*b + 3*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + ((2*(49*a^2*A + 75*A*b^2 + 135*a*b*B)*Sqrt[a + b*Cos[c + d* x]]*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (((2*(a - b)*Sqrt[a + b]*(147 *a^4*A + 279*a^2*A*b^2 - 10*A*b^4 + 435*a^3*b*B + 45*a*b^3*B)*Cot[c + d*x] *EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]] )], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d) - (2*(a - b)*Sqrt[a + b]*(10*A*b^3 - 6*a^2* b*(19*A - 60*B) + 3*a^3*(49*A - 25*B) + 15*a*b^2*(11*A - 3*B))*Cot[c + d*x ]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x] ])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d)/a + (2*(163*a^2*A*b + 5*A*b^3 + 75*a^3*B + 13 5*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)))/ (5*a))/7)/9
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f _.)*(x_)]]), x_Symbol] :> Simp[-2*(Tan[e + f*x]/(a*f))*Rt[(a + b)/d, 2]*Sqr t[a*((1 - Csc[e + f*x])/(a + b))]*Sqrt[a*((1 + Csc[e + f*x])/(a - b))]*Elli pticF[ArcSin[Sqrt[a + b*Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]]/Rt[(a + b)/d, 2] ], -(a + b)/(a - b)], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && PosQ[(a + b)/d]
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_)])/(((b_.)*sin[(e_.) + (f_.)*(x_)]) ^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[-2*A* (c - d)*(Tan[e + f*x]/(f*b*c^2))*Rt[(c + d)/b, 2]*Sqrt[c*((1 + Csc[e + f*x] )/(c - d))]*Sqrt[c*((1 - Csc[e + f*x])/(c + d))]*EllipticE[ArcSin[Sqrt[c + d*Sin[e + f*x]]/Sqrt[b*Sin[e + f*x]]/Rt[(c + d)/b, 2]], -(c + d)/(c - d)], x] /; FreeQ[{b, c, d, e, f, A, B}, x] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && PosQ[(c + d)/b]
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_ .)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> S imp[(A - B)/(a - b) Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f* x]]), x], x] - Simp[(A*b - a*B)/(a - b) Int[(1 + Sin[e + f*x])/((a + b*Si n[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*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] && NeQ[A, B]
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[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b* c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)) Int [(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A *b^2 - a*b*B + a^2*C)*(m + n + 3)*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] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ [n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) | | EqQ[a, 0])))
Leaf count of result is larger than twice the leaf count of optimal. \(2682\) vs. \(2(472)=944\).
Time = 80.11 (sec) , antiderivative size = 2683, normalized size of antiderivative = 5.14
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2683\) |
| parts | \(\text {Expression too large to display}\) | \(2699\) |
Input:
int((a+cos(d*x+c)*b)^(5/2)*(A+B*cos(d*x+c))/cos(d*x+c)^(11/2),x,method=_RE TURNVERBOSE)
Output:
2/315/d*(B*((a+cos(d*x+c)*b)/(cos(d*x+c)+1)/(a+b))^(1/2)*(cos(d*x+c)/(cos( d*x+c)+1))^(1/2)*a^2*b^3*EllipticF(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1 /2))*(-45*cos(d*x+c)^6-90*cos(d*x+c)^5-45*cos(d*x+c)^4)+A*((a+cos(d*x+c)*b )/(cos(d*x+c)+1)/(a+b))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*a^3*b^2*El lipticE(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*(279*cos(d*x+c)^6+558* cos(d*x+c)^5+279*cos(d*x+c)^4)+A*((a+cos(d*x+c)*b)/(cos(d*x+c)+1)/(a+b))^( 1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*a^2*b^3*EllipticE(cot(d*x+c)-csc(d* x+c),(-(a-b)/(a+b))^(1/2))*(279*cos(d*x+c)^6+558*cos(d*x+c)^5+279*cos(d*x+ c)^4)+A*((a+cos(d*x+c)*b)/(cos(d*x+c)+1)/(a+b))^(1/2)*(cos(d*x+c)/(cos(d*x +c)+1))^(1/2)*a*b^4*EllipticE(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))* (-10*cos(d*x+c)^6-20*cos(d*x+c)^5-10*cos(d*x+c)^4)+B*((a+cos(d*x+c)*b)/(co s(d*x+c)+1)/(a+b))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*a^4*b*EllipticE (cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*(435*cos(d*x+c)^6+870*cos(d*x +c)^5+435*cos(d*x+c)^4)+B*((a+cos(d*x+c)*b)/(cos(d*x+c)+1)/(a+b))^(1/2)*(c os(d*x+c)/(cos(d*x+c)+1))^(1/2)*a^3*b^2*EllipticE(cot(d*x+c)-csc(d*x+c),(- (a-b)/(a+b))^(1/2))*(435*cos(d*x+c)^6+870*cos(d*x+c)^5+435*cos(d*x+c)^4)+B *((a+cos(d*x+c)*b)/(cos(d*x+c)+1)/(a+b))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1)) ^(1/2)*a^2*b^3*EllipticE(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*(45*c os(d*x+c)^6+90*cos(d*x+c)^5+45*cos(d*x+c)^4)+B*((a+cos(d*x+c)*b)/(cos(d*x+ c)+1)/(a+b))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*a*b^4*EllipticE(co...
\[ \int \frac {(a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\int { \frac {{\left (B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\cos \left (d x + c\right )^{\frac {11}{2}}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c))/cos(d*x+c)^(11/2),x, alg orithm="fricas")
Output:
integral((B*b^2*cos(d*x + c)^3 + A*a^2 + (2*B*a*b + A*b^2)*cos(d*x + c)^2 + (B*a^2 + 2*A*a*b)*cos(d*x + c))*sqrt(b*cos(d*x + c) + a)/cos(d*x + c)^(1 1/2), x)
Timed out. \[ \int \frac {(a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(d*x+c))**(5/2)*(A+B*cos(d*x+c))/cos(d*x+c)**(11/2),x)
Output:
Timed out
\[ \int \frac {(a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\int { \frac {{\left (B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\cos \left (d x + c\right )^{\frac {11}{2}}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c))/cos(d*x+c)^(11/2),x, alg orithm="maxima")
Output:
integrate((B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^(5/2)/cos(d*x + c)^(11 /2), x)
\[ \int \frac {(a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\int { \frac {{\left (B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\cos \left (d x + c\right )^{\frac {11}{2}}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c))/cos(d*x+c)^(11/2),x, alg orithm="giac")
Output:
integrate((B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^(5/2)/cos(d*x + c)^(11 /2), x)
Timed out. \[ \int \frac {(a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\int \frac {\left (A+B\,\cos \left (c+d\,x\right )\right )\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}}{{\cos \left (c+d\,x\right )}^{11/2}} \,d x \] Input:
int(((A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(5/2))/cos(c + d*x)^(11/2), x)
Output:
int(((A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(5/2))/cos(c + d*x)^(11/2), x)
\[ \int \frac {(a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\left (\int \frac {\sqrt {\cos \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{6}}d x \right ) a^{3}+3 \left (\int \frac {\sqrt {\cos \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{5}}d x \right ) a^{2} b +3 \left (\int \frac {\sqrt {\cos \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{4}}d x \right ) a \,b^{2}+\left (\int \frac {\sqrt {\cos \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{3}}d x \right ) b^{3} \] Input:
int((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c))/cos(d*x+c)^(11/2),x)
Output:
int((sqrt(cos(c + d*x)*b + a)*sqrt(cos(c + d*x)))/cos(c + d*x)**6,x)*a**3 + 3*int((sqrt(cos(c + d*x)*b + a)*sqrt(cos(c + d*x)))/cos(c + d*x)**5,x)*a **2*b + 3*int((sqrt(cos(c + d*x)*b + a)*sqrt(cos(c + d*x)))/cos(c + d*x)** 4,x)*a*b**2 + int((sqrt(cos(c + d*x)*b + a)*sqrt(cos(c + d*x)))/cos(c + d* x)**3,x)*b**3