Integrand size = 23, antiderivative size = 159 \[ \int \cos ^6(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {4096 a^5 \cos ^7(c+d x)}{45045 d (a+a \sin (c+d x))^{7/2}}-\frac {1024 a^4 \cos ^7(c+d x)}{6435 d (a+a \sin (c+d x))^{5/2}}-\frac {128 a^3 \cos ^7(c+d x)}{715 d (a+a \sin (c+d x))^{3/2}}-\frac {32 a^2 \cos ^7(c+d x)}{195 d \sqrt {a+a \sin (c+d x)}}-\frac {2 a \cos ^7(c+d x) \sqrt {a+a \sin (c+d x)}}{15 d} \] Output:
-4096/45045*a^5*cos(d*x+c)^7/d/(a+a*sin(d*x+c))^(7/2)-1024/6435*a^4*cos(d* x+c)^7/d/(a+a*sin(d*x+c))^(5/2)-128/715*a^3*cos(d*x+c)^7/d/(a+a*sin(d*x+c) )^(3/2)-32/195*a^2*cos(d*x+c)^7/d/(a+a*sin(d*x+c))^(1/2)-2/15*a*cos(d*x+c) ^7*(a+a*sin(d*x+c))^(1/2)/d
Time = 0.64 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.50 \[ \int \cos ^6(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {2 \cos ^7(c+d x) (a (1+\sin (c+d x)))^{3/2} \left (16363+34748 \sin (c+d x)+33138 \sin ^2(c+d x)+15708 \sin ^3(c+d x)+3003 \sin ^4(c+d x)\right )}{45045 d (1+\sin (c+d x))^5} \] Input:
Integrate[Cos[c + d*x]^6*(a + a*Sin[c + d*x])^(3/2),x]
Output:
(-2*Cos[c + d*x]^7*(a*(1 + Sin[c + d*x]))^(3/2)*(16363 + 34748*Sin[c + d*x ] + 33138*Sin[c + d*x]^2 + 15708*Sin[c + d*x]^3 + 3003*Sin[c + d*x]^4))/(4 5045*d*(1 + Sin[c + d*x])^5)
Time = 0.80 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.08, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {3042, 3153, 3042, 3153, 3042, 3153, 3042, 3153, 3042, 3152}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \cos ^6(c+d x) (a \sin (c+d x)+a)^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (c+d x)^6 (a \sin (c+d x)+a)^{3/2}dx\) |
| \(\Big \downarrow \) 3153 |
| \(\displaystyle \frac {16}{15} a \int \cos ^6(c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {2 a \cos ^7(c+d x) \sqrt {a \sin (c+d x)+a}}{15 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {16}{15} a \int \cos (c+d x)^6 \sqrt {\sin (c+d x) a+a}dx-\frac {2 a \cos ^7(c+d x) \sqrt {a \sin (c+d x)+a}}{15 d}\) |
| \(\Big \downarrow \) 3153 |
| \(\displaystyle \frac {16}{15} a \left (\frac {12}{13} a \int \frac {\cos ^6(c+d x)}{\sqrt {\sin (c+d x) a+a}}dx-\frac {2 a \cos ^7(c+d x)}{13 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {2 a \cos ^7(c+d x) \sqrt {a \sin (c+d x)+a}}{15 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {16}{15} a \left (\frac {12}{13} a \int \frac {\cos (c+d x)^6}{\sqrt {\sin (c+d x) a+a}}dx-\frac {2 a \cos ^7(c+d x)}{13 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {2 a \cos ^7(c+d x) \sqrt {a \sin (c+d x)+a}}{15 d}\) |
| \(\Big \downarrow \) 3153 |
| \(\displaystyle \frac {16}{15} a \left (\frac {12}{13} a \left (\frac {8}{11} a \int \frac {\cos ^6(c+d x)}{(\sin (c+d x) a+a)^{3/2}}dx-\frac {2 a \cos ^7(c+d x)}{11 d (a \sin (c+d x)+a)^{3/2}}\right )-\frac {2 a \cos ^7(c+d x)}{13 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {2 a \cos ^7(c+d x) \sqrt {a \sin (c+d x)+a}}{15 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {16}{15} a \left (\frac {12}{13} a \left (\frac {8}{11} a \int \frac {\cos (c+d x)^6}{(\sin (c+d x) a+a)^{3/2}}dx-\frac {2 a \cos ^7(c+d x)}{11 d (a \sin (c+d x)+a)^{3/2}}\right )-\frac {2 a \cos ^7(c+d x)}{13 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {2 a \cos ^7(c+d x) \sqrt {a \sin (c+d x)+a}}{15 d}\) |
| \(\Big \downarrow \) 3153 |
| \(\displaystyle \frac {16}{15} a \left (\frac {12}{13} a \left (\frac {8}{11} a \left (\frac {4}{9} a \int \frac {\cos ^6(c+d x)}{(\sin (c+d x) a+a)^{5/2}}dx-\frac {2 a \cos ^7(c+d x)}{9 d (a \sin (c+d x)+a)^{5/2}}\right )-\frac {2 a \cos ^7(c+d x)}{11 d (a \sin (c+d x)+a)^{3/2}}\right )-\frac {2 a \cos ^7(c+d x)}{13 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {2 a \cos ^7(c+d x) \sqrt {a \sin (c+d x)+a}}{15 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {16}{15} a \left (\frac {12}{13} a \left (\frac {8}{11} a \left (\frac {4}{9} a \int \frac {\cos (c+d x)^6}{(\sin (c+d x) a+a)^{5/2}}dx-\frac {2 a \cos ^7(c+d x)}{9 d (a \sin (c+d x)+a)^{5/2}}\right )-\frac {2 a \cos ^7(c+d x)}{11 d (a \sin (c+d x)+a)^{3/2}}\right )-\frac {2 a \cos ^7(c+d x)}{13 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {2 a \cos ^7(c+d x) \sqrt {a \sin (c+d x)+a}}{15 d}\) |
| \(\Big \downarrow \) 3152 |
| \(\displaystyle \frac {16}{15} a \left (\frac {12}{13} a \left (\frac {8}{11} a \left (-\frac {8 a^2 \cos ^7(c+d x)}{63 d (a \sin (c+d x)+a)^{7/2}}-\frac {2 a \cos ^7(c+d x)}{9 d (a \sin (c+d x)+a)^{5/2}}\right )-\frac {2 a \cos ^7(c+d x)}{11 d (a \sin (c+d x)+a)^{3/2}}\right )-\frac {2 a \cos ^7(c+d x)}{13 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {2 a \cos ^7(c+d x) \sqrt {a \sin (c+d x)+a}}{15 d}\) |
Input:
Int[Cos[c + d*x]^6*(a + a*Sin[c + d*x])^(3/2),x]
Output:
(-2*a*Cos[c + d*x]^7*Sqrt[a + a*Sin[c + d*x]])/(15*d) + (16*a*((-2*a*Cos[c + d*x]^7)/(13*d*Sqrt[a + a*Sin[c + d*x]]) + (12*a*((-2*a*Cos[c + d*x]^7)/ (11*d*(a + a*Sin[c + d*x])^(3/2)) + (8*a*((-8*a^2*Cos[c + d*x]^7)/(63*d*(a + a*Sin[c + d*x])^(7/2)) - (2*a*Cos[c + d*x]^7)/(9*d*(a + a*Sin[c + d*x]) ^(5/2))))/11))/13))/15
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[b*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x ])^(m - 1)/(f*g*(m - 1))), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && EqQ[2*m + p - 1, 0] && NeQ[m, 1]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[(-b)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m - 1)/(f*g*(m + p))), x] + Simp[a*((2*m + p - 1)/(m + p)) Int[(g* Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[Simplify[(2*m + p - 1)/2], 0] && NeQ[m + p, 0]
Time = 0.54 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.55
| method | result | size |
| default | \(-\frac {2 \left (1+\sin \left (d x +c \right )\right ) a^{2} \left (\sin \left (d x +c \right )-1\right )^{4} \left (3003 \sin \left (d x +c \right )^{4}+15708 \sin \left (d x +c \right )^{3}+33138 \sin \left (d x +c \right )^{2}+34748 \sin \left (d x +c \right )+16363\right )}{45045 \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(87\) |
Input:
int(cos(d*x+c)^6*(a+a*sin(d*x+c))^(3/2),x,method=_RETURNVERBOSE)
Output:
-2/45045*(1+sin(d*x+c))*a^2*(sin(d*x+c)-1)^4*(3003*sin(d*x+c)^4+15708*sin( d*x+c)^3+33138*sin(d*x+c)^2+34748*sin(d*x+c)+16363)/cos(d*x+c)/(a+a*sin(d* x+c))^(1/2)/d
Time = 0.10 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.32 \[ \int \cos ^6(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {2 \, {\left (3003 \, a \cos \left (d x + c\right )^{8} + 6699 \, a \cos \left (d x + c\right )^{7} - 336 \, a \cos \left (d x + c\right )^{6} + 448 \, a \cos \left (d x + c\right )^{5} - 640 \, a \cos \left (d x + c\right )^{4} + 1024 \, a \cos \left (d x + c\right )^{3} - 2048 \, a \cos \left (d x + c\right )^{2} + 8192 \, a \cos \left (d x + c\right ) + {\left (3003 \, a \cos \left (d x + c\right )^{7} - 3696 \, a \cos \left (d x + c\right )^{6} - 4032 \, a \cos \left (d x + c\right )^{5} - 4480 \, a \cos \left (d x + c\right )^{4} - 5120 \, a \cos \left (d x + c\right )^{3} - 6144 \, a \cos \left (d x + c\right )^{2} - 8192 \, a \cos \left (d x + c\right ) - 16384 \, a\right )} \sin \left (d x + c\right ) + 16384 \, a\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{45045 \, {\left (d \cos \left (d x + c\right ) + d \sin \left (d x + c\right ) + d\right )}} \] Input:
integrate(cos(d*x+c)^6*(a+a*sin(d*x+c))^(3/2),x, algorithm="fricas")
Output:
-2/45045*(3003*a*cos(d*x + c)^8 + 6699*a*cos(d*x + c)^7 - 336*a*cos(d*x + c)^6 + 448*a*cos(d*x + c)^5 - 640*a*cos(d*x + c)^4 + 1024*a*cos(d*x + c)^3 - 2048*a*cos(d*x + c)^2 + 8192*a*cos(d*x + c) + (3003*a*cos(d*x + c)^7 - 3696*a*cos(d*x + c)^6 - 4032*a*cos(d*x + c)^5 - 4480*a*cos(d*x + c)^4 - 51 20*a*cos(d*x + c)^3 - 6144*a*cos(d*x + c)^2 - 8192*a*cos(d*x + c) - 16384* a)*sin(d*x + c) + 16384*a)*sqrt(a*sin(d*x + c) + a)/(d*cos(d*x + c) + d*si n(d*x + c) + d)
Timed out. \[ \int \cos ^6(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**6*(a+a*sin(d*x+c))**(3/2),x)
Output:
Timed out
\[ \int \cos ^6(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{6} \,d x } \] Input:
integrate(cos(d*x+c)^6*(a+a*sin(d*x+c))^(3/2),x, algorithm="maxima")
Output:
integrate((a*sin(d*x + c) + a)^(3/2)*cos(d*x + c)^6, x)
Time = 0.14 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.02 \[ \int \cos ^6(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\frac {256 \, \sqrt {2} {\left (3003 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} - 13860 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 24570 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 20020 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 6435 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}\right )} \sqrt {a}}{45045 \, d} \] Input:
integrate(cos(d*x+c)^6*(a+a*sin(d*x+c))^(3/2),x, algorithm="giac")
Output:
256/45045*sqrt(2)*(3003*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^15 - 13860*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1 /4*pi + 1/2*d*x + 1/2*c)^13 + 24570*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))* sin(-1/4*pi + 1/2*d*x + 1/2*c)^11 - 20020*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/ 2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^9 + 6435*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^7)*sqrt(a)/d
Timed out. \[ \int \cos ^6(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\int {\cos \left (c+d\,x\right )}^6\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2} \,d x \] Input:
int(cos(c + d*x)^6*(a + a*sin(c + d*x))^(3/2),x)
Output:
int(cos(c + d*x)^6*(a + a*sin(c + d*x))^(3/2), x)
\[ \int \cos ^6(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\sqrt {a}\, a \left (\int \sqrt {\sin \left (d x +c \right )+1}\, \cos \left (d x +c \right )^{6} \sin \left (d x +c \right )d x +\int \sqrt {\sin \left (d x +c \right )+1}\, \cos \left (d x +c \right )^{6}d x \right ) \] Input:
int(cos(d*x+c)^6*(a+a*sin(d*x+c))^(3/2),x)
Output:
sqrt(a)*a*(int(sqrt(sin(c + d*x) + 1)*cos(c + d*x)**6*sin(c + d*x),x) + in t(sqrt(sin(c + d*x) + 1)*cos(c + d*x)**6,x))