3.2.40 \(\int \cos ^6(c+d x) (a+a \sin (c+d x))^{7/2} \, dx\) [140]

Optimal. Leaf size=223 \[ -\frac {131072 a^7 \cos ^7(c+d x)}{969969 d (a+a \sin (c+d x))^{7/2}}-\frac {32768 a^6 \cos ^7(c+d x)}{138567 d (a+a \sin (c+d x))^{5/2}}-\frac {12288 a^5 \cos ^7(c+d x)}{46189 d (a+a \sin (c+d x))^{3/2}}-\frac {1024 a^4 \cos ^7(c+d x)}{4199 d \sqrt {a+a \sin (c+d x)}}-\frac {64 a^3 \cos ^7(c+d x) \sqrt {a+a \sin (c+d x)}}{323 d}-\frac {48 a^2 \cos ^7(c+d x) (a+a \sin (c+d x))^{3/2}}{323 d}-\frac {2 a \cos ^7(c+d x) (a+a \sin (c+d x))^{5/2}}{19 d} \]

[Out]

-131072/969969*a^7*cos(d*x+c)^7/d/(a+a*sin(d*x+c))^(7/2)-32768/138567*a^6*cos(d*x+c)^7/d/(a+a*sin(d*x+c))^(5/2
)-12288/46189*a^5*cos(d*x+c)^7/d/(a+a*sin(d*x+c))^(3/2)-48/323*a^2*cos(d*x+c)^7*(a+a*sin(d*x+c))^(3/2)/d-2/19*
a*cos(d*x+c)^7*(a+a*sin(d*x+c))^(5/2)/d-1024/4199*a^4*cos(d*x+c)^7/d/(a+a*sin(d*x+c))^(1/2)-64/323*a^3*cos(d*x
+c)^7*(a+a*sin(d*x+c))^(1/2)/d

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Rubi [A]
time = 0.29, antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2753, 2752} \begin {gather*} -\frac {131072 a^7 \cos ^7(c+d x)}{969969 d (a \sin (c+d x)+a)^{7/2}}-\frac {32768 a^6 \cos ^7(c+d x)}{138567 d (a \sin (c+d x)+a)^{5/2}}-\frac {12288 a^5 \cos ^7(c+d x)}{46189 d (a \sin (c+d x)+a)^{3/2}}-\frac {1024 a^4 \cos ^7(c+d x)}{4199 d \sqrt {a \sin (c+d x)+a}}-\frac {64 a^3 \cos ^7(c+d x) \sqrt {a \sin (c+d x)+a}}{323 d}-\frac {48 a^2 \cos ^7(c+d x) (a \sin (c+d x)+a)^{3/2}}{323 d}-\frac {2 a \cos ^7(c+d x) (a \sin (c+d x)+a)^{5/2}}{19 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^6*(a + a*Sin[c + d*x])^(7/2),x]

[Out]

(-131072*a^7*Cos[c + d*x]^7)/(969969*d*(a + a*Sin[c + d*x])^(7/2)) - (32768*a^6*Cos[c + d*x]^7)/(138567*d*(a +
a*Sin[c + d*x])^(5/2)) - (12288*a^5*Cos[c + d*x]^7)/(46189*d*(a + a*Sin[c + d*x])^(3/2)) - (1024*a^4*Cos[c +
d*x]^7)/(4199*d*Sqrt[a + a*Sin[c + d*x]]) - (64*a^3*Cos[c + d*x]^7*Sqrt[a + a*Sin[c + d*x]])/(323*d) - (48*a^2
*Cos[c + d*x]^7*(a + a*Sin[c + d*x])^(3/2))/(323*d) - (2*a*Cos[c + d*x]^7*(a + a*Sin[c + d*x])^(5/2))/(19*d)

Rule 2752

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[b*(g*C
os[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] && Eq
Q[a^2 - b^2, 0] && EqQ[2*m + p - 1, 0] && NeQ[m, 1]

Rule 2753

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] + Dist[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]

Rubi steps

\begin {align*} \int \cos ^6(c+d x) (a+a \sin (c+d x))^{7/2} \, dx &=-\frac {2 a \cos ^7(c+d x) (a+a \sin (c+d x))^{5/2}}{19 d}+\frac {1}{19} (24 a) \int \cos ^6(c+d x) (a+a \sin (c+d x))^{5/2} \, dx\\ &=-\frac {48 a^2 \cos ^7(c+d x) (a+a \sin (c+d x))^{3/2}}{323 d}-\frac {2 a \cos ^7(c+d x) (a+a \sin (c+d x))^{5/2}}{19 d}+\frac {1}{323} \left (480 a^2\right ) \int \cos ^6(c+d x) (a+a \sin (c+d x))^{3/2} \, dx\\ &=-\frac {64 a^3 \cos ^7(c+d x) \sqrt {a+a \sin (c+d x)}}{323 d}-\frac {48 a^2 \cos ^7(c+d x) (a+a \sin (c+d x))^{3/2}}{323 d}-\frac {2 a \cos ^7(c+d x) (a+a \sin (c+d x))^{5/2}}{19 d}+\frac {1}{323} \left (512 a^3\right ) \int \cos ^6(c+d x) \sqrt {a+a \sin (c+d x)} \, dx\\ &=-\frac {1024 a^4 \cos ^7(c+d x)}{4199 d \sqrt {a+a \sin (c+d x)}}-\frac {64 a^3 \cos ^7(c+d x) \sqrt {a+a \sin (c+d x)}}{323 d}-\frac {48 a^2 \cos ^7(c+d x) (a+a \sin (c+d x))^{3/2}}{323 d}-\frac {2 a \cos ^7(c+d x) (a+a \sin (c+d x))^{5/2}}{19 d}+\frac {\left (6144 a^4\right ) \int \frac {\cos ^6(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx}{4199}\\ &=-\frac {12288 a^5 \cos ^7(c+d x)}{46189 d (a+a \sin (c+d x))^{3/2}}-\frac {1024 a^4 \cos ^7(c+d x)}{4199 d \sqrt {a+a \sin (c+d x)}}-\frac {64 a^3 \cos ^7(c+d x) \sqrt {a+a \sin (c+d x)}}{323 d}-\frac {48 a^2 \cos ^7(c+d x) (a+a \sin (c+d x))^{3/2}}{323 d}-\frac {2 a \cos ^7(c+d x) (a+a \sin (c+d x))^{5/2}}{19 d}+\frac {\left (49152 a^5\right ) \int \frac {\cos ^6(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx}{46189}\\ &=-\frac {32768 a^6 \cos ^7(c+d x)}{138567 d (a+a \sin (c+d x))^{5/2}}-\frac {12288 a^5 \cos ^7(c+d x)}{46189 d (a+a \sin (c+d x))^{3/2}}-\frac {1024 a^4 \cos ^7(c+d x)}{4199 d \sqrt {a+a \sin (c+d x)}}-\frac {64 a^3 \cos ^7(c+d x) \sqrt {a+a \sin (c+d x)}}{323 d}-\frac {48 a^2 \cos ^7(c+d x) (a+a \sin (c+d x))^{3/2}}{323 d}-\frac {2 a \cos ^7(c+d x) (a+a \sin (c+d x))^{5/2}}{19 d}+\frac {\left (65536 a^6\right ) \int \frac {\cos ^6(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx}{138567}\\ &=-\frac {131072 a^7 \cos ^7(c+d x)}{969969 d (a+a \sin (c+d x))^{7/2}}-\frac {32768 a^6 \cos ^7(c+d x)}{138567 d (a+a \sin (c+d x))^{5/2}}-\frac {12288 a^5 \cos ^7(c+d x)}{46189 d (a+a \sin (c+d x))^{3/2}}-\frac {1024 a^4 \cos ^7(c+d x)}{4199 d \sqrt {a+a \sin (c+d x)}}-\frac {64 a^3 \cos ^7(c+d x) \sqrt {a+a \sin (c+d x)}}{323 d}-\frac {48 a^2 \cos ^7(c+d x) (a+a \sin (c+d x))^{3/2}}{323 d}-\frac {2 a \cos ^7(c+d x) (a+a \sin (c+d x))^{5/2}}{19 d}\\ \end {align*}

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Mathematica [A]
time = 0.34, size = 102, normalized size = 0.46 \begin {gather*} -\frac {2 a^3 \cos ^7(c+d x) \sqrt {a (1+\sin (c+d x))} \left (646739+1778602 \sin (c+d x)+2546901 \sin ^2(c+d x)+2244396 \sin ^3(c+d x)+1222221 \sin ^4(c+d x)+378378 \sin ^5(c+d x)+51051 \sin ^6(c+d x)\right )}{969969 d (1+\sin (c+d x))^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^6*(a + a*Sin[c + d*x])^(7/2),x]

[Out]

(-2*a^3*Cos[c + d*x]^7*Sqrt[a*(1 + Sin[c + d*x])]*(646739 + 1778602*Sin[c + d*x] + 2546901*Sin[c + d*x]^2 + 22
44396*Sin[c + d*x]^3 + 1222221*Sin[c + d*x]^4 + 378378*Sin[c + d*x]^5 + 51051*Sin[c + d*x]^6))/(969969*d*(1 +
Sin[c + d*x])^4)

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Maple [A]
time = 0.40, size = 107, normalized size = 0.48

method	result	size
default	\(-\frac {2 \left (1+\sin \left (d x +c \right )\right ) a^{4} \left (\sin \left (d x +c \right )-1\right )^{4} \left (51051 \left (\sin ^{6}\left (d x +c \right )\right )+378378 \left (\sin ^{5}\left (d x +c \right )\right )+1222221 \left (\sin ^{4}\left (d x +c \right )\right )+2244396 \left (\sin ^{3}\left (d x +c \right )\right )+2546901 \left (\sin ^{2}\left (d x +c \right )\right )+1778602 \sin \left (d x +c \right )+646739\right )}{969969 \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\)	\(107\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*(a+a*sin(d*x+c))^(7/2),x,method=_RETURNVERBOSE)

[Out]

-2/969969*(1+sin(d*x+c))*a^4*(sin(d*x+c)-1)^4*(51051*sin(d*x+c)^6+378378*sin(d*x+c)^5+1222221*sin(d*x+c)^4+224
4396*sin(d*x+c)^3+2546901*sin(d*x+c)^2+1778602*sin(d*x+c)+646739)/cos(d*x+c)/(a+a*sin(d*x+c))^(1/2)/d

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*(a+a*sin(d*x+c))^(7/2),x, algorithm="maxima")

[Out]

integrate((a*sin(d*x + c) + a)^(7/2)*cos(d*x + c)^6, x)

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Fricas [A]
time = 0.36, size = 296, normalized size = 1.33 \begin {gather*} \frac {2 \, {\left (51051 \, a^{3} \cos \left (d x + c\right )^{10} + 225225 \, a^{3} \cos \left (d x + c\right )^{9} - 270270 \, a^{3} \cos \left (d x + c\right )^{8} - 562716 \, a^{3} \cos \left (d x + c\right )^{7} + 10752 \, a^{3} \cos \left (d x + c\right )^{6} - 14336 \, a^{3} \cos \left (d x + c\right )^{5} + 20480 \, a^{3} \cos \left (d x + c\right )^{4} - 32768 \, a^{3} \cos \left (d x + c\right )^{3} + 65536 \, a^{3} \cos \left (d x + c\right )^{2} - 262144 \, a^{3} \cos \left (d x + c\right ) - 524288 \, a^{3} + {\left (51051 \, a^{3} \cos \left (d x + c\right )^{9} - 174174 \, a^{3} \cos \left (d x + c\right )^{8} - 444444 \, a^{3} \cos \left (d x + c\right )^{7} + 118272 \, a^{3} \cos \left (d x + c\right )^{6} + 129024 \, a^{3} \cos \left (d x + c\right )^{5} + 143360 \, a^{3} \cos \left (d x + c\right )^{4} + 163840 \, a^{3} \cos \left (d x + c\right )^{3} + 196608 \, a^{3} \cos \left (d x + c\right )^{2} + 262144 \, a^{3} \cos \left (d x + c\right ) + 524288 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{969969 \, {\left (d \cos \left (d x + c\right ) + d \sin \left (d x + c\right ) + d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*(a+a*sin(d*x+c))^(7/2),x, algorithm="fricas")

[Out]

2/969969*(51051*a^3*cos(d*x + c)^10 + 225225*a^3*cos(d*x + c)^9 - 270270*a^3*cos(d*x + c)^8 - 562716*a^3*cos(d
*x + c)^7 + 10752*a^3*cos(d*x + c)^6 - 14336*a^3*cos(d*x + c)^5 + 20480*a^3*cos(d*x + c)^4 - 32768*a^3*cos(d*x
+ c)^3 + 65536*a^3*cos(d*x + c)^2 - 262144*a^3*cos(d*x + c) - 524288*a^3 + (51051*a^3*cos(d*x + c)^9 - 174174
*a^3*cos(d*x + c)^8 - 444444*a^3*cos(d*x + c)^7 + 118272*a^3*cos(d*x + c)^6 + 129024*a^3*cos(d*x + c)^5 + 1433
60*a^3*cos(d*x + c)^4 + 163840*a^3*cos(d*x + c)^3 + 196608*a^3*cos(d*x + c)^2 + 262144*a^3*cos(d*x + c) + 5242
88*a^3)*sin(d*x + c))*sqrt(a*sin(d*x + c) + a)/(d*cos(d*x + c) + d*sin(d*x + c) + d)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**6*(a+a*sin(d*x+c))**(7/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 10626 deep

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Giac [A]
time = 6.41, size = 236, normalized size = 1.06 \begin {gather*} \frac {1024 \, \sqrt {2} {\left (51051 \, a^{3} \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 )^{19} - 342342 \, a^{3} \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 )^{17} + 969969 \, a^{3} \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} - 1492260 \, a^{3} \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} + 1322685 \, a^{3} \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} - 646646 \, a^{3} \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} + 138567 \, a^{3} \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}}{969969 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*(a+a*sin(d*x+c))^(7/2),x, algorithm="giac")

[Out]

1024/969969*sqrt(2)*(51051*a^3*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^19 - 342342*
a^3*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^17 + 969969*a^3*sgn(cos(-1/4*pi + 1/2*d
*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^15 - 1492260*a^3*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi +
1/2*d*x + 1/2*c)^13 + 1322685*a^3*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^11 - 646
646*a^3*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^9 + 138567*a^3*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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\cos \left (c+d\,x\right )}^6\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{7/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^6*(a + a*sin(c + d*x))^(7/2),x)

[Out]

int(cos(c + d*x)^6*(a + a*sin(c + d*x))^(7/2), x)

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