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

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

[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)

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Rubi [A]  time = 0.429509, antiderivative size = 223, normalized size of antiderivative = 1., 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 = {2674, 2673} \[ -\frac{48 a^2 \cos ^7(c+d x) (a \sin (c+d x)+a)^{3/2}}{323 d}-\frac{64 a^3 \cos ^7(c+d x) \sqrt{a \sin (c+d x)+a}}{323 d}-\frac{1024 a^4 \cos ^7(c+d x)}{4199 d \sqrt{a \sin (c+d x)+a}}-\frac{12288 a^5 \cos ^7(c+d x)}{46189 d (a \sin (c+d x)+a)^{3/2}}-\frac{32768 a^6 \cos ^7(c+d x)}{138567 d (a \sin (c+d x)+a)^{5/2}}-\frac{131072 a^7 \cos ^7(c+d x)}{969969 d (a \sin (c+d x)+a)^{7/2}}-\frac{2 a \cos ^7(c+d x) (a \sin (c+d x)+a)^{5/2}}{19 d} \]

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 2674

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]

Rule 2673

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] && Eq
Q[a^2 - b^2, 0] && EqQ[2*m + p - 1, 0] && NeQ[m, 1]

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*}

Mathematica [A]  time = 0.548844, size = 102, normalized size = 0.46 \[ -\frac{2 a^3 \left (51051 \sin ^6(c+d x)+378378 \sin ^5(c+d x)+1222221 \sin ^4(c+d x)+2244396 \sin ^3(c+d x)+2546901 \sin ^2(c+d x)+1778602 \sin (c+d x)+646739\right ) \cos ^7(c+d x) \sqrt{a (\sin (c+d x)+1)}}{969969 d (\sin (c+d x)+1)^4} \]

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 = 0.119, size = 107, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 2+2\,\sin \left ( dx+c \right ) \right ){a}^{4} \left ( \sin \left ( dx+c \right ) -1 \right ) ^{4} \left ( 51051\, \left ( \sin \left ( dx+c \right ) \right ) ^{6}+378378\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}+1222221\, \left ( \sin \left ( dx+c \right ) \right ) ^{4}+2244396\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}+2546901\, \left ( \sin \left ( dx+c \right ) \right ) ^{2}+1778602\,\sin \left ( dx+c \right ) +646739 \right ) }{969969\,d\cos \left ( dx+c \right ) }{\frac{1}{\sqrt{a+a\sin \left ( dx+c \right ) }}}} \end{align*}

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)

[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., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{7}{2}} \cos \left (d x + c\right )^{6}\,{d x} \end{align*}

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 = 1.76278, size = 859, normalized size = 3.85 \begin{align*} \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{align*}

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(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{7}{2}} \cos \left (d x + c\right )^{6}\,{d x} \end{align*}

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]

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