3.3.85 \(\int \cos ^6(e+f x) (a+b \sin ^2(e+f x)) \, dx\) [285]

Optimal. Leaf size=109 \[ \frac {5}{128} (8 a+b) x+\frac {5 (8 a+b) \cos (e+f x) \sin (e+f x)}{128 f}+\frac {5 (8 a+b) \cos ^3(e+f x) \sin (e+f x)}{192 f}+\frac {(8 a+b) \cos ^5(e+f x) \sin (e+f x)}{48 f}-\frac {b \cos ^7(e+f x) \sin (e+f x)}{8 f} \]

[Out]

5/128*(8*a+b)*x+5/128*(8*a+b)*cos(f*x+e)*sin(f*x+e)/f+5/192*(8*a+b)*cos(f*x+e)^3*sin(f*x+e)/f+1/48*(8*a+b)*cos
(f*x+e)^5*sin(f*x+e)/f-1/8*b*cos(f*x+e)^7*sin(f*x+e)/f

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Rubi [A]
time = 0.04, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3270, 393, 205, 209} \begin {gather*} \frac {(8 a+b) \sin (e+f x) \cos ^5(e+f x)}{48 f}+\frac {5 (8 a+b) \sin (e+f x) \cos ^3(e+f x)}{192 f}+\frac {5 (8 a+b) \sin (e+f x) \cos (e+f x)}{128 f}+\frac {5}{128} x (8 a+b)-\frac {b \sin (e+f x) \cos ^7(e+f x)}{8 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[e + f*x]^6*(a + b*Sin[e + f*x]^2),x]

[Out]

(5*(8*a + b)*x)/128 + (5*(8*a + b)*Cos[e + f*x]*Sin[e + f*x])/(128*f) + (5*(8*a + b)*Cos[e + f*x]^3*Sin[e + f*
x])/(192*f) + ((8*a + b)*Cos[e + f*x]^5*Sin[e + f*x])/(48*f) - (b*Cos[e + f*x]^7*Sin[e + f*x])/(8*f)

Rule 205

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 393

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*c - a*d))*x*((a + b*x^n)^(p
+ 1)/(a*b*n*(p + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x]
/; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])

Rule 3270

Int[cos[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = FreeF
actors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[(a + (a + b)*ff^2*x^2)^p/(1 + ff^2*x^2)^(m/2 + p + 1), x], x, T
an[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]

Rubi steps

\begin {align*} \int \cos ^6(e+f x) \left (a+b \sin ^2(e+f x)\right ) \, dx &=\frac {\text {Subst}\left (\int \frac {a+(a+b) x^2}{\left (1+x^2\right )^5} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {b \cos ^7(e+f x) \sin (e+f x)}{8 f}+\frac {(8 a+b) \text {Subst}\left (\int \frac {1}{\left (1+x^2\right )^4} \, dx,x,\tan (e+f x)\right )}{8 f}\\ &=\frac {(8 a+b) \cos ^5(e+f x) \sin (e+f x)}{48 f}-\frac {b \cos ^7(e+f x) \sin (e+f x)}{8 f}+\frac {(5 (8 a+b)) \text {Subst}\left (\int \frac {1}{\left (1+x^2\right )^3} \, dx,x,\tan (e+f x)\right )}{48 f}\\ &=\frac {5 (8 a+b) \cos ^3(e+f x) \sin (e+f x)}{192 f}+\frac {(8 a+b) \cos ^5(e+f x) \sin (e+f x)}{48 f}-\frac {b \cos ^7(e+f x) \sin (e+f x)}{8 f}+\frac {(5 (8 a+b)) \text {Subst}\left (\int \frac {1}{\left (1+x^2\right )^2} \, dx,x,\tan (e+f x)\right )}{64 f}\\ &=\frac {5 (8 a+b) \cos (e+f x) \sin (e+f x)}{128 f}+\frac {5 (8 a+b) \cos ^3(e+f x) \sin (e+f x)}{192 f}+\frac {(8 a+b) \cos ^5(e+f x) \sin (e+f x)}{48 f}-\frac {b \cos ^7(e+f x) \sin (e+f x)}{8 f}+\frac {(5 (8 a+b)) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{128 f}\\ &=\frac {5}{128} (8 a+b) x+\frac {5 (8 a+b) \cos (e+f x) \sin (e+f x)}{128 f}+\frac {5 (8 a+b) \cos ^3(e+f x) \sin (e+f x)}{192 f}+\frac {(8 a+b) \cos ^5(e+f x) \sin (e+f x)}{48 f}-\frac {b \cos ^7(e+f x) \sin (e+f x)}{8 f}\\ \end {align*}

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Mathematica [A]
time = 0.22, size = 87, normalized size = 0.80 \begin {gather*} \frac {960 a e+960 a f x+120 b f x+48 (15 a+b) \sin (2 (e+f x))+24 (6 a-b) \sin (4 (e+f x))+16 a \sin (6 (e+f x))-16 b \sin (6 (e+f x))-3 b \sin (8 (e+f x))}{3072 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[e + f*x]^6*(a + b*Sin[e + f*x]^2),x]

[Out]

(960*a*e + 960*a*f*x + 120*b*f*x + 48*(15*a + b)*Sin[2*(e + f*x)] + 24*(6*a - b)*Sin[4*(e + f*x)] + 16*a*Sin[6
*(e + f*x)] - 16*b*Sin[6*(e + f*x)] - 3*b*Sin[8*(e + f*x)])/(3072*f)

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Maple [A]
time = 0.48, size = 112, normalized size = 1.03

method	result	size
derivativedivides	\(\frac {b \left (-\frac {\sin \left (f x +e \right ) \left (\cos ^{7}\left (f x +e \right )\right )}{8}+\frac {\left (\cos ^{5}\left (f x +e \right )+\frac {5 \left (\cos ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \cos \left (f x +e \right )}{8}\right ) \sin \left (f x +e \right )}{48}+\frac {5 f x}{128}+\frac {5 e}{128}\right )+a \left (\frac {\left (\cos ^{5}\left (f x +e \right )+\frac {5 \left (\cos ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \cos \left (f x +e \right )}{8}\right ) \sin \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )}{f}\)	\(112\)
default	\(\frac {b \left (-\frac {\sin \left (f x +e \right ) \left (\cos ^{7}\left (f x +e \right )\right )}{8}+\frac {\left (\cos ^{5}\left (f x +e \right )+\frac {5 \left (\cos ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \cos \left (f x +e \right )}{8}\right ) \sin \left (f x +e \right )}{48}+\frac {5 f x}{128}+\frac {5 e}{128}\right )+a \left (\frac {\left (\cos ^{5}\left (f x +e \right )+\frac {5 \left (\cos ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \cos \left (f x +e \right )}{8}\right ) \sin \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )}{f}\)	\(112\)
risch	\(\frac {5 a x}{16}+\frac {5 b x}{128}-\frac {b \sin \left (8 f x +8 e \right )}{1024 f}+\frac {\sin \left (6 f x +6 e \right ) a}{192 f}-\frac {\sin \left (6 f x +6 e \right ) b}{192 f}+\frac {3 \sin \left (4 f x +4 e \right ) a}{64 f}-\frac {\sin \left (4 f x +4 e \right ) b}{128 f}+\frac {15 \sin \left (2 f x +2 e \right ) a}{64 f}+\frac {\sin \left (2 f x +2 e \right ) b}{64 f}\)	\(115\)
norman	\(\frac {\left (\frac {5 a}{16}+\frac {5 b}{128}\right ) x +\left (\frac {5 a}{2}+\frac {5 b}{16}\right ) x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {5 a}{2}+\frac {5 b}{16}\right ) x \left (\tan ^{14}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {5 a}{16}+\frac {5 b}{128}\right ) x \left (\tan ^{16}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {35 a}{2}+\frac {35 b}{16}\right ) x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {35 a}{2}+\frac {35 b}{16}\right ) x \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {35 a}{4}+\frac {35 b}{32}\right ) x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {35 a}{4}+\frac {35 b}{32}\right ) x \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {175 a}{8}+\frac {175 b}{64}\right ) x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {\left (88 a -5 b \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{64 f}-\frac {\left (88 a -5 b \right ) \left (\tan ^{15}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{64 f}+\frac {5 \left (136 a +353 b \right ) \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{192 f}-\frac {5 \left (136 a +353 b \right ) \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{192 f}+\frac {\left (488 a +397 b \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{192 f}-\frac {\left (488 a +397 b \right ) \left (\tan ^{13}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{192 f}+\frac {\left (904 a -895 b \right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{192 f}-\frac {\left (904 a -895 b \right ) \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{192 f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{8}}\)	\(369\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(f*x+e)^6*(a+b*sin(f*x+e)^2),x,method=_RETURNVERBOSE)

[Out]

1/f*(b*(-1/8*sin(f*x+e)*cos(f*x+e)^7+1/48*(cos(f*x+e)^5+5/4*cos(f*x+e)^3+15/8*cos(f*x+e))*sin(f*x+e)+5/128*f*x
+5/128*e)+a*(1/6*(cos(f*x+e)^5+5/4*cos(f*x+e)^3+15/8*cos(f*x+e))*sin(f*x+e)+5/16*f*x+5/16*e))

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Maxima [A]
time = 0.53, size = 131, normalized size = 1.20 \begin {gather*} \frac {15 \, {\left (f x + e\right )} {\left (8 \, a + b\right )} + \frac {15 \, {\left (8 \, a + b\right )} \tan \left (f x + e\right )^{7} + 55 \, {\left (8 \, a + b\right )} \tan \left (f x + e\right )^{5} + 73 \, {\left (8 \, a + b\right )} \tan \left (f x + e\right )^{3} + 3 \, {\left (88 \, a - 5 \, b\right )} \tan \left (f x + e\right )}{\tan \left (f x + e\right )^{8} + 4 \, \tan \left (f x + e\right )^{6} + 6 \, \tan \left (f x + e\right )^{4} + 4 \, \tan \left (f x + e\right )^{2} + 1}}{384 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)^6*(a+b*sin(f*x+e)^2),x, algorithm="maxima")

[Out]

1/384*(15*(f*x + e)*(8*a + b) + (15*(8*a + b)*tan(f*x + e)^7 + 55*(8*a + b)*tan(f*x + e)^5 + 73*(8*a + b)*tan(
f*x + e)^3 + 3*(88*a - 5*b)*tan(f*x + e))/(tan(f*x + e)^8 + 4*tan(f*x + e)^6 + 6*tan(f*x + e)^4 + 4*tan(f*x +
e)^2 + 1))/f

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Fricas [A]
time = 0.40, size = 78, normalized size = 0.72 \begin {gather*} \frac {15 \, {\left (8 \, a + b\right )} f x - {\left (48 \, b \cos \left (f x + e\right )^{7} - 8 \, {\left (8 \, a + b\right )} \cos \left (f x + e\right )^{5} - 10 \, {\left (8 \, a + b\right )} \cos \left (f x + e\right )^{3} - 15 \, {\left (8 \, a + b\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{384 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)^6*(a+b*sin(f*x+e)^2),x, algorithm="fricas")

[Out]

1/384*(15*(8*a + b)*f*x - (48*b*cos(f*x + e)^7 - 8*(8*a + b)*cos(f*x + e)^5 - 10*(8*a + b)*cos(f*x + e)^3 - 15
*(8*a + b)*cos(f*x + e))*sin(f*x + e))/f

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 354 vs. \(2 (107) = 214\).
time = 0.92, size = 354, normalized size = 3.25 \begin {gather*} \begin {cases} \frac {5 a x \sin ^{6}{\left (e + f x \right )}}{16} + \frac {15 a x \sin ^{4}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{16} + \frac {15 a x \sin ^{2}{\left (e + f x \right )} \cos ^{4}{\left (e + f x \right )}}{16} + \frac {5 a x \cos ^{6}{\left (e + f x \right )}}{16} + \frac {5 a \sin ^{5}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{16 f} + \frac {5 a \sin ^{3}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{6 f} + \frac {11 a \sin {\left (e + f x \right )} \cos ^{5}{\left (e + f x \right )}}{16 f} + \frac {5 b x \sin ^{8}{\left (e + f x \right )}}{128} + \frac {5 b x \sin ^{6}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{32} + \frac {15 b x \sin ^{4}{\left (e + f x \right )} \cos ^{4}{\left (e + f x \right )}}{64} + \frac {5 b x \sin ^{2}{\left (e + f x \right )} \cos ^{6}{\left (e + f x \right )}}{32} + \frac {5 b x \cos ^{8}{\left (e + f x \right )}}{128} + \frac {5 b \sin ^{7}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{128 f} + \frac {55 b \sin ^{5}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{384 f} + \frac {73 b \sin ^{3}{\left (e + f x \right )} \cos ^{5}{\left (e + f x \right )}}{384 f} - \frac {5 b \sin {\left (e + f x \right )} \cos ^{7}{\left (e + f x \right )}}{128 f} & \text {for}\: f \neq 0 \\x \left (a + b \sin ^{2}{\left (e \right )}\right ) \cos ^{6}{\left (e \right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)**6*(a+b*sin(f*x+e)**2),x)

[Out]

Piecewise((5*a*x*sin(e + f*x)**6/16 + 15*a*x*sin(e + f*x)**4*cos(e + f*x)**2/16 + 15*a*x*sin(e + f*x)**2*cos(e
+ f*x)**4/16 + 5*a*x*cos(e + f*x)**6/16 + 5*a*sin(e + f*x)**5*cos(e + f*x)/(16*f) + 5*a*sin(e + f*x)**3*cos(e
+ f*x)**3/(6*f) + 11*a*sin(e + f*x)*cos(e + f*x)**5/(16*f) + 5*b*x*sin(e + f*x)**8/128 + 5*b*x*sin(e + f*x)**
6*cos(e + f*x)**2/32 + 15*b*x*sin(e + f*x)**4*cos(e + f*x)**4/64 + 5*b*x*sin(e + f*x)**2*cos(e + f*x)**6/32 +
5*b*x*cos(e + f*x)**8/128 + 5*b*sin(e + f*x)**7*cos(e + f*x)/(128*f) + 55*b*sin(e + f*x)**5*cos(e + f*x)**3/(3
84*f) + 73*b*sin(e + f*x)**3*cos(e + f*x)**5/(384*f) - 5*b*sin(e + f*x)*cos(e + f*x)**7/(128*f), Ne(f, 0)), (x
*(a + b*sin(e)**2)*cos(e)**6, True))

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Giac [A]
time = 0.52, size = 87, normalized size = 0.80 \begin {gather*} \frac {5}{128} \, {\left (8 \, a + b\right )} x - \frac {b \sin \left (8 \, f x + 8 \, e\right )}{1024 \, f} + \frac {{\left (a - b\right )} \sin \left (6 \, f x + 6 \, e\right )}{192 \, f} + \frac {{\left (6 \, a - b\right )} \sin \left (4 \, f x + 4 \, e\right )}{128 \, f} + \frac {{\left (15 \, a + b\right )} \sin \left (2 \, f x + 2 \, e\right )}{64 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)^6*(a+b*sin(f*x+e)^2),x, algorithm="giac")

[Out]

5/128*(8*a + b)*x - 1/1024*b*sin(8*f*x + 8*e)/f + 1/192*(a - b)*sin(6*f*x + 6*e)/f + 1/128*(6*a - b)*sin(4*f*x
+ 4*e)/f + 1/64*(15*a + b)*sin(2*f*x + 2*e)/f

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Mupad [B]
time = 15.22, size = 119, normalized size = 1.09 \begin {gather*} x\,\left (\frac {5\,a}{16}+\frac {5\,b}{128}\right )+\frac {\left (\frac {5\,a}{16}+\frac {5\,b}{128}\right )\,{\mathrm {tan}\left (e+f\,x\right )}^7+\left (\frac {55\,a}{48}+\frac {55\,b}{384}\right )\,{\mathrm {tan}\left (e+f\,x\right )}^5+\left (\frac {73\,a}{48}+\frac {73\,b}{384}\right )\,{\mathrm {tan}\left (e+f\,x\right )}^3+\left (\frac {11\,a}{16}-\frac {5\,b}{128}\right )\,\mathrm {tan}\left (e+f\,x\right )}{f\,\left ({\mathrm {tan}\left (e+f\,x\right )}^8+4\,{\mathrm {tan}\left (e+f\,x\right )}^6+6\,{\mathrm {tan}\left (e+f\,x\right )}^4+4\,{\mathrm {tan}\left (e+f\,x\right )}^2+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(e + f*x)^6*(a + b*sin(e + f*x)^2),x)

[Out]

x*((5*a)/16 + (5*b)/128) + (tan(e + f*x)^7*((5*a)/16 + (5*b)/128) + tan(e + f*x)^5*((55*a)/48 + (55*b)/384) +
tan(e + f*x)^3*((73*a)/48 + (73*b)/384) + tan(e + f*x)*((11*a)/16 - (5*b)/128))/(f*(4*tan(e + f*x)^2 + 6*tan(e
+ f*x)^4 + 4*tan(e + f*x)^6 + tan(e + f*x)^8 + 1))

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