3.1.21 \(\int (a+a \sec (c+d x))^2 \sin ^5(c+d x) \, dx\) [21]

Optimal. Leaf size=112 \[ \frac {a^2 \cos (c+d x)}{d}+\frac {2 a^2 \cos ^2(c+d x)}{d}+\frac {a^2 \cos ^3(c+d x)}{3 d}-\frac {a^2 \cos ^4(c+d x)}{2 d}-\frac {a^2 \cos ^5(c+d x)}{5 d}-\frac {2 a^2 \log (\cos (c+d x))}{d}+\frac {a^2 \sec (c+d x)}{d} \]

[Out]

a^2*cos(d*x+c)/d+2*a^2*cos(d*x+c)^2/d+1/3*a^2*cos(d*x+c)^3/d-1/2*a^2*cos(d*x+c)^4/d-1/5*a^2*cos(d*x+c)^5/d-2*a
^2*ln(cos(d*x+c))/d+a^2*sec(d*x+c)/d

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Rubi [A]
time = 0.11, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3957, 2915, 12, 90} \begin {gather*} -\frac {a^2 \cos ^5(c+d x)}{5 d}-\frac {a^2 \cos ^4(c+d x)}{2 d}+\frac {a^2 \cos ^3(c+d x)}{3 d}+\frac {2 a^2 \cos ^2(c+d x)}{d}+\frac {a^2 \cos (c+d x)}{d}+\frac {a^2 \sec (c+d x)}{d}-\frac {2 a^2 \log (\cos (c+d x))}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + a*Sec[c + d*x])^2*Sin[c + d*x]^5,x]

[Out]

(a^2*Cos[c + d*x])/d + (2*a^2*Cos[c + d*x]^2)/d + (a^2*Cos[c + d*x]^3)/(3*d) - (a^2*Cos[c + d*x]^4)/(2*d) - (a
^2*Cos[c + d*x]^5)/(5*d) - (2*a^2*Log[Cos[c + d*x]])/d + (a^2*Sec[c + d*x])/d

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 2915

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)
*(x_)])^(n_.), x_Symbol] :> Dist[1/(b^p*f), Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d/b)*x
)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2,
0]

Rule 3957

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Co
s[e + f*x])^p*((b + a*Sin[e + f*x])^m/Sin[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rubi steps

\begin {align*} \int (a+a \sec (c+d x))^2 \sin ^5(c+d x) \, dx &=\int (-a-a \cos (c+d x))^2 \sin ^3(c+d x) \tan ^2(c+d x) \, dx\\ &=\frac {\text {Subst}\left (\int \frac {a^2 (-a-x)^2 (-a+x)^4}{x^2} \, dx,x,-a \cos (c+d x)\right )}{a^5 d}\\ &=\frac {\text {Subst}\left (\int \frac {(-a-x)^2 (-a+x)^4}{x^2} \, dx,x,-a \cos (c+d x)\right )}{a^3 d}\\ &=\frac {\text {Subst}\left (\int \left (-a^4+\frac {a^6}{x^2}-\frac {2 a^5}{x}+4 a^3 x-a^2 x^2-2 a x^3+x^4\right ) \, dx,x,-a \cos (c+d x)\right )}{a^3 d}\\ &=\frac {a^2 \cos (c+d x)}{d}+\frac {2 a^2 \cos ^2(c+d x)}{d}+\frac {a^2 \cos ^3(c+d x)}{3 d}-\frac {a^2 \cos ^4(c+d x)}{2 d}-\frac {a^2 \cos ^5(c+d x)}{5 d}-\frac {2 a^2 \log (\cos (c+d x))}{d}+\frac {a^2 \sec (c+d x)}{d}\\ \end {align*}

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Mathematica [A]
time = 0.21, size = 87, normalized size = 0.78 \begin {gather*} -\frac {a^2 (-750-275 \cos (2 (c+d x))-165 \cos (3 (c+d x))-2 \cos (4 (c+d x))+15 \cos (5 (c+d x))+3 \cos (6 (c+d x))+30 \cos (c+d x) (-3+32 \log (\cos (c+d x)))) \sec (c+d x)}{480 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sec[c + d*x])^2*Sin[c + d*x]^5,x]

[Out]

-1/480*(a^2*(-750 - 275*Cos[2*(c + d*x)] - 165*Cos[3*(c + d*x)] - 2*Cos[4*(c + d*x)] + 15*Cos[5*(c + d*x)] + 3
*Cos[6*(c + d*x)] + 30*Cos[c + d*x]*(-3 + 32*Log[Cos[c + d*x]]))*Sec[c + d*x])/d

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Maple [A]
time = 0.12, size = 121, normalized size = 1.08

method	result	size
derivativedivides	\(\frac {a^{2} \left (\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )+2 a^{2} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-\frac {a^{2} \left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )}{5}}{d}\)	\(121\)
default	\(\frac {a^{2} \left (\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )+2 a^{2} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-\frac {a^{2} \left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )}{5}}{d}\)	\(121\)
risch	\(2 i a^{2} x +\frac {3 a^{2} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}+\frac {9 a^{2} {\mathrm e}^{i \left (d x +c \right )}}{16 d}+\frac {9 a^{2} {\mathrm e}^{-i \left (d x +c \right )}}{16 d}+\frac {3 a^{2} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}+\frac {4 i a^{2} c}{d}+\frac {2 a^{2} {\mathrm e}^{i \left (d x +c \right )}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {2 a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}-\frac {a^{2} \cos \left (5 d x +5 c \right )}{80 d}-\frac {a^{2} \cos \left (4 d x +4 c \right )}{16 d}+\frac {a^{2} \cos \left (3 d x +3 c \right )}{48 d}\)	\(188\)
norman	\(\frac {-\frac {64 a^{2}}{15 d}-\frac {64 a^{2} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {4 a^{2} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {16 a^{2} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {16 a^{2} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {196 a^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {2 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {2 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}+\frac {2 a^{2} \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\)	\(199\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sec(d*x+c))^2*sin(d*x+c)^5,x,method=_RETURNVERBOSE)

[Out]

1/d*(a^2*(sin(d*x+c)^6/cos(d*x+c)+(8/3+sin(d*x+c)^4+4/3*sin(d*x+c)^2)*cos(d*x+c))+2*a^2*(-1/4*sin(d*x+c)^4-1/2
*sin(d*x+c)^2-ln(cos(d*x+c)))-1/5*a^2*(8/3+sin(d*x+c)^4+4/3*sin(d*x+c)^2)*cos(d*x+c))

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Maxima [A]
time = 0.27, size = 94, normalized size = 0.84 \begin {gather*} -\frac {6 \, a^{2} \cos \left (d x + c\right )^{5} + 15 \, a^{2} \cos \left (d x + c\right )^{4} - 10 \, a^{2} \cos \left (d x + c\right )^{3} - 60 \, a^{2} \cos \left (d x + c\right )^{2} - 30 \, a^{2} \cos \left (d x + c\right ) + 60 \, a^{2} \log \left (\cos \left (d x + c\right )\right ) - \frac {30 \, a^{2}}{\cos \left (d x + c\right )}}{30 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^2*sin(d*x+c)^5,x, algorithm="maxima")

[Out]

-1/30*(6*a^2*cos(d*x + c)^5 + 15*a^2*cos(d*x + c)^4 - 10*a^2*cos(d*x + c)^3 - 60*a^2*cos(d*x + c)^2 - 30*a^2*c
os(d*x + c) + 60*a^2*log(cos(d*x + c)) - 30*a^2/cos(d*x + c))/d

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Fricas [A]
time = 4.04, size = 115, normalized size = 1.03 \begin {gather*} -\frac {48 \, a^{2} \cos \left (d x + c\right )^{6} + 120 \, a^{2} \cos \left (d x + c\right )^{5} - 80 \, a^{2} \cos \left (d x + c\right )^{4} - 480 \, a^{2} \cos \left (d x + c\right )^{3} - 240 \, a^{2} \cos \left (d x + c\right )^{2} + 480 \, a^{2} \cos \left (d x + c\right ) \log \left (-\cos \left (d x + c\right )\right ) + 195 \, a^{2} \cos \left (d x + c\right ) - 240 \, a^{2}}{240 \, d \cos \left (d x + c\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^2*sin(d*x+c)^5,x, algorithm="fricas")

[Out]

-1/240*(48*a^2*cos(d*x + c)^6 + 120*a^2*cos(d*x + c)^5 - 80*a^2*cos(d*x + c)^4 - 480*a^2*cos(d*x + c)^3 - 240*
a^2*cos(d*x + c)^2 + 480*a^2*cos(d*x + c)*log(-cos(d*x + c)) + 195*a^2*cos(d*x + c) - 240*a^2)/(d*cos(d*x + c)
)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} a^{2} \left (\int 2 \sin ^{5}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \sin ^{5}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \sin ^{5}{\left (c + d x \right )}\, dx\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))**2*sin(d*x+c)**5,x)

[Out]

a**2*(Integral(2*sin(c + d*x)**5*sec(c + d*x), x) + Integral(sin(c + d*x)**5*sec(c + d*x)**2, x) + Integral(si
n(c + d*x)**5, x))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 270 vs. \(2 (106) = 212\).
time = 0.68, size = 270, normalized size = 2.41 \begin {gather*} \frac {60 \, a^{2} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 60 \, a^{2} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {60 \, {\left (2 \, a^{2} + \frac {a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )}}{\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1} + \frac {69 \, a^{2} - \frac {525 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {1650 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {1610 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {745 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {137 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{5}}}{30 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^2*sin(d*x+c)^5,x, algorithm="giac")

[Out]

1/30*(60*a^2*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)) - 60*a^2*log(abs(-(cos(d*x + c) - 1)/(cos(d*
x + c) + 1) - 1)) + 60*(2*a^2 + a^2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1))/((cos(d*x + c) - 1)/(cos(d*x + c) +
1) + 1) + (69*a^2 - 525*a^2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1650*a^2*(cos(d*x + c) - 1)^2/(cos(d*x +
c) + 1)^2 - 1610*a^2*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 745*a^2*(cos(d*x + c) - 1)^4/(cos(d*x + c) +
1)^4 - 137*a^2*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5)/((cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1)^5)/d

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Mupad [B]
time = 0.89, size = 91, normalized size = 0.81 \begin {gather*} \frac {a^2\,\cos \left (c+d\,x\right )+\frac {a^2}{\cos \left (c+d\,x\right )}+2\,a^2\,{\cos \left (c+d\,x\right )}^2+\frac {a^2\,{\cos \left (c+d\,x\right )}^3}{3}-\frac {a^2\,{\cos \left (c+d\,x\right )}^4}{2}-\frac {a^2\,{\cos \left (c+d\,x\right )}^5}{5}-2\,a^2\,\ln \left (\cos \left (c+d\,x\right )\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(c + d*x)^5*(a + a/cos(c + d*x))^2,x)

[Out]

(a^2*cos(c + d*x) + a^2/cos(c + d*x) + 2*a^2*cos(c + d*x)^2 + (a^2*cos(c + d*x)^3)/3 - (a^2*cos(c + d*x)^4)/2
- (a^2*cos(c + d*x)^5)/5 - 2*a^2*log(cos(c + d*x)))/d

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