3.2 \(\int (a+a \sec (c+d x)) \sin ^7(c+d x) \, dx\)

Optimal. Leaf size=119 \[ \frac{a \cos ^7(c+d x)}{7 d}+\frac{a \cos ^6(c+d x)}{6 d}-\frac{3 a \cos ^5(c+d x)}{5 d}-\frac{3 a \cos ^4(c+d x)}{4 d}+\frac{a \cos ^3(c+d x)}{d}+\frac{3 a \cos ^2(c+d x)}{2 d}-\frac{a \cos (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d} \]

[Out]

-((a*Cos[c + d*x])/d) + (3*a*Cos[c + d*x]^2)/(2*d) + (a*Cos[c + d*x]^3)/d - (3*a*Cos[c + d*x]^4)/(4*d) - (3*a*
Cos[c + d*x]^5)/(5*d) + (a*Cos[c + d*x]^6)/(6*d) + (a*Cos[c + d*x]^7)/(7*d) - (a*Log[Cos[c + d*x]])/d

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Rubi [A]  time = 0.0974775, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {3872, 2836, 12, 88} \[ \frac{a \cos ^7(c+d x)}{7 d}+\frac{a \cos ^6(c+d x)}{6 d}-\frac{3 a \cos ^5(c+d x)}{5 d}-\frac{3 a \cos ^4(c+d x)}{4 d}+\frac{a \cos ^3(c+d x)}{d}+\frac{3 a \cos ^2(c+d x)}{2 d}-\frac{a \cos (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sec[c + d*x])*Sin[c + d*x]^7,x]

[Out]

-((a*Cos[c + d*x])/d) + (3*a*Cos[c + d*x]^2)/(2*d) + (a*Cos[c + d*x]^3)/d - (3*a*Cos[c + d*x]^4)/(4*d) - (3*a*
Cos[c + d*x]^5)/(5*d) + (a*Cos[c + d*x]^6)/(6*d) + (a*Cos[c + d*x]^7)/(7*d) - (a*Log[Cos[c + d*x]])/d

Rule 3872

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[((g*C
os[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]

Rule 2836

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*x)/b
)^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 12

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

Rule 88

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]))

Rubi steps

\begin{align*} \int (a+a \sec (c+d x)) \sin ^7(c+d x) \, dx &=-\int (-a-a \cos (c+d x)) \sin ^6(c+d x) \tan (c+d x) \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{a (-a-x)^3 (-a+x)^4}{x} \, dx,x,-a \cos (c+d x)\right )}{a^7 d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(-a-x)^3 (-a+x)^4}{x} \, dx,x,-a \cos (c+d x)\right )}{a^6 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^6-\frac{a^7}{x}+3 a^5 x-3 a^4 x^2-3 a^3 x^3+3 a^2 x^4+a x^5-x^6\right ) \, dx,x,-a \cos (c+d x)\right )}{a^6 d}\\ &=-\frac{a \cos (c+d x)}{d}+\frac{3 a \cos ^2(c+d x)}{2 d}+\frac{a \cos ^3(c+d x)}{d}-\frac{3 a \cos ^4(c+d x)}{4 d}-\frac{3 a \cos ^5(c+d x)}{5 d}+\frac{a \cos ^6(c+d x)}{6 d}+\frac{a \cos ^7(c+d x)}{7 d}-\frac{a \log (\cos (c+d x))}{d}\\ \end{align*}

Mathematica [A]  time = 0.130659, size = 86, normalized size = 0.72 \[ \frac{a \left (1120 \cos ^6(c+d x)-5040 \cos ^4(c+d x)+10080 \cos ^2(c+d x)-3675 \cos (c+d x)+735 \cos (3 (c+d x))-147 \cos (5 (c+d x))+15 \cos (7 (c+d x))-6720 \log (\cos (c+d x))\right )}{6720 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sec[c + d*x])*Sin[c + d*x]^7,x]

[Out]

(a*(-3675*Cos[c + d*x] + 10080*Cos[c + d*x]^2 - 5040*Cos[c + d*x]^4 + 1120*Cos[c + d*x]^6 + 735*Cos[3*(c + d*x
)] - 147*Cos[5*(c + d*x)] + 15*Cos[7*(c + d*x)] - 6720*Log[Cos[c + d*x]]))/(6720*d)

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Maple [A]  time = 0.088, size = 129, normalized size = 1.1 \begin{align*} -{\frac{16\,a\cos \left ( dx+c \right ) }{35\,d}}-{\frac{a\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{7\,d}}-{\frac{6\,a\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{35\,d}}-{\frac{8\,a\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{35\,d}}-{\frac{a \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{6\,d}}-{\frac{a \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{4\,d}}-{\frac{a \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-{\frac{a\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sec(d*x+c))*sin(d*x+c)^7,x)

[Out]

-16/35*a*cos(d*x+c)/d-1/7/d*a*cos(d*x+c)*sin(d*x+c)^6-6/35/d*a*cos(d*x+c)*sin(d*x+c)^4-8/35/d*a*cos(d*x+c)*sin
(d*x+c)^2-1/6/d*a*sin(d*x+c)^6-1/4/d*a*sin(d*x+c)^4-1/2/d*a*sin(d*x+c)^2-a*ln(cos(d*x+c))/d

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Maxima [A]  time = 1.06778, size = 123, normalized size = 1.03 \begin{align*} \frac{60 \, a \cos \left (d x + c\right )^{7} + 70 \, a \cos \left (d x + c\right )^{6} - 252 \, a \cos \left (d x + c\right )^{5} - 315 \, a \cos \left (d x + c\right )^{4} + 420 \, a \cos \left (d x + c\right )^{3} + 630 \, a \cos \left (d x + c\right )^{2} - 420 \, a \cos \left (d x + c\right ) - 420 \, a \log \left (\cos \left (d x + c\right )\right )}{420 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))*sin(d*x+c)^7,x, algorithm="maxima")

[Out]

1/420*(60*a*cos(d*x + c)^7 + 70*a*cos(d*x + c)^6 - 252*a*cos(d*x + c)^5 - 315*a*cos(d*x + c)^4 + 420*a*cos(d*x
+ c)^3 + 630*a*cos(d*x + c)^2 - 420*a*cos(d*x + c) - 420*a*log(cos(d*x + c)))/d

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Fricas [A]  time = 1.83266, size = 261, normalized size = 2.19 \begin{align*} \frac{60 \, a \cos \left (d x + c\right )^{7} + 70 \, a \cos \left (d x + c\right )^{6} - 252 \, a \cos \left (d x + c\right )^{5} - 315 \, a \cos \left (d x + c\right )^{4} + 420 \, a \cos \left (d x + c\right )^{3} + 630 \, a \cos \left (d x + c\right )^{2} - 420 \, a \cos \left (d x + c\right ) - 420 \, a \log \left (-\cos \left (d x + c\right )\right )}{420 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))*sin(d*x+c)^7,x, algorithm="fricas")

[Out]

1/420*(60*a*cos(d*x + c)^7 + 70*a*cos(d*x + c)^6 - 252*a*cos(d*x + c)^5 - 315*a*cos(d*x + c)^4 + 420*a*cos(d*x
+ c)^3 + 630*a*cos(d*x + c)^2 - 420*a*cos(d*x + c) - 420*a*log(-cos(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((a+a*sec(d*x+c))*sin(d*x+c)**7,x)

[Out]

Timed out

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Giac [B]  time = 1.50592, size = 333, normalized size = 2.8 \begin{align*} \frac{420 \, a \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 420 \, a \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac{1473 \, a - \frac{11151 \, a{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{36813 \, a{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{69475 \, a{\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{56035 \, a{\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{28749 \, a{\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{8463 \, a{\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac{1089 \, a{\left (\cos \left (d x + c\right ) - 1\right )}^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{{\left (\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{7}}}{420 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))*sin(d*x+c)^7,x, algorithm="giac")

[Out]

1/420*(420*a*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)) - 420*a*log(abs(-(cos(d*x + c) - 1)/(cos(d*x
+ c) + 1) - 1)) + (1473*a - 11151*a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 36813*a*(cos(d*x + c) - 1)^2/(cos
(d*x + c) + 1)^2 - 69475*a*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 56035*a*(cos(d*x + c) - 1)^4/(cos(d*x +
c) + 1)^4 - 28749*a*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5 + 8463*a*(cos(d*x + c) - 1)^6/(cos(d*x + c) + 1
)^6 - 1089*a*(cos(d*x + c) - 1)^7/(cos(d*x + c) + 1)^7)/((cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1)^7)/d