\(\int (a+a \sec (c+d x)) \sin ^7(c+d x) \, dx\) [2]

Optimal result

Integrand size = 19, antiderivative size = 119 \[ \int (a+a \sec (c+d x)) \sin ^7(c+d x) \, dx=-\frac {a \cos (c+d x)}{d}+\frac {3 a \cos ^2(c+d x)}{2 d}+\frac {a \cos ^3(c+d x)}{d}-\frac {3 a \cos ^4(c+d x)}{4 d}-\frac {3 a \cos ^5(c+d x)}{5 d}+\frac {a \cos ^6(c+d x)}{6 d}+\frac {a \cos ^7(c+d x)}{7 d}-\frac {a \log (\cos (c+d x))}{d} \]

[Out]

-a*cos(d*x+c)/d+3/2*a*cos(d*x+c)^2/d+a*cos(d*x+c)^3/d-3/4*a*cos(d*x+c)^4/d-3/5*a*cos(d*x+c)^5/d+1/6*a*cos(d*x+
c)^6/d+1/7*a*cos(d*x+c)^7/d-a*ln(cos(d*x+c))/d

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {3957, 2915, 12, 90} \[ \int (a+a \sec (c+d x)) \sin ^7(c+d x) \, dx=\frac {a \cos ^7(c+d x)}{7 d}+\frac {a \cos ^6(c+d x)}{6 d}-\frac {3 a \cos ^5(c+d x)}{5 d}-\frac {3 a \cos ^4(c+d x)}{4 d}+\frac {a \cos ^3(c+d x)}{d}+\frac {3 a \cos ^2(c+d x)}{2 d}-\frac {a \cos (c+d x)}{d}-\frac {a \log (\cos (c+d x))}{d} \]

[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 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*} \text {integral}& = -\int (-a-a \cos (c+d x)) \sin ^6(c+d x) \tan (c+d x) \, dx \\ & = \frac {\text {Subst}\left (\int \frac {a (-a-x)^3 (-a+x)^4}{x} \, dx,x,-a \cos (c+d x)\right )}{a^7 d} \\ & = \frac {\text {Subst}\left (\int \frac {(-a-x)^3 (-a+x)^4}{x} \, dx,x,-a \cos (c+d x)\right )}{a^6 d} \\ & = \frac {\text {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] (verified)

Time = 0.18 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.95 \[ \int (a+a \sec (c+d x)) \sin ^7(c+d x) \, dx=-\frac {35 a \cos (c+d x)}{64 d}+\frac {7 a \cos (3 (c+d x))}{64 d}-\frac {7 a \cos (5 (c+d x))}{320 d}+\frac {a \cos (7 (c+d x))}{448 d}-\frac {a \left (-\frac {3}{2} \cos ^2(c+d x)+\frac {3}{4} \cos ^4(c+d x)-\frac {1}{6} \cos ^6(c+d x)+\log (\cos (c+d x))\right )}{d} \]

[In]

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

[Out]

(-35*a*Cos[c + d*x])/(64*d) + (7*a*Cos[3*(c + d*x)])/(64*d) - (7*a*Cos[5*(c + d*x)])/(320*d) + (a*Cos[7*(c + d
*x)])/(448*d) - (a*((-3*Cos[c + d*x]^2)/2 + (3*Cos[c + d*x]^4)/4 - Cos[c + d*x]^6/6 + Log[Cos[c + d*x]]))/d

Maple [A] (verified)

Time = 1.46 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.73

method	result	size
derivativedivides	\(\frac {a \left (-\frac {\sin \left (d x +c \right )^{6}}{6}-\frac {\sin \left (d x +c \right )^{4}}{4}-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-\frac {a \left (\frac {16}{5}+\sin \left (d x +c \right )^{6}+\frac {6 \sin \left (d x +c \right )^{4}}{5}+\frac {8 \sin \left (d x +c \right )^{2}}{5}\right ) \cos \left (d x +c \right )}{7}}{d}\)	\(87\)
default	\(\frac {a \left (-\frac {\sin \left (d x +c \right )^{6}}{6}-\frac {\sin \left (d x +c \right )^{4}}{4}-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-\frac {a \left (\frac {16}{5}+\sin \left (d x +c \right )^{6}+\frac {6 \sin \left (d x +c \right )^{4}}{5}+\frac {8 \sin \left (d x +c \right )^{2}}{5}\right ) \cos \left (d x +c \right )}{7}}{d}\)	\(87\)
parts	\(-\frac {a \left (\frac {16}{5}+\sin \left (d x +c \right )^{6}+\frac {6 \sin \left (d x +c \right )^{4}}{5}+\frac {8 \sin \left (d x +c \right )^{2}}{5}\right ) \cos \left (d x +c \right )}{7 d}+\frac {a \left (-\frac {\sin \left (d x +c \right )^{6}}{6}-\frac {\sin \left (d x +c \right )^{4}}{4}-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )}{d}\)	\(89\)
parallelrisch	\(\frac {\left (192 \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )-192 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-192 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {5732}{35}-12 \cos \left (4 d x +4 c \right )+21 \cos \left (3 d x +3 c \right )-105 \cos \left (d x +c \right )+\frac {3 \cos \left (7 d x +7 c \right )}{7}+\cos \left (6 d x +6 c \right )-\frac {21 \cos \left (5 d x +5 c \right )}{5}+87 \cos \left (2 d x +2 c \right )\right ) a}{192 d}\)	\(123\)
risch	\(i a x +\frac {2 i a c}{d}+\frac {29 a \,{\mathrm e}^{2 i \left (d x +c \right )}}{128 d}+\frac {29 a \,{\mathrm e}^{-2 i \left (d x +c \right )}}{128 d}-\frac {a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}-\frac {35 a \cos \left (d x +c \right )}{64 d}+\frac {a \cos \left (7 d x +7 c \right )}{448 d}+\frac {a \cos \left (6 d x +6 c \right )}{192 d}-\frac {7 a \cos \left (5 d x +5 c \right )}{320 d}-\frac {a \cos \left (4 d x +4 c \right )}{16 d}+\frac {7 a \cos \left (3 d x +3 c \right )}{64 d}\)	\(150\)
norman	\(\frac {-\frac {32 a}{35 d}-\frac {128 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{3 d}-\frac {166 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{5 d}-\frac {224 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3 d}-\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{d}-\frac {42 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{5 d}-\frac {14 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{7}}+\frac {a \ln \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{d}-\frac {a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\)	\(182\)

[In]

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

[Out]

1/d*(a*(-1/6*sin(d*x+c)^6-1/4*sin(d*x+c)^4-1/2*sin(d*x+c)^2-ln(cos(d*x+c)))-1/7*a*(16/5+sin(d*x+c)^6+6/5*sin(d
*x+c)^4+8/5*sin(d*x+c)^2)*cos(d*x+c))

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.78 \[ \int (a+a \sec (c+d x)) \sin ^7(c+d x) \, dx=\frac {60 \, a \cos \left (d x + c\right )^{7} + 70 \, a \cos \left (d x + c\right )^{6} - 252 \, a \cos \left (d x + c\right )^{5} - 315 \, a \cos \left (d x + c\right )^{4} + 420 \, a \cos \left (d x + c\right )^{3} + 630 \, a \cos \left (d x + c\right )^{2} - 420 \, a \cos \left (d x + c\right ) - 420 \, a \log \left (-\cos \left (d x + c\right )\right )}{420 \, d} \]

[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

Sympy [F(-1)]

Timed out. \[ \int (a+a \sec (c+d x)) \sin ^7(c+d x) \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.76 \[ \int (a+a \sec (c+d x)) \sin ^7(c+d x) \, dx=\frac {60 \, a \cos \left (d x + c\right )^{7} + 70 \, a \cos \left (d x + c\right )^{6} - 252 \, a \cos \left (d x + c\right )^{5} - 315 \, a \cos \left (d x + c\right )^{4} + 420 \, a \cos \left (d x + c\right )^{3} + 630 \, a \cos \left (d x + c\right )^{2} - 420 \, a \cos \left (d x + c\right ) - 420 \, a \log \left (\cos \left (d x + c\right )\right )}{420 \, d} \]

[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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 247 vs. \(2 (109) = 218\).

Time = 0.31 (sec) , antiderivative size = 247, normalized size of antiderivative = 2.08 \[ \int (a+a \sec (c+d x)) \sin ^7(c+d x) \, dx=\frac {420 \, a \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 420 \, a \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {1473 \, a - \frac {11151 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {36813 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {69475 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {56035 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {28749 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {8463 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {1089 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{7}}}{420 \, d} \]

[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

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.75 \[ \int (a+a \sec (c+d x)) \sin ^7(c+d x) \, dx=-\frac {a\,\cos \left (c+d\,x\right )-\frac {3\,a\,{\cos \left (c+d\,x\right )}^2}{2}-a\,{\cos \left (c+d\,x\right )}^3+\frac {3\,a\,{\cos \left (c+d\,x\right )}^4}{4}+\frac {3\,a\,{\cos \left (c+d\,x\right )}^5}{5}-\frac {a\,{\cos \left (c+d\,x\right )}^6}{6}-\frac {a\,{\cos \left (c+d\,x\right )}^7}{7}+a\,\ln \left (\cos \left (c+d\,x\right )\right )}{d} \]

[In]

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

[Out]

-(a*cos(c + d*x) - (3*a*cos(c + d*x)^2)/2 - a*cos(c + d*x)^3 + (3*a*cos(c + d*x)^4)/4 + (3*a*cos(c + d*x)^5)/5
- (a*cos(c + d*x)^6)/6 - (a*cos(c + d*x)^7)/7 + a*log(cos(c + d*x)))/d