\(\int \frac {\sin ^7(c+d x)}{a+b \sec (c+d x)} \, dx\) [196]

Optimal result

Integrand size = 21, antiderivative size = 223 \[ \int \frac {\sin ^7(c+d x)}{a+b \sec (c+d x)} \, dx=-\frac {\left (a^2-b^2\right )^3 \cos (c+d x)}{a^7 d}-\frac {b \left (3 a^4-3 a^2 b^2+b^4\right ) \cos ^2(c+d x)}{2 a^6 d}+\frac {\left (3 a^4-3 a^2 b^2+b^4\right ) \cos ^3(c+d x)}{3 a^5 d}+\frac {b \left (3 a^2-b^2\right ) \cos ^4(c+d x)}{4 a^4 d}-\frac {\left (3 a^2-b^2\right ) \cos ^5(c+d x)}{5 a^3 d}-\frac {b \cos ^6(c+d x)}{6 a^2 d}+\frac {\cos ^7(c+d x)}{7 a d}+\frac {b \left (a^2-b^2\right )^3 \log (b+a \cos (c+d x))}{a^8 d} \]

[Out]

-(a^2-b^2)^3*cos(d*x+c)/a^7/d-1/2*b*(3*a^4-3*a^2*b^2+b^4)*cos(d*x+c)^2/a^6/d+1/3*(3*a^4-3*a^2*b^2+b^4)*cos(d*x
+c)^3/a^5/d+1/4*b*(3*a^2-b^2)*cos(d*x+c)^4/a^4/d-1/5*(3*a^2-b^2)*cos(d*x+c)^5/a^3/d-1/6*b*cos(d*x+c)^6/a^2/d+1
/7*cos(d*x+c)^7/a/d+b*(a^2-b^2)^3*ln(b+a*cos(d*x+c))/a^8/d

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3957, 2916, 12, 786} \[ \int \frac {\sin ^7(c+d x)}{a+b \sec (c+d x)} \, dx=-\frac {b \cos ^6(c+d x)}{6 a^2 d}+\frac {b \left (a^2-b^2\right )^3 \log (a \cos (c+d x)+b)}{a^8 d}-\frac {\left (a^2-b^2\right )^3 \cos (c+d x)}{a^7 d}+\frac {b \left (3 a^2-b^2\right ) \cos ^4(c+d x)}{4 a^4 d}-\frac {\left (3 a^2-b^2\right ) \cos ^5(c+d x)}{5 a^3 d}-\frac {b \left (3 a^4-3 a^2 b^2+b^4\right ) \cos ^2(c+d x)}{2 a^6 d}+\frac {\left (3 a^4-3 a^2 b^2+b^4\right ) \cos ^3(c+d x)}{3 a^5 d}+\frac {\cos ^7(c+d x)}{7 a d} \]

[In]

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

[Out]

-(((a^2 - b^2)^3*Cos[c + d*x])/(a^7*d)) - (b*(3*a^4 - 3*a^2*b^2 + b^4)*Cos[c + d*x]^2)/(2*a^6*d) + ((3*a^4 - 3
*a^2*b^2 + b^4)*Cos[c + d*x]^3)/(3*a^5*d) + (b*(3*a^2 - b^2)*Cos[c + d*x]^4)/(4*a^4*d) - ((3*a^2 - b^2)*Cos[c
+ d*x]^5)/(5*a^3*d) - (b*Cos[c + d*x]^6)/(6*a^2*d) + Cos[c + d*x]^7/(7*a*d) + (b*(a^2 - b^2)^3*Log[b + a*Cos[c
+ d*x]])/(a^8*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 786

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]

Rule 2916

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*(c + (d/b)*x)^n*(b^2 - x^2)^((p - 1)/2), x], x
, b*Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IntegerQ[(p - 1)/2] && NeQ[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 \frac {\cos (c+d x) \sin ^7(c+d x)}{-b-a \cos (c+d x)} \, dx \\ & = \frac {\text {Subst}\left (\int \frac {x \left (a^2-x^2\right )^3}{a (-b+x)} \, dx,x,-a \cos (c+d x)\right )}{a^7 d} \\ & = \frac {\text {Subst}\left (\int \frac {x \left (a^2-x^2\right )^3}{-b+x} \, dx,x,-a \cos (c+d x)\right )}{a^8 d} \\ & = \frac {\text {Subst}\left (\int \left (\left (a^2-b^2\right )^3+\frac {b \left (-a^2+b^2\right )^3}{b-x}-b \left (3 a^4-3 a^2 b^2+b^4\right ) x-\left (3 a^4-3 a^2 b^2+b^4\right ) x^2-b \left (-3 a^2+b^2\right ) x^3+\left (3 a^2-b^2\right ) x^4-b x^5-x^6\right ) \, dx,x,-a \cos (c+d x)\right )}{a^8 d} \\ & = -\frac {\left (a^2-b^2\right )^3 \cos (c+d x)}{a^7 d}-\frac {b \left (3 a^4-3 a^2 b^2+b^4\right ) \cos ^2(c+d x)}{2 a^6 d}+\frac {\left (3 a^4-3 a^2 b^2+b^4\right ) \cos ^3(c+d x)}{3 a^5 d}+\frac {b \left (3 a^2-b^2\right ) \cos ^4(c+d x)}{4 a^4 d}-\frac {\left (3 a^2-b^2\right ) \cos ^5(c+d x)}{5 a^3 d}-\frac {b \cos ^6(c+d x)}{6 a^2 d}+\frac {\cos ^7(c+d x)}{7 a d}+\frac {b \left (a^2-b^2\right )^3 \log (b+a \cos (c+d x))}{a^8 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.81 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.26 \[ \int \frac {\sin ^7(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {-105 a \left (35 a^6-152 a^4 b^2+176 a^2 b^4-64 b^6\right ) \cos (c+d x)-105 \left (29 a^6 b-40 a^4 b^3+16 a^2 b^5\right ) \cos (2 (c+d x))+735 a^7 \cos (3 (c+d x))-1260 a^5 b^2 \cos (3 (c+d x))+560 a^3 b^4 \cos (3 (c+d x))+420 a^6 b \cos (4 (c+d x))-210 a^4 b^3 \cos (4 (c+d x))-147 a^7 \cos (5 (c+d x))+84 a^5 b^2 \cos (5 (c+d x))-35 a^6 b \cos (6 (c+d x))+15 a^7 \cos (7 (c+d x))+6720 a^6 b \log (b+a \cos (c+d x))-20160 a^4 b^3 \log (b+a \cos (c+d x))+20160 a^2 b^5 \log (b+a \cos (c+d x))-6720 b^7 \log (b+a \cos (c+d x))}{6720 a^8 d} \]

[In]

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

[Out]

(-105*a*(35*a^6 - 152*a^4*b^2 + 176*a^2*b^4 - 64*b^6)*Cos[c + d*x] - 105*(29*a^6*b - 40*a^4*b^3 + 16*a^2*b^5)*
Cos[2*(c + d*x)] + 735*a^7*Cos[3*(c + d*x)] - 1260*a^5*b^2*Cos[3*(c + d*x)] + 560*a^3*b^4*Cos[3*(c + d*x)] + 4
20*a^6*b*Cos[4*(c + d*x)] - 210*a^4*b^3*Cos[4*(c + d*x)] - 147*a^7*Cos[5*(c + d*x)] + 84*a^5*b^2*Cos[5*(c + d*
x)] - 35*a^6*b*Cos[6*(c + d*x)] + 15*a^7*Cos[7*(c + d*x)] + 6720*a^6*b*Log[b + a*Cos[c + d*x]] - 20160*a^4*b^3
*Log[b + a*Cos[c + d*x]] + 20160*a^2*b^5*Log[b + a*Cos[c + d*x]] - 6720*b^7*Log[b + a*Cos[c + d*x]])/(6720*a^8
*d)

Maple [A] (verified)

Time = 1.36 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.23

method	result	size
derivativedivides	\(\frac {\frac {\frac {\cos \left (d x +c \right )^{7} a^{6}}{7}-\frac {b \cos \left (d x +c \right )^{6} a^{5}}{6}-\frac {3 a^{6} \cos \left (d x +c \right )^{5}}{5}+\frac {a^{4} b^{2} \cos \left (d x +c \right )^{5}}{5}+\frac {3 \cos \left (d x +c \right )^{4} a^{5} b}{4}-\frac {\cos \left (d x +c \right )^{4} a^{3} b^{3}}{4}+\cos \left (d x +c \right )^{3} a^{6}-\cos \left (d x +c \right )^{3} a^{4} b^{2}+\frac {\cos \left (d x +c \right )^{3} a^{2} b^{4}}{3}-\frac {3 \cos \left (d x +c \right )^{2} a^{5} b}{2}+\frac {3 \cos \left (d x +c \right )^{2} a^{3} b^{3}}{2}-\frac {\cos \left (d x +c \right )^{2} a \,b^{5}}{2}-\cos \left (d x +c \right ) a^{6}+3 \cos \left (d x +c \right ) a^{4} b^{2}-3 \cos \left (d x +c \right ) a^{2} b^{4}+b^{6} \cos \left (d x +c \right )}{a^{7}}+\frac {b \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{a^{8}}}{d}\)	\(275\)
default	\(\frac {\frac {\frac {\cos \left (d x +c \right )^{7} a^{6}}{7}-\frac {b \cos \left (d x +c \right )^{6} a^{5}}{6}-\frac {3 a^{6} \cos \left (d x +c \right )^{5}}{5}+\frac {a^{4} b^{2} \cos \left (d x +c \right )^{5}}{5}+\frac {3 \cos \left (d x +c \right )^{4} a^{5} b}{4}-\frac {\cos \left (d x +c \right )^{4} a^{3} b^{3}}{4}+\cos \left (d x +c \right )^{3} a^{6}-\cos \left (d x +c \right )^{3} a^{4} b^{2}+\frac {\cos \left (d x +c \right )^{3} a^{2} b^{4}}{3}-\frac {3 \cos \left (d x +c \right )^{2} a^{5} b}{2}+\frac {3 \cos \left (d x +c \right )^{2} a^{3} b^{3}}{2}-\frac {\cos \left (d x +c \right )^{2} a \,b^{5}}{2}-\cos \left (d x +c \right ) a^{6}+3 \cos \left (d x +c \right ) a^{4} b^{2}-3 \cos \left (d x +c \right ) a^{2} b^{4}+b^{6} \cos \left (d x +c \right )}{a^{7}}+\frac {b \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{a^{8}}}{d}\)	\(275\)
parallelrisch	\(\frac {320 b \left (a -b \right )^{3} \left (a +b \right )^{3} \ln \left (-2 a +\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (a -b \right )\right )-320 b \left (a -b \right )^{3} \left (a +b \right )^{3} \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )-7 \left (\frac {5 \left (29 a^{5} b -40 a^{3} b^{3}+16 a \,b^{5}\right ) \cos \left (2 d x +2 c \right )}{7}+5 \left (-a^{6}+\frac {12}{7} a^{4} b^{2}-\frac {16}{21} a^{2} b^{4}\right ) \cos \left (3 d x +3 c \right )+\frac {10 \left (-2 a^{5} b +a^{3} b^{3}\right ) \cos \left (4 d x +4 c \right )}{7}+\left (a^{6}-\frac {4}{7} a^{4} b^{2}\right ) \cos \left (5 d x +5 c \right )+\frac {5 a^{5} b \cos \left (6 d x +6 c \right )}{21}-\frac {5 a^{6} \cos \left (7 d x +7 c \right )}{49}+5 \left (5 a^{6}-\frac {152}{7} a^{4} b^{2}+\frac {176}{7} a^{2} b^{4}-\frac {64}{7} b^{6}\right ) \cos \left (d x +c \right )+\frac {1024 a^{6}}{49}-\frac {380 a^{5} b}{21}-\frac {704 a^{4} b^{2}}{7}+\frac {190 a^{3} b^{3}}{7}+\frac {2560 a^{2} b^{4}}{21}-\frac {80 a \,b^{5}}{7}-\frac {320 b^{6}}{7}\right ) a}{320 a^{8} d}\)	\(300\)
norman	\(\frac {\frac {\left (2 a^{5} b +2 a^{4} b^{2}-4 a^{3} b^{3}-4 a^{2} b^{4}+2 a \,b^{5}+2 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{a^{7} d}+\frac {\left (14 a^{5} b +16 a^{4} b^{2}-24 a^{3} b^{3}-28 a^{2} b^{4}+10 a \,b^{5}+12 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{a^{7} d}+\frac {-96 a^{6}+462 a^{4} b^{2}-560 a^{2} b^{4}+210 b^{6}}{105 a^{7} d}+\frac {\left (128 a^{5} b +174 a^{4} b^{2}-156 a^{3} b^{3}-232 a^{2} b^{4}+60 a \,b^{5}+90 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{3 a^{7} d}+\frac {\left (-96 a^{6}+30 a^{5} b +432 a^{4} b^{2}-60 a^{3} b^{3}-500 a^{2} b^{4}+30 a \,b^{5}+180 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{15 a^{7} d}+\frac {\left (-96 a^{6}+70 a^{5} b +382 a^{4} b^{2}-120 a^{3} b^{3}-420 a^{2} b^{4}+50 a \,b^{5}+150 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{5 a^{7} d}+\frac {\left (-96 a^{6}+128 a^{5} b +288 a^{4} b^{2}-156 a^{3} b^{3}-328 a^{2} b^{4}+60 a \,b^{5}+120 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3 a^{7} d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{7}}+\frac {\left (a +b \right ) b \left (a^{5}-a^{4} b -2 a^{3} b^{2}+2 a^{2} b^{3}+a \,b^{4}-b^{5}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )}{a^{8} d}-\frac {\left (a +b \right ) b \left (a^{5}-a^{4} b -2 a^{3} b^{2}+2 a^{2} b^{3}+a \,b^{4}-b^{5}\right ) \ln \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{a^{8} d}\)	\(577\)
risch	\(-\frac {7 \cos \left (5 d x +5 c \right )}{320 d a}-\frac {b \cos \left (6 d x +6 c \right )}{192 d \,a^{2}}+\frac {7 \cos \left (3 d x +3 c \right )}{64 a d}+\frac {\cos \left (7 d x +7 c \right )}{448 a d}+\frac {\cos \left (5 d x +5 c \right ) b^{2}}{80 d \,a^{3}}+\frac {b \cos \left (4 d x +4 c \right )}{16 d \,a^{2}}-\frac {b^{3} \cos \left (4 d x +4 c \right )}{32 d \,a^{4}}-\frac {3 \cos \left (3 d x +3 c \right ) b^{2}}{16 a^{3} d}+\frac {\cos \left (3 d x +3 c \right ) b^{4}}{12 a^{5} d}+\frac {b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{a^{2} d}-\frac {3 b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{a^{4} d}+\frac {3 b^{5} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{a^{6} d}-\frac {b^{7} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{a^{8} d}-\frac {29 b \,{\mathrm e}^{-2 i \left (d x +c \right )}}{128 a^{2} d}+\frac {5 b^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{16 a^{4} d}-\frac {b^{5} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 a^{6} d}+\frac {19 \,{\mathrm e}^{i \left (d x +c \right )} b^{2}}{16 a^{3} d}-\frac {11 \,{\mathrm e}^{i \left (d x +c \right )} b^{4}}{8 a^{5} d}+\frac {{\mathrm e}^{i \left (d x +c \right )} b^{6}}{2 a^{7} d}+\frac {19 \,{\mathrm e}^{-i \left (d x +c \right )} b^{2}}{16 a^{3} d}-\frac {11 \,{\mathrm e}^{-i \left (d x +c \right )} b^{4}}{8 a^{5} d}+\frac {{\mathrm e}^{-i \left (d x +c \right )} b^{6}}{2 a^{7} d}-\frac {i x b}{a^{2}}+\frac {3 i x \,b^{3}}{a^{4}}-\frac {3 i x \,b^{5}}{a^{6}}+\frac {i x \,b^{7}}{a^{8}}-\frac {29 b \,{\mathrm e}^{2 i \left (d x +c \right )}}{128 a^{2} d}+\frac {5 b^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{16 a^{4} d}-\frac {b^{5} {\mathrm e}^{2 i \left (d x +c \right )}}{8 a^{6} d}-\frac {35 \,{\mathrm e}^{-i \left (d x +c \right )}}{128 a d}-\frac {35 \,{\mathrm e}^{i \left (d x +c \right )}}{128 d a}-\frac {2 i b c}{a^{2} d}+\frac {6 i b^{3} c}{a^{4} d}-\frac {6 i b^{5} c}{a^{6} d}+\frac {2 i b^{7} c}{a^{8} d}\)	\(676\)

[In]

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

[Out]

1/d*(1/a^7*(1/7*cos(d*x+c)^7*a^6-1/6*b*cos(d*x+c)^6*a^5-3/5*a^6*cos(d*x+c)^5+1/5*a^4*b^2*cos(d*x+c)^5+3/4*cos(
d*x+c)^4*a^5*b-1/4*cos(d*x+c)^4*a^3*b^3+cos(d*x+c)^3*a^6-cos(d*x+c)^3*a^4*b^2+1/3*cos(d*x+c)^3*a^2*b^4-3/2*cos
(d*x+c)^2*a^5*b+3/2*cos(d*x+c)^2*a^3*b^3-1/2*cos(d*x+c)^2*a*b^5-cos(d*x+c)*a^6+3*cos(d*x+c)*a^4*b^2-3*cos(d*x+
c)*a^2*b^4+b^6*cos(d*x+c))+b*(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/a^8*ln(b+a*cos(d*x+c)))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.00 \[ \int \frac {\sin ^7(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {60 \, a^{7} \cos \left (d x + c\right )^{7} - 70 \, a^{6} b \cos \left (d x + c\right )^{6} - 84 \, {\left (3 \, a^{7} - a^{5} b^{2}\right )} \cos \left (d x + c\right )^{5} + 105 \, {\left (3 \, a^{6} b - a^{4} b^{3}\right )} \cos \left (d x + c\right )^{4} + 140 \, {\left (3 \, a^{7} - 3 \, a^{5} b^{2} + a^{3} b^{4}\right )} \cos \left (d x + c\right )^{3} - 210 \, {\left (3 \, a^{6} b - 3 \, a^{4} b^{3} + a^{2} b^{5}\right )} \cos \left (d x + c\right )^{2} - 420 \, {\left (a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right )} \cos \left (d x + c\right ) + 420 \, {\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{420 \, a^{8} d} \]

[In]

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

[Out]

1/420*(60*a^7*cos(d*x + c)^7 - 70*a^6*b*cos(d*x + c)^6 - 84*(3*a^7 - a^5*b^2)*cos(d*x + c)^5 + 105*(3*a^6*b -
a^4*b^3)*cos(d*x + c)^4 + 140*(3*a^7 - 3*a^5*b^2 + a^3*b^4)*cos(d*x + c)^3 - 210*(3*a^6*b - 3*a^4*b^3 + a^2*b^
5)*cos(d*x + c)^2 - 420*(a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cos(d*x + c) + 420*(a^6*b - 3*a^4*b^3 + 3*a^2*b^
5 - b^7)*log(a*cos(d*x + c) + b))/(a^8*d)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.00 \[ \int \frac {\sin ^7(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {\frac {60 \, a^{6} \cos \left (d x + c\right )^{7} - 70 \, a^{5} b \cos \left (d x + c\right )^{6} - 84 \, {\left (3 \, a^{6} - a^{4} b^{2}\right )} \cos \left (d x + c\right )^{5} + 105 \, {\left (3 \, a^{5} b - a^{3} b^{3}\right )} \cos \left (d x + c\right )^{4} + 140 \, {\left (3 \, a^{6} - 3 \, a^{4} b^{2} + a^{2} b^{4}\right )} \cos \left (d x + c\right )^{3} - 210 \, {\left (3 \, a^{5} b - 3 \, a^{3} b^{3} + a b^{5}\right )} \cos \left (d x + c\right )^{2} - 420 \, {\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \cos \left (d x + c\right )}{a^{7}} + \frac {420 \, {\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{8}}}{420 \, d} \]

[In]

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

[Out]

1/420*((60*a^6*cos(d*x + c)^7 - 70*a^5*b*cos(d*x + c)^6 - 84*(3*a^6 - a^4*b^2)*cos(d*x + c)^5 + 105*(3*a^5*b -
a^3*b^3)*cos(d*x + c)^4 + 140*(3*a^6 - 3*a^4*b^2 + a^2*b^4)*cos(d*x + c)^3 - 210*(3*a^5*b - 3*a^3*b^3 + a*b^5
)*cos(d*x + c)^2 - 420*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cos(d*x + c))/a^7 + 420*(a^6*b - 3*a^4*b^3 + 3*a^2*
b^5 - b^7)*log(a*cos(d*x + c) + b)/a^8)/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1559 vs. \(2 (211) = 422\).

Time = 0.36 (sec) , antiderivative size = 1559, normalized size of antiderivative = 6.99 \[ \int \frac {\sin ^7(c+d x)}{a+b \sec (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/420*(420*(a^7*b - a^6*b^2 - 3*a^5*b^3 + 3*a^4*b^4 + 3*a^3*b^5 - 3*a^2*b^6 - a*b^7 + b^8)*log(abs(a + b + a*(
cos(d*x + c) - 1)/(cos(d*x + c) + 1) - b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1)))/(a^9 - a^8*b) - 420*(a^6*b -
3*a^4*b^3 + 3*a^2*b^5 - b^7)*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1))/a^8 + (384*a^7 - 1089*a^6*b
- 1848*a^5*b^2 + 3267*a^4*b^3 + 2240*a^3*b^4 - 3267*a^2*b^5 - 840*a*b^6 + 1089*b^7 - 2688*a^7*(cos(d*x + c) -
1)/(cos(d*x + c) + 1) + 8463*a^6*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 12096*a^5*b^2*(cos(d*x + c) - 1)/(c
os(d*x + c) + 1) - 24549*a^4*b^3*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 14000*a^3*b^4*(cos(d*x + c) - 1)/(cos
(d*x + c) + 1) + 23709*a^2*b^5*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 5040*a*b^6*(cos(d*x + c) - 1)/(cos(d*x
+ c) + 1) - 7623*b^7*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 8064*a^7*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^
2 - 28749*a^6*b*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 32088*a^5*b^2*(cos(d*x + c) - 1)^2/(cos(d*x + c) +
1)^2 + 78687*a^4*b^3*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 35280*a^3*b^4*(cos(d*x + c) - 1)^2/(cos(d*x
+ c) + 1)^2 - 72807*a^2*b^5*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 12600*a*b^6*(cos(d*x + c) - 1)^2/(cos(
d*x + c) + 1)^2 + 22869*b^7*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 13440*a^7*(cos(d*x + c) - 1)^3/(cos(d*
x + c) + 1)^3 + 56035*a^6*b*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 40320*a^5*b^2*(cos(d*x + c) - 1)^3/(co
s(d*x + c) + 1)^3 - 136185*a^4*b^3*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 - 45920*a^3*b^4*(cos(d*x + c) - 1
)^3/(cos(d*x + c) + 1)^3 + 122745*a^2*b^5*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 16800*a*b^6*(cos(d*x + c
) - 1)^3/(cos(d*x + c) + 1)^3 - 38115*b^7*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 - 56035*a^6*b*(cos(d*x + c
) - 1)^4/(cos(d*x + c) + 1)^4 - 24360*a^5*b^2*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 + 136185*a^4*b^3*(cos(
d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 + 32480*a^3*b^4*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 - 122745*a^2*b^
5*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 - 12600*a*b^6*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 + 38115*b^
7*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 + 28749*a^6*b*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5 + 6720*a^5
*b^2*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5 - 78687*a^4*b^3*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5 - 117
60*a^3*b^4*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5 + 72807*a^2*b^5*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5
+ 5040*a*b^6*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5 - 22869*b^7*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5
- 8463*a^6*b*(cos(d*x + c) - 1)^6/(cos(d*x + c) + 1)^6 - 840*a^5*b^2*(cos(d*x + c) - 1)^6/(cos(d*x + c) + 1)^6
+ 24549*a^4*b^3*(cos(d*x + c) - 1)^6/(cos(d*x + c) + 1)^6 + 1680*a^3*b^4*(cos(d*x + c) - 1)^6/(cos(d*x + c) +
1)^6 - 23709*a^2*b^5*(cos(d*x + c) - 1)^6/(cos(d*x + c) + 1)^6 - 840*a*b^6*(cos(d*x + c) - 1)^6/(cos(d*x + c)
+ 1)^6 + 7623*b^7*(cos(d*x + c) - 1)^6/(cos(d*x + c) + 1)^6 + 1089*a^6*b*(cos(d*x + c) - 1)^7/(cos(d*x + c) +
1)^7 - 3267*a^4*b^3*(cos(d*x + c) - 1)^7/(cos(d*x + c) + 1)^7 + 3267*a^2*b^5*(cos(d*x + c) - 1)^7/(cos(d*x +
c) + 1)^7 - 1089*b^7*(cos(d*x + c) - 1)^7/(cos(d*x + c) + 1)^7)/(a^8*((cos(d*x + c) - 1)/(cos(d*x + c) + 1) -
1)^7))/d

Mupad [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.12 \[ \int \frac {\sin ^7(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {{\cos \left (c+d\,x\right )}^3\,\left (\frac {1}{a}-\frac {b^2\,\left (\frac {1}{a}-\frac {b^2}{3\,a^3}\right )}{a^2}\right )-{\cos \left (c+d\,x\right )}^5\,\left (\frac {3}{5\,a}-\frac {b^2}{5\,a^3}\right )-\cos \left (c+d\,x\right )\,\left (\frac {1}{a}-\frac {b^2\,\left (\frac {3}{a}-\frac {b^2\,\left (\frac {3}{a}-\frac {b^2}{a^3}\right )}{a^2}\right )}{a^2}\right )+\frac {{\cos \left (c+d\,x\right )}^7}{7\,a}+\frac {\ln \left (b+a\,\cos \left (c+d\,x\right )\right )\,\left (a^6\,b-3\,a^4\,b^3+3\,a^2\,b^5-b^7\right )}{a^8}-\frac {b\,{\cos \left (c+d\,x\right )}^6}{6\,a^2}-\frac {b\,{\cos \left (c+d\,x\right )}^2\,\left (\frac {3}{a}-\frac {b^2\,\left (\frac {3}{a}-\frac {b^2}{a^3}\right )}{a^2}\right )}{2\,a}+\frac {b\,{\cos \left (c+d\,x\right )}^4\,\left (\frac {3}{a}-\frac {b^2}{a^3}\right )}{4\,a}}{d} \]

[In]

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

[Out]

(cos(c + d*x)^3*(1/a - (b^2*(1/a - b^2/(3*a^3)))/a^2) - cos(c + d*x)^5*(3/(5*a) - b^2/(5*a^3)) - cos(c + d*x)*
(1/a - (b^2*(3/a - (b^2*(3/a - b^2/a^3))/a^2))/a^2) + cos(c + d*x)^7/(7*a) + (log(b + a*cos(c + d*x))*(a^6*b -
b^7 + 3*a^2*b^5 - 3*a^4*b^3))/a^8 - (b*cos(c + d*x)^6)/(6*a^2) - (b*cos(c + d*x)^2*(3/a - (b^2*(3/a - b^2/a^3
))/a^2))/(2*a) + (b*cos(c + d*x)^4*(3/a - b^2/a^3))/(4*a))/d