\(\int \csc ^7(c+d x) (a+a \sec (c+d x)) \, dx\) [9]

Optimal result

Integrand size = 19, antiderivative size = 163 \[ \int \csc ^7(c+d x) (a+a \sec (c+d x)) \, dx=-\frac {a^4}{24 d (a-a \cos (c+d x))^3}-\frac {5 a^3}{32 d (a-a \cos (c+d x))^2}-\frac {a^2}{2 d (a-a \cos (c+d x))}-\frac {a^3}{32 d (a+a \cos (c+d x))^2}-\frac {3 a^2}{16 d (a+a \cos (c+d x))}+\frac {21 a \log (1-\cos (c+d x))}{32 d}-\frac {a \log (\cos (c+d x))}{d}+\frac {11 a \log (1+\cos (c+d x))}{32 d} \]

[Out]

-1/24*a^4/d/(a-a*cos(d*x+c))^3-5/32*a^3/d/(a-a*cos(d*x+c))^2-1/2*a^2/d/(a-a*cos(d*x+c))-1/32*a^3/d/(a+a*cos(d*
x+c))^2-3/16*a^2/d/(a+a*cos(d*x+c))+21/32*a*ln(1-cos(d*x+c))/d-a*ln(cos(d*x+c))/d+11/32*a*ln(1+cos(d*x+c))/d

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 163, 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} \[ \int \csc ^7(c+d x) (a+a \sec (c+d x)) \, dx=-\frac {a^4}{24 d (a-a \cos (c+d x))^3}-\frac {5 a^3}{32 d (a-a \cos (c+d x))^2}-\frac {a^3}{32 d (a \cos (c+d x)+a)^2}-\frac {a^2}{2 d (a-a \cos (c+d x))}-\frac {3 a^2}{16 d (a \cos (c+d x)+a)}+\frac {21 a \log (1-\cos (c+d x))}{32 d}-\frac {a \log (\cos (c+d x))}{d}+\frac {11 a \log (\cos (c+d x)+1)}{32 d} \]

[In]

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

[Out]

-1/24*a^4/(d*(a - a*Cos[c + d*x])^3) - (5*a^3)/(32*d*(a - a*Cos[c + d*x])^2) - a^2/(2*d*(a - a*Cos[c + d*x]))
- a^3/(32*d*(a + a*Cos[c + d*x])^2) - (3*a^2)/(16*d*(a + a*Cos[c + d*x])) + (21*a*Log[1 - Cos[c + d*x]])/(32*d
) - (a*Log[Cos[c + d*x]])/d + (11*a*Log[1 + Cos[c + d*x]])/(32*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)) \csc ^7(c+d x) \sec (c+d x) \, dx \\ & = \frac {a^7 \text {Subst}\left (\int \frac {a}{(-a-x)^4 x (-a+x)^3} \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = \frac {a^8 \text {Subst}\left (\int \frac {1}{(-a-x)^4 x (-a+x)^3} \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = \frac {a^8 \text {Subst}\left (\int \left (-\frac {1}{16 a^5 (a-x)^3}-\frac {3}{16 a^6 (a-x)^2}-\frac {11}{32 a^7 (a-x)}-\frac {1}{a^7 x}+\frac {1}{8 a^4 (a+x)^4}+\frac {5}{16 a^5 (a+x)^3}+\frac {1}{2 a^6 (a+x)^2}+\frac {21}{32 a^7 (a+x)}\right ) \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = -\frac {a^4}{24 d (a-a \cos (c+d x))^3}-\frac {5 a^3}{32 d (a-a \cos (c+d x))^2}-\frac {a^2}{2 d (a-a \cos (c+d x))}-\frac {a^3}{32 d (a+a \cos (c+d x))^2}-\frac {3 a^2}{16 d (a+a \cos (c+d x))}+\frac {21 a \log (1-\cos (c+d x))}{32 d}-\frac {a \log (\cos (c+d x))}{d}+\frac {11 a \log (1+\cos (c+d x))}{32 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.42 \[ \int \csc ^7(c+d x) (a+a \sec (c+d x)) \, dx=-\frac {5 a \csc ^2\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {a \csc ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {a \csc ^6\left (\frac {1}{2} (c+d x)\right )}{384 d}-\frac {a \csc ^2(c+d x)}{2 d}-\frac {a \csc ^4(c+d x)}{4 d}-\frac {a \csc ^6(c+d x)}{6 d}-\frac {5 a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}-\frac {a \log (\cos (c+d x))}{d}+\frac {5 a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}+\frac {a \log (\sin (c+d x))}{d}+\frac {5 a \sec ^2\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {a \sec ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {a \sec ^6\left (\frac {1}{2} (c+d x)\right )}{384 d} \]

[In]

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

[Out]

(-5*a*Csc[(c + d*x)/2]^2)/(64*d) - (a*Csc[(c + d*x)/2]^4)/(64*d) - (a*Csc[(c + d*x)/2]^6)/(384*d) - (a*Csc[c +
d*x]^2)/(2*d) - (a*Csc[c + d*x]^4)/(4*d) - (a*Csc[c + d*x]^6)/(6*d) - (5*a*Log[Cos[(c + d*x)/2]])/(16*d) - (a
*Log[Cos[c + d*x]])/d + (5*a*Log[Sin[(c + d*x)/2]])/(16*d) + (a*Log[Sin[c + d*x]])/d + (5*a*Sec[(c + d*x)/2]^2
)/(64*d) + (a*Sec[(c + d*x)/2]^4)/(64*d) + (a*Sec[(c + d*x)/2]^6)/(384*d)

Maple [A] (verified)

Time = 1.12 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.63

method	result	size
derivativedivides	\(\frac {a \left (-\frac {1}{6 \sin \left (d x +c \right )^{6}}-\frac {1}{4 \sin \left (d x +c \right )^{4}}-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+a \left (\left (-\frac {\csc \left (d x +c \right )^{5}}{6}-\frac {5 \csc \left (d x +c \right )^{3}}{24}-\frac {5 \csc \left (d x +c \right )}{16}\right ) \cot \left (d x +c \right )+\frac {5 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{16}\right )}{d}\)	\(103\)
default	\(\frac {a \left (-\frac {1}{6 \sin \left (d x +c \right )^{6}}-\frac {1}{4 \sin \left (d x +c \right )^{4}}-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+a \left (\left (-\frac {\csc \left (d x +c \right )^{5}}{6}-\frac {5 \csc \left (d x +c \right )^{3}}{24}-\frac {5 \csc \left (d x +c \right )}{16}\right ) \cot \left (d x +c \right )+\frac {5 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{16}\right )}{d}\)	\(103\)
parallelrisch	\(-\frac {a \left (\cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\frac {21 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{2}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{2}+66 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+21 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-252 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+192 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+192 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )\right )}{192 d}\)	\(111\)
norman	\(\frac {-\frac {a}{192 d}-\frac {7 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{128 d}-\frac {11 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{32 d}-\frac {7 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{64 d}-\frac {a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{128 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}+\frac {21 a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}-\frac {a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\)	\(141\)
risch	\(\frac {a \left (15 \,{\mathrm e}^{9 i \left (d x +c \right )}+18 \,{\mathrm e}^{8 i \left (d x +c \right )}-136 \,{\mathrm e}^{7 i \left (d x +c \right )}-34 \,{\mathrm e}^{6 i \left (d x +c \right )}+402 \,{\mathrm e}^{5 i \left (d x +c \right )}-34 \,{\mathrm e}^{4 i \left (d x +c \right )}-136 \,{\mathrm e}^{3 i \left (d x +c \right )}+18 \,{\mathrm e}^{2 i \left (d x +c \right )}+15 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{24 d \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{6} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{4}}+\frac {21 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{16 d}+\frac {11 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{16 d}-\frac {a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\)	\(188\)

[In]

int(csc(d*x+c)^7*(a+a*sec(d*x+c)),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(tan(d*x+c)))+a*((-1/6*csc(d*x+c)^5-5/24*csc(d*x
+c)^3-5/16*csc(d*x+c))*cot(d*x+c)+5/16*ln(-cot(d*x+c)+csc(d*x+c))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 307 vs. \(2 (152) = 304\).

Time = 0.30 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.88 \[ \int \csc ^7(c+d x) (a+a \sec (c+d x)) \, dx=\frac {30 \, a \cos \left (d x + c\right )^{4} + 18 \, a \cos \left (d x + c\right )^{3} - 98 \, a \cos \left (d x + c\right )^{2} - 22 \, a \cos \left (d x + c\right ) - 96 \, {\left (a \cos \left (d x + c\right )^{5} - a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{3} + 2 \, a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right ) - a\right )} \log \left (-\cos \left (d x + c\right )\right ) + 33 \, {\left (a \cos \left (d x + c\right )^{5} - a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{3} + 2 \, a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right ) - a\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 63 \, {\left (a \cos \left (d x + c\right )^{5} - a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{3} + 2 \, a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right ) - a\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 88 \, a}{96 \, {\left (d \cos \left (d x + c\right )^{5} - d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{3} + 2 \, d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right ) - d\right )}} \]

[In]

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

[Out]

1/96*(30*a*cos(d*x + c)^4 + 18*a*cos(d*x + c)^3 - 98*a*cos(d*x + c)^2 - 22*a*cos(d*x + c) - 96*(a*cos(d*x + c)
^5 - a*cos(d*x + c)^4 - 2*a*cos(d*x + c)^3 + 2*a*cos(d*x + c)^2 + a*cos(d*x + c) - a)*log(-cos(d*x + c)) + 33*
(a*cos(d*x + c)^5 - a*cos(d*x + c)^4 - 2*a*cos(d*x + c)^3 + 2*a*cos(d*x + c)^2 + a*cos(d*x + c) - a)*log(1/2*c
os(d*x + c) + 1/2) + 63*(a*cos(d*x + c)^5 - a*cos(d*x + c)^4 - 2*a*cos(d*x + c)^3 + 2*a*cos(d*x + c)^2 + a*cos
(d*x + c) - a)*log(-1/2*cos(d*x + c) + 1/2) + 88*a)/(d*cos(d*x + c)^5 - d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^3
+ 2*d*cos(d*x + c)^2 + d*cos(d*x + c) - d)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.83 \[ \int \csc ^7(c+d x) (a+a \sec (c+d x)) \, dx=\frac {33 \, a \log \left (\cos \left (d x + c\right ) + 1\right ) + 63 \, a \log \left (\cos \left (d x + c\right ) - 1\right ) - 96 \, a \log \left (\cos \left (d x + c\right )\right ) + \frac {2 \, {\left (15 \, a \cos \left (d x + c\right )^{4} + 9 \, a \cos \left (d x + c\right )^{3} - 49 \, a \cos \left (d x + c\right )^{2} - 11 \, a \cos \left (d x + c\right ) + 44 \, a\right )}}{\cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} + 2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right ) - 1}}{96 \, d} \]

[In]

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

[Out]

1/96*(33*a*log(cos(d*x + c) + 1) + 63*a*log(cos(d*x + c) - 1) - 96*a*log(cos(d*x + c)) + 2*(15*a*cos(d*x + c)^
4 + 9*a*cos(d*x + c)^3 - 49*a*cos(d*x + c)^2 - 11*a*cos(d*x + c) + 44*a)/(cos(d*x + c)^5 - cos(d*x + c)^4 - 2*
cos(d*x + c)^3 + 2*cos(d*x + c)^2 + cos(d*x + c) - 1))/d

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.20 \[ \int \csc ^7(c+d x) (a+a \sec (c+d x)) \, dx=\frac {252 \, a \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 384 \, a \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {{\left (2 \, a - \frac {21 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {132 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {462 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) - 1\right )}^{3}} + \frac {42 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {3 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}{384 \, d} \]

[In]

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

[Out]

1/384*(252*a*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1)) - 384*a*log(abs(-(cos(d*x + c) - 1)/(cos(d*x +
c) + 1) - 1)) + (2*a - 21*a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 132*a*(cos(d*x + c) - 1)^2/(cos(d*x + c) +
1)^2 - 462*a*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3)*(cos(d*x + c) + 1)^3/(cos(d*x + c) - 1)^3 + 42*a*(cos
(d*x + c) - 1)/(cos(d*x + c) + 1) - 3*a*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2)/d

Mupad [B] (verification not implemented)

Time = 13.58 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.87 \[ \int \csc ^7(c+d x) (a+a \sec (c+d x)) \, dx=\frac {\frac {5\,a\,{\cos \left (c+d\,x\right )}^4}{16}+\frac {3\,a\,{\cos \left (c+d\,x\right )}^3}{16}-\frac {49\,a\,{\cos \left (c+d\,x\right )}^2}{48}-\frac {11\,a\,\cos \left (c+d\,x\right )}{48}+\frac {11\,a}{12}}{d\,\left ({\cos \left (c+d\,x\right )}^5-{\cos \left (c+d\,x\right )}^4-2\,{\cos \left (c+d\,x\right )}^3+2\,{\cos \left (c+d\,x\right )}^2+\cos \left (c+d\,x\right )-1\right )}-\frac {a\,\ln \left (\cos \left (c+d\,x\right )\right )}{d}+\frac {21\,a\,\ln \left (\cos \left (c+d\,x\right )-1\right )}{32\,d}+\frac {11\,a\,\ln \left (\cos \left (c+d\,x\right )+1\right )}{32\,d} \]

[In]

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

[Out]

((11*a)/12 - (11*a*cos(c + d*x))/48 - (49*a*cos(c + d*x)^2)/48 + (3*a*cos(c + d*x)^3)/16 + (5*a*cos(c + d*x)^4
)/16)/(d*(cos(c + d*x) + 2*cos(c + d*x)^2 - 2*cos(c + d*x)^3 - cos(c + d*x)^4 + cos(c + d*x)^5 - 1)) - (a*log(
cos(c + d*x)))/d + (21*a*log(cos(c + d*x) - 1))/(32*d) + (11*a*log(cos(c + d*x) + 1))/(32*d)