\(\int \csc ^{10}(c+d x) (a+a \sec (c+d x))^2 \, dx\) [37]

Optimal result

Integrand size = 21, antiderivative size = 201 \[ \int \csc ^{10}(c+d x) (a+a \sec (c+d x))^2 \, dx=\frac {2 a^2 \text {arctanh}(\sin (c+d x))}{d}-\frac {6 a^2 \cot (c+d x)}{d}-\frac {14 a^2 \cot ^3(c+d x)}{3 d}-\frac {16 a^2 \cot ^5(c+d x)}{5 d}-\frac {9 a^2 \cot ^7(c+d x)}{7 d}-\frac {2 a^2 \cot ^9(c+d x)}{9 d}-\frac {2 a^2 \csc (c+d x)}{d}-\frac {2 a^2 \csc ^3(c+d x)}{3 d}-\frac {2 a^2 \csc ^5(c+d x)}{5 d}-\frac {2 a^2 \csc ^7(c+d x)}{7 d}-\frac {2 a^2 \csc ^9(c+d x)}{9 d}+\frac {a^2 \tan (c+d x)}{d} \]

[Out]

2*a^2*arctanh(sin(d*x+c))/d-6*a^2*cot(d*x+c)/d-14/3*a^2*cot(d*x+c)^3/d-16/5*a^2*cot(d*x+c)^5/d-9/7*a^2*cot(d*x
+c)^7/d-2/9*a^2*cot(d*x+c)^9/d-2*a^2*csc(d*x+c)/d-2/3*a^2*csc(d*x+c)^3/d-2/5*a^2*csc(d*x+c)^5/d-2/7*a^2*csc(d*
x+c)^7/d-2/9*a^2*csc(d*x+c)^9/d+a^2*tan(d*x+c)/d

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3957, 2952, 3852, 2701, 308, 213, 2700, 276} \[ \int \csc ^{10}(c+d x) (a+a \sec (c+d x))^2 \, dx=\frac {2 a^2 \text {arctanh}(\sin (c+d x))}{d}+\frac {a^2 \tan (c+d x)}{d}-\frac {2 a^2 \cot ^9(c+d x)}{9 d}-\frac {9 a^2 \cot ^7(c+d x)}{7 d}-\frac {16 a^2 \cot ^5(c+d x)}{5 d}-\frac {14 a^2 \cot ^3(c+d x)}{3 d}-\frac {6 a^2 \cot (c+d x)}{d}-\frac {2 a^2 \csc ^9(c+d x)}{9 d}-\frac {2 a^2 \csc ^7(c+d x)}{7 d}-\frac {2 a^2 \csc ^5(c+d x)}{5 d}-\frac {2 a^2 \csc ^3(c+d x)}{3 d}-\frac {2 a^2 \csc (c+d x)}{d} \]

[In]

Int[Csc[c + d*x]^10*(a + a*Sec[c + d*x])^2,x]

[Out]

(2*a^2*ArcTanh[Sin[c + d*x]])/d - (6*a^2*Cot[c + d*x])/d - (14*a^2*Cot[c + d*x]^3)/(3*d) - (16*a^2*Cot[c + d*x
]^5)/(5*d) - (9*a^2*Cot[c + d*x]^7)/(7*d) - (2*a^2*Cot[c + d*x]^9)/(9*d) - (2*a^2*Csc[c + d*x])/d - (2*a^2*Csc
[c + d*x]^3)/(3*d) - (2*a^2*Csc[c + d*x]^5)/(5*d) - (2*a^2*Csc[c + d*x]^7)/(7*d) - (2*a^2*Csc[c + d*x]^9)/(9*d
) + (a^2*Tan[c + d*x])/d

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 308

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 2700

Int[csc[(e_.) + (f_.)*(x_)]^(m_.)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(1 + x^2)^((
m + n)/2 - 1)/x^m, x], x, Tan[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n)/2]

Rule 2701

Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[-(f*a^n)^(-1), Subst
[Int[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Csc[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && Integer
Q[(n + 1)/2] &&  !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])

Rule 2952

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*
(x_)])^(m_), x_Symbol] :> Int[ExpandTrig[(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x]
/; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/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))^2 \csc ^{10}(c+d x) \sec ^2(c+d x) \, dx \\ & = \int \left (a^2 \csc ^{10}(c+d x)+2 a^2 \csc ^{10}(c+d x) \sec (c+d x)+a^2 \csc ^{10}(c+d x) \sec ^2(c+d x)\right ) \, dx \\ & = a^2 \int \csc ^{10}(c+d x) \, dx+a^2 \int \csc ^{10}(c+d x) \sec ^2(c+d x) \, dx+\left (2 a^2\right ) \int \csc ^{10}(c+d x) \sec (c+d x) \, dx \\ & = \frac {a^2 \text {Subst}\left (\int \frac {\left (1+x^2\right )^5}{x^{10}} \, dx,x,\tan (c+d x)\right )}{d}-\frac {a^2 \text {Subst}\left (\int \left (1+4 x^2+6 x^4+4 x^6+x^8\right ) \, dx,x,\cot (c+d x)\right )}{d}-\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {x^{10}}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d} \\ & = -\frac {a^2 \cot (c+d x)}{d}-\frac {4 a^2 \cot ^3(c+d x)}{3 d}-\frac {6 a^2 \cot ^5(c+d x)}{5 d}-\frac {4 a^2 \cot ^7(c+d x)}{7 d}-\frac {a^2 \cot ^9(c+d x)}{9 d}+\frac {a^2 \text {Subst}\left (\int \left (1+\frac {1}{x^{10}}+\frac {5}{x^8}+\frac {10}{x^6}+\frac {10}{x^4}+\frac {5}{x^2}\right ) \, dx,x,\tan (c+d x)\right )}{d}-\frac {\left (2 a^2\right ) \text {Subst}\left (\int \left (1+x^2+x^4+x^6+x^8+\frac {1}{-1+x^2}\right ) \, dx,x,\csc (c+d x)\right )}{d} \\ & = -\frac {6 a^2 \cot (c+d x)}{d}-\frac {14 a^2 \cot ^3(c+d x)}{3 d}-\frac {16 a^2 \cot ^5(c+d x)}{5 d}-\frac {9 a^2 \cot ^7(c+d x)}{7 d}-\frac {2 a^2 \cot ^9(c+d x)}{9 d}-\frac {2 a^2 \csc (c+d x)}{d}-\frac {2 a^2 \csc ^3(c+d x)}{3 d}-\frac {2 a^2 \csc ^5(c+d x)}{5 d}-\frac {2 a^2 \csc ^7(c+d x)}{7 d}-\frac {2 a^2 \csc ^9(c+d x)}{9 d}+\frac {a^2 \tan (c+d x)}{d}-\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d} \\ & = \frac {2 a^2 \text {arctanh}(\sin (c+d x))}{d}-\frac {6 a^2 \cot (c+d x)}{d}-\frac {14 a^2 \cot ^3(c+d x)}{3 d}-\frac {16 a^2 \cot ^5(c+d x)}{5 d}-\frac {9 a^2 \cot ^7(c+d x)}{7 d}-\frac {2 a^2 \cot ^9(c+d x)}{9 d}-\frac {2 a^2 \csc (c+d x)}{d}-\frac {2 a^2 \csc ^3(c+d x)}{3 d}-\frac {2 a^2 \csc ^5(c+d x)}{5 d}-\frac {2 a^2 \csc ^7(c+d x)}{7 d}-\frac {2 a^2 \csc ^9(c+d x)}{9 d}+\frac {a^2 \tan (c+d x)}{d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1050\) vs. \(2(201)=402\).

Time = 10.92 (sec) , antiderivative size = 1050, normalized size of antiderivative = 5.22 \[ \int \csc ^{10}(c+d x) (a+a \sec (c+d x))^2 \, dx=-\frac {6899 \cos ^2(c+d x) \cot \left (\frac {c}{2}\right ) \csc ^2\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^2}{80640 d}-\frac {193 \cos ^2(c+d x) \cot \left (\frac {c}{2}\right ) \csc ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^2}{13440 d}-\frac {71 \cos ^2(c+d x) \cot \left (\frac {c}{2}\right ) \csc ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^2}{32256 d}-\frac {\cos ^2(c+d x) \cot \left (\frac {c}{2}\right ) \csc ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^2}{4608 d}-\frac {\cos ^2(c+d x) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^2}{2 d}+\frac {\cos ^2(c+d x) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^2}{2 d}+\frac {123041 \cos ^2(c+d x) \csc \left (\frac {c}{2}\right ) \csc \left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^2 \sin \left (\frac {d x}{2}\right )}{161280 d}+\frac {6899 \cos ^2(c+d x) \csc \left (\frac {c}{2}\right ) \csc ^3\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^2 \sin \left (\frac {d x}{2}\right )}{80640 d}+\frac {193 \cos ^2(c+d x) \csc \left (\frac {c}{2}\right ) \csc ^5\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^2 \sin \left (\frac {d x}{2}\right )}{13440 d}+\frac {71 \cos ^2(c+d x) \csc \left (\frac {c}{2}\right ) \csc ^7\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^2 \sin \left (\frac {d x}{2}\right )}{32256 d}+\frac {\cos ^2(c+d x) \csc \left (\frac {c}{2}\right ) \csc ^9\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^2 \sin \left (\frac {d x}{2}\right )}{4608 d}+\frac {803 \cos ^2(c+d x) \sec \left (\frac {c}{2}\right ) \sec ^5\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^2 \sin \left (\frac {d x}{2}\right )}{7680 d}+\frac {49 \cos ^2(c+d x) \sec \left (\frac {c}{2}\right ) \sec ^7\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^2 \sin \left (\frac {d x}{2}\right )}{7680 d}+\frac {\cos ^2(c+d x) \sec \left (\frac {c}{2}\right ) \sec ^9\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^2 \sin \left (\frac {d x}{2}\right )}{2560 d}+\frac {\cos (c+d x) \sec (c) \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^2 \sin (d x)}{4 d}+\frac {49 \cos ^2(c+d x) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^2 \tan \left (\frac {c}{2}\right )}{7680 d}+\frac {\cos ^2(c+d x) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^2 \tan \left (\frac {c}{2}\right )}{2560 d} \]

[In]

Integrate[Csc[c + d*x]^10*(a + a*Sec[c + d*x])^2,x]

[Out]

(-6899*Cos[c + d*x]^2*Cot[c/2]*Csc[c/2 + (d*x)/2]^2*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2)/(80640*d) - (
193*Cos[c + d*x]^2*Cot[c/2]*Csc[c/2 + (d*x)/2]^4*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2)/(13440*d) - (71*
Cos[c + d*x]^2*Cot[c/2]*Csc[c/2 + (d*x)/2]^6*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2)/(32256*d) - (Cos[c +
d*x]^2*Cot[c/2]*Csc[c/2 + (d*x)/2]^8*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2)/(4608*d) - (Cos[c + d*x]^2*
Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2)/(2*d) + (Cos[c + d*x
]^2*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2)/(2*d) + (123041*
Cos[c + d*x]^2*Csc[c/2]*Csc[c/2 + (d*x)/2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*Sin[(d*x)/2])/(161280*d
) + (6899*Cos[c + d*x]^2*Csc[c/2]*Csc[c/2 + (d*x)/2]^3*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*Sin[(d*x)/2
])/(80640*d) + (193*Cos[c + d*x]^2*Csc[c/2]*Csc[c/2 + (d*x)/2]^5*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*S
in[(d*x)/2])/(13440*d) + (71*Cos[c + d*x]^2*Csc[c/2]*Csc[c/2 + (d*x)/2]^7*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c +
d*x])^2*Sin[(d*x)/2])/(32256*d) + (Cos[c + d*x]^2*Csc[c/2]*Csc[c/2 + (d*x)/2]^9*Sec[c/2 + (d*x)/2]^4*(a + a*Se
c[c + d*x])^2*Sin[(d*x)/2])/(4608*d) + (803*Cos[c + d*x]^2*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(a + a*Sec[c + d*x])^
2*Sin[(d*x)/2])/(7680*d) + (49*Cos[c + d*x]^2*Sec[c/2]*Sec[c/2 + (d*x)/2]^7*(a + a*Sec[c + d*x])^2*Sin[(d*x)/2
])/(7680*d) + (Cos[c + d*x]^2*Sec[c/2]*Sec[c/2 + (d*x)/2]^9*(a + a*Sec[c + d*x])^2*Sin[(d*x)/2])/(2560*d) + (C
os[c + d*x]*Sec[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*Sin[d*x])/(4*d) + (49*Cos[c + d*x]^2*Sec[c/2 +
(d*x)/2]^6*(a + a*Sec[c + d*x])^2*Tan[c/2])/(7680*d) + (Cos[c + d*x]^2*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x
])^2*Tan[c/2])/(2560*d)

Maple [A] (verified)

Time = 1.48 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.08

method	result	size
parallelrisch	\(\frac {a^{2} \left (35 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}+63 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+460 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+1092 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+3096 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+16800 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-80640 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+80640 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+16044 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-238140 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+80640 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-80640 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+119910 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{40320 d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-40320 d}\)	\(217\)
derivativedivides	\(\frac {a^{2} \left (-\frac {1}{9 \sin \left (d x +c \right )^{9} \cos \left (d x +c \right )}-\frac {10}{63 \sin \left (d x +c \right )^{7} \cos \left (d x +c \right )}-\frac {16}{63 \sin \left (d x +c \right )^{5} \cos \left (d x +c \right )}-\frac {32}{63 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )}+\frac {128}{63 \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {256 \cot \left (d x +c \right )}{63}\right )+2 a^{2} \left (-\frac {1}{9 \sin \left (d x +c \right )^{9}}-\frac {1}{7 \sin \left (d x +c \right )^{7}}-\frac {1}{5 \sin \left (d x +c \right )^{5}}-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{\sin \left (d x +c \right )}+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{2} \left (-\frac {128}{315}-\frac {\csc \left (d x +c \right )^{8}}{9}-\frac {8 \csc \left (d x +c \right )^{6}}{63}-\frac {16 \csc \left (d x +c \right )^{4}}{105}-\frac {64 \csc \left (d x +c \right )^{2}}{315}\right ) \cot \left (d x +c \right )}{d}\)	\(231\)
default	\(\frac {a^{2} \left (-\frac {1}{9 \sin \left (d x +c \right )^{9} \cos \left (d x +c \right )}-\frac {10}{63 \sin \left (d x +c \right )^{7} \cos \left (d x +c \right )}-\frac {16}{63 \sin \left (d x +c \right )^{5} \cos \left (d x +c \right )}-\frac {32}{63 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )}+\frac {128}{63 \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {256 \cot \left (d x +c \right )}{63}\right )+2 a^{2} \left (-\frac {1}{9 \sin \left (d x +c \right )^{9}}-\frac {1}{7 \sin \left (d x +c \right )^{7}}-\frac {1}{5 \sin \left (d x +c \right )^{5}}-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{\sin \left (d x +c \right )}+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{2} \left (-\frac {128}{315}-\frac {\csc \left (d x +c \right )^{8}}{9}-\frac {8 \csc \left (d x +c \right )^{6}}{63}-\frac {16 \csc \left (d x +c \right )^{4}}{105}-\frac {64 \csc \left (d x +c \right )^{2}}{315}\right ) \cot \left (d x +c \right )}{d}\)	\(231\)
risch	\(-\frac {4 i a^{2} \left (315 \,{\mathrm e}^{15 i \left (d x +c \right )}-1260 \,{\mathrm e}^{14 i \left (d x +c \right )}+525 \,{\mathrm e}^{13 i \left (d x +c \right )}+4200 \,{\mathrm e}^{12 i \left (d x +c \right )}-5817 \,{\mathrm e}^{11 i \left (d x +c \right )}-2772 \,{\mathrm e}^{10 i \left (d x +c \right )}+10161 \,{\mathrm e}^{9 i \left (d x +c \right )}-4944 \,{\mathrm e}^{8 i \left (d x +c \right )}-3919 \,{\mathrm e}^{7 i \left (d x +c \right )}+15532 \,{\mathrm e}^{6 i \left (d x +c \right )}-8633 \,{\mathrm e}^{5 i \left (d x +c \right )}-8472 \,{\mathrm e}^{4 i \left (d x +c \right )}+8973 \,{\mathrm e}^{3 i \left (d x +c \right )}+148 \,{\mathrm e}^{2 i \left (d x +c \right )}-2501 \,{\mathrm e}^{i \left (d x +c \right )}+704\right )}{315 d \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{9} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{5} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {2 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {2 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}\)	\(259\)

[In]

int(csc(d*x+c)^10*(a+a*sec(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

a^2*(35*cot(1/2*d*x+1/2*c)^9+63*tan(1/2*d*x+1/2*c)^7+460*cot(1/2*d*x+1/2*c)^7+1092*tan(1/2*d*x+1/2*c)^5+3096*c
ot(1/2*d*x+1/2*c)^5+16800*tan(1/2*d*x+1/2*c)^3-80640*ln(tan(1/2*d*x+1/2*c)-1)*tan(1/2*d*x+1/2*c)^2+80640*ln(ta
n(1/2*d*x+1/2*c)+1)*tan(1/2*d*x+1/2*c)^2+16044*cot(1/2*d*x+1/2*c)^3-238140*tan(1/2*d*x+1/2*c)+80640*ln(tan(1/2
*d*x+1/2*c)-1)-80640*ln(tan(1/2*d*x+1/2*c)+1)+119910*cot(1/2*d*x+1/2*c))/(40320*d*tan(1/2*d*x+1/2*c)^2-40320*d
)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 406 vs. \(2 (185) = 370\).

Time = 0.28 (sec) , antiderivative size = 406, normalized size of antiderivative = 2.02 \[ \int \csc ^{10}(c+d x) (a+a \sec (c+d x))^2 \, dx=-\frac {1408 \, a^{2} \cos \left (d x + c\right )^{8} - 2186 \, a^{2} \cos \left (d x + c\right )^{7} - 3372 \, a^{2} \cos \left (d x + c\right )^{6} + 6200 \, a^{2} \cos \left (d x + c\right )^{5} + 2060 \, a^{2} \cos \left (d x + c\right )^{4} - 5784 \, a^{2} \cos \left (d x + c\right )^{3} + 268 \, a^{2} \cos \left (d x + c\right )^{2} + 1756 \, a^{2} \cos \left (d x + c\right ) - 315 \, {\left (a^{2} \cos \left (d x + c\right )^{7} - 2 \, a^{2} \cos \left (d x + c\right )^{6} - a^{2} \cos \left (d x + c\right )^{5} + 4 \, a^{2} \cos \left (d x + c\right )^{4} - a^{2} \cos \left (d x + c\right )^{3} - 2 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2} \cos \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 315 \, {\left (a^{2} \cos \left (d x + c\right )^{7} - 2 \, a^{2} \cos \left (d x + c\right )^{6} - a^{2} \cos \left (d x + c\right )^{5} + 4 \, a^{2} \cos \left (d x + c\right )^{4} - a^{2} \cos \left (d x + c\right )^{3} - 2 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2} \cos \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) - 315 \, a^{2}}{315 \, {\left (d \cos \left (d x + c\right )^{7} - 2 \, d \cos \left (d x + c\right )^{6} - d \cos \left (d x + c\right )^{5} + 4 \, d \cos \left (d x + c\right )^{4} - d \cos \left (d x + c\right )^{3} - 2 \, d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )} \]

[In]

integrate(csc(d*x+c)^10*(a+a*sec(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/315*(1408*a^2*cos(d*x + c)^8 - 2186*a^2*cos(d*x + c)^7 - 3372*a^2*cos(d*x + c)^6 + 6200*a^2*cos(d*x + c)^5
+ 2060*a^2*cos(d*x + c)^4 - 5784*a^2*cos(d*x + c)^3 + 268*a^2*cos(d*x + c)^2 + 1756*a^2*cos(d*x + c) - 315*(a^
2*cos(d*x + c)^7 - 2*a^2*cos(d*x + c)^6 - a^2*cos(d*x + c)^5 + 4*a^2*cos(d*x + c)^4 - a^2*cos(d*x + c)^3 - 2*a
^2*cos(d*x + c)^2 + a^2*cos(d*x + c))*log(sin(d*x + c) + 1)*sin(d*x + c) + 315*(a^2*cos(d*x + c)^7 - 2*a^2*cos
(d*x + c)^6 - a^2*cos(d*x + c)^5 + 4*a^2*cos(d*x + c)^4 - a^2*cos(d*x + c)^3 - 2*a^2*cos(d*x + c)^2 + a^2*cos(
d*x + c))*log(-sin(d*x + c) + 1)*sin(d*x + c) - 315*a^2)/((d*cos(d*x + c)^7 - 2*d*cos(d*x + c)^6 - d*cos(d*x +
c)^5 + 4*d*cos(d*x + c)^4 - d*cos(d*x + c)^3 - 2*d*cos(d*x + c)^2 + d*cos(d*x + c))*sin(d*x + c))

Sympy [F(-1)]

Timed out. \[ \int \csc ^{10}(c+d x) (a+a \sec (c+d x))^2 \, dx=\text {Timed out} \]

[In]

integrate(csc(d*x+c)**10*(a+a*sec(d*x+c))**2,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.01 \[ \int \csc ^{10}(c+d x) (a+a \sec (c+d x))^2 \, dx=-\frac {a^{2} {\left (\frac {2 \, {\left (315 \, \sin \left (d x + c\right )^{8} + 105 \, \sin \left (d x + c\right )^{6} + 63 \, \sin \left (d x + c\right )^{4} + 45 \, \sin \left (d x + c\right )^{2} + 35\right )}}{\sin \left (d x + c\right )^{9}} - 315 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 315 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 5 \, a^{2} {\left (\frac {315 \, \tan \left (d x + c\right )^{8} + 210 \, \tan \left (d x + c\right )^{6} + 126 \, \tan \left (d x + c\right )^{4} + 45 \, \tan \left (d x + c\right )^{2} + 7}{\tan \left (d x + c\right )^{9}} - 63 \, \tan \left (d x + c\right )\right )} + \frac {{\left (315 \, \tan \left (d x + c\right )^{8} + 420 \, \tan \left (d x + c\right )^{6} + 378 \, \tan \left (d x + c\right )^{4} + 180 \, \tan \left (d x + c\right )^{2} + 35\right )} a^{2}}{\tan \left (d x + c\right )^{9}}}{315 \, d} \]

[In]

integrate(csc(d*x+c)^10*(a+a*sec(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/315*(a^2*(2*(315*sin(d*x + c)^8 + 105*sin(d*x + c)^6 + 63*sin(d*x + c)^4 + 45*sin(d*x + c)^2 + 35)/sin(d*x
+ c)^9 - 315*log(sin(d*x + c) + 1) + 315*log(sin(d*x + c) - 1)) + 5*a^2*((315*tan(d*x + c)^8 + 210*tan(d*x + c
)^6 + 126*tan(d*x + c)^4 + 45*tan(d*x + c)^2 + 7)/tan(d*x + c)^9 - 63*tan(d*x + c)) + (315*tan(d*x + c)^8 + 42
0*tan(d*x + c)^6 + 378*tan(d*x + c)^4 + 180*tan(d*x + c)^2 + 35)*a^2/tan(d*x + c)^9)/d

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.00 \[ \int \csc ^{10}(c+d x) (a+a \sec (c+d x))^2 \, dx=\frac {63 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1155 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 80640 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 80640 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 17955 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {80640 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} - \frac {139545 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 19635 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 3591 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 495 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 35 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9}}}{40320 \, d} \]

[In]

integrate(csc(d*x+c)^10*(a+a*sec(d*x+c))^2,x, algorithm="giac")

[Out]

1/40320*(63*a^2*tan(1/2*d*x + 1/2*c)^5 + 1155*a^2*tan(1/2*d*x + 1/2*c)^3 + 80640*a^2*log(abs(tan(1/2*d*x + 1/2
*c) + 1)) - 80640*a^2*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + 17955*a^2*tan(1/2*d*x + 1/2*c) - 80640*a^2*tan(1/2*
d*x + 1/2*c)/(tan(1/2*d*x + 1/2*c)^2 - 1) - (139545*a^2*tan(1/2*d*x + 1/2*c)^8 + 19635*a^2*tan(1/2*d*x + 1/2*c
)^6 + 3591*a^2*tan(1/2*d*x + 1/2*c)^4 + 495*a^2*tan(1/2*d*x + 1/2*c)^2 + 35*a^2)/tan(1/2*d*x + 1/2*c)^9)/d

Mupad [B] (verification not implemented)

Time = 13.10 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.97 \[ \int \csc ^{10}(c+d x) (a+a \sec (c+d x))^2 \, dx=\frac {11\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{384\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{640\,d}+\frac {4\,a^2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {-699\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\frac {1142\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{3}+\frac {764\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{15}+\frac {344\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{35}+\frac {92\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{63}+\frac {a^2}{9}}{d\,\left (128\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-128\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\right )}+\frac {57\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128\,d} \]

[In]

int((a + a/cos(c + d*x))^2/sin(c + d*x)^10,x)

[Out]

(11*a^2*tan(c/2 + (d*x)/2)^3)/(384*d) + (a^2*tan(c/2 + (d*x)/2)^5)/(640*d) + (4*a^2*atanh(tan(c/2 + (d*x)/2)))
/d - ((92*a^2*tan(c/2 + (d*x)/2)^2)/63 + (344*a^2*tan(c/2 + (d*x)/2)^4)/35 + (764*a^2*tan(c/2 + (d*x)/2)^6)/15
+ (1142*a^2*tan(c/2 + (d*x)/2)^8)/3 - 699*a^2*tan(c/2 + (d*x)/2)^10 + a^2/9)/(d*(128*tan(c/2 + (d*x)/2)^9 - 1
28*tan(c/2 + (d*x)/2)^11)) + (57*a^2*tan(c/2 + (d*x)/2))/(128*d)