3.50 \(\int \csc ^7(c+d x) (a+b \tan (c+d x))^4 \, dx\)

Optimal. Leaf size=402 \[ \frac{45 a^2 b^2 \sec (c+d x)}{4 d}-\frac{45 a^2 b^2 \tanh ^{-1}(\cos (c+d x))}{4 d}-\frac{3 a^2 b^2 \csc ^4(c+d x) \sec (c+d x)}{2 d}-\frac{15 a^2 b^2 \csc ^2(c+d x) \sec (c+d x)}{4 d}-\frac{4 a^3 b \csc ^5(c+d x)}{5 d}-\frac{4 a^3 b \csc ^3(c+d x)}{3 d}-\frac{4 a^3 b \csc (c+d x)}{d}+\frac{4 a^3 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{5 a^4 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a^4 \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac{5 a^4 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac{5 a^4 \cot (c+d x) \csc (c+d x)}{16 d}-\frac{10 a b^3 \csc ^3(c+d x)}{3 d}-\frac{10 a b^3 \csc (c+d x)}{d}+\frac{10 a b^3 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{2 a b^3 \csc ^3(c+d x) \sec ^2(c+d x)}{d}+\frac{5 b^4 \sec ^3(c+d x)}{6 d}+\frac{5 b^4 \sec (c+d x)}{2 d}-\frac{5 b^4 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{b^4 \csc ^2(c+d x) \sec ^3(c+d x)}{2 d} \]

[Out]

(-5*a^4*ArcTanh[Cos[c + d*x]])/(16*d) - (45*a^2*b^2*ArcTanh[Cos[c + d*x]])/(4*d) - (5*b^4*ArcTanh[Cos[c + d*x]
])/(2*d) + (4*a^3*b*ArcTanh[Sin[c + d*x]])/d + (10*a*b^3*ArcTanh[Sin[c + d*x]])/d - (4*a^3*b*Csc[c + d*x])/d -
(10*a*b^3*Csc[c + d*x])/d - (5*a^4*Cot[c + d*x]*Csc[c + d*x])/(16*d) - (4*a^3*b*Csc[c + d*x]^3)/(3*d) - (10*a
*b^3*Csc[c + d*x]^3)/(3*d) - (5*a^4*Cot[c + d*x]*Csc[c + d*x]^3)/(24*d) - (4*a^3*b*Csc[c + d*x]^5)/(5*d) - (a^
4*Cot[c + d*x]*Csc[c + d*x]^5)/(6*d) + (45*a^2*b^2*Sec[c + d*x])/(4*d) + (5*b^4*Sec[c + d*x])/(2*d) - (15*a^2*
b^2*Csc[c + d*x]^2*Sec[c + d*x])/(4*d) - (3*a^2*b^2*Csc[c + d*x]^4*Sec[c + d*x])/(2*d) + (2*a*b^3*Csc[c + d*x]
^3*Sec[c + d*x]^2)/d + (5*b^4*Sec[c + d*x]^3)/(6*d) - (b^4*Csc[c + d*x]^2*Sec[c + d*x]^3)/(2*d)

________________________________________________________________________________________

Rubi [A]  time = 0.305542, antiderivative size = 402, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {3517, 3768, 3770, 2621, 302, 207, 2622, 288, 321} \[ \frac{45 a^2 b^2 \sec (c+d x)}{4 d}-\frac{45 a^2 b^2 \tanh ^{-1}(\cos (c+d x))}{4 d}-\frac{3 a^2 b^2 \csc ^4(c+d x) \sec (c+d x)}{2 d}-\frac{15 a^2 b^2 \csc ^2(c+d x) \sec (c+d x)}{4 d}-\frac{4 a^3 b \csc ^5(c+d x)}{5 d}-\frac{4 a^3 b \csc ^3(c+d x)}{3 d}-\frac{4 a^3 b \csc (c+d x)}{d}+\frac{4 a^3 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{5 a^4 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a^4 \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac{5 a^4 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac{5 a^4 \cot (c+d x) \csc (c+d x)}{16 d}-\frac{10 a b^3 \csc ^3(c+d x)}{3 d}-\frac{10 a b^3 \csc (c+d x)}{d}+\frac{10 a b^3 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{2 a b^3 \csc ^3(c+d x) \sec ^2(c+d x)}{d}+\frac{5 b^4 \sec ^3(c+d x)}{6 d}+\frac{5 b^4 \sec (c+d x)}{2 d}-\frac{5 b^4 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{b^4 \csc ^2(c+d x) \sec ^3(c+d x)}{2 d} \]

Antiderivative was successfully verified.

[In]

Int[Csc[c + d*x]^7*(a + b*Tan[c + d*x])^4,x]

[Out]

(-5*a^4*ArcTanh[Cos[c + d*x]])/(16*d) - (45*a^2*b^2*ArcTanh[Cos[c + d*x]])/(4*d) - (5*b^4*ArcTanh[Cos[c + d*x]
])/(2*d) + (4*a^3*b*ArcTanh[Sin[c + d*x]])/d + (10*a*b^3*ArcTanh[Sin[c + d*x]])/d - (4*a^3*b*Csc[c + d*x])/d -
(10*a*b^3*Csc[c + d*x])/d - (5*a^4*Cot[c + d*x]*Csc[c + d*x])/(16*d) - (4*a^3*b*Csc[c + d*x]^3)/(3*d) - (10*a
*b^3*Csc[c + d*x]^3)/(3*d) - (5*a^4*Cot[c + d*x]*Csc[c + d*x]^3)/(24*d) - (4*a^3*b*Csc[c + d*x]^5)/(5*d) - (a^
4*Cot[c + d*x]*Csc[c + d*x]^5)/(6*d) + (45*a^2*b^2*Sec[c + d*x])/(4*d) + (5*b^4*Sec[c + d*x])/(2*d) - (15*a^2*
b^2*Csc[c + d*x]^2*Sec[c + d*x])/(4*d) - (3*a^2*b^2*Csc[c + d*x]^4*Sec[c + d*x])/(2*d) + (2*a*b^3*Csc[c + d*x]
^3*Sec[c + d*x]^2)/d + (5*b^4*Sec[c + d*x]^3)/(6*d) - (b^4*Csc[c + d*x]^2*Sec[c + d*x]^3)/(2*d)

Rule 3517

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Int[Expand[Sin[e
+ f*x]^m*(a + b*Tan[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] && IGtQ[n, 0]

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 2621

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 302

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 207

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

Rule 2622

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

Rule 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rubi steps

\begin{align*} \int \csc ^7(c+d x) (a+b \tan (c+d x))^4 \, dx &=\int \left (a^4 \csc ^7(c+d x)+4 a^3 b \csc ^6(c+d x) \sec (c+d x)+6 a^2 b^2 \csc ^5(c+d x) \sec ^2(c+d x)+4 a b^3 \csc ^4(c+d x) \sec ^3(c+d x)+b^4 \csc ^3(c+d x) \sec ^4(c+d x)\right ) \, dx\\ &=a^4 \int \csc ^7(c+d x) \, dx+\left (4 a^3 b\right ) \int \csc ^6(c+d x) \sec (c+d x) \, dx+\left (6 a^2 b^2\right ) \int \csc ^5(c+d x) \sec ^2(c+d x) \, dx+\left (4 a b^3\right ) \int \csc ^4(c+d x) \sec ^3(c+d x) \, dx+b^4 \int \csc ^3(c+d x) \sec ^4(c+d x) \, dx\\ &=-\frac{a^4 \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac{1}{6} \left (5 a^4\right ) \int \csc ^5(c+d x) \, dx-\frac{\left (4 a^3 b\right ) \operatorname{Subst}\left (\int \frac{x^6}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}+\frac{\left (6 a^2 b^2\right ) \operatorname{Subst}\left (\int \frac{x^6}{\left (-1+x^2\right )^3} \, dx,x,\sec (c+d x)\right )}{d}-\frac{\left (4 a b^3\right ) \operatorname{Subst}\left (\int \frac{x^6}{\left (-1+x^2\right )^2} \, dx,x,\csc (c+d x)\right )}{d}+\frac{b^4 \operatorname{Subst}\left (\int \frac{x^6}{\left (-1+x^2\right )^2} \, dx,x,\sec (c+d x)\right )}{d}\\ &=-\frac{5 a^4 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac{a^4 \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac{3 a^2 b^2 \csc ^4(c+d x) \sec (c+d x)}{2 d}+\frac{2 a b^3 \csc ^3(c+d x) \sec ^2(c+d x)}{d}-\frac{b^4 \csc ^2(c+d x) \sec ^3(c+d x)}{2 d}+\frac{1}{8} \left (5 a^4\right ) \int \csc ^3(c+d x) \, dx-\frac{\left (4 a^3 b\right ) \operatorname{Subst}\left (\int \left (1+x^2+x^4+\frac{1}{-1+x^2}\right ) \, dx,x,\csc (c+d x)\right )}{d}+\frac{\left (15 a^2 b^2\right ) \operatorname{Subst}\left (\int \frac{x^4}{\left (-1+x^2\right )^2} \, dx,x,\sec (c+d x)\right )}{2 d}-\frac{\left (10 a b^3\right ) \operatorname{Subst}\left (\int \frac{x^4}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}+\frac{\left (5 b^4\right ) \operatorname{Subst}\left (\int \frac{x^4}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{2 d}\\ &=-\frac{4 a^3 b \csc (c+d x)}{d}-\frac{5 a^4 \cot (c+d x) \csc (c+d x)}{16 d}-\frac{4 a^3 b \csc ^3(c+d x)}{3 d}-\frac{5 a^4 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac{4 a^3 b \csc ^5(c+d x)}{5 d}-\frac{a^4 \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac{15 a^2 b^2 \csc ^2(c+d x) \sec (c+d x)}{4 d}-\frac{3 a^2 b^2 \csc ^4(c+d x) \sec (c+d x)}{2 d}+\frac{2 a b^3 \csc ^3(c+d x) \sec ^2(c+d x)}{d}-\frac{b^4 \csc ^2(c+d x) \sec ^3(c+d x)}{2 d}+\frac{1}{16} \left (5 a^4\right ) \int \csc (c+d x) \, dx-\frac{\left (4 a^3 b\right ) \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}+\frac{\left (45 a^2 b^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{4 d}-\frac{\left (10 a b^3\right ) \operatorname{Subst}\left (\int \left (1+x^2+\frac{1}{-1+x^2}\right ) \, dx,x,\csc (c+d x)\right )}{d}+\frac{\left (5 b^4\right ) \operatorname{Subst}\left (\int \left (1+x^2+\frac{1}{-1+x^2}\right ) \, dx,x,\sec (c+d x)\right )}{2 d}\\ &=-\frac{5 a^4 \tanh ^{-1}(\cos (c+d x))}{16 d}+\frac{4 a^3 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{4 a^3 b \csc (c+d x)}{d}-\frac{10 a b^3 \csc (c+d x)}{d}-\frac{5 a^4 \cot (c+d x) \csc (c+d x)}{16 d}-\frac{4 a^3 b \csc ^3(c+d x)}{3 d}-\frac{10 a b^3 \csc ^3(c+d x)}{3 d}-\frac{5 a^4 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac{4 a^3 b \csc ^5(c+d x)}{5 d}-\frac{a^4 \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac{45 a^2 b^2 \sec (c+d x)}{4 d}+\frac{5 b^4 \sec (c+d x)}{2 d}-\frac{15 a^2 b^2 \csc ^2(c+d x) \sec (c+d x)}{4 d}-\frac{3 a^2 b^2 \csc ^4(c+d x) \sec (c+d x)}{2 d}+\frac{2 a b^3 \csc ^3(c+d x) \sec ^2(c+d x)}{d}+\frac{5 b^4 \sec ^3(c+d x)}{6 d}-\frac{b^4 \csc ^2(c+d x) \sec ^3(c+d x)}{2 d}+\frac{\left (45 a^2 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{4 d}-\frac{\left (10 a b^3\right ) \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}+\frac{\left (5 b^4\right ) \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{2 d}\\ &=-\frac{5 a^4 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{45 a^2 b^2 \tanh ^{-1}(\cos (c+d x))}{4 d}-\frac{5 b^4 \tanh ^{-1}(\cos (c+d x))}{2 d}+\frac{4 a^3 b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{10 a b^3 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{4 a^3 b \csc (c+d x)}{d}-\frac{10 a b^3 \csc (c+d x)}{d}-\frac{5 a^4 \cot (c+d x) \csc (c+d x)}{16 d}-\frac{4 a^3 b \csc ^3(c+d x)}{3 d}-\frac{10 a b^3 \csc ^3(c+d x)}{3 d}-\frac{5 a^4 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac{4 a^3 b \csc ^5(c+d x)}{5 d}-\frac{a^4 \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac{45 a^2 b^2 \sec (c+d x)}{4 d}+\frac{5 b^4 \sec (c+d x)}{2 d}-\frac{15 a^2 b^2 \csc ^2(c+d x) \sec (c+d x)}{4 d}-\frac{3 a^2 b^2 \csc ^4(c+d x) \sec (c+d x)}{2 d}+\frac{2 a b^3 \csc ^3(c+d x) \sec ^2(c+d x)}{d}+\frac{5 b^4 \sec ^3(c+d x)}{6 d}-\frac{b^4 \csc ^2(c+d x) \sec ^3(c+d x)}{2 d}\\ \end{align*}

Mathematica [A]  time = 6.25887, size = 660, normalized size = 1.64 \[ -\frac{5 \left (36 a^2 b^2+a^4+8 b^4\right ) \cos ^4(c+d x) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^4}{16 d (a \cos (c+d x)+b \sin (c+d x))^4}-\frac{2 \left (2 a^3 b+5 a b^3\right ) \cos ^4(c+d x) (a+b \tan (c+d x))^4 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}{d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{5 \left (36 a^2 b^2+a^4+8 b^4\right ) \cos ^4(c+d x) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^4}{16 d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{2 \left (2 a^3 b+5 a b^3\right ) \cos ^4(c+d x) (a+b \tan (c+d x))^4 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{\cot (c+d x) \csc ^5(c+d x) (a+b \tan (c+d x))^4 \left (-7200 a^2 b^2 \cos (2 (c+d x))+2160 a^2 b^2 \cos (4 (c+d x))+7200 a^2 b^2 \cos (6 (c+d x))-2700 a^2 b^2 \cos (8 (c+d x))+540 a^2 b^2-15744 a^3 b \sin (2 (c+d x))-1152 a^3 b \sin (4 (c+d x))+3200 a^3 b \sin (6 (c+d x))-960 a^3 b \sin (8 (c+d x))-2760 a^4 \cos (2 (c+d x))+60 a^4 \cos (4 (c+d x))+200 a^4 \cos (6 (c+d x))-75 a^4 \cos (8 (c+d x))-2545 a^4-8640 a b^3 \sin (2 (c+d x))-2880 a b^3 \sin (4 (c+d x))+8000 a b^3 \sin (6 (c+d x))-2400 a b^3 \sin (8 (c+d x))-6720 b^4 \cos (2 (c+d x))+480 b^4 \cos (4 (c+d x))+1600 b^4 \cos (6 (c+d x))-600 b^4 \cos (8 (c+d x))+5240 b^4\right )}{30720 d (a \cos (c+d x)+b \sin (c+d x))^4} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[c + d*x]^7*(a + b*Tan[c + d*x])^4,x]

[Out]

(-5*(a^4 + 36*a^2*b^2 + 8*b^4)*Cos[c + d*x]^4*Log[Cos[(c + d*x)/2]]*(a + b*Tan[c + d*x])^4)/(16*d*(a*Cos[c + d
*x] + b*Sin[c + d*x])^4) - (2*(2*a^3*b + 5*a*b^3)*Cos[c + d*x]^4*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(a +
b*Tan[c + d*x])^4)/(d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + (5*(a^4 + 36*a^2*b^2 + 8*b^4)*Cos[c + d*x]^4*Log
[Sin[(c + d*x)/2]]*(a + b*Tan[c + d*x])^4)/(16*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + (2*(2*a^3*b + 5*a*b^3)
*Cos[c + d*x]^4*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(a + b*Tan[c + d*x])^4)/(d*(a*Cos[c + d*x] + b*Sin[c
+ d*x])^4) + (Cot[c + d*x]*Csc[c + d*x]^5*(-2545*a^4 + 540*a^2*b^2 + 5240*b^4 - 2760*a^4*Cos[2*(c + d*x)] - 72
00*a^2*b^2*Cos[2*(c + d*x)] - 6720*b^4*Cos[2*(c + d*x)] + 60*a^4*Cos[4*(c + d*x)] + 2160*a^2*b^2*Cos[4*(c + d*
x)] + 480*b^4*Cos[4*(c + d*x)] + 200*a^4*Cos[6*(c + d*x)] + 7200*a^2*b^2*Cos[6*(c + d*x)] + 1600*b^4*Cos[6*(c
+ d*x)] - 75*a^4*Cos[8*(c + d*x)] - 2700*a^2*b^2*Cos[8*(c + d*x)] - 600*b^4*Cos[8*(c + d*x)] - 15744*a^3*b*Sin
[2*(c + d*x)] - 8640*a*b^3*Sin[2*(c + d*x)] - 1152*a^3*b*Sin[4*(c + d*x)] - 2880*a*b^3*Sin[4*(c + d*x)] + 3200
*a^3*b*Sin[6*(c + d*x)] + 8000*a*b^3*Sin[6*(c + d*x)] - 960*a^3*b*Sin[8*(c + d*x)] - 2400*a*b^3*Sin[8*(c + d*x
)])*(a + b*Tan[c + d*x])^4)/(30720*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4)

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Maple [A]  time = 0.127, size = 442, normalized size = 1.1 \begin{align*}{\frac{{b}^{4}}{3\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}-{\frac{5\,{b}^{4}}{6\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}\cos \left ( dx+c \right ) }}+{\frac{5\,{b}^{4}}{2\,d\cos \left ( dx+c \right ) }}+{\frac{5\,{b}^{4}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{2\,d}}-{\frac{4\,{b}^{3}a}{3\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{10\,{b}^{3}a}{3\,d\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}-10\,{\frac{{b}^{3}a}{d\sin \left ( dx+c \right ) }}+10\,{\frac{{b}^{3}a\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}-{\frac{3\,{a}^{2}{b}^{2}}{2\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}\cos \left ( dx+c \right ) }}-{\frac{15\,{a}^{2}{b}^{2}}{4\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}\cos \left ( dx+c \right ) }}+{\frac{45\,{a}^{2}{b}^{2}}{4\,d\cos \left ( dx+c \right ) }}+{\frac{45\,{a}^{2}{b}^{2}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{4\,d}}-{\frac{4\,b{a}^{3}}{5\,d \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{4\,b{a}^{3}}{3\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-4\,{\frac{b{a}^{3}}{d\sin \left ( dx+c \right ) }}+4\,{\frac{b{a}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}-{\frac{{a}^{4}\cot \left ( dx+c \right ) \left ( \csc \left ( dx+c \right ) \right ) ^{5}}{6\,d}}-{\frac{5\,{a}^{4}\cot \left ( dx+c \right ) \left ( \csc \left ( dx+c \right ) \right ) ^{3}}{24\,d}}-{\frac{5\,{a}^{4}\cot \left ( dx+c \right ) \csc \left ( dx+c \right ) }{16\,d}}+{\frac{5\,{a}^{4}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{16\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(d*x+c)^7*(a+b*tan(d*x+c))^4,x)

[Out]

1/3/d*b^4/sin(d*x+c)^2/cos(d*x+c)^3-5/6/d*b^4/sin(d*x+c)^2/cos(d*x+c)+5/2/d*b^4/cos(d*x+c)+5/2/d*b^4*ln(csc(d*
x+c)-cot(d*x+c))-4/3/d*b^3*a/sin(d*x+c)^3/cos(d*x+c)^2+10/3/d*b^3*a/sin(d*x+c)/cos(d*x+c)^2-10/d*b^3*a/sin(d*x
+c)+10/d*b^3*a*ln(sec(d*x+c)+tan(d*x+c))-3/2/d*a^2*b^2/sin(d*x+c)^4/cos(d*x+c)-15/4/d*a^2*b^2/sin(d*x+c)^2/cos
(d*x+c)+45/4/d*a^2*b^2/cos(d*x+c)+45/4/d*a^2*b^2*ln(csc(d*x+c)-cot(d*x+c))-4/5/d*b*a^3/sin(d*x+c)^5-4/3/d*b*a^
3/sin(d*x+c)^3-4/d*b*a^3/sin(d*x+c)+4/d*b*a^3*ln(sec(d*x+c)+tan(d*x+c))-1/6*a^4*cot(d*x+c)*csc(d*x+c)^5/d-5/24
*a^4*cot(d*x+c)*csc(d*x+c)^3/d-5/16*a^4*cot(d*x+c)*csc(d*x+c)/d+5/16/d*a^4*ln(csc(d*x+c)-cot(d*x+c))

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Maxima [A]  time = 1.15415, size = 522, normalized size = 1.3 \begin{align*} \frac{5 \, a^{4}{\left (\frac{2 \,{\left (15 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{3} + 33 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 40 \, b^{4}{\left (\frac{2 \,{\left (15 \, \cos \left (d x + c\right )^{4} - 10 \, \cos \left (d x + c\right )^{2} - 2\right )}}{\cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{3}} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 180 \, a^{2} b^{2}{\left (\frac{2 \,{\left (15 \, \cos \left (d x + c\right )^{4} - 25 \, \cos \left (d x + c\right )^{2} + 8\right )}}{\cos \left (d x + c\right )^{5} - 2 \, \cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 160 \, a b^{3}{\left (\frac{2 \,{\left (15 \, \sin \left (d x + c\right )^{4} - 10 \, \sin \left (d x + c\right )^{2} - 2\right )}}{\sin \left (d x + c\right )^{5} - \sin \left (d x + c\right )^{3}} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 64 \, a^{3} b{\left (\frac{2 \,{\left (15 \, \sin \left (d x + c\right )^{4} + 5 \, \sin \left (d x + c\right )^{2} + 3\right )}}{\sin \left (d x + c\right )^{5}} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{480 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^7*(a+b*tan(d*x+c))^4,x, algorithm="maxima")

[Out]

1/480*(5*a^4*(2*(15*cos(d*x + c)^5 - 40*cos(d*x + c)^3 + 33*cos(d*x + c))/(cos(d*x + c)^6 - 3*cos(d*x + c)^4 +
3*cos(d*x + c)^2 - 1) - 15*log(cos(d*x + c) + 1) + 15*log(cos(d*x + c) - 1)) + 40*b^4*(2*(15*cos(d*x + c)^4 -
10*cos(d*x + c)^2 - 2)/(cos(d*x + c)^5 - cos(d*x + c)^3) - 15*log(cos(d*x + c) + 1) + 15*log(cos(d*x + c) - 1
)) + 180*a^2*b^2*(2*(15*cos(d*x + c)^4 - 25*cos(d*x + c)^2 + 8)/(cos(d*x + c)^5 - 2*cos(d*x + c)^3 + cos(d*x +
c)) - 15*log(cos(d*x + c) + 1) + 15*log(cos(d*x + c) - 1)) - 160*a*b^3*(2*(15*sin(d*x + c)^4 - 10*sin(d*x + c
)^2 - 2)/(sin(d*x + c)^5 - sin(d*x + c)^3) - 15*log(sin(d*x + c) + 1) + 15*log(sin(d*x + c) - 1)) - 64*a^3*b*(
2*(15*sin(d*x + c)^4 + 5*sin(d*x + c)^2 + 3)/sin(d*x + c)^5 - 15*log(sin(d*x + c) + 1) + 15*log(sin(d*x + c) -
1)))/d

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Fricas [A]  time = 4.56755, size = 1661, normalized size = 4.13 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^7*(a+b*tan(d*x+c))^4,x, algorithm="fricas")

[Out]

1/480*(150*(a^4 + 36*a^2*b^2 + 8*b^4)*cos(d*x + c)^8 - 400*(a^4 + 36*a^2*b^2 + 8*b^4)*cos(d*x + c)^6 + 330*(a^
4 + 36*a^2*b^2 + 8*b^4)*cos(d*x + c)^4 - 160*b^4 - 480*(6*a^2*b^2 + b^4)*cos(d*x + c)^2 - 75*((a^4 + 36*a^2*b^
2 + 8*b^4)*cos(d*x + c)^9 - 3*(a^4 + 36*a^2*b^2 + 8*b^4)*cos(d*x + c)^7 + 3*(a^4 + 36*a^2*b^2 + 8*b^4)*cos(d*x
+ c)^5 - (a^4 + 36*a^2*b^2 + 8*b^4)*cos(d*x + c)^3)*log(1/2*cos(d*x + c) + 1/2) + 75*((a^4 + 36*a^2*b^2 + 8*b
^4)*cos(d*x + c)^9 - 3*(a^4 + 36*a^2*b^2 + 8*b^4)*cos(d*x + c)^7 + 3*(a^4 + 36*a^2*b^2 + 8*b^4)*cos(d*x + c)^5
- (a^4 + 36*a^2*b^2 + 8*b^4)*cos(d*x + c)^3)*log(-1/2*cos(d*x + c) + 1/2) + 480*((2*a^3*b + 5*a*b^3)*cos(d*x
+ c)^9 - 3*(2*a^3*b + 5*a*b^3)*cos(d*x + c)^7 + 3*(2*a^3*b + 5*a*b^3)*cos(d*x + c)^5 - (2*a^3*b + 5*a*b^3)*cos
(d*x + c)^3)*log(sin(d*x + c) + 1) - 480*((2*a^3*b + 5*a*b^3)*cos(d*x + c)^9 - 3*(2*a^3*b + 5*a*b^3)*cos(d*x +
c)^7 + 3*(2*a^3*b + 5*a*b^3)*cos(d*x + c)^5 - (2*a^3*b + 5*a*b^3)*cos(d*x + c)^3)*log(-sin(d*x + c) + 1) + 64
*(15*(2*a^3*b + 5*a*b^3)*cos(d*x + c)^7 - 35*(2*a^3*b + 5*a*b^3)*cos(d*x + c)^5 - 15*a*b^3*cos(d*x + c) + 23*(
2*a^3*b + 5*a*b^3)*cos(d*x + c)^3)*sin(d*x + c))/(d*cos(d*x + c)^9 - 3*d*cos(d*x + c)^7 + 3*d*cos(d*x + c)^5 -
d*cos(d*x + c)^3)

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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(csc(d*x+c)**7*(a+b*tan(d*x+c))**4,x)

[Out]

Timed out

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Giac [A]  time = 2.80009, size = 873, normalized size = 2.17 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^7*(a+b*tan(d*x+c))^4,x, algorithm="giac")

[Out]

1/1920*(5*a^4*tan(1/2*d*x + 1/2*c)^6 - 48*a^3*b*tan(1/2*d*x + 1/2*c)^5 + 45*a^4*tan(1/2*d*x + 1/2*c)^4 + 180*a
^2*b^2*tan(1/2*d*x + 1/2*c)^4 - 560*a^3*b*tan(1/2*d*x + 1/2*c)^3 - 320*a*b^3*tan(1/2*d*x + 1/2*c)^3 + 225*a^4*
tan(1/2*d*x + 1/2*c)^2 + 2880*a^2*b^2*tan(1/2*d*x + 1/2*c)^2 + 240*b^4*tan(1/2*d*x + 1/2*c)^2 - 5280*a^3*b*tan
(1/2*d*x + 1/2*c) - 8640*a*b^3*tan(1/2*d*x + 1/2*c) + 3840*(2*a^3*b + 5*a*b^3)*log(abs(tan(1/2*d*x + 1/2*c) +
1)) - 3840*(2*a^3*b + 5*a*b^3)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + 600*(a^4 + 36*a^2*b^2 + 8*b^4)*log(abs(tan
(1/2*d*x + 1/2*c))) + 1280*(6*a*b^3*tan(1/2*d*x + 1/2*c)^5 - 18*a^2*b^2*tan(1/2*d*x + 1/2*c)^4 - 9*b^4*tan(1/2
*d*x + 1/2*c)^4 + 36*a^2*b^2*tan(1/2*d*x + 1/2*c)^2 + 12*b^4*tan(1/2*d*x + 1/2*c)^2 - 6*a*b^3*tan(1/2*d*x + 1/
2*c) - 18*a^2*b^2 - 7*b^4)/(tan(1/2*d*x + 1/2*c)^2 - 1)^3 - (1470*a^4*tan(1/2*d*x + 1/2*c)^6 + 52920*a^2*b^2*t
an(1/2*d*x + 1/2*c)^6 + 11760*b^4*tan(1/2*d*x + 1/2*c)^6 + 5280*a^3*b*tan(1/2*d*x + 1/2*c)^5 + 8640*a*b^3*tan(
1/2*d*x + 1/2*c)^5 + 225*a^4*tan(1/2*d*x + 1/2*c)^4 + 2880*a^2*b^2*tan(1/2*d*x + 1/2*c)^4 + 240*b^4*tan(1/2*d*
x + 1/2*c)^4 + 560*a^3*b*tan(1/2*d*x + 1/2*c)^3 + 320*a*b^3*tan(1/2*d*x + 1/2*c)^3 + 45*a^4*tan(1/2*d*x + 1/2*
c)^2 + 180*a^2*b^2*tan(1/2*d*x + 1/2*c)^2 + 48*a^3*b*tan(1/2*d*x + 1/2*c) + 5*a^4)/tan(1/2*d*x + 1/2*c)^6)/d