\(\int \csc ^7(c+d x) (a+b \tan (c+d x))^4 \, dx\) [50]

Optimal result

Integrand size = 21, antiderivative size = 402 \[ \int \csc ^7(c+d x) (a+b \tan (c+d x))^4 \, dx=-\frac {5 a^4 \text {arctanh}(\cos (c+d x))}{16 d}-\frac {45 a^2 b^2 \text {arctanh}(\cos (c+d x))}{4 d}-\frac {5 b^4 \text {arctanh}(\cos (c+d x))}{2 d}+\frac {4 a^3 b \text {arctanh}(\sin (c+d x))}{d}+\frac {10 a b^3 \text {arctanh}(\sin (c+d x))}{d}-\frac {4 a^3 b \csc (c+d x)}{d}-\frac {10 a b^3 \csc (c+d x)}{d}-\frac {5 a^4 \cot (c+d x) \csc (c+d x)}{16 d}-\frac {4 a^3 b \csc ^3(c+d x)}{3 d}-\frac {10 a b^3 \csc ^3(c+d x)}{3 d}-\frac {5 a^4 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {4 a^3 b \csc ^5(c+d x)}{5 d}-\frac {a^4 \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac {45 a^2 b^2 \sec (c+d x)}{4 d}+\frac {5 b^4 \sec (c+d x)}{2 d}-\frac {15 a^2 b^2 \csc ^2(c+d x) \sec (c+d x)}{4 d}-\frac {3 a^2 b^2 \csc ^4(c+d x) \sec (c+d x)}{2 d}+\frac {2 a b^3 \csc ^3(c+d x) \sec ^2(c+d x)}{d}+\frac {5 b^4 \sec ^3(c+d x)}{6 d}-\frac {b^4 \csc ^2(c+d x) \sec ^3(c+d x)}{2 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3598, 3853, 3855, 2701, 308, 213, 2702, 294, 327} \[ \int \csc ^7(c+d x) (a+b \tan (c+d x))^4 \, dx=-\frac {5 a^4 \text {arctanh}(\cos (c+d x))}{16 d}-\frac {a^4 \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac {5 a^4 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {5 a^4 \cot (c+d x) \csc (c+d x)}{16 d}+\frac {4 a^3 b \text {arctanh}(\sin (c+d x))}{d}-\frac {4 a^3 b \csc ^5(c+d x)}{5 d}-\frac {4 a^3 b \csc ^3(c+d x)}{3 d}-\frac {4 a^3 b \csc (c+d x)}{d}-\frac {45 a^2 b^2 \text {arctanh}(\cos (c+d x))}{4 d}+\frac {45 a^2 b^2 \sec (c+d x)}{4 d}-\frac {3 a^2 b^2 \csc ^4(c+d x) \sec (c+d x)}{2 d}-\frac {15 a^2 b^2 \csc ^2(c+d x) \sec (c+d x)}{4 d}+\frac {10 a b^3 \text {arctanh}(\sin (c+d x))}{d}-\frac {10 a b^3 \csc ^3(c+d x)}{3 d}-\frac {10 a b^3 \csc (c+d x)}{d}+\frac {2 a b^3 \csc ^3(c+d x) \sec ^2(c+d x)}{d}-\frac {5 b^4 \text {arctanh}(\cos (c+d x))}{2 d}+\frac {5 b^4 \sec ^3(c+d x)}{6 d}+\frac {5 b^4 \sec (c+d x)}{2 d}-\frac {b^4 \csc ^2(c+d x) \sec ^3(c+d x)}{2 d} \]

[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 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 294

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 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 327

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]

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 2702

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 3598

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 3853

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 3855

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

Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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 ) \text {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 ) \text {Subst}\left (\int \frac {x^6}{\left (-1+x^2\right )^2} \, dx,x,\csc (c+d x)\right )}{d}+\frac {b^4 \text {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 ) \text {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 ) \text {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 ) \text {Subst}\left (\int \frac {x^4}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}+\frac {\left (5 b^4\right ) \text {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 ) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}+\frac {\left (45 a^2 b^2\right ) \text {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 ) \text {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 ) \text {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 \text {arctanh}(\cos (c+d x))}{16 d}+\frac {4 a^3 b \text {arctanh}(\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 ) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{4 d}-\frac {\left (10 a b^3\right ) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}+\frac {\left (5 b^4\right ) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{2 d} \\ & = -\frac {5 a^4 \text {arctanh}(\cos (c+d x))}{16 d}-\frac {45 a^2 b^2 \text {arctanh}(\cos (c+d x))}{4 d}-\frac {5 b^4 \text {arctanh}(\cos (c+d x))}{2 d}+\frac {4 a^3 b \text {arctanh}(\sin (c+d x))}{d}+\frac {10 a b^3 \text {arctanh}(\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] (verified)

Time = 8.99 (sec) , antiderivative size = 660, normalized size of antiderivative = 1.64 \[ \int \csc ^7(c+d x) (a+b \tan (c+d x))^4 \, dx=-\frac {5 \left (a^4+36 a^2 b^2+8 b^4\right ) \cos ^4(c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^4}{16 d (a \cos (c+d x)+b \sin (c+d x))^4}-\frac {2 \left (2 a^3 b+5 a b^3\right ) \cos ^4(c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^4}{d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac {5 \left (a^4+36 a^2 b^2+8 b^4\right ) \cos ^4(c+d x) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^4}{16 d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac {2 \left (2 a^3 b+5 a b^3\right ) \cos ^4(c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^4}{d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac {\cot (c+d x) \csc ^5(c+d x) \left (-2545 a^4+540 a^2 b^2+5240 b^4-2760 a^4 \cos (2 (c+d x))-7200 a^2 b^2 \cos (2 (c+d x))-6720 b^4 \cos (2 (c+d x))+60 a^4 \cos (4 (c+d x))+2160 a^2 b^2 \cos (4 (c+d x))+480 b^4 \cos (4 (c+d x))+200 a^4 \cos (6 (c+d x))+7200 a^2 b^2 \cos (6 (c+d x))+1600 b^4 \cos (6 (c+d x))-75 a^4 \cos (8 (c+d x))-2700 a^2 b^2 \cos (8 (c+d x))-600 b^4 \cos (8 (c+d x))-15744 a^3 b \sin (2 (c+d x))-8640 a b^3 \sin (2 (c+d x))-1152 a^3 b \sin (4 (c+d x))-2880 a b^3 \sin (4 (c+d x))+3200 a^3 b \sin (6 (c+d x))+8000 a b^3 \sin (6 (c+d x))-960 a^3 b \sin (8 (c+d x))-2400 a b^3 \sin (8 (c+d x))\right ) (a+b \tan (c+d x))^4}{30720 d (a \cos (c+d x)+b \sin (c+d x))^4} \]

[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)

Maple [A] (verified)

Time = 50.86 (sec) , antiderivative size = 327, normalized size of antiderivative = 0.81

method	result	size
derivativedivides	\(\frac {b^{4} \left (\frac {1}{3 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{3}}-\frac {5}{6 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {5}{2 \cos \left (d x +c \right )}+\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+4 a \,b^{3} \left (-\frac {1}{3 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{2}}+\frac {5}{6 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {5}{2 \sin \left (d x +c \right )}+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 a^{2} b^{2} \left (-\frac {1}{4 \sin \left (d x +c \right )^{4} \cos \left (d x +c \right )}-\frac {5}{8 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {15}{8 \cos \left (d x +c \right )}+\frac {15 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )+4 a^{3} b \left (-\frac {1}{5 \sin \left (d x +c \right )^{5}}-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{\sin \left (d x +c \right )}+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{4} \left (\left (-\frac {\left (\csc ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\csc ^{3}\left (d x +c \right )\right )}{24}-\frac {5 \csc \left (d x +c \right )}{16}\right ) \cot \left (d x +c \right )+\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )}{d}\)	\(327\)
default	\(\frac {b^{4} \left (\frac {1}{3 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{3}}-\frac {5}{6 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {5}{2 \cos \left (d x +c \right )}+\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+4 a \,b^{3} \left (-\frac {1}{3 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{2}}+\frac {5}{6 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {5}{2 \sin \left (d x +c \right )}+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 a^{2} b^{2} \left (-\frac {1}{4 \sin \left (d x +c \right )^{4} \cos \left (d x +c \right )}-\frac {5}{8 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {15}{8 \cos \left (d x +c \right )}+\frac {15 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )+4 a^{3} b \left (-\frac {1}{5 \sin \left (d x +c \right )^{5}}-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{\sin \left (d x +c \right )}+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{4} \left (\left (-\frac {\left (\csc ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\csc ^{3}\left (d x +c \right )\right )}{24}-\frac {5 \csc \left (d x +c \right )}{16}\right ) \cot \left (d x +c \right )+\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )}{d}\)	\(327\)
risch	\(\text {Expression too large to display}\)	\(953\)

[In]

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

[Out]

1/d*(b^4*(1/3/sin(d*x+c)^2/cos(d*x+c)^3-5/6/sin(d*x+c)^2/cos(d*x+c)+5/2/cos(d*x+c)+5/2*ln(csc(d*x+c)-cot(d*x+c
)))+4*a*b^3*(-1/3/sin(d*x+c)^3/cos(d*x+c)^2+5/6/sin(d*x+c)/cos(d*x+c)^2-5/2/sin(d*x+c)+5/2*ln(sec(d*x+c)+tan(d
*x+c)))+6*a^2*b^2*(-1/4/sin(d*x+c)^4/cos(d*x+c)-5/8/sin(d*x+c)^2/cos(d*x+c)+15/8/cos(d*x+c)+15/8*ln(csc(d*x+c)
-cot(d*x+c)))+4*a^3*b*(-1/5/sin(d*x+c)^5-1/3/sin(d*x+c)^3-1/sin(d*x+c)+ln(sec(d*x+c)+tan(d*x+c)))+a^4*((-1/6*c
sc(d*x+c)^5-5/24*csc(d*x+c)^3-5/16*csc(d*x+c))*cot(d*x+c)+5/16*ln(csc(d*x+c)-cot(d*x+c))))

Fricas [A] (verification not implemented)

none

Time = 0.42 (sec) , antiderivative size = 697, normalized size of antiderivative = 1.73 \[ \int \csc ^7(c+d x) (a+b \tan (c+d x))^4 \, dx=\frac {150 \, {\left (a^{4} + 36 \, a^{2} b^{2} + 8 \, b^{4}\right )} \cos \left (d x + c\right )^{8} - 400 \, {\left (a^{4} + 36 \, a^{2} b^{2} + 8 \, b^{4}\right )} \cos \left (d x + c\right )^{6} + 330 \, {\left (a^{4} + 36 \, a^{2} b^{2} + 8 \, b^{4}\right )} \cos \left (d x + c\right )^{4} - 160 \, b^{4} - 480 \, {\left (6 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{2} - 75 \, {\left ({\left (a^{4} + 36 \, a^{2} b^{2} + 8 \, b^{4}\right )} \cos \left (d x + c\right )^{9} - 3 \, {\left (a^{4} + 36 \, a^{2} b^{2} + 8 \, b^{4}\right )} \cos \left (d x + c\right )^{7} + 3 \, {\left (a^{4} + 36 \, a^{2} b^{2} + 8 \, b^{4}\right )} \cos \left (d x + c\right )^{5} - {\left (a^{4} + 36 \, a^{2} b^{2} + 8 \, b^{4}\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 75 \, {\left ({\left (a^{4} + 36 \, a^{2} b^{2} + 8 \, b^{4}\right )} \cos \left (d x + c\right )^{9} - 3 \, {\left (a^{4} + 36 \, a^{2} b^{2} + 8 \, b^{4}\right )} \cos \left (d x + c\right )^{7} + 3 \, {\left (a^{4} + 36 \, a^{2} b^{2} + 8 \, b^{4}\right )} \cos \left (d x + c\right )^{5} - {\left (a^{4} + 36 \, a^{2} b^{2} + 8 \, b^{4}\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 480 \, {\left ({\left (2 \, a^{3} b + 5 \, a b^{3}\right )} \cos \left (d x + c\right )^{9} - 3 \, {\left (2 \, a^{3} b + 5 \, a b^{3}\right )} \cos \left (d x + c\right )^{7} + 3 \, {\left (2 \, a^{3} b + 5 \, a b^{3}\right )} \cos \left (d x + c\right )^{5} - {\left (2 \, a^{3} b + 5 \, a b^{3}\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 480 \, {\left ({\left (2 \, a^{3} b + 5 \, a b^{3}\right )} \cos \left (d x + c\right )^{9} - 3 \, {\left (2 \, a^{3} b + 5 \, a b^{3}\right )} \cos \left (d x + c\right )^{7} + 3 \, {\left (2 \, a^{3} b + 5 \, a b^{3}\right )} \cos \left (d x + c\right )^{5} - {\left (2 \, a^{3} b + 5 \, a b^{3}\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 64 \, {\left (15 \, {\left (2 \, a^{3} b + 5 \, a b^{3}\right )} \cos \left (d x + c\right )^{7} - 35 \, {\left (2 \, a^{3} b + 5 \, a b^{3}\right )} \cos \left (d x + c\right )^{5} - 15 \, a b^{3} \cos \left (d x + c\right ) + 23 \, {\left (2 \, a^{3} b + 5 \, a b^{3}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{480 \, {\left (d \cos \left (d x + c\right )^{9} - 3 \, d \cos \left (d x + c\right )^{7} + 3 \, d \cos \left (d x + c\right )^{5} - d \cos \left (d x + c\right )^{3}\right )}} \]

[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)

Sympy [F(-1)]

Timed out. \[ \int \csc ^7(c+d x) (a+b \tan (c+d x))^4 \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 387, normalized size of antiderivative = 0.96 \[ \int \csc ^7(c+d x) (a+b \tan (c+d x))^4 \, dx=\frac {5 \, a^{4} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{3} + 33 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 40 \, b^{4} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{4} - 10 \, \cos \left (d x + c\right )^{2} - 2\right )}}{\cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{3}} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 180 \, a^{2} b^{2} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{4} - 25 \, \cos \left (d x + c\right )^{2} + 8\right )}}{\cos \left (d x + c\right )^{5} - 2 \, \cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 160 \, a b^{3} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{4} - 10 \, \sin \left (d x + c\right )^{2} - 2\right )}}{\sin \left (d x + c\right )^{5} - \sin \left (d x + c\right )^{3}} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 64 \, a^{3} b {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{4} + 5 \, \sin \left (d x + c\right )^{2} + 3\right )}}{\sin \left (d x + c\right )^{5}} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{480 \, d} \]

[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

Giac [A] (verification not implemented)

none

Time = 1.15 (sec) , antiderivative size = 647, normalized size of antiderivative = 1.61 \[ \int \csc ^7(c+d x) (a+b \tan (c+d x))^4 \, dx=\frac {5 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 48 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 45 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 180 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 560 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 320 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 225 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2880 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 240 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 5280 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 8640 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3840 \, {\left (2 \, a^{3} b + 5 \, a b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3840 \, {\left (2 \, a^{3} b + 5 \, a b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 600 \, {\left (a^{4} + 36 \, a^{2} b^{2} + 8 \, b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + \frac {1280 \, {\left (6 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 18 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 9 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 36 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 12 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 6 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 18 \, a^{2} b^{2} - 7 \, b^{4}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}} - \frac {1470 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 52920 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 11760 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 5280 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 8640 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 225 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 2880 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 240 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 560 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 320 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 45 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 180 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 48 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, a^{4}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}}}{1920 \, d} \]

[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

Mupad [B] (verification not implemented)

Time = 4.70 (sec) , antiderivative size = 990, normalized size of antiderivative = 2.46 \[ \int \csc ^7(c+d x) (a+b \tan (c+d x))^4 \, dx=\text {Too large to display} \]

[In]

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

[Out]

(a^4*tan(c/2 + (d*x)/2)^6)/(384*d) - (atan(-((10*a*b^3 + 4*a^3*b)*(20*a*b^3 + 8*a^3*b - 6*tan(c/2 + (d*x)/2)*(
10*a*b^3 + 4*a^3*b) - tan(c/2 + (d*x)/2)*((5*a^4)/8 + 5*b^4 + (45*a^2*b^2)/2))*1i + (10*a*b^3 + 4*a^3*b)*(20*a
*b^3 + 8*a^3*b + 6*tan(c/2 + (d*x)/2)*(10*a*b^3 + 4*a^3*b) - tan(c/2 + (d*x)/2)*((5*a^4)/8 + 5*b^4 + (45*a^2*b
^2)/2))*1i)/(2*tan(c/2 + (d*x)/2)*(400*a^2*b^6 + 320*a^4*b^4 + 64*a^6*b^2) + (10*a*b^3 + 4*a^3*b)*(20*a*b^3 +
8*a^3*b - 6*tan(c/2 + (d*x)/2)*(10*a*b^3 + 4*a^3*b) - tan(c/2 + (d*x)/2)*((5*a^4)/8 + 5*b^4 + (45*a^2*b^2)/2))
- (10*a*b^3 + 4*a^3*b)*(20*a*b^3 + 8*a^3*b + 6*tan(c/2 + (d*x)/2)*(10*a*b^3 + 4*a^3*b) - tan(c/2 + (d*x)/2)*(
(5*a^4)/8 + 5*b^4 + (45*a^2*b^2)/2)) + 100*a*b^7 + 5*a^7*b + 490*a^3*b^5 + (385*a^5*b^3)/2))*(a*b^3*20i + a^3*
b*8i))/d + (tan(c/2 + (d*x)/2)^4*((a^2*(a^2 + 12*b^2))/128 + a^4/64))/d + (tan(c/2 + (d*x)/2)^2*((a^2*(a^2 + 1
2*b^2))/16 + (7*a^4)/128 + b^4/8 + (3*a^2*b^2)/4))/d - (tan(c/2 + (d*x)/2)*((9*a*b^3)/2 + (11*a^3*b)/4))/d - (
tan(c/2 + (d*x)/2)^4*((7*a^4)/2 + 8*b^4 + 78*a^2*b^2) - tan(c/2 + (d*x)/2)^10*((15*a^4)/2 + 392*b^4 + 864*a^2*
b^2) - tan(c/2 + (d*x)/2)^6*((109*a^4)/6 + (968*b^4)/3 + 1038*a^2*b^2) + tan(c/2 + (d*x)/2)^8*(21*a^4 + 536*b^
4 + 1818*a^2*b^2) + tan(c/2 + (d*x)/2)^2*(a^4 + 6*a^2*b^2) + a^4/6 - tan(c/2 + (d*x)/2)^11*(32*a*b^3 + 176*a^3
*b) + tan(c/2 + (d*x)/2)^3*((32*a*b^3)/3 + (208*a^3*b)/15) + tan(c/2 + (d*x)/2)^5*(256*a*b^3 + (624*a^3*b)/5)
- tan(c/2 + (d*x)/2)^7*(1088*a*b^3 + (2368*a^3*b)/5) + tan(c/2 + (d*x)/2)^9*((2560*a*b^3)/3 + (1528*a^3*b)/3)
+ (8*a^3*b*tan(c/2 + (d*x)/2))/5)/(d*(64*tan(c/2 + (d*x)/2)^6 - 192*tan(c/2 + (d*x)/2)^8 + 192*tan(c/2 + (d*x)
/2)^10 - 64*tan(c/2 + (d*x)/2)^12)) - (tan(c/2 + (d*x)/2)^3*((a*b^3)/6 + (7*a^3*b)/24))/d + (log(tan(c/2 + (d*
x)/2))*((5*a^4)/16 + (5*b^4)/2 + (45*a^2*b^2)/4))/d - (a^3*b*tan(c/2 + (d*x)/2)^5)/(40*d)