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

Optimal result

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

-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*arctanh(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)*c 
sc(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*co 
t(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*csc(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
 

Mathematica [A] (warning: unable to verify)

Time = 7.81 (sec) , antiderivative size = 660, normalized size of antiderivative = 1.64 \[ \int \csc ^7(c+d x) (a+b \tan (c+d x))^4 \, dx=-\frac {5 \left (a^4+36 a^2 b^2+8 b^4\right ) \cos ^4(c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^4}{16 d (a \cos (c+d x)+b \sin (c+d x))^4}-\frac {2 \left (2 a^3 b+5 a b^3\right ) \cos ^4(c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^4}{d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac {5 \left (a^4+36 a^2 b^2+8 b^4\right ) \cos ^4(c+d x) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^4}{16 d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac {2 \left (2 a^3 b+5 a b^3\right ) \cos ^4(c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^4}{d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac {\cot (c+d x) \csc ^5(c+d x) \left (-2545 a^4+540 a^2 b^2+5240 b^4-2760 a^4 \cos (2 (c+d x))-7200 a^2 b^2 \cos (2 (c+d x))-6720 b^4 \cos (2 (c+d x))+60 a^4 \cos (4 (c+d x))+2160 a^2 b^2 \cos (4 (c+d x))+480 b^4 \cos (4 (c+d x))+200 a^4 \cos (6 (c+d x))+7200 a^2 b^2 \cos (6 (c+d x))+1600 b^4 \cos (6 (c+d x))-75 a^4 \cos (8 (c+d x))-2700 a^2 b^2 \cos (8 (c+d x))-600 b^4 \cos (8 (c+d x))-15744 a^3 b \sin (2 (c+d x))-8640 a b^3 \sin (2 (c+d x))-1152 a^3 b \sin (4 (c+d x))-2880 a b^3 \sin (4 (c+d x))+3200 a^3 b \sin (6 (c+d x))+8000 a b^3 \sin (6 (c+d x))-960 a^3 b \sin (8 (c+d x))-2400 a b^3 \sin (8 (c+d x))\right ) (a+b \tan (c+d x))^4}{30720 d (a \cos (c+d x)+b \sin (c+d x))^4} \] Input:

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

Output:

(-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)] - 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)])*(a + b*Tan[c + d*x])^4)/(30720*d*(a*Cos[c + 
d*x] + b*Sin[c + d*x])^4)
 

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3042, 4000, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a+b \tan (c+d x))^4}{\sin (c+d x)^7}dx\)

\(\Big \downarrow \) 4000

\(\displaystyle \int \left (a^4 \csc ^7(c+d x)+4 a^3 b \csc ^6(c+d x) \sec (c+d x)+6 a^2 b^2 \csc ^5(c+d x) \sec ^2(c+d x)+4 a b^3 \csc ^4(c+d x) \sec ^3(c+d x)+b^4 \csc ^3(c+d x) \sec ^4(c+d x)\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {5 a^4 \text {arctanh}(\cos (c+d x))}{16 d}-\frac {a^4 \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac {5 a^4 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {5 a^4 \cot (c+d x) \csc (c+d x)}{16 d}+\frac {4 a^3 b \text {arctanh}(\sin (c+d x))}{d}-\frac {4 a^3 b \csc ^5(c+d x)}{5 d}-\frac {4 a^3 b \csc ^3(c+d x)}{3 d}-\frac {4 a^3 b \csc (c+d x)}{d}-\frac {45 a^2 b^2 \text {arctanh}(\cos (c+d x))}{4 d}+\frac {45 a^2 b^2 \sec (c+d x)}{4 d}-\frac {3 a^2 b^2 \csc ^4(c+d x) \sec (c+d x)}{2 d}-\frac {15 a^2 b^2 \csc ^2(c+d x) \sec (c+d x)}{4 d}+\frac {10 a b^3 \text {arctanh}(\sin (c+d x))}{d}-\frac {10 a b^3 \csc ^3(c+d x)}{3 d}-\frac {10 a b^3 \csc (c+d x)}{d}+\frac {2 a b^3 \csc ^3(c+d x) \sec ^2(c+d x)}{d}-\frac {5 b^4 \text {arctanh}(\cos (c+d x))}{2 d}+\frac {5 b^4 \sec ^3(c+d x)}{6 d}+\frac {5 b^4 \sec (c+d x)}{2 d}-\frac {b^4 \csc ^2(c+d x) \sec ^3(c+d x)}{2 d}\)

Input:

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

Output:

(-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*Co 
t[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)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4000

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]
 

Maple [A] (verified)

Time = 80.34 (sec) , antiderivative size = 327, normalized size of antiderivative = 0.81

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

Input:

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

Output:

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/co 
s(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*b^2*a^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*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(csc(d*x+c)-c 
ot(d*x+c))))
 

Fricas [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 697, normalized size of antiderivative = 1.73 \[ \int \csc ^7(c+d x) (a+b \tan (c+d x))^4 \, dx =\text {Too large to display} \] Input:

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

Output:

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

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 387, normalized size of antiderivative = 0.96 \[ \int \csc ^7(c+d x) (a+b \tan (c+d x))^4 \, dx=\frac {5 \, a^{4} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{3} + 33 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 40 \, b^{4} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{4} - 10 \, \cos \left (d x + c\right )^{2} - 2\right )}}{\cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{3}} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 180 \, a^{2} b^{2} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{4} - 25 \, \cos \left (d x + c\right )^{2} + 8\right )}}{\cos \left (d x + c\right )^{5} - 2 \, \cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 160 \, a b^{3} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{4} - 10 \, \sin \left (d x + c\right )^{2} - 2\right )}}{\sin \left (d x + c\right )^{5} - \sin \left (d x + c\right )^{3}} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 64 \, a^{3} b {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{4} + 5 \, \sin \left (d x + c\right )^{2} + 3\right )}}{\sin \left (d x + c\right )^{5}} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{480 \, d} \] Input:

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

Output:

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 - 1 
0*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)

Time = 0.43 (sec) , antiderivative size = 647, normalized size of antiderivative = 1.61 \[ \int \csc ^7(c+d x) (a+b \tan (c+d x))^4 \, dx =\text {Too large to display} \] Input:

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

Output:

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 + 4 
5*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*ta 
n(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*tan(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 = 1.24 (sec) , antiderivative size = 990, normalized size of antiderivative = 2.46 \[ \int \csc ^7(c+d x) (a+b \tan (c+d x))^4 \, dx=\text {Too large to display} \] Input:

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

Output:

(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 + 12*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*...
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 888, normalized size of antiderivative = 2.21 \[ \int \csc ^7(c+d x) (a+b \tan (c+d x))^4 \, dx =\text {Too large to display} \] Input:

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

Output:

( - 7680*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**8*a**3*b - 1 
9200*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**8*a*b**3 + 7680* 
cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**6*a**3*b + 19200*cos( 
c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**6*a*b**3 + 7680*cos(c + d 
*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**8*a**3*b + 19200*cos(c + d*x)* 
log(tan((c + d*x)/2) + 1)*sin(c + d*x)**8*a*b**3 - 7680*cos(c + d*x)*log(t 
an((c + d*x)/2) + 1)*sin(c + d*x)**6*a**3*b - 19200*cos(c + d*x)*log(tan(( 
c + d*x)/2) + 1)*sin(c + d*x)**6*a*b**3 + 600*cos(c + d*x)*log(tan((c + d* 
x)/2))*sin(c + d*x)**8*a**4 + 21600*cos(c + d*x)*log(tan((c + d*x)/2))*sin 
(c + d*x)**8*a**2*b**2 + 4800*cos(c + d*x)*log(tan((c + d*x)/2))*sin(c + d 
*x)**8*b**4 - 600*cos(c + d*x)*log(tan((c + d*x)/2))*sin(c + d*x)**6*a**4 
- 21600*cos(c + d*x)*log(tan((c + d*x)/2))*sin(c + d*x)**6*a**2*b**2 - 480 
0*cos(c + d*x)*log(tan((c + d*x)/2))*sin(c + d*x)**6*b**4 - 545*cos(c + d* 
x)*sin(c + d*x)**8*a**4 - 19620*cos(c + d*x)*sin(c + d*x)**8*a**2*b**2 - 5 
200*cos(c + d*x)*sin(c + d*x)**8*b**4 - 7680*cos(c + d*x)*sin(c + d*x)**7* 
a**3*b - 19200*cos(c + d*x)*sin(c + d*x)**7*a*b**3 + 545*cos(c + d*x)*sin( 
c + d*x)**6*a**4 + 19620*cos(c + d*x)*sin(c + d*x)**6*a**2*b**2 + 5200*cos 
(c + d*x)*sin(c + d*x)**6*b**4 + 5120*cos(c + d*x)*sin(c + d*x)**5*a**3*b 
+ 12800*cos(c + d*x)*sin(c + d*x)**5*a*b**3 + 1024*cos(c + d*x)*sin(c + d* 
x)**3*a**3*b + 2560*cos(c + d*x)*sin(c + d*x)**3*a*b**3 + 1536*cos(c + ...