\(\int \csc ^5(c+d x) (a+b \tan (c+d x))^4 \, dx\) [48]

Optimal result

Integrand size = 21, antiderivative size = 274 \[ \int \csc ^5(c+d x) (a+b \tan (c+d x))^4 \, dx=-\frac {3 a^4 \text {arctanh}(\cos (c+d x))}{8 d}-\frac {9 a^2 b^2 \text {arctanh}(\cos (c+d x))}{d}-\frac {b^4 \text {arctanh}(\cos (c+d x))}{d}+\frac {4 a^3 b \text {arctanh}(\sin (c+d x))}{d}+\frac {6 a b^3 \text {arctanh}(\sin (c+d x))}{d}-\frac {4 a^3 b \csc (c+d x)}{d}-\frac {6 a b^3 \csc (c+d x)}{d}-\frac {3 a^4 \cot (c+d x) \csc (c+d x)}{8 d}-\frac {4 a^3 b \csc ^3(c+d x)}{3 d}-\frac {a^4 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {9 a^2 b^2 \sec (c+d x)}{d}+\frac {b^4 \sec (c+d x)}{d}-\frac {3 a^2 b^2 \csc ^2(c+d x) \sec (c+d x)}{d}+\frac {2 a b^3 \csc (c+d x) \sec ^2(c+d x)}{d}+\frac {b^4 \sec ^3(c+d x)}{3 d} \] Output:

-3/8*a^4*arctanh(cos(d*x+c))/d-9*a^2*b^2*arctanh(cos(d*x+c))/d-b^4*arctanh 
(cos(d*x+c))/d+4*a^3*b*arctanh(sin(d*x+c))/d+6*a*b^3*arctanh(sin(d*x+c))/d 
-4*a^3*b*csc(d*x+c)/d-6*a*b^3*csc(d*x+c)/d-3/8*a^4*cot(d*x+c)*csc(d*x+c)/d 
-4/3*a^3*b*csc(d*x+c)^3/d-1/4*a^4*cot(d*x+c)*csc(d*x+c)^3/d+9*a^2*b^2*sec( 
d*x+c)/d+b^4*sec(d*x+c)/d-3*a^2*b^2*csc(d*x+c)^2*sec(d*x+c)/d+2*a*b^3*csc( 
d*x+c)*sec(d*x+c)^2/d+1/3*b^4*sec(d*x+c)^3/d
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1491\) vs. \(2(274)=548\).

Time = 7.20 (sec) , antiderivative size = 1491, normalized size of antiderivative = 5.44 \[ \int \csc ^5(c+d x) (a+b \tan (c+d x))^4 \, dx =\text {Too large to display} \] Input:

Integrate[Csc[c + d*x]^5*(a + b*Tan[c + d*x])^4,x]
 

Output:

(b^2*(36*a^2 + 7*b^2)*Cos[c + d*x]^4*(a + b*Tan[c + d*x])^4)/(6*d*(a*Cos[c 
 + d*x] + b*Sin[c + d*x])^4) + ((-7*a^3*b*Cos[(c + d*x)/2] - 6*a*b^3*Cos[( 
c + d*x)/2])*Cos[c + d*x]^4*Csc[(c + d*x)/2]*(a + b*Tan[c + d*x])^4)/(3*d* 
(a*Cos[c + d*x] + b*Sin[c + d*x])^4) - (3*(a^4 + 8*a^2*b^2)*Cos[c + d*x]^4 
*Csc[(c + d*x)/2]^2*(a + b*Tan[c + d*x])^4)/(32*d*(a*Cos[c + d*x] + b*Sin[ 
c + d*x])^4) - (a^3*b*Cos[c + d*x]^4*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2*( 
a + b*Tan[c + d*x])^4)/(6*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) - (a^4*Co 
s[c + d*x]^4*Csc[(c + d*x)/2]^4*(a + b*Tan[c + d*x])^4)/(64*d*(a*Cos[c + d 
*x] + b*Sin[c + d*x])^4) + ((-3*a^4 - 72*a^2*b^2 - 8*b^4)*Cos[c + d*x]^4*L 
og[Cos[(c + d*x)/2]]*(a + b*Tan[c + d*x])^4)/(8*d*(a*Cos[c + d*x] + b*Sin[ 
c + d*x])^4) - (2*(2*a^3*b + 3*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) + ((3*a^4 + 72*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)/(8*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + ( 
2*(2*a^3*b + 3*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) + (3*( 
a^4 + 8*a^2*b^2)*Cos[c + d*x]^4*Sec[(c + d*x)/2]^2*(a + b*Tan[c + d*x])^4) 
/(32*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + (a^4*Cos[c + d*x]^4*Sec[(c + 
 d*x)/2]^4*(a + b*Tan[c + d*x])^4)/(64*d*(a*Cos[c + d*x] + b*Sin[c + d*x]) 
^4) + ((12*a*b^3 + b^4)*Cos[c + d*x]^4*(a + b*Tan[c + d*x])^4)/(12*d*(C...
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 274, 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.

\(\displaystyle \int \csc ^5(c+d x) (a+b \tan (c+d x))^4 \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a+b \tan (c+d x))^4}{\sin (c+d x)^5}dx\)

\(\Big \downarrow \) 4000

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 a^4 \text {arctanh}(\cos (c+d x))}{8 d}-\frac {a^4 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {3 a^4 \cot (c+d x) \csc (c+d x)}{8 d}+\frac {4 a^3 b \text {arctanh}(\sin (c+d x))}{d}-\frac {4 a^3 b \csc ^3(c+d x)}{3 d}-\frac {4 a^3 b \csc (c+d x)}{d}-\frac {9 a^2 b^2 \text {arctanh}(\cos (c+d x))}{d}+\frac {9 a^2 b^2 \sec (c+d x)}{d}-\frac {3 a^2 b^2 \csc ^2(c+d x) \sec (c+d x)}{d}+\frac {6 a b^3 \text {arctanh}(\sin (c+d x))}{d}-\frac {6 a b^3 \csc (c+d x)}{d}+\frac {2 a b^3 \csc (c+d x) \sec ^2(c+d x)}{d}-\frac {b^4 \text {arctanh}(\cos (c+d x))}{d}+\frac {b^4 \sec ^3(c+d x)}{3 d}+\frac {b^4 \sec (c+d x)}{d}\)

Input:

Int[Csc[c + d*x]^5*(a + b*Tan[c + d*x])^4,x]
 

Output:

(-3*a^4*ArcTanh[Cos[c + d*x]])/(8*d) - (9*a^2*b^2*ArcTanh[Cos[c + d*x]])/d 
 - (b^4*ArcTanh[Cos[c + d*x]])/d + (4*a^3*b*ArcTanh[Sin[c + d*x]])/d + (6* 
a*b^3*ArcTanh[Sin[c + d*x]])/d - (4*a^3*b*Csc[c + d*x])/d - (6*a*b^3*Csc[c 
 + d*x])/d - (3*a^4*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (4*a^3*b*Csc[c + d* 
x]^3)/(3*d) - (a^4*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d) + (9*a^2*b^2*Sec[c + 
 d*x])/d + (b^4*Sec[c + d*x])/d - (3*a^2*b^2*Csc[c + d*x]^2*Sec[c + d*x])/ 
d + (2*a*b^3*Csc[c + d*x]*Sec[c + d*x]^2)/d + (b^4*Sec[c + d*x]^3)/(3*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 = 27.00 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.88

method	result	size
derivativedivides	\(\frac {b^{4} \left (\frac {1}{3 \cos \left (d x +c \right )^{3}}+\frac {1}{\cos \left (d x +c \right )}+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+4 a \,b^{3} \left (\frac {1}{2 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {3}{2 \sin \left (d x +c \right )}+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 b^{2} a^{2} \left (-\frac {1}{2 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {3}{2 \cos \left (d x +c \right )}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+4 a^{3} b \left (-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{\sin \left (d x +c \right )}+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{4} \left (\left (-\frac {\csc \left (d x +c \right )^{3}}{4}-\frac {3 \csc \left (d x +c \right )}{8}\right ) \cot \left (d x +c \right )+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )}{d}\)	\(241\)
default	\(\frac {b^{4} \left (\frac {1}{3 \cos \left (d x +c \right )^{3}}+\frac {1}{\cos \left (d x +c \right )}+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+4 a \,b^{3} \left (\frac {1}{2 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {3}{2 \sin \left (d x +c \right )}+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 b^{2} a^{2} \left (-\frac {1}{2 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {3}{2 \cos \left (d x +c \right )}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+4 a^{3} b \left (-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{\sin \left (d x +c \right )}+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{4} \left (\left (-\frac {\csc \left (d x +c \right )^{3}}{4}-\frac {3 \csc \left (d x +c \right )}{8}\right ) \cot \left (d x +c \right )+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )}{d}\)	\(241\)
risch	\(\frac {{\mathrm e}^{i \left (d x +c \right )} \left (24 b^{4}+216 b^{2} a^{2}+9 a^{4}-192 i a \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+288 a^{2} b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-216 a^{2} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-144 a^{2} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-180 a^{4} {\mathrm e}^{6 i \left (d x +c \right )}+288 b^{4} {\mathrm e}^{6 i \left (d x +c \right )}-105 a^{4} {\mathrm e}^{4 i \left (d x +c \right )}-152 b^{4} {\mathrm e}^{4 i \left (d x +c \right )}-6 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}-16 b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-48 i a \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-128 i a^{3} b \,{\mathrm e}^{2 i \left (d x +c \right )}+128 i a^{3} b \,{\mathrm e}^{10 i \left (d x +c \right )}+192 i a \,b^{3} {\mathrm e}^{10 i \left (d x +c \right )}+544 i a^{3} b \,{\mathrm e}^{8 i \left (d x +c \right )}-96 i a^{3} b \,{\mathrm e}^{12 i \left (d x +c \right )}-144 i a \,b^{3} {\mathrm e}^{12 i \left (d x +c \right )}+48 i a \,b^{3} {\mathrm e}^{8 i \left (d x +c \right )}-544 i a^{3} b \,{\mathrm e}^{4 i \left (d x +c \right )}+9 a^{4} {\mathrm e}^{12 i \left (d x +c \right )}-6 a^{4} {\mathrm e}^{10 i \left (d x +c \right )}-105 a^{4} {\mathrm e}^{8 i \left (d x +c \right )}+24 b^{4} {\mathrm e}^{12 i \left (d x +c \right )}-152 b^{4} {\mathrm e}^{8 i \left (d x +c \right )}-16 b^{4} {\mathrm e}^{10 i \left (d x +c \right )}+216 a^{2} b^{2} {\mathrm e}^{12 i \left (d x +c \right )}+96 i a^{3} b +144 i a \,b^{3}-216 a^{2} b^{2} {\mathrm e}^{8 i \left (d x +c \right )}-144 a^{2} b^{2} {\mathrm e}^{10 i \left (d x +c \right )}\right )}{12 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}+\frac {4 a^{3} b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}+\frac {6 a \,b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {3 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d}-\frac {9 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{2}}{d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{4}}{d}-\frac {4 a^{3} b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}-\frac {6 a \,b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {3 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d}+\frac {9 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{2}}{d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{4}}{d}\)	\(778\)

Input:

int(csc(d*x+c)^5*(a+b*tan(d*x+c))^4,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 547 vs. \(2 (264) = 528\).

Time = 0.22 (sec) , antiderivative size = 547, normalized size of antiderivative = 2.00 \[ \int \csc ^5(c+d x) (a+b \tan (c+d x))^4 \, dx=\frac {6 \, {\left (3 \, a^{4} + 72 \, a^{2} b^{2} + 8 \, b^{4}\right )} \cos \left (d x + c\right )^{6} - 10 \, {\left (3 \, a^{4} + 72 \, a^{2} b^{2} + 8 \, b^{4}\right )} \cos \left (d x + c\right )^{4} + 16 \, b^{4} + 16 \, {\left (18 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{2} - 3 \, {\left ({\left (3 \, a^{4} + 72 \, a^{2} b^{2} + 8 \, b^{4}\right )} \cos \left (d x + c\right )^{7} - 2 \, {\left (3 \, a^{4} + 72 \, a^{2} b^{2} + 8 \, b^{4}\right )} \cos \left (d x + c\right )^{5} + {\left (3 \, a^{4} + 72 \, 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 ) + 3 \, {\left ({\left (3 \, a^{4} + 72 \, a^{2} b^{2} + 8 \, b^{4}\right )} \cos \left (d x + c\right )^{7} - 2 \, {\left (3 \, a^{4} + 72 \, a^{2} b^{2} + 8 \, b^{4}\right )} \cos \left (d x + c\right )^{5} + {\left (3 \, a^{4} + 72 \, 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 ) + 48 \, {\left ({\left (2 \, a^{3} b + 3 \, a b^{3}\right )} \cos \left (d x + c\right )^{7} - 2 \, {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} \cos \left (d x + c\right )^{5} + {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 48 \, {\left ({\left (2 \, a^{3} b + 3 \, a b^{3}\right )} \cos \left (d x + c\right )^{7} - 2 \, {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} \cos \left (d x + c\right )^{5} + {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 32 \, {\left (3 \, {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} \cos \left (d x + c\right )^{5} + 3 \, a b^{3} \cos \left (d x + c\right ) - 4 \, {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{48 \, {\left (d \cos \left (d x + c\right )^{7} - 2 \, d \cos \left (d x + c\right )^{5} + d \cos \left (d x + c\right )^{3}\right )}} \] Input:

integrate(csc(d*x+c)^5*(a+b*tan(d*x+c))^4,x, algorithm="fricas")
 

Output:

1/48*(6*(3*a^4 + 72*a^2*b^2 + 8*b^4)*cos(d*x + c)^6 - 10*(3*a^4 + 72*a^2*b 
^2 + 8*b^4)*cos(d*x + c)^4 + 16*b^4 + 16*(18*a^2*b^2 + b^4)*cos(d*x + c)^2 
 - 3*((3*a^4 + 72*a^2*b^2 + 8*b^4)*cos(d*x + c)^7 - 2*(3*a^4 + 72*a^2*b^2 
+ 8*b^4)*cos(d*x + c)^5 + (3*a^4 + 72*a^2*b^2 + 8*b^4)*cos(d*x + c)^3)*log 
(1/2*cos(d*x + c) + 1/2) + 3*((3*a^4 + 72*a^2*b^2 + 8*b^4)*cos(d*x + c)^7 
- 2*(3*a^4 + 72*a^2*b^2 + 8*b^4)*cos(d*x + c)^5 + (3*a^4 + 72*a^2*b^2 + 8* 
b^4)*cos(d*x + c)^3)*log(-1/2*cos(d*x + c) + 1/2) + 48*((2*a^3*b + 3*a*b^3 
)*cos(d*x + c)^7 - 2*(2*a^3*b + 3*a*b^3)*cos(d*x + c)^5 + (2*a^3*b + 3*a*b 
^3)*cos(d*x + c)^3)*log(sin(d*x + c) + 1) - 48*((2*a^3*b + 3*a*b^3)*cos(d* 
x + c)^7 - 2*(2*a^3*b + 3*a*b^3)*cos(d*x + c)^5 + (2*a^3*b + 3*a*b^3)*cos( 
d*x + c)^3)*log(-sin(d*x + c) + 1) + 32*(3*(2*a^3*b + 3*a*b^3)*cos(d*x + c 
)^5 + 3*a*b^3*cos(d*x + c) - 4*(2*a^3*b + 3*a*b^3)*cos(d*x + c)^3)*sin(d*x 
 + c))/(d*cos(d*x + c)^7 - 2*d*cos(d*x + c)^5 + d*cos(d*x + c)^3)
 

Sympy [F]

\[ \int \csc ^5(c+d x) (a+b \tan (c+d x))^4 \, dx=\int \left (a + b \tan {\left (c + d x \right )}\right )^{4} \csc ^{5}{\left (c + d x \right )}\, dx \] Input:

integrate(csc(d*x+c)**5*(a+b*tan(d*x+c))**4,x)
 

Output:

Integral((a + b*tan(c + d*x))**4*csc(c + d*x)**5, x)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.11 \[ \int \csc ^5(c+d x) (a+b \tan (c+d x))^4 \, dx=\frac {3 \, a^{4} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{3} - 5 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 72 \, a^{2} b^{2} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{2} - 2\right )}}{\cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 48 \, a b^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{2} - 2\right )}}{\sin \left (d x + c\right )^{3} - \sin \left (d x + c\right )} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 8 \, b^{4} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{2} + 1\right )}}{\cos \left (d x + c\right )^{3}} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 32 \, a^{3} b {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{2} + 1\right )}}{\sin \left (d x + c\right )^{3}} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{48 \, d} \] Input:

integrate(csc(d*x+c)^5*(a+b*tan(d*x+c))^4,x, algorithm="maxima")
 

Output:

1/48*(3*a^4*(2*(3*cos(d*x + c)^3 - 5*cos(d*x + c))/(cos(d*x + c)^4 - 2*cos 
(d*x + c)^2 + 1) - 3*log(cos(d*x + c) + 1) + 3*log(cos(d*x + c) - 1)) + 72 
*a^2*b^2*(2*(3*cos(d*x + c)^2 - 2)/(cos(d*x + c)^3 - cos(d*x + c)) - 3*log 
(cos(d*x + c) + 1) + 3*log(cos(d*x + c) - 1)) - 48*a*b^3*(2*(3*sin(d*x + c 
)^2 - 2)/(sin(d*x + c)^3 - sin(d*x + c)) - 3*log(sin(d*x + c) + 1) + 3*log 
(sin(d*x + c) - 1)) + 8*b^4*(2*(3*cos(d*x + c)^2 + 1)/cos(d*x + c)^3 - 3*l 
og(cos(d*x + c) + 1) + 3*log(cos(d*x + c) - 1)) - 32*a^3*b*(2*(3*sin(d*x + 
 c)^2 + 1)/sin(d*x + c)^3 - 3*log(sin(d*x + c) + 1) + 3*log(sin(d*x + c) - 
 1)))/d
 

Giac [A] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 479, normalized size of antiderivative = 1.75 \[ \int \csc ^5(c+d x) (a+b \tan (c+d x))^4 \, dx =\text {Too large to display} \] Input:

integrate(csc(d*x+c)^5*(a+b*tan(d*x+c))^4,x, algorithm="giac")
 

Output:

1/192*(3*a^4*tan(1/2*d*x + 1/2*c)^4 - 32*a^3*b*tan(1/2*d*x + 1/2*c)^3 + 24 
*a^4*tan(1/2*d*x + 1/2*c)^2 + 144*a^2*b^2*tan(1/2*d*x + 1/2*c)^2 - 480*a^3 
*b*tan(1/2*d*x + 1/2*c) - 384*a*b^3*tan(1/2*d*x + 1/2*c) + 384*(2*a^3*b + 
3*a*b^3)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 384*(2*a^3*b + 3*a*b^3)*log( 
abs(tan(1/2*d*x + 1/2*c) - 1)) + 24*(3*a^4 + 72*a^2*b^2 + 8*b^4)*log(abs(t 
an(1/2*d*x + 1/2*c))) + 256*(3*a*b^3*tan(1/2*d*x + 1/2*c)^5 - 9*a^2*b^2*ta 
n(1/2*d*x + 1/2*c)^4 - 3*b^4*tan(1/2*d*x + 1/2*c)^4 + 18*a^2*b^2*tan(1/2*d 
*x + 1/2*c)^2 + 3*b^4*tan(1/2*d*x + 1/2*c)^2 - 3*a*b^3*tan(1/2*d*x + 1/2*c 
) - 9*a^2*b^2 - 2*b^4)/(tan(1/2*d*x + 1/2*c)^2 - 1)^3 - (150*a^4*tan(1/2*d 
*x + 1/2*c)^4 + 3600*a^2*b^2*tan(1/2*d*x + 1/2*c)^4 + 400*b^4*tan(1/2*d*x 
+ 1/2*c)^4 + 480*a^3*b*tan(1/2*d*x + 1/2*c)^3 + 384*a*b^3*tan(1/2*d*x + 1/ 
2*c)^3 + 24*a^4*tan(1/2*d*x + 1/2*c)^2 + 144*a^2*b^2*tan(1/2*d*x + 1/2*c)^ 
2 + 32*a^3*b*tan(1/2*d*x + 1/2*c) + 3*a^4)/tan(1/2*d*x + 1/2*c)^4)/d
 

Mupad [B] (verification not implemented)

Time = 1.24 (sec) , antiderivative size = 857, normalized size of antiderivative = 3.13 \[ \int \csc ^5(c+d x) (a+b \tan (c+d x))^4 \, dx=\text {Too large to display} \] Input:

int((a + b*tan(c + d*x))^4/sin(c + d*x)^5,x)
 

Output:

(a^4*tan(c/2 + (d*x)/2)^4)/(64*d) - (atan(-((6*a*b^3 + 4*a^3*b)*(12*a*b^3 
+ 8*a^3*b - 6*tan(c/2 + (d*x)/2)*(6*a*b^3 + 4*a^3*b) - tan(c/2 + (d*x)/2)* 
((3*a^4)/4 + 2*b^4 + 18*a^2*b^2))*1i + (6*a*b^3 + 4*a^3*b)*(12*a*b^3 + 8*a 
^3*b + 6*tan(c/2 + (d*x)/2)*(6*a*b^3 + 4*a^3*b) - tan(c/2 + (d*x)/2)*((3*a 
^4)/4 + 2*b^4 + 18*a^2*b^2))*1i)/(2*tan(c/2 + (d*x)/2)*(144*a^2*b^6 + 192* 
a^4*b^4 + 64*a^6*b^2) + (6*a*b^3 + 4*a^3*b)*(12*a*b^3 + 8*a^3*b - 6*tan(c/ 
2 + (d*x)/2)*(6*a*b^3 + 4*a^3*b) - tan(c/2 + (d*x)/2)*((3*a^4)/4 + 2*b^4 + 
 18*a^2*b^2)) - (6*a*b^3 + 4*a^3*b)*(12*a*b^3 + 8*a^3*b + 6*tan(c/2 + (d*x 
)/2)*(6*a*b^3 + 4*a^3*b) - tan(c/2 + (d*x)/2)*((3*a^4)/4 + 2*b^4 + 18*a^2* 
b^2)) + 24*a*b^7 + 6*a^7*b + 232*a^3*b^5 + 153*a^5*b^3))*(a*b^3*12i + a^3* 
b*8i))/d - (tan(c/2 + (d*x)/2)*(2*a^3*b + (a*b*(a^2 + 4*b^2))/2))/d + (log 
(tan(c/2 + (d*x)/2))*((3*a^4)/8 + b^4 + 9*a^2*b^2))/d + (tan(c/2 + (d*x)/2 
)^2*(a^4/8 + (3*a^2*b^2)/4))/d - (tan(c/2 + (d*x)/2)^6*((23*a^4)/4 + 64*b^ 
4 + 420*a^2*b^2) - tan(c/2 + (d*x)/2)^4*((21*a^4)/4 + (128*b^4)/3 + 228*a^ 
2*b^2) - tan(c/2 + (d*x)/2)^8*(2*a^4 + 64*b^4 + 204*a^2*b^2) + a^4/4 + tan 
(c/2 + (d*x)/2)^2*((5*a^4)/4 + 12*a^2*b^2) + tan(c/2 + (d*x)/2)^3*(32*a*b^ 
3 + 32*a^3*b) + tan(c/2 + (d*x)/2)^9*(32*a*b^3 - 40*a^3*b) - tan(c/2 + (d* 
x)/2)^5*(160*a*b^3 + 112*a^3*b) + tan(c/2 + (d*x)/2)^7*(96*a*b^3 + (352*a^ 
3*b)/3) + (8*a^3*b*tan(c/2 + (d*x)/2))/3)/(d*(16*tan(c/2 + (d*x)/2)^4 - 48 
*tan(c/2 + (d*x)/2)^6 + 48*tan(c/2 + (d*x)/2)^8 - 16*tan(c/2 + (d*x)/2)...
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 806, normalized size of antiderivative = 2.94 \[ \int \csc ^5(c+d x) (a+b \tan (c+d x))^4 \, dx =\text {Too large to display} \] Input:

int(csc(d*x+c)^5*(a+b*tan(d*x+c))^4,x)
 

Output:

( - 768*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**6*a**3*b - 11 
52*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**6*a*b**3 + 768*cos 
(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4*a**3*b + 1152*cos(c + 
d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4*a*b**3 + 768*cos(c + d*x)*l 
og(tan((c + d*x)/2) + 1)*sin(c + d*x)**6*a**3*b + 1152*cos(c + d*x)*log(ta 
n((c + d*x)/2) + 1)*sin(c + d*x)**6*a*b**3 - 768*cos(c + d*x)*log(tan((c + 
 d*x)/2) + 1)*sin(c + d*x)**4*a**3*b - 1152*cos(c + d*x)*log(tan((c + d*x) 
/2) + 1)*sin(c + d*x)**4*a*b**3 + 72*cos(c + d*x)*log(tan((c + d*x)/2))*si 
n(c + d*x)**6*a**4 + 1728*cos(c + d*x)*log(tan((c + d*x)/2))*sin(c + d*x)* 
*6*a**2*b**2 + 192*cos(c + d*x)*log(tan((c + d*x)/2))*sin(c + d*x)**6*b**4 
 - 72*cos(c + d*x)*log(tan((c + d*x)/2))*sin(c + d*x)**4*a**4 - 1728*cos(c 
 + d*x)*log(tan((c + d*x)/2))*sin(c + d*x)**4*a**2*b**2 - 192*cos(c + d*x) 
*log(tan((c + d*x)/2))*sin(c + d*x)**4*b**4 - 63*cos(c + d*x)*sin(c + d*x) 
**6*a**4 - 1584*cos(c + d*x)*sin(c + d*x)**6*a**2*b**2 - 256*cos(c + d*x)* 
sin(c + d*x)**6*b**4 - 768*cos(c + d*x)*sin(c + d*x)**5*a**3*b - 1152*cos( 
c + d*x)*sin(c + d*x)**5*a*b**3 + 63*cos(c + d*x)*sin(c + d*x)**4*a**4 + 1 
584*cos(c + d*x)*sin(c + d*x)**4*a**2*b**2 + 256*cos(c + d*x)*sin(c + d*x) 
**4*b**4 + 512*cos(c + d*x)*sin(c + d*x)**3*a**3*b + 768*cos(c + d*x)*sin( 
c + d*x)**3*a*b**3 + 256*cos(c + d*x)*sin(c + d*x)*a**3*b + 72*sin(c + d*x 
)**6*a**4 + 1728*sin(c + d*x)**6*a**2*b**2 + 192*sin(c + d*x)**6*b**4 -...