\(\int \sec ^9(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx\) [88]

Optimal result

Integrand size = 28, antiderivative size = 330 \[ \int \sec ^9(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {3 a^4 \text {arctanh}(\sin (c+d x))}{8 d}-\frac {3 a^2 b^2 \text {arctanh}(\sin (c+d x))}{8 d}+\frac {3 b^4 \text {arctanh}(\sin (c+d x))}{128 d}+\frac {4 a^3 b \sec ^5(c+d x)}{5 d}-\frac {4 a b^3 \sec ^5(c+d x)}{5 d}+\frac {4 a b^3 \sec ^7(c+d x)}{7 d}+\frac {3 a^4 \sec (c+d x) \tan (c+d x)}{8 d}-\frac {3 a^2 b^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {3 b^4 \sec (c+d x) \tan (c+d x)}{128 d}+\frac {a^4 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {a^2 b^2 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {b^4 \sec ^3(c+d x) \tan (c+d x)}{64 d}+\frac {a^2 b^2 \sec ^5(c+d x) \tan (c+d x)}{d}-\frac {b^4 \sec ^5(c+d x) \tan (c+d x)}{16 d}+\frac {b^4 \sec ^5(c+d x) \tan ^3(c+d x)}{8 d} \] Output:

3/8*a^4*arctanh(sin(d*x+c))/d-3/8*a^2*b^2*arctanh(sin(d*x+c))/d+3/128*b^4* 
arctanh(sin(d*x+c))/d+4/5*a^3*b*sec(d*x+c)^5/d-4/5*a*b^3*sec(d*x+c)^5/d+4/ 
7*a*b^3*sec(d*x+c)^7/d+3/8*a^4*sec(d*x+c)*tan(d*x+c)/d-3/8*a^2*b^2*sec(d*x 
+c)*tan(d*x+c)/d+3/128*b^4*sec(d*x+c)*tan(d*x+c)/d+1/4*a^4*sec(d*x+c)^3*ta 
n(d*x+c)/d-1/4*a^2*b^2*sec(d*x+c)^3*tan(d*x+c)/d+1/64*b^4*sec(d*x+c)^3*tan 
(d*x+c)/d+a^2*b^2*sec(d*x+c)^5*tan(d*x+c)/d-1/16*b^4*sec(d*x+c)^5*tan(d*x+ 
c)/d+1/8*b^4*sec(d*x+c)^5*tan(d*x+c)^3/d
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1732\) vs. \(2(330)=660\).

Time = 7.28 (sec) , antiderivative size = 1732, normalized size of antiderivative = 5.25 \[ \int \sec ^9(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx =\text {Too large to display} \] Input:

Integrate[Sec[c + d*x]^9*(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]
 

Output:

(a*b*(42*a^2 - 17*b^2)*Cos[c + d*x]^4*(a + b*Tan[c + d*x])^4)/(140*d*(a*Co 
s[c + d*x] + b*Sin[c + d*x])^4) - (3*(16*a^4 - 16*a^2*b^2 + b^4)*Cos[c + d 
*x]^4*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(a + b*Tan[c + d*x])^4)/(12 
8*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + (3*(16*a^4 - 16*a^2*b^2 + b^4)* 
Cos[c + d*x]^4*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(a + b*Tan[c + d*x 
])^4)/(128*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + (b^4*Cos[c + d*x]^4*(a 
 + b*Tan[c + d*x])^4)/(128*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^8*(a*Co 
s[c + d*x] + b*Sin[c + d*x])^4) + ((56*a^2*b^2 + 16*a*b^3 - 7*b^4)*Cos[c + 
 d*x]^4*(a + b*Tan[c + d*x])^4)/(448*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2 
])^6*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + ((560*a^4 + 896*a^3*b - 256*a* 
b^3 - 35*b^4)*Cos[c + d*x]^4*(a + b*Tan[c + d*x])^4)/(8960*d*(Cos[(c + d*x 
)/2] - Sin[(c + d*x)/2])^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + ((1680*a 
^4 + 1344*a^3*b - 1680*a^2*b^2 - 544*a*b^3 + 105*b^4)*Cos[c + d*x]^4*(a + 
b*Tan[c + d*x])^4)/(8960*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2*(a*Cos[ 
c + d*x] + b*Sin[c + d*x])^4) + (a*b^3*Cos[c + d*x]^4*Sin[(c + d*x)/2]*(a 
+ b*Tan[c + d*x])^4)/(14*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^7*(a*Cos[ 
c + d*x] + b*Sin[c + d*x])^4) - (b^4*Cos[c + d*x]^4*(a + b*Tan[c + d*x])^4 
)/(128*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8*(a*Cos[c + d*x] + b*Sin[c 
 + d*x])^4) - (a*b^3*Cos[c + d*x]^4*Sin[(c + d*x)/2]*(a + b*Tan[c + d*x])^ 
4)/(14*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^7*(a*Cos[c + d*x] + b*Si...
 

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {3042, 3569, 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.

\(\displaystyle \int \sec ^9(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a \cos (c+d x)+b \sin (c+d x))^4}{\cos (c+d x)^9}dx\)

\(\Big \downarrow \) 3569

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3 a^4 \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a^4 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac {3 a^4 \tan (c+d x) \sec (c+d x)}{8 d}+\frac {4 a^3 b \sec ^5(c+d x)}{5 d}-\frac {3 a^2 b^2 \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a^2 b^2 \tan (c+d x) \sec ^5(c+d x)}{d}-\frac {a^2 b^2 \tan (c+d x) \sec ^3(c+d x)}{4 d}-\frac {3 a^2 b^2 \tan (c+d x) \sec (c+d x)}{8 d}+\frac {4 a b^3 \sec ^7(c+d x)}{7 d}-\frac {4 a b^3 \sec ^5(c+d x)}{5 d}+\frac {3 b^4 \text {arctanh}(\sin (c+d x))}{128 d}+\frac {b^4 \tan ^3(c+d x) \sec ^5(c+d x)}{8 d}-\frac {b^4 \tan (c+d x) \sec ^5(c+d x)}{16 d}+\frac {b^4 \tan (c+d x) \sec ^3(c+d x)}{64 d}+\frac {3 b^4 \tan (c+d x) \sec (c+d x)}{128 d}\)

Input:

Int[Sec[c + d*x]^9*(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]
 

Output:

(3*a^4*ArcTanh[Sin[c + d*x]])/(8*d) - (3*a^2*b^2*ArcTanh[Sin[c + d*x]])/(8 
*d) + (3*b^4*ArcTanh[Sin[c + d*x]])/(128*d) + (4*a^3*b*Sec[c + d*x]^5)/(5* 
d) - (4*a*b^3*Sec[c + d*x]^5)/(5*d) + (4*a*b^3*Sec[c + d*x]^7)/(7*d) + (3* 
a^4*Sec[c + d*x]*Tan[c + d*x])/(8*d) - (3*a^2*b^2*Sec[c + d*x]*Tan[c + d*x 
])/(8*d) + (3*b^4*Sec[c + d*x]*Tan[c + d*x])/(128*d) + (a^4*Sec[c + d*x]^3 
*Tan[c + d*x])/(4*d) - (a^2*b^2*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (b^4* 
Sec[c + d*x]^3*Tan[c + d*x])/(64*d) + (a^2*b^2*Sec[c + d*x]^5*Tan[c + d*x] 
)/d - (b^4*Sec[c + d*x]^5*Tan[c + d*x])/(16*d) + (b^4*Sec[c + d*x]^5*Tan[c 
 + d*x]^3)/(8*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 3569

Int[cos[(c_.) + (d_.)*(x_)]^(m_.)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*si 
n[(c_.) + (d_.)*(x_)])^(n_.), x_Symbol] :> Int[ExpandTrig[cos[c + d*x]^m*(a 
*cos[c + d*x] + b*sin[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d}, x] && Inte 
gerQ[m] && IGtQ[n, 0]
 

Maple [A] (verified)

Time = 1.61 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.92

method	result	size
parts	\(\frac {a^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}+\frac {b^{4} \left (\frac {\sin \left (d x +c \right )^{5}}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin \left (d x +c \right )^{5}}{16 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{5}}{64 \cos \left (d x +c \right )^{4}}-\frac {\sin \left (d x +c \right )^{5}}{128 \cos \left (d x +c \right )^{2}}-\frac {\sin \left (d x +c \right )^{3}}{128}-\frac {3 \sin \left (d x +c \right )}{128}+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{128}\right )}{d}+\frac {4 a^{3} b \sec \left (d x +c \right )^{5}}{5 d}+\frac {6 a^{2} b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{6 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{3}}{8 \cos \left (d x +c \right )^{4}}+\frac {\sin \left (d x +c \right )^{3}}{16 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{16}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}+\frac {4 b^{3} a \left (\frac {\sec \left (d x +c \right )^{7}}{7}-\frac {\sec \left (d x +c \right )^{5}}{5}\right )}{d}\)	\(304\)
derivativedivides	\(\frac {a^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+\frac {4 a^{3} b}{5 \cos \left (d x +c \right )^{5}}+6 a^{2} b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{6 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{3}}{8 \cos \left (d x +c \right )^{4}}+\frac {\sin \left (d x +c \right )^{3}}{16 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{16}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+4 b^{3} a \left (\frac {\sin \left (d x +c \right )^{4}}{7 \cos \left (d x +c \right )^{7}}+\frac {3 \sin \left (d x +c \right )^{4}}{35 \cos \left (d x +c \right )^{5}}+\frac {\sin \left (d x +c \right )^{4}}{35 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{4}}{35 \cos \left (d x +c \right )}-\frac {\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{35}\right )+b^{4} \left (\frac {\sin \left (d x +c \right )^{5}}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin \left (d x +c \right )^{5}}{16 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{5}}{64 \cos \left (d x +c \right )^{4}}-\frac {\sin \left (d x +c \right )^{5}}{128 \cos \left (d x +c \right )^{2}}-\frac {\sin \left (d x +c \right )^{3}}{128}-\frac {3 \sin \left (d x +c \right )}{128}+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{128}\right )}{d}\)	\(363\)
default	\(\frac {a^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+\frac {4 a^{3} b}{5 \cos \left (d x +c \right )^{5}}+6 a^{2} b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{6 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{3}}{8 \cos \left (d x +c \right )^{4}}+\frac {\sin \left (d x +c \right )^{3}}{16 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{16}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+4 b^{3} a \left (\frac {\sin \left (d x +c \right )^{4}}{7 \cos \left (d x +c \right )^{7}}+\frac {3 \sin \left (d x +c \right )^{4}}{35 \cos \left (d x +c \right )^{5}}+\frac {\sin \left (d x +c \right )^{4}}{35 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{4}}{35 \cos \left (d x +c \right )}-\frac {\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{35}\right )+b^{4} \left (\frac {\sin \left (d x +c \right )^{5}}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin \left (d x +c \right )^{5}}{16 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{5}}{64 \cos \left (d x +c \right )^{4}}-\frac {\sin \left (d x +c \right )^{5}}{128 \cos \left (d x +c \right )^{2}}-\frac {\sin \left (d x +c \right )^{3}}{128}-\frac {3 \sin \left (d x +c \right )}{128}+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{128}\right )}{d}\)	\(363\)
parallelrisch	\(\frac {-13440 \left (\frac {35}{8}+\frac {\cos \left (8 d x +8 c \right )}{8}+\cos \left (6 d x +6 c \right )+\frac {7 \cos \left (4 d x +4 c \right )}{2}+7 \cos \left (2 d x +2 c \right )\right ) \left (a^{4}-a^{2} b^{2}+\frac {1}{16} b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+13440 \left (\frac {35}{8}+\frac {\cos \left (8 d x +8 c \right )}{8}+\cos \left (6 d x +6 c \right )+\frac {7 \cos \left (4 d x +4 c \right )}{2}+7 \cos \left (2 d x +2 c \right )\right ) \left (a^{4}-a^{2} b^{2}+\frac {1}{16} b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (200704 a^{3} b -57344 b^{3} a \right ) \cos \left (2 d x +2 c \right )+\left (100352 a^{3} b -28672 b^{3} a \right ) \cos \left (4 d x +4 c \right )+\left (28672 a^{3} b -8192 b^{3} a \right ) \cos \left (6 d x +6 c \right )+\left (3584 a^{3} b -1024 b^{3} a \right ) \cos \left (8 d x +8 c \right )+\left (57120 a^{4}+86240 a^{2} b^{2}-23310 b^{4}\right ) \sin \left (3 d x +3 c \right )+\left (25760 a^{4}-25760 a^{2} b^{2}+1610 b^{4}\right ) \sin \left (5 d x +5 c \right )+\left (3360 a^{4}-3360 a^{2} b^{2}+210 b^{4}\right ) \sin \left (7 d x +7 c \right )+\left (114688 a^{3} b -114688 b^{3} a \right ) \cos \left (3 d x +3 c \right )+\left (34720 a^{4}+108640 a^{2} b^{2}+46970 b^{4}\right ) \sin \left (d x +c \right )+344064 \left (\left (a^{2}-\frac {b^{2}}{21}\right ) \cos \left (d x +c \right )+\frac {35 a^{2}}{96}-\frac {5 b^{2}}{48}\right ) a b}{4480 d \left (\cos \left (8 d x +8 c \right )+8 \cos \left (6 d x +6 c \right )+28 \cos \left (4 d x +4 c \right )+56 \cos \left (2 d x +2 c \right )+35\right )}\)	\(462\)
risch	\(-\frac {i {\mathrm e}^{i \left (d x +c \right )} \left (-1680 a^{4}-105 b^{4}+172032 i a^{3} b \,{\mathrm e}^{6 i \left (d x +c \right )}-8192 i a \,b^{3} {\mathrm e}^{8 i \left (d x +c \right )}+172032 i a^{3} b \,{\mathrm e}^{8 i \left (d x +c \right )}-57344 i a \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+57344 i a^{3} b \,{\mathrm e}^{4 i \left (d x +c \right )}-8192 i a \,b^{3} {\mathrm e}^{6 i \left (d x +c \right )}+1680 a^{2} b^{2}+54320 a^{2} b^{2} {\mathrm e}^{8 i \left (d x +c \right )}-54320 a^{2} b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-43120 a^{2} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+12880 a^{2} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-12880 a^{2} b^{2} {\mathrm e}^{12 i \left (d x +c \right )}-1680 a^{2} b^{2} {\mathrm e}^{14 i \left (d x +c \right )}-23485 b^{4} {\mathrm e}^{6 i \left (d x +c \right )}-11655 b^{4} {\mathrm e}^{10 i \left (d x +c \right )}+28560 a^{4} {\mathrm e}^{10 i \left (d x +c \right )}-805 b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-57344 i a \,b^{3} {\mathrm e}^{10 i \left (d x +c \right )}+57344 i a^{3} b \,{\mathrm e}^{10 i \left (d x +c \right )}+12880 a^{4} {\mathrm e}^{12 i \left (d x +c \right )}+805 b^{4} {\mathrm e}^{12 i \left (d x +c \right )}+1680 a^{4} {\mathrm e}^{14 i \left (d x +c \right )}+105 b^{4} {\mathrm e}^{14 i \left (d x +c \right )}+17360 a^{4} {\mathrm e}^{8 i \left (d x +c \right )}+23485 b^{4} {\mathrm e}^{8 i \left (d x +c \right )}-17360 a^{4} {\mathrm e}^{6 i \left (d x +c \right )}-28560 a^{4} {\mathrm e}^{4 i \left (d x +c \right )}+11655 b^{4} {\mathrm e}^{4 i \left (d x +c \right )}-12880 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}+43120 a^{2} b^{2} {\mathrm e}^{10 i \left (d x +c \right )}\right )}{2240 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{8}}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a^{4}}{8 d}-\frac {3 b^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a^{2}}{8 d}+\frac {3 b^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{128 d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{4}}{8 d}+\frac {3 b^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{2}}{8 d}-\frac {3 b^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{128 d}\)	\(684\)

Input:

int(sec(d*x+c)^9*(a*cos(d*x+c)+b*sin(d*x+c))^4,x,method=_RETURNVERBOSE)
 

Output:

a^4/d*(-(-1/4*sec(d*x+c)^3-3/8*sec(d*x+c))*tan(d*x+c)+3/8*ln(sec(d*x+c)+ta 
n(d*x+c)))+b^4/d*(1/8*sin(d*x+c)^5/cos(d*x+c)^8+1/16*sin(d*x+c)^5/cos(d*x+ 
c)^6+1/64*sin(d*x+c)^5/cos(d*x+c)^4-1/128*sin(d*x+c)^5/cos(d*x+c)^2-1/128* 
sin(d*x+c)^3-3/128*sin(d*x+c)+3/128*ln(sec(d*x+c)+tan(d*x+c)))+4/5*a^3*b*s 
ec(d*x+c)^5/d+6*a^2*b^2/d*(1/6*sin(d*x+c)^3/cos(d*x+c)^6+1/8*sin(d*x+c)^3/ 
cos(d*x+c)^4+1/16*sin(d*x+c)^3/cos(d*x+c)^2+1/16*sin(d*x+c)-1/16*ln(sec(d* 
x+c)+tan(d*x+c)))+4*b^3*a/d*(1/7*sec(d*x+c)^7-1/5*sec(d*x+c)^5)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.65 \[ \int \sec ^9(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {105 \, {\left (16 \, a^{4} - 16 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{8} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left (16 \, a^{4} - 16 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{8} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 5120 \, a b^{3} \cos \left (d x + c\right ) + 7168 \, {\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{3} + 70 \, {\left (3 \, {\left (16 \, a^{4} - 16 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{6} + 2 \, {\left (16 \, a^{4} - 16 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{4} + 16 \, b^{4} + 8 \, {\left (16 \, a^{2} b^{2} - 3 \, b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{8960 \, d \cos \left (d x + c\right )^{8}} \] Input:

integrate(sec(d*x+c)^9*(a*cos(d*x+c)+b*sin(d*x+c))^4,x, algorithm="fricas" 
)
 

Output:

1/8960*(105*(16*a^4 - 16*a^2*b^2 + b^4)*cos(d*x + c)^8*log(sin(d*x + c) + 
1) - 105*(16*a^4 - 16*a^2*b^2 + b^4)*cos(d*x + c)^8*log(-sin(d*x + c) + 1) 
 + 5120*a*b^3*cos(d*x + c) + 7168*(a^3*b - a*b^3)*cos(d*x + c)^3 + 70*(3*( 
16*a^4 - 16*a^2*b^2 + b^4)*cos(d*x + c)^6 + 2*(16*a^4 - 16*a^2*b^2 + b^4)* 
cos(d*x + c)^4 + 16*b^4 + 8*(16*a^2*b^2 - 3*b^4)*cos(d*x + c)^2)*sin(d*x + 
 c))/(d*cos(d*x + c)^8)
 

Sympy [F(-1)]

Timed out. \[ \int \sec ^9(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\text {Timed out} \] Input:

integrate(sec(d*x+c)**9*(a*cos(d*x+c)+b*sin(d*x+c))**4,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 322, normalized size of antiderivative = 0.98 \[ \int \sec ^9(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=-\frac {35 \, b^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{7} - 11 \, \sin \left (d x + c\right )^{5} - 11 \, \sin \left (d x + c\right )^{3} + 3 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{8} - 4 \, \sin \left (d x + c\right )^{6} + 6 \, \sin \left (d x + c\right )^{4} - 4 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 560 \, a^{2} b^{2} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 8 \, \sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 560 \, a^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - \frac {7168 \, a^{3} b}{\cos \left (d x + c\right )^{5}} + \frac {1024 \, {\left (7 \, \cos \left (d x + c\right )^{2} - 5\right )} a b^{3}}{\cos \left (d x + c\right )^{7}}}{8960 \, d} \] Input:

integrate(sec(d*x+c)^9*(a*cos(d*x+c)+b*sin(d*x+c))^4,x, algorithm="maxima" 
)
 

Output:

-1/8960*(35*b^4*(2*(3*sin(d*x + c)^7 - 11*sin(d*x + c)^5 - 11*sin(d*x + c) 
^3 + 3*sin(d*x + c))/(sin(d*x + c)^8 - 4*sin(d*x + c)^6 + 6*sin(d*x + c)^4 
 - 4*sin(d*x + c)^2 + 1) - 3*log(sin(d*x + c) + 1) + 3*log(sin(d*x + c) - 
1)) - 560*a^2*b^2*(2*(3*sin(d*x + c)^5 - 8*sin(d*x + c)^3 - 3*sin(d*x + c) 
)/(sin(d*x + c)^6 - 3*sin(d*x + c)^4 + 3*sin(d*x + c)^2 - 1) - 3*log(sin(d 
*x + c) + 1) + 3*log(sin(d*x + c) - 1)) + 560*a^4*(2*(3*sin(d*x + c)^3 - 5 
*sin(d*x + c))/(sin(d*x + c)^4 - 2*sin(d*x + c)^2 + 1) - 3*log(sin(d*x + c 
) + 1) + 3*log(sin(d*x + c) - 1)) - 7168*a^3*b/cos(d*x + c)^5 + 1024*(7*co 
s(d*x + c)^2 - 5)*a*b^3/cos(d*x + c)^7)/d
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 706 vs. \(2 (302) = 604\).

Time = 0.27 (sec) , antiderivative size = 706, normalized size of antiderivative = 2.14 \[ \int \sec ^9(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\text {Too large to display} \] Input:

integrate(sec(d*x+c)^9*(a*cos(d*x+c)+b*sin(d*x+c))^4,x, algorithm="giac")
 

Output:

1/4480*(105*(16*a^4 - 16*a^2*b^2 + b^4)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) 
 - 105*(16*a^4 - 16*a^2*b^2 + b^4)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + 2* 
(2800*a^4*tan(1/2*d*x + 1/2*c)^15 + 1680*a^2*b^2*tan(1/2*d*x + 1/2*c)^15 - 
 105*b^4*tan(1/2*d*x + 1/2*c)^15 - 17920*a^3*b*tan(1/2*d*x + 1/2*c)^14 - 9 
520*a^4*tan(1/2*d*x + 1/2*c)^13 + 22960*a^2*b^2*tan(1/2*d*x + 1/2*c)^13 + 
805*b^4*tan(1/2*d*x + 1/2*c)^13 + 53760*a^3*b*tan(1/2*d*x + 1/2*c)^12 - 35 
840*a*b^3*tan(1/2*d*x + 1/2*c)^12 + 11760*a^4*tan(1/2*d*x + 1/2*c)^11 - 72 
80*a^2*b^2*tan(1/2*d*x + 1/2*c)^11 + 11655*b^4*tan(1/2*d*x + 1/2*c)^11 - 8 
9600*a^3*b*tan(1/2*d*x + 1/2*c)^10 - 5040*a^4*tan(1/2*d*x + 1/2*c)^9 - 173 
60*a^2*b^2*tan(1/2*d*x + 1/2*c)^9 + 23485*b^4*tan(1/2*d*x + 1/2*c)^9 + 125 
440*a^3*b*tan(1/2*d*x + 1/2*c)^8 - 35840*a*b^3*tan(1/2*d*x + 1/2*c)^8 - 50 
40*a^4*tan(1/2*d*x + 1/2*c)^7 - 17360*a^2*b^2*tan(1/2*d*x + 1/2*c)^7 + 234 
85*b^4*tan(1/2*d*x + 1/2*c)^7 - 111104*a^3*b*tan(1/2*d*x + 1/2*c)^6 + 5734 
4*a*b^3*tan(1/2*d*x + 1/2*c)^6 + 11760*a^4*tan(1/2*d*x + 1/2*c)^5 - 7280*a 
^2*b^2*tan(1/2*d*x + 1/2*c)^5 + 11655*b^4*tan(1/2*d*x + 1/2*c)^5 + 46592*a 
^3*b*tan(1/2*d*x + 1/2*c)^4 + 7168*a*b^3*tan(1/2*d*x + 1/2*c)^4 - 9520*a^4 
*tan(1/2*d*x + 1/2*c)^3 + 22960*a^2*b^2*tan(1/2*d*x + 1/2*c)^3 + 805*b^4*t 
an(1/2*d*x + 1/2*c)^3 - 10752*a^3*b*tan(1/2*d*x + 1/2*c)^2 + 8192*a*b^3*ta 
n(1/2*d*x + 1/2*c)^2 + 2800*a^4*tan(1/2*d*x + 1/2*c) + 1680*a^2*b^2*tan(1/ 
2*d*x + 1/2*c) - 105*b^4*tan(1/2*d*x + 1/2*c) + 3584*a^3*b - 1024*a*b^3...
 

Mupad [B] (verification not implemented)

Time = 19.87 (sec) , antiderivative size = 566, normalized size of antiderivative = 1.72 \[ \int \sec ^9(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx =\text {Too large to display} \] Input:

int((a*cos(c + d*x) + b*sin(c + d*x))^4/cos(c + d*x)^9,x)
 

Output:

(tan(c/2 + (d*x)/2)^15*((5*a^4)/4 - (3*b^4)/64 + (3*a^2*b^2)/4) + tan(c/2 
+ (d*x)/2)^3*((23*b^4)/64 - (17*a^4)/4 + (41*a^2*b^2)/4) + tan(c/2 + (d*x) 
/2)^13*((23*b^4)/64 - (17*a^4)/4 + (41*a^2*b^2)/4) + tan(c/2 + (d*x)/2)^5* 
((21*a^4)/4 + (333*b^4)/64 - (13*a^2*b^2)/4) + tan(c/2 + (d*x)/2)^11*((21* 
a^4)/4 + (333*b^4)/64 - (13*a^2*b^2)/4) - tan(c/2 + (d*x)/2)^7*((9*a^4)/4 
- (671*b^4)/64 + (31*a^2*b^2)/4) - tan(c/2 + (d*x)/2)^9*((9*a^4)/4 - (671* 
b^4)/64 + (31*a^2*b^2)/4) - (16*a*b^3)/35 + (8*a^3*b)/5 + tan(c/2 + (d*x)/ 
2)*((5*a^4)/4 - (3*b^4)/64 + (3*a^2*b^2)/4) - tan(c/2 + (d*x)/2)^12*(16*a* 
b^3 - 24*a^3*b) - tan(c/2 + (d*x)/2)^8*(16*a*b^3 - 56*a^3*b) + tan(c/2 + ( 
d*x)/2)^4*((16*a*b^3)/5 + (104*a^3*b)/5) + tan(c/2 + (d*x)/2)^2*((128*a*b^ 
3)/35 - (24*a^3*b)/5) + tan(c/2 + (d*x)/2)^6*((128*a*b^3)/5 - (248*a^3*b)/ 
5) - 40*a^3*b*tan(c/2 + (d*x)/2)^10 - 8*a^3*b*tan(c/2 + (d*x)/2)^14)/(d*(2 
8*tan(c/2 + (d*x)/2)^4 - 8*tan(c/2 + (d*x)/2)^2 - 56*tan(c/2 + (d*x)/2)^6 
+ 70*tan(c/2 + (d*x)/2)^8 - 56*tan(c/2 + (d*x)/2)^10 + 28*tan(c/2 + (d*x)/ 
2)^12 - 8*tan(c/2 + (d*x)/2)^14 + tan(c/2 + (d*x)/2)^16 + 1)) + (atanh(tan 
(c/2 + (d*x)/2))*((3*a^4)/4 + (3*b^4)/64 - (3*a^2*b^2)/4))/d
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 1130, normalized size of antiderivative = 3.42 \[ \int \sec ^9(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx =\text {Too large to display} \] Input:

int(sec(d*x+c)^9*(a*cos(d*x+c)+b*sin(d*x+c))^4,x)
 

Output:

( - 3584*cos(c + d*x)*sin(c + d*x)**2*a**3*b + 3584*cos(c + d*x)*sin(c + d 
*x)**2*a*b**3 + 3584*cos(c + d*x)*a**3*b - 1024*cos(c + d*x)*a*b**3 - 1680 
*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**8*a**4 + 1680*log(tan((c + d*x)/2 
) - 1)*sin(c + d*x)**8*a**2*b**2 - 105*log(tan((c + d*x)/2) - 1)*sin(c + d 
*x)**8*b**4 + 6720*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**6*a**4 - 6720*l 
og(tan((c + d*x)/2) - 1)*sin(c + d*x)**6*a**2*b**2 + 420*log(tan((c + d*x) 
/2) - 1)*sin(c + d*x)**6*b**4 - 10080*log(tan((c + d*x)/2) - 1)*sin(c + d* 
x)**4*a**4 + 10080*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4*a**2*b**2 - 6 
30*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4*b**4 + 6720*log(tan((c + d*x) 
/2) - 1)*sin(c + d*x)**2*a**4 - 6720*log(tan((c + d*x)/2) - 1)*sin(c + d*x 
)**2*a**2*b**2 + 420*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*b**4 - 1680 
*log(tan((c + d*x)/2) - 1)*a**4 + 1680*log(tan((c + d*x)/2) - 1)*a**2*b**2 
 - 105*log(tan((c + d*x)/2) - 1)*b**4 + 1680*log(tan((c + d*x)/2) + 1)*sin 
(c + d*x)**8*a**4 - 1680*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**8*a**2*b* 
*2 + 105*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**8*b**4 - 6720*log(tan((c 
+ d*x)/2) + 1)*sin(c + d*x)**6*a**4 + 6720*log(tan((c + d*x)/2) + 1)*sin(c 
 + d*x)**6*a**2*b**2 - 420*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**6*b**4 
+ 10080*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**4*a**4 - 10080*log(tan((c 
+ d*x)/2) + 1)*sin(c + d*x)**4*a**2*b**2 + 630*log(tan((c + d*x)/2) + 1)*s 
in(c + d*x)**4*b**4 - 6720*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**2*a*...