\(\int (b \sec (c+d x)+a \sin (c+d x))^3 (a \cos (c+d x)+b \sec (c+d x) \tan (c+d x)) \, dx\) [638]

Optimal result

Integrand size = 43, antiderivative size = 26 \[ \int (b \sec (c+d x)+a \sin (c+d x))^3 (a \cos (c+d x)+b \sec (c+d x) \tan (c+d x)) \, dx=\frac {(b \sec (c+d x)+a \sin (c+d x))^4}{4 d} \]

[Out]

1/4*(b*sec(d*x+c)+a*sin(d*x+c))^4/d

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {4470} \[ \int (b \sec (c+d x)+a \sin (c+d x))^3 (a \cos (c+d x)+b \sec (c+d x) \tan (c+d x)) \, dx=\frac {(a \sin (c+d x)+b \sec (c+d x))^4}{4 d} \]

[In]

Int[(b*Sec[c + d*x] + a*Sin[c + d*x])^3*(a*Cos[c + d*x] + b*Sec[c + d*x]*Tan[c + d*x]),x]

[Out]

(b*Sec[c + d*x] + a*Sin[c + d*x])^4/(4*d)

Rule 4470

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[ActivateTrig[y], ActivateTrig[u], x]}, Simp[q*(A
ctivateTrig[y^(m + 1)]/(m + 1)), x] /;  !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1] &&  !InertTrigFreeQ[u]

Rubi steps \begin{align*} \text {integral}& = \frac {(b \sec (c+d x)+a \sin (c+d x))^4}{4 d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(938\) vs. \(2(26)=52\).

Time = 12.23 (sec) , antiderivative size = 938, normalized size of antiderivative = 36.08 \[ \int (b \sec (c+d x)+a \sin (c+d x))^3 (a \cos (c+d x)+b \sec (c+d x) \tan (c+d x)) \, dx=\frac {8 b^4 \cos (c+d x) (b \sec (c+d x)+a \sin (c+d x))^3 (a \cos (c+d x)+b \sec (c+d x) \tan (c+d x))}{d (3 a \cos (c+d x)+a \cos (3 c+3 d x)+4 b \sin (c+d x)) (2 b+a \sin (2 c+2 d x))^3}+\frac {a^4 \cos (4 c) \cos (4 d x) \cos ^5(c+d x) (b \sec (c+d x)+a \sin (c+d x))^3 (a \cos (c+d x)+b \sec (c+d x) \tan (c+d x))}{d (3 a \cos (c+d x)+a \cos (3 c+3 d x)+4 b \sin (c+d x)) (2 b+a \sin (2 c+2 d x))^3}+\frac {16 a b^2 \cos ^3(c+d x) \sec (c) (3 a \cos (c)+2 b \sin (c)) (b \sec (c+d x)+a \sin (c+d x))^3 (a \cos (c+d x)+b \sec (c+d x) \tan (c+d x))}{d (3 a \cos (c+d x)+a \cos (3 c+3 d x)+4 b \sin (c+d x)) (2 b+a \sin (2 c+2 d x))^3}-\frac {4 a^3 \cos (2 d x) \cos ^5(c+d x) (a \cos (2 c)+4 b \sin (2 c)) (b \sec (c+d x)+a \sin (c+d x))^3 (a \cos (c+d x)+b \sec (c+d x) \tan (c+d x))}{d (3 a \cos (c+d x)+a \cos (3 c+3 d x)+4 b \sin (c+d x)) (2 b+a \sin (2 c+2 d x))^3}+\frac {32 a b^3 \cos ^2(c+d x) \sec (c) \sin (d x) (b \sec (c+d x)+a \sin (c+d x))^3 (a \cos (c+d x)+b \sec (c+d x) \tan (c+d x))}{d (3 a \cos (c+d x)+a \cos (3 c+3 d x)+4 b \sin (c+d x)) (2 b+a \sin (2 c+2 d x))^3}+\frac {32 a^3 b \cos ^4(c+d x) \sec (c) \sin (d x) (b \sec (c+d x)+a \sin (c+d x))^3 (a \cos (c+d x)+b \sec (c+d x) \tan (c+d x))}{d (3 a \cos (c+d x)+a \cos (3 c+3 d x)+4 b \sin (c+d x)) (2 b+a \sin (2 c+2 d x))^3}+\frac {4 a^3 \cos ^5(c+d x) (-4 b \cos (2 c)+a \sin (2 c)) \sin (2 d x) (b \sec (c+d x)+a \sin (c+d x))^3 (a \cos (c+d x)+b \sec (c+d x) \tan (c+d x))}{d (3 a \cos (c+d x)+a \cos (3 c+3 d x)+4 b \sin (c+d x)) (2 b+a \sin (2 c+2 d x))^3}-\frac {a^4 \cos ^5(c+d x) \sin (4 c) \sin (4 d x) (b \sec (c+d x)+a \sin (c+d x))^3 (a \cos (c+d x)+b \sec (c+d x) \tan (c+d x))}{d (3 a \cos (c+d x)+a \cos (3 c+3 d x)+4 b \sin (c+d x)) (2 b+a \sin (2 c+2 d x))^3} \]

[In]

Integrate[(b*Sec[c + d*x] + a*Sin[c + d*x])^3*(a*Cos[c + d*x] + b*Sec[c + d*x]*Tan[c + d*x]),x]

[Out]

(8*b^4*Cos[c + d*x]*(b*Sec[c + d*x] + a*Sin[c + d*x])^3*(a*Cos[c + d*x] + b*Sec[c + d*x]*Tan[c + d*x]))/(d*(3*
a*Cos[c + d*x] + a*Cos[3*c + 3*d*x] + 4*b*Sin[c + d*x])*(2*b + a*Sin[2*c + 2*d*x])^3) + (a^4*Cos[4*c]*Cos[4*d*
x]*Cos[c + d*x]^5*(b*Sec[c + d*x] + a*Sin[c + d*x])^3*(a*Cos[c + d*x] + b*Sec[c + d*x]*Tan[c + d*x]))/(d*(3*a*
Cos[c + d*x] + a*Cos[3*c + 3*d*x] + 4*b*Sin[c + d*x])*(2*b + a*Sin[2*c + 2*d*x])^3) + (16*a*b^2*Cos[c + d*x]^3
*Sec[c]*(3*a*Cos[c] + 2*b*Sin[c])*(b*Sec[c + d*x] + a*Sin[c + d*x])^3*(a*Cos[c + d*x] + b*Sec[c + d*x]*Tan[c +
d*x]))/(d*(3*a*Cos[c + d*x] + a*Cos[3*c + 3*d*x] + 4*b*Sin[c + d*x])*(2*b + a*Sin[2*c + 2*d*x])^3) - (4*a^3*C
os[2*d*x]*Cos[c + d*x]^5*(a*Cos[2*c] + 4*b*Sin[2*c])*(b*Sec[c + d*x] + a*Sin[c + d*x])^3*(a*Cos[c + d*x] + b*S
ec[c + d*x]*Tan[c + d*x]))/(d*(3*a*Cos[c + d*x] + a*Cos[3*c + 3*d*x] + 4*b*Sin[c + d*x])*(2*b + a*Sin[2*c + 2*
d*x])^3) + (32*a*b^3*Cos[c + d*x]^2*Sec[c]*Sin[d*x]*(b*Sec[c + d*x] + a*Sin[c + d*x])^3*(a*Cos[c + d*x] + b*Se
c[c + d*x]*Tan[c + d*x]))/(d*(3*a*Cos[c + d*x] + a*Cos[3*c + 3*d*x] + 4*b*Sin[c + d*x])*(2*b + a*Sin[2*c + 2*d
*x])^3) + (32*a^3*b*Cos[c + d*x]^4*Sec[c]*Sin[d*x]*(b*Sec[c + d*x] + a*Sin[c + d*x])^3*(a*Cos[c + d*x] + b*Sec
[c + d*x]*Tan[c + d*x]))/(d*(3*a*Cos[c + d*x] + a*Cos[3*c + 3*d*x] + 4*b*Sin[c + d*x])*(2*b + a*Sin[2*c + 2*d*
x])^3) + (4*a^3*Cos[c + d*x]^5*(-4*b*Cos[2*c] + a*Sin[2*c])*Sin[2*d*x]*(b*Sec[c + d*x] + a*Sin[c + d*x])^3*(a*
Cos[c + d*x] + b*Sec[c + d*x]*Tan[c + d*x]))/(d*(3*a*Cos[c + d*x] + a*Cos[3*c + 3*d*x] + 4*b*Sin[c + d*x])*(2*
b + a*Sin[2*c + 2*d*x])^3) - (a^4*Cos[c + d*x]^5*Sin[4*c]*Sin[4*d*x]*(b*Sec[c + d*x] + a*Sin[c + d*x])^3*(a*Co
s[c + d*x] + b*Sec[c + d*x]*Tan[c + d*x]))/(d*(3*a*Cos[c + d*x] + a*Cos[3*c + 3*d*x] + 4*b*Sin[c + d*x])*(2*b
+ a*Sin[2*c + 2*d*x])^3)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(186\) vs. \(2(24)=48\).

Time = 222.00 (sec) , antiderivative size = 187, normalized size of antiderivative = 7.19

method	result	size
derivativedivides	\(\frac {\frac {a^{4} \sin \left (d x +c \right )^{4}}{4}+a^{3} b \left (\frac {\sin \left (d x +c \right )^{5}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{3}+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )+3 a^{3} b \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 a^{2} b^{2} \left (\frac {\tan \left (d x +c \right )^{2}}{2}+\ln \left (\cos \left (d x +c \right )\right )\right )-3 a^{2} b^{2} \ln \left (\cos \left (d x +c \right )\right )+\frac {a \,b^{3} \sin \left (d x +c \right )^{3}}{\cos \left (d x +c \right )^{3}}+a \,b^{3} \tan \left (d x +c \right )+\frac {b^{4}}{4 \cos \left (d x +c \right )^{4}}}{d}\)	\(187\)
default	\(\frac {\frac {a^{4} \sin \left (d x +c \right )^{4}}{4}+a^{3} b \left (\frac {\sin \left (d x +c \right )^{5}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{3}+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )+3 a^{3} b \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 a^{2} b^{2} \left (\frac {\tan \left (d x +c \right )^{2}}{2}+\ln \left (\cos \left (d x +c \right )\right )\right )-3 a^{2} b^{2} \ln \left (\cos \left (d x +c \right )\right )+\frac {a \,b^{3} \sin \left (d x +c \right )^{3}}{\cos \left (d x +c \right )^{3}}+a \,b^{3} \tan \left (d x +c \right )+\frac {b^{4}}{4 \cos \left (d x +c \right )^{4}}}{d}\)	\(187\)
parts	\(\frac {\frac {a^{4} \sin \left (d x +c \right )^{4}}{4}+3 a^{3} b \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )-3 a^{2} b^{2} \ln \left (\cos \left (d x +c \right )\right )+a \,b^{3} \tan \left (d x +c \right )}{d}+\frac {b^{4} \sec \left (d x +c \right )^{4}}{4 d}+\frac {a^{3} b \left (\frac {\sin \left (d x +c \right )^{5}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{3}+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )}{d}+\frac {3 a^{2} b^{2} \left (\frac {\tan \left (d x +c \right )^{2}}{2}+\ln \left (\cos \left (d x +c \right )\right )\right )}{d}+\frac {a \,b^{3} \tan \left (d x +c \right )^{3}}{d}\)	\(192\)
risch	\(\frac {a^{4} {\mathrm e}^{4 i \left (d x +c \right )}}{64 d}+\frac {i a^{3} {\mathrm e}^{2 i \left (d x +c \right )} b}{4 d}-\frac {a^{4} {\mathrm e}^{2 i \left (d x +c \right )}}{16 d}-\frac {i a^{3} {\mathrm e}^{-2 i \left (d x +c \right )} b}{4 d}-\frac {a^{4} {\mathrm e}^{-2 i \left (d x +c \right )}}{16 d}+\frac {a^{4} {\mathrm e}^{-4 i \left (d x +c \right )}}{64 d}+\frac {2 b \left (i a^{3} {\mathrm e}^{6 i \left (d x +c \right )}-2 i a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+3 a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}+3 i a^{3} {\mathrm e}^{4 i \left (d x +c \right )}+6 a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}+2 b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+3 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+2 i a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+3 a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}+i a^{3}\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}\)	\(270\)

[In]

int((b*sec(d*x+c)+sin(d*x+c)*a)^3*(cos(d*x+c)*a+b*sec(d*x+c)*tan(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d*(1/4*a^4*sin(d*x+c)^4+a^3*b*(sin(d*x+c)^5/cos(d*x+c)+(sin(d*x+c)^3+3/2*sin(d*x+c))*cos(d*x+c)-3/2*d*x-3/2*
c)+3*a^3*b*(-1/2*sin(d*x+c)*cos(d*x+c)+1/2*d*x+1/2*c)+3*a^2*b^2*(1/2*tan(d*x+c)^2+ln(cos(d*x+c)))-3*a^2*b^2*ln
(cos(d*x+c))+a*b^3*sin(d*x+c)^3/cos(d*x+c)^3+a*b^3*tan(d*x+c)+1/4*b^4/cos(d*x+c)^4)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (24) = 48\).

Time = 0.27 (sec) , antiderivative size = 122, normalized size of antiderivative = 4.69 \[ \int (b \sec (c+d x)+a \sin (c+d x))^3 (a \cos (c+d x)+b \sec (c+d x) \tan (c+d x)) \, dx=\frac {8 \, a^{4} \cos \left (d x + c\right )^{8} - 16 \, a^{4} \cos \left (d x + c\right )^{6} + 5 \, a^{4} \cos \left (d x + c\right )^{4} + 48 \, a^{2} b^{2} \cos \left (d x + c\right )^{2} + 8 \, b^{4} - 32 \, {\left (a^{3} b \cos \left (d x + c\right )^{5} - a^{3} b \cos \left (d x + c\right )^{3} - a b^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{32 \, d \cos \left (d x + c\right )^{4}} \]

[In]

integrate((b*sec(d*x+c)+a*sin(d*x+c))^3*(a*cos(d*x+c)+b*sec(d*x+c)*tan(d*x+c)),x, algorithm="fricas")

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (20) = 40\).

Time = 2.51 (sec) , antiderivative size = 129, normalized size of antiderivative = 4.96 \[ \int (b \sec (c+d x)+a \sin (c+d x))^3 (a \cos (c+d x)+b \sec (c+d x) \tan (c+d x)) \, dx=\begin {cases} \frac {a^{4} \sin ^{4}{\left (c + d x \right )}}{4 d} + \frac {a^{3} b \sin ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{d} + \frac {3 a^{2} b^{2} \sin ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{2 d} + \frac {a b^{3} \sin {\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}}{d} + \frac {b^{4} \sec ^{4}{\left (c + d x \right )}}{4 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + b \sec {\left (c \right )}\right )^{3} \left (a \cos {\left (c \right )} + b \tan {\left (c \right )} \sec {\left (c \right )}\right ) & \text {otherwise} \end {cases} \]

[In]

integrate((b*sec(d*x+c)+a*sin(d*x+c))**3*(a*cos(d*x+c)+b*sec(d*x+c)*tan(d*x+c)),x)

[Out]

Piecewise((a**4*sin(c + d*x)**4/(4*d) + a**3*b*sin(c + d*x)**3*sec(c + d*x)/d + 3*a**2*b**2*sin(c + d*x)**2*se
c(c + d*x)**2/(2*d) + a*b**3*sin(c + d*x)*sec(c + d*x)**3/d + b**4*sec(c + d*x)**4/(4*d), Ne(d, 0)), (x*(a*sin
(c) + b*sec(c))**3*(a*cos(c) + b*tan(c)*sec(c)), True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int (b \sec (c+d x)+a \sin (c+d x))^3 (a \cos (c+d x)+b \sec (c+d x) \tan (c+d x)) \, dx=\frac {{\left (b \sec \left (d x + c\right ) + a \sin \left (d x + c\right )\right )}^{4}}{4 \, d} \]

[In]

integrate((b*sec(d*x+c)+a*sin(d*x+c))^3*(a*cos(d*x+c)+b*sec(d*x+c)*tan(d*x+c)),x, algorithm="maxima")

[Out]

1/4*(b*sec(d*x + c) + a*sin(d*x + c))^4/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 142 vs. \(2 (24) = 48\).

Time = 0.81 (sec) , antiderivative size = 142, normalized size of antiderivative = 5.46 \[ \int (b \sec (c+d x)+a \sin (c+d x))^3 (a \cos (c+d x)+b \sec (c+d x) \tan (c+d x)) \, dx=\frac {b^{4} \tan \left (d x + c\right )^{4} + 4 \, a b^{3} \tan \left (d x + c\right )^{3} + 6 \, a^{2} b^{2} \tan \left (d x + c\right )^{2} + 2 \, b^{4} \tan \left (d x + c\right )^{2} + 4 \, a^{3} b \tan \left (d x + c\right ) + 4 \, a b^{3} \tan \left (d x + c\right ) - \frac {4 \, a^{3} b \tan \left (d x + c\right )^{3} + 2 \, a^{4} \tan \left (d x + c\right )^{2} + 4 \, a^{3} b \tan \left (d x + c\right ) + a^{4}}{{\left (\tan \left (d x + c\right )^{2} + 1\right )}^{2}}}{4 \, d} \]

[In]

integrate((b*sec(d*x+c)+a*sin(d*x+c))^3*(a*cos(d*x+c)+b*sec(d*x+c)*tan(d*x+c)),x, algorithm="giac")

[Out]

1/4*(b^4*tan(d*x + c)^4 + 4*a*b^3*tan(d*x + c)^3 + 6*a^2*b^2*tan(d*x + c)^2 + 2*b^4*tan(d*x + c)^2 + 4*a^3*b*t
an(d*x + c) + 4*a*b^3*tan(d*x + c) - (4*a^3*b*tan(d*x + c)^3 + 2*a^4*tan(d*x + c)^2 + 4*a^3*b*tan(d*x + c) + a
^4)/(tan(d*x + c)^2 + 1)^2)/d

Mupad [B] (verification not implemented)

Time = 27.19 (sec) , antiderivative size = 185, normalized size of antiderivative = 7.12 \[ \int (b \sec (c+d x)+a \sin (c+d x))^3 (a \cos (c+d x)+b \sec (c+d x) \tan (c+d x)) \, dx=\frac {a^4\,{\cos \left (2\,c+2\,d\,x\right )}^4-2\,a^4\,{\cos \left (2\,c+2\,d\,x\right )}^2+a^4-8\,\sin \left (2\,c+2\,d\,x\right )\,a^3\,b\,{\cos \left (2\,c+2\,d\,x\right )}^2+8\,\sin \left (2\,c+2\,d\,x\right )\,a^3\,b-24\,a^2\,b^2\,{\cos \left (2\,c+2\,d\,x\right )}^2+24\,a^2\,b^2+32\,\sin \left (2\,c+2\,d\,x\right )\,a\,b^3-4\,b^4\,{\cos \left (2\,c+2\,d\,x\right )}^2-8\,b^4\,\cos \left (2\,c+2\,d\,x\right )+12\,b^4}{d\,\left (16\,{\cos \left (2\,c+2\,d\,x\right )}^2+32\,\cos \left (2\,c+2\,d\,x\right )+16\right )} \]

[In]

int((a*sin(c + d*x) + b/cos(c + d*x))^3*(a*cos(c + d*x) + (b*tan(c + d*x))/cos(c + d*x)),x)

[Out]

(a^4*cos(2*c + 2*d*x)^4 - 2*a^4*cos(2*c + 2*d*x)^2 - 4*b^4*cos(2*c + 2*d*x)^2 + a^4 + 12*b^4 + 24*a^2*b^2 - 8*
b^4*cos(2*c + 2*d*x) - 24*a^2*b^2*cos(2*c + 2*d*x)^2 + 32*a*b^3*sin(2*c + 2*d*x) + 8*a^3*b*sin(2*c + 2*d*x) -
8*a^3*b*cos(2*c + 2*d*x)^2*sin(2*c + 2*d*x))/(d*(32*cos(2*c + 2*d*x) + 16*cos(2*c + 2*d*x)^2 + 16))