3.1.20 \(\int (a+a \sec (c+d x))^2 \sin ^7(c+d x) \, dx\) [20]

Optimal. Leaf size=131 \[ \frac {2 a^2 \cos (c+d x)}{d}+\frac {3 a^2 \cos ^2(c+d x)}{d}-\frac {3 a^2 \cos ^4(c+d x)}{2 d}-\frac {2 a^2 \cos ^5(c+d x)}{5 d}+\frac {a^2 \cos ^6(c+d x)}{3 d}+\frac {a^2 \cos ^7(c+d x)}{7 d}-\frac {2 a^2 \log (\cos (c+d x))}{d}+\frac {a^2 \sec (c+d x)}{d} \]

[Out]

2*a^2*cos(d*x+c)/d+3*a^2*cos(d*x+c)^2/d-3/2*a^2*cos(d*x+c)^4/d-2/5*a^2*cos(d*x+c)^5/d+1/3*a^2*cos(d*x+c)^6/d+1
/7*a^2*cos(d*x+c)^7/d-2*a^2*ln(cos(d*x+c))/d+a^2*sec(d*x+c)/d

________________________________________________________________________________________

Rubi [A]
time = 0.12, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3957, 2915, 12, 90} \begin {gather*} \frac {a^2 \cos ^7(c+d x)}{7 d}+\frac {a^2 \cos ^6(c+d x)}{3 d}-\frac {2 a^2 \cos ^5(c+d x)}{5 d}-\frac {3 a^2 \cos ^4(c+d x)}{2 d}+\frac {3 a^2 \cos ^2(c+d x)}{d}+\frac {2 a^2 \cos (c+d x)}{d}+\frac {a^2 \sec (c+d x)}{d}-\frac {2 a^2 \log (\cos (c+d x))}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + a*Sec[c + d*x])^2*Sin[c + d*x]^7,x]

[Out]

(2*a^2*Cos[c + d*x])/d + (3*a^2*Cos[c + d*x]^2)/d - (3*a^2*Cos[c + d*x]^4)/(2*d) - (2*a^2*Cos[c + d*x]^5)/(5*d
) + (a^2*Cos[c + d*x]^6)/(3*d) + (a^2*Cos[c + d*x]^7)/(7*d) - (2*a^2*Log[Cos[c + d*x]])/d + (a^2*Sec[c + d*x])
/d

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 2915

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)
*(x_)])^(n_.), x_Symbol] :> Dist[1/(b^p*f), Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d/b)*x
)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2,
0]

Rule 3957

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Co
s[e + f*x])^p*((b + a*Sin[e + f*x])^m/Sin[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rubi steps

\begin {align*} \int (a+a \sec (c+d x))^2 \sin ^7(c+d x) \, dx &=\int (-a-a \cos (c+d x))^2 \sin ^5(c+d x) \tan ^2(c+d x) \, dx\\ &=\frac {\text {Subst}\left (\int \frac {a^2 (-a-x)^3 (-a+x)^5}{x^2} \, dx,x,-a \cos (c+d x)\right )}{a^7 d}\\ &=\frac {\text {Subst}\left (\int \frac {(-a-x)^3 (-a+x)^5}{x^2} \, dx,x,-a \cos (c+d x)\right )}{a^5 d}\\ &=\frac {\text {Subst}\left (\int \left (-2 a^6+\frac {a^8}{x^2}-\frac {2 a^7}{x}+6 a^5 x-6 a^3 x^3+2 a^2 x^4+2 a x^5-x^6\right ) \, dx,x,-a \cos (c+d x)\right )}{a^5 d}\\ &=\frac {2 a^2 \cos (c+d x)}{d}+\frac {3 a^2 \cos ^2(c+d x)}{d}-\frac {3 a^2 \cos ^4(c+d x)}{2 d}-\frac {2 a^2 \cos ^5(c+d x)}{5 d}+\frac {a^2 \cos ^6(c+d x)}{3 d}+\frac {a^2 \cos ^7(c+d x)}{7 d}-\frac {2 a^2 \log (\cos (c+d x))}{d}+\frac {a^2 \sec (c+d x)}{d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.38, size = 107, normalized size = 0.82 \begin {gather*} \frac {a^2 (25725+11760 \cos (2 (c+d x))+5250 \cos (3 (c+d x))-588 \cos (4 (c+d x))-770 \cos (5 (c+d x))-48 \cos (6 (c+d x))+70 \cos (7 (c+d x))+15 \cos (8 (c+d x))-70 \cos (c+d x) (5+384 \log (\cos (c+d x)))) \sec (c+d x)}{13440 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sec[c + d*x])^2*Sin[c + d*x]^7,x]

[Out]

(a^2*(25725 + 11760*Cos[2*(c + d*x)] + 5250*Cos[3*(c + d*x)] - 588*Cos[4*(c + d*x)] - 770*Cos[5*(c + d*x)] - 4
8*Cos[6*(c + d*x)] + 70*Cos[7*(c + d*x)] + 15*Cos[8*(c + d*x)] - 70*Cos[c + d*x]*(5 + 384*Log[Cos[c + d*x]]))*
Sec[c + d*x])/(13440*d)

________________________________________________________________________________________

Maple [A]
time = 0.14, size = 151, normalized size = 1.15

method	result	size
derivativedivides	\(\frac {a^{2} \left (\frac {\sin ^{8}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\frac {16}{5}+\sin ^{6}\left (d x +c \right )+\frac {6 \left (\sin ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (d x +c \right )\right )}{5}\right ) \cos \left (d x +c \right )\right )+2 a^{2} \left (-\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{6}-\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-\frac {a^{2} \left (\frac {16}{5}+\sin ^{6}\left (d x +c \right )+\frac {6 \left (\sin ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (d x +c \right )\right )}{5}\right ) \cos \left (d x +c \right )}{7}}{d}\)	\(151\)
default	\(\frac {a^{2} \left (\frac {\sin ^{8}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\frac {16}{5}+\sin ^{6}\left (d x +c \right )+\frac {6 \left (\sin ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (d x +c \right )\right )}{5}\right ) \cos \left (d x +c \right )\right )+2 a^{2} \left (-\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{6}-\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-\frac {a^{2} \left (\frac {16}{5}+\sin ^{6}\left (d x +c \right )+\frac {6 \left (\sin ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (d x +c \right )\right )}{5}\right ) \cos \left (d x +c \right )}{7}}{d}\)	\(151\)
risch	\(2 i a^{2} x +\frac {29 a^{2} {\mathrm e}^{2 i \left (d x +c \right )}}{64 d}+\frac {117 a^{2} {\mathrm e}^{i \left (d x +c \right )}}{128 d}+\frac {117 a^{2} {\mathrm e}^{-i \left (d x +c \right )}}{128 d}+\frac {29 a^{2} {\mathrm e}^{-2 i \left (d x +c \right )}}{64 d}-\frac {2 a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}+\frac {2 a^{2} {\mathrm e}^{i \left (d x +c \right )}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+\frac {4 i a^{2} c}{d}+\frac {a^{2} \cos \left (7 d x +7 c \right )}{448 d}+\frac {a^{2} \cos \left (6 d x +6 c \right )}{96 d}-\frac {3 a^{2} \cos \left (5 d x +5 c \right )}{320 d}-\frac {a^{2} \cos \left (4 d x +4 c \right )}{8 d}-\frac {5 a^{2} \cos \left (3 d x +3 c \right )}{64 d}\)	\(222\)
norman	\(\frac {-\frac {192 a^{2}}{35 d}-\frac {64 a^{2} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {4 a^{2} \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {24 a^{2} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {172 a^{2} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {264 a^{2} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}-\frac {292 a^{2} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}-\frac {1012 a^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {2 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {2 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}+\frac {2 a^{2} \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\)	\(237\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sec(d*x+c))^2*sin(d*x+c)^7,x,method=_RETURNVERBOSE)

[Out]

1/d*(a^2*(sin(d*x+c)^8/cos(d*x+c)+(16/5+sin(d*x+c)^6+6/5*sin(d*x+c)^4+8/5*sin(d*x+c)^2)*cos(d*x+c))+2*a^2*(-1/
6*sin(d*x+c)^6-1/4*sin(d*x+c)^4-1/2*sin(d*x+c)^2-ln(cos(d*x+c)))-1/7*a^2*(16/5+sin(d*x+c)^6+6/5*sin(d*x+c)^4+8
/5*sin(d*x+c)^2)*cos(d*x+c))

________________________________________________________________________________________

Maxima [A]
time = 0.29, size = 107, normalized size = 0.82 \begin {gather*} \frac {30 \, a^{2} \cos \left (d x + c\right )^{7} + 70 \, a^{2} \cos \left (d x + c\right )^{6} - 84 \, a^{2} \cos \left (d x + c\right )^{5} - 315 \, a^{2} \cos \left (d x + c\right )^{4} + 630 \, a^{2} \cos \left (d x + c\right )^{2} + 420 \, a^{2} \cos \left (d x + c\right ) - 420 \, a^{2} \log \left (\cos \left (d x + c\right )\right ) + \frac {210 \, a^{2}}{\cos \left (d x + c\right )}}{210 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^2*sin(d*x+c)^7,x, algorithm="maxima")

[Out]

1/210*(30*a^2*cos(d*x + c)^7 + 70*a^2*cos(d*x + c)^6 - 84*a^2*cos(d*x + c)^5 - 315*a^2*cos(d*x + c)^4 + 630*a^
2*cos(d*x + c)^2 + 420*a^2*cos(d*x + c) - 420*a^2*log(cos(d*x + c)) + 210*a^2/cos(d*x + c))/d

________________________________________________________________________________________

Fricas [A]
time = 3.01, size = 128, normalized size = 0.98 \begin {gather*} \frac {120 \, a^{2} \cos \left (d x + c\right )^{8} + 280 \, a^{2} \cos \left (d x + c\right )^{7} - 336 \, a^{2} \cos \left (d x + c\right )^{6} - 1260 \, a^{2} \cos \left (d x + c\right )^{5} + 2520 \, a^{2} \cos \left (d x + c\right )^{3} + 1680 \, a^{2} \cos \left (d x + c\right )^{2} - 1680 \, a^{2} \cos \left (d x + c\right ) \log \left (-\cos \left (d x + c\right )\right ) - 875 \, a^{2} \cos \left (d x + c\right ) + 840 \, a^{2}}{840 \, d \cos \left (d x + c\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^2*sin(d*x+c)^7,x, algorithm="fricas")

[Out]

1/840*(120*a^2*cos(d*x + c)^8 + 280*a^2*cos(d*x + c)^7 - 336*a^2*cos(d*x + c)^6 - 1260*a^2*cos(d*x + c)^5 + 25
20*a^2*cos(d*x + c)^3 + 1680*a^2*cos(d*x + c)^2 - 1680*a^2*cos(d*x + c)*log(-cos(d*x + c)) - 875*a^2*cos(d*x +
c) + 840*a^2)/(d*cos(d*x + c))

________________________________________________________________________________________

Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))**2*sin(d*x+c)**7,x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3005 deep

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 320 vs. \(2 (123) = 246\).
time = 0.58, size = 320, normalized size = 2.44 \begin {gather*} \frac {420 \, a^{2} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 420 \, a^{2} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {420 \, {\left (2 \, a^{2} + \frac {a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )}}{\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1} + \frac {357 \, a^{2} - \frac {3759 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {16737 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {42595 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {43855 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {25389 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {8043 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {1089 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{7}}}{210 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^2*sin(d*x+c)^7,x, algorithm="giac")

[Out]

1/210*(420*a^2*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)) - 420*a^2*log(abs(-(cos(d*x + c) - 1)/(cos
(d*x + c) + 1) - 1)) + 420*(2*a^2 + a^2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1))/((cos(d*x + c) - 1)/(cos(d*x +
c) + 1) + 1) + (357*a^2 - 3759*a^2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 16737*a^2*(cos(d*x + c) - 1)^2/(cos
(d*x + c) + 1)^2 - 42595*a^2*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 43855*a^2*(cos(d*x + c) - 1)^4/(cos(d
*x + c) + 1)^4 - 25389*a^2*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5 + 8043*a^2*(cos(d*x + c) - 1)^6/(cos(d*x
+ c) + 1)^6 - 1089*a^2*(cos(d*x + c) - 1)^7/(cos(d*x + c) + 1)^7)/((cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1)^
7)/d

________________________________________________________________________________________

Mupad [B]
time = 0.95, size = 105, normalized size = 0.80 \begin {gather*} \frac {2\,a^2\,\cos \left (c+d\,x\right )+\frac {a^2}{\cos \left (c+d\,x\right )}+3\,a^2\,{\cos \left (c+d\,x\right )}^2-\frac {3\,a^2\,{\cos \left (c+d\,x\right )}^4}{2}-\frac {2\,a^2\,{\cos \left (c+d\,x\right )}^5}{5}+\frac {a^2\,{\cos \left (c+d\,x\right )}^6}{3}+\frac {a^2\,{\cos \left (c+d\,x\right )}^7}{7}-2\,a^2\,\ln \left (\cos \left (c+d\,x\right )\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(c + d*x)^7*(a + a/cos(c + d*x))^2,x)

[Out]

(2*a^2*cos(c + d*x) + a^2/cos(c + d*x) + 3*a^2*cos(c + d*x)^2 - (3*a^2*cos(c + d*x)^4)/2 - (2*a^2*cos(c + d*x)
^5)/5 + (a^2*cos(c + d*x)^6)/3 + (a^2*cos(c + d*x)^7)/7 - 2*a^2*log(cos(c + d*x)))/d

________________________________________________________________________________________