3.416 \(\int \sec (c+d x) (a+b \sin (c+d x))^8 \, dx\)

Optimal. Leaf size=245 \[ -\frac{b^6 \left (28 a^2+b^2\right ) \sin ^5(c+d x)}{5 d}-\frac{2 a b^5 \left (7 a^2+b^2\right ) \sin ^4(c+d x)}{d}-\frac{b^4 \left (28 a^2 b^2+70 a^4+b^4\right ) \sin ^3(c+d x)}{3 d}-\frac{4 a b^3 \left (7 a^2 b^2+7 a^4+b^4\right ) \sin ^2(c+d x)}{d}-\frac{b^2 \left (70 a^4 b^2+28 a^2 b^4+28 a^6+b^6\right ) \sin (c+d x)}{d}-\frac{4 a b^7 \sin ^6(c+d x)}{3 d}+\frac{(a-b)^8 \log (\sin (c+d x)+1)}{2 d}-\frac{(a+b)^8 \log (1-\sin (c+d x))}{2 d}-\frac{b^8 \sin ^7(c+d x)}{7 d} \]

[Out]

-((a + b)^8*Log[1 - Sin[c + d*x]])/(2*d) + ((a - b)^8*Log[1 + Sin[c + d*x]])/(2*d) - (b^2*(28*a^6 + 70*a^4*b^2
+ 28*a^2*b^4 + b^6)*Sin[c + d*x])/d - (4*a*b^3*(7*a^4 + 7*a^2*b^2 + b^4)*Sin[c + d*x]^2)/d - (b^4*(70*a^4 + 2
8*a^2*b^2 + b^4)*Sin[c + d*x]^3)/(3*d) - (2*a*b^5*(7*a^2 + b^2)*Sin[c + d*x]^4)/d - (b^6*(28*a^2 + b^2)*Sin[c
+ d*x]^5)/(5*d) - (4*a*b^7*Sin[c + d*x]^6)/(3*d) - (b^8*Sin[c + d*x]^7)/(7*d)

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Rubi [A]  time = 0.182292, antiderivative size = 245, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {2668, 702, 633, 31} \[ -\frac{b^6 \left (28 a^2+b^2\right ) \sin ^5(c+d x)}{5 d}-\frac{2 a b^5 \left (7 a^2+b^2\right ) \sin ^4(c+d x)}{d}-\frac{b^4 \left (28 a^2 b^2+70 a^4+b^4\right ) \sin ^3(c+d x)}{3 d}-\frac{4 a b^3 \left (7 a^2 b^2+7 a^4+b^4\right ) \sin ^2(c+d x)}{d}-\frac{b^2 \left (70 a^4 b^2+28 a^2 b^4+28 a^6+b^6\right ) \sin (c+d x)}{d}-\frac{4 a b^7 \sin ^6(c+d x)}{3 d}+\frac{(a-b)^8 \log (\sin (c+d x)+1)}{2 d}-\frac{(a+b)^8 \log (1-\sin (c+d x))}{2 d}-\frac{b^8 \sin ^7(c+d x)}{7 d} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]*(a + b*Sin[c + d*x])^8,x]

[Out]

-((a + b)^8*Log[1 - Sin[c + d*x]])/(2*d) + ((a - b)^8*Log[1 + Sin[c + d*x]])/(2*d) - (b^2*(28*a^6 + 70*a^4*b^2
+ 28*a^2*b^4 + b^6)*Sin[c + d*x])/d - (4*a*b^3*(7*a^4 + 7*a^2*b^2 + b^4)*Sin[c + d*x]^2)/d - (b^4*(70*a^4 + 2
8*a^2*b^2 + b^4)*Sin[c + d*x]^3)/(3*d) - (2*a*b^5*(7*a^2 + b^2)*Sin[c + d*x]^4)/d - (b^6*(28*a^2 + b^2)*Sin[c
+ d*x]^5)/(5*d) - (4*a*b^7*Sin[c + d*x]^6)/(3*d) - (b^8*Sin[c + d*x]^7)/(7*d)

Rule 2668

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 702

Int[((d_) + (e_.)*(x_))^(m_)/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[(d + e*x)^m, a + c*x^2,
x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[m, 1] && (NeQ[d, 0] || GtQ[m, 2])

Rule 633

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[-(a*c), 2]}, Dist[e/2 + (c*d)/(2*q),
Int[1/(-q + c*x), x], x] + Dist[e/2 - (c*d)/(2*q), Int[1/(q + c*x), x], x]] /; FreeQ[{a, c, d, e}, x] && NiceS
qrtQ[-(a*c)]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \sec (c+d x) (a+b \sin (c+d x))^8 \, dx &=\frac{b \operatorname{Subst}\left (\int \frac{(a+x)^8}{b^2-x^2} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac{b \operatorname{Subst}\left (\int \left (-28 a^6-70 a^4 b^2-28 a^2 b^4-b^6-8 a \left (7 a^4+7 a^2 b^2+b^4\right ) x-\left (70 a^4+28 a^2 b^2+b^4\right ) x^2-8 a \left (7 a^2+b^2\right ) x^3-\left (28 a^2+b^2\right ) x^4-8 a x^5-x^6+\frac{a^8+28 a^6 b^2+70 a^4 b^4+28 a^2 b^6+b^8+8 a \left (a^2+b^2\right ) \left (a^4+6 a^2 b^2+b^4\right ) x}{b^2-x^2}\right ) \, dx,x,b \sin (c+d x)\right )}{d}\\ &=-\frac{b^2 \left (28 a^6+70 a^4 b^2+28 a^2 b^4+b^6\right ) \sin (c+d x)}{d}-\frac{4 a b^3 \left (7 a^4+7 a^2 b^2+b^4\right ) \sin ^2(c+d x)}{d}-\frac{b^4 \left (70 a^4+28 a^2 b^2+b^4\right ) \sin ^3(c+d x)}{3 d}-\frac{2 a b^5 \left (7 a^2+b^2\right ) \sin ^4(c+d x)}{d}-\frac{b^6 \left (28 a^2+b^2\right ) \sin ^5(c+d x)}{5 d}-\frac{4 a b^7 \sin ^6(c+d x)}{3 d}-\frac{b^8 \sin ^7(c+d x)}{7 d}+\frac{b \operatorname{Subst}\left (\int \frac{a^8+28 a^6 b^2+70 a^4 b^4+28 a^2 b^6+b^8+8 a \left (a^2+b^2\right ) \left (a^4+6 a^2 b^2+b^4\right ) x}{b^2-x^2} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=-\frac{b^2 \left (28 a^6+70 a^4 b^2+28 a^2 b^4+b^6\right ) \sin (c+d x)}{d}-\frac{4 a b^3 \left (7 a^4+7 a^2 b^2+b^4\right ) \sin ^2(c+d x)}{d}-\frac{b^4 \left (70 a^4+28 a^2 b^2+b^4\right ) \sin ^3(c+d x)}{3 d}-\frac{2 a b^5 \left (7 a^2+b^2\right ) \sin ^4(c+d x)}{d}-\frac{b^6 \left (28 a^2+b^2\right ) \sin ^5(c+d x)}{5 d}-\frac{4 a b^7 \sin ^6(c+d x)}{3 d}-\frac{b^8 \sin ^7(c+d x)}{7 d}-\frac{(a-b)^8 \operatorname{Subst}\left (\int \frac{1}{-b-x} \, dx,x,b \sin (c+d x)\right )}{2 d}+\frac{(a+b)^8 \operatorname{Subst}\left (\int \frac{1}{b-x} \, dx,x,b \sin (c+d x)\right )}{2 d}\\ &=-\frac{(a+b)^8 \log (1-\sin (c+d x))}{2 d}+\frac{(a-b)^8 \log (1+\sin (c+d x))}{2 d}-\frac{b^2 \left (28 a^6+70 a^4 b^2+28 a^2 b^4+b^6\right ) \sin (c+d x)}{d}-\frac{4 a b^3 \left (7 a^4+7 a^2 b^2+b^4\right ) \sin ^2(c+d x)}{d}-\frac{b^4 \left (70 a^4+28 a^2 b^2+b^4\right ) \sin ^3(c+d x)}{3 d}-\frac{2 a b^5 \left (7 a^2+b^2\right ) \sin ^4(c+d x)}{d}-\frac{b^6 \left (28 a^2+b^2\right ) \sin ^5(c+d x)}{5 d}-\frac{4 a b^7 \sin ^6(c+d x)}{3 d}-\frac{b^8 \sin ^7(c+d x)}{7 d}\\ \end{align*}

Mathematica [A]  time = 0.216993, size = 227, normalized size = 0.93 \[ \frac{b \left (-\frac{1}{5} b^5 \left (28 a^2+b^2\right ) \sin ^5(c+d x)-2 a b^4 \left (7 a^2+b^2\right ) \sin ^4(c+d x)-\frac{1}{3} b^3 \left (28 a^2 b^2+70 a^4+b^4\right ) \sin ^3(c+d x)-4 a b^2 \left (7 a^2 b^2+7 a^4+b^4\right ) \sin ^2(c+d x)-b \left (70 a^4 b^2+28 a^2 b^4+28 a^6+b^6\right ) \sin (c+d x)-\frac{4}{3} a b^6 \sin ^6(c+d x)+\frac{(a-b)^8 \log (\sin (c+d x)+1)}{2 b}-\frac{(a+b)^8 \log (1-\sin (c+d x))}{2 b}-\frac{1}{7} b^7 \sin ^7(c+d x)\right )}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]*(a + b*Sin[c + d*x])^8,x]

[Out]

(b*(-((a + b)^8*Log[1 - Sin[c + d*x]])/(2*b) + ((a - b)^8*Log[1 + Sin[c + d*x]])/(2*b) - b*(28*a^6 + 70*a^4*b^
2 + 28*a^2*b^4 + b^6)*Sin[c + d*x] - 4*a*b^2*(7*a^4 + 7*a^2*b^2 + b^4)*Sin[c + d*x]^2 - (b^3*(70*a^4 + 28*a^2*
b^2 + b^4)*Sin[c + d*x]^3)/3 - 2*a*b^4*(7*a^2 + b^2)*Sin[c + d*x]^4 - (b^5*(28*a^2 + b^2)*Sin[c + d*x]^5)/5 -
(4*a*b^6*Sin[c + d*x]^6)/3 - (b^7*Sin[c + d*x]^7)/7))/d

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Maple [A]  time = 0.103, size = 465, normalized size = 1.9 \begin{align*}{\frac{{a}^{8}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}-{\frac{{b}^{8}\sin \left ( dx+c \right ) }{d}}-{\frac{{b}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+{\frac{{b}^{8}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}-{\frac{{b}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{5\,d}}-{\frac{{b}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{7\,d}}-70\,{\frac{{a}^{4}{b}^{4}\sin \left ( dx+c \right ) }{d}}+70\,{\frac{{a}^{4}{b}^{4}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}-14\,{\frac{{a}^{3}{b}^{5} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{d}}-28\,{\frac{{a}^{3}{b}^{5} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{d}}-56\,{\frac{{a}^{3}{b}^{5}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}-{\frac{28\,{a}^{2}{b}^{6} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{5\,d}}-{\frac{28\,{a}^{2}{b}^{6} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-28\,{\frac{{a}^{2}{b}^{6}\sin \left ( dx+c \right ) }{d}}+28\,{\frac{{a}^{2}{b}^{6}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}-2\,{\frac{a{b}^{7} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{d}}-4\,{\frac{a{b}^{7} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{d}}-8\,{\frac{a{b}^{7}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}-8\,{\frac{{a}^{7}b\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}-28\,{\frac{{a}^{6}{b}^{2}\sin \left ( dx+c \right ) }{d}}+28\,{\frac{{a}^{6}{b}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}-28\,{\frac{{a}^{5}{b}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{d}}-56\,{\frac{{a}^{5}{b}^{3}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}-{\frac{70\,{a}^{4}{b}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-{\frac{4\,a{b}^{7} \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{3\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)*(a+b*sin(d*x+c))^8,x)

[Out]

1/d*a^8*ln(sec(d*x+c)+tan(d*x+c))-1/d*b^8*sin(d*x+c)-1/3/d*b^8*sin(d*x+c)^3+1/d*b^8*ln(sec(d*x+c)+tan(d*x+c))-
1/5/d*b^8*sin(d*x+c)^5-1/7*b^8*sin(d*x+c)^7/d-70/d*a^4*b^4*sin(d*x+c)+70/d*a^4*b^4*ln(sec(d*x+c)+tan(d*x+c))-1
4/d*a^3*b^5*sin(d*x+c)^4-28/d*a^3*b^5*sin(d*x+c)^2-56/d*a^3*b^5*ln(cos(d*x+c))-28/5/d*a^2*b^6*sin(d*x+c)^5-28/
3/d*a^2*b^6*sin(d*x+c)^3-28/d*a^2*b^6*sin(d*x+c)+28/d*a^2*b^6*ln(sec(d*x+c)+tan(d*x+c))-2/d*a*b^7*sin(d*x+c)^4
-4/d*a*b^7*sin(d*x+c)^2-8/d*a*b^7*ln(cos(d*x+c))-8/d*a^7*b*ln(cos(d*x+c))-28/d*a^6*b^2*sin(d*x+c)+28/d*a^6*b^2
*ln(sec(d*x+c)+tan(d*x+c))-28/d*a^5*b^3*sin(d*x+c)^2-56/d*a^5*b^3*ln(cos(d*x+c))-70/3/d*a^4*b^4*sin(d*x+c)^3-4
/3*a*b^7*sin(d*x+c)^6/d

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Maxima [A]  time = 0.955783, size = 428, normalized size = 1.75 \begin{align*} -\frac{30 \, b^{8} \sin \left (d x + c\right )^{7} + 280 \, a b^{7} \sin \left (d x + c\right )^{6} + 42 \,{\left (28 \, a^{2} b^{6} + b^{8}\right )} \sin \left (d x + c\right )^{5} + 420 \,{\left (7 \, a^{3} b^{5} + a b^{7}\right )} \sin \left (d x + c\right )^{4} + 70 \,{\left (70 \, a^{4} b^{4} + 28 \, a^{2} b^{6} + b^{8}\right )} \sin \left (d x + c\right )^{3} + 840 \,{\left (7 \, a^{5} b^{3} + 7 \, a^{3} b^{5} + a b^{7}\right )} \sin \left (d x + c\right )^{2} - 105 \,{\left (a^{8} - 8 \, a^{7} b + 28 \, a^{6} b^{2} - 56 \, a^{5} b^{3} + 70 \, a^{4} b^{4} - 56 \, a^{3} b^{5} + 28 \, a^{2} b^{6} - 8 \, a b^{7} + b^{8}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 105 \,{\left (a^{8} + 8 \, a^{7} b + 28 \, a^{6} b^{2} + 56 \, a^{5} b^{3} + 70 \, a^{4} b^{4} + 56 \, a^{3} b^{5} + 28 \, a^{2} b^{6} + 8 \, a b^{7} + b^{8}\right )} \log \left (\sin \left (d x + c\right ) - 1\right ) + 210 \,{\left (28 \, a^{6} b^{2} + 70 \, a^{4} b^{4} + 28 \, a^{2} b^{6} + b^{8}\right )} \sin \left (d x + c\right )}{210 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)*(a+b*sin(d*x+c))^8,x, algorithm="maxima")

[Out]

-1/210*(30*b^8*sin(d*x + c)^7 + 280*a*b^7*sin(d*x + c)^6 + 42*(28*a^2*b^6 + b^8)*sin(d*x + c)^5 + 420*(7*a^3*b
^5 + a*b^7)*sin(d*x + c)^4 + 70*(70*a^4*b^4 + 28*a^2*b^6 + b^8)*sin(d*x + c)^3 + 840*(7*a^5*b^3 + 7*a^3*b^5 +
a*b^7)*sin(d*x + c)^2 - 105*(a^8 - 8*a^7*b + 28*a^6*b^2 - 56*a^5*b^3 + 70*a^4*b^4 - 56*a^3*b^5 + 28*a^2*b^6 -
8*a*b^7 + b^8)*log(sin(d*x + c) + 1) + 105*(a^8 + 8*a^7*b + 28*a^6*b^2 + 56*a^5*b^3 + 70*a^4*b^4 + 56*a^3*b^5
+ 28*a^2*b^6 + 8*a*b^7 + b^8)*log(sin(d*x + c) - 1) + 210*(28*a^6*b^2 + 70*a^4*b^4 + 28*a^2*b^6 + b^8)*sin(d*x
+ c))/d

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Fricas [A]  time = 3.42162, size = 784, normalized size = 3.2 \begin{align*} \frac{280 \, a b^{7} \cos \left (d x + c\right )^{6} - 420 \,{\left (7 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{4} + 840 \,{\left (7 \, a^{5} b^{3} + 14 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{2} + 105 \,{\left (a^{8} - 8 \, a^{7} b + 28 \, a^{6} b^{2} - 56 \, a^{5} b^{3} + 70 \, a^{4} b^{4} - 56 \, a^{3} b^{5} + 28 \, a^{2} b^{6} - 8 \, a b^{7} + b^{8}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \,{\left (a^{8} + 8 \, a^{7} b + 28 \, a^{6} b^{2} + 56 \, a^{5} b^{3} + 70 \, a^{4} b^{4} + 56 \, a^{3} b^{5} + 28 \, a^{2} b^{6} + 8 \, a b^{7} + b^{8}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (15 \, b^{8} \cos \left (d x + c\right )^{6} - 2940 \, a^{6} b^{2} - 9800 \, a^{4} b^{4} - 4508 \, a^{2} b^{6} - 176 \, b^{8} - 6 \,{\left (98 \, a^{2} b^{6} + 11 \, b^{8}\right )} \cos \left (d x + c\right )^{4} + 2 \,{\left (1225 \, a^{4} b^{4} + 1078 \, a^{2} b^{6} + 61 \, b^{8}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{210 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)*(a+b*sin(d*x+c))^8,x, algorithm="fricas")

[Out]

1/210*(280*a*b^7*cos(d*x + c)^6 - 420*(7*a^3*b^5 + 3*a*b^7)*cos(d*x + c)^4 + 840*(7*a^5*b^3 + 14*a^3*b^5 + 3*a
*b^7)*cos(d*x + c)^2 + 105*(a^8 - 8*a^7*b + 28*a^6*b^2 - 56*a^5*b^3 + 70*a^4*b^4 - 56*a^3*b^5 + 28*a^2*b^6 - 8
*a*b^7 + b^8)*log(sin(d*x + c) + 1) - 105*(a^8 + 8*a^7*b + 28*a^6*b^2 + 56*a^5*b^3 + 70*a^4*b^4 + 56*a^3*b^5 +
28*a^2*b^6 + 8*a*b^7 + b^8)*log(-sin(d*x + c) + 1) + 2*(15*b^8*cos(d*x + c)^6 - 2940*a^6*b^2 - 9800*a^4*b^4 -
4508*a^2*b^6 - 176*b^8 - 6*(98*a^2*b^6 + 11*b^8)*cos(d*x + c)^4 + 2*(1225*a^4*b^4 + 1078*a^2*b^6 + 61*b^8)*co
s(d*x + c)^2)*sin(d*x + c))/d

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)*(a+b*sin(d*x+c))**8,x)

[Out]

Timed out

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Giac [A]  time = 1.1722, size = 510, normalized size = 2.08 \begin{align*} -\frac{30 \, b^{8} \sin \left (d x + c\right )^{7} + 280 \, a b^{7} \sin \left (d x + c\right )^{6} + 1176 \, a^{2} b^{6} \sin \left (d x + c\right )^{5} + 42 \, b^{8} \sin \left (d x + c\right )^{5} + 2940 \, a^{3} b^{5} \sin \left (d x + c\right )^{4} + 420 \, a b^{7} \sin \left (d x + c\right )^{4} + 4900 \, a^{4} b^{4} \sin \left (d x + c\right )^{3} + 1960 \, a^{2} b^{6} \sin \left (d x + c\right )^{3} + 70 \, b^{8} \sin \left (d x + c\right )^{3} + 5880 \, a^{5} b^{3} \sin \left (d x + c\right )^{2} + 5880 \, a^{3} b^{5} \sin \left (d x + c\right )^{2} + 840 \, a b^{7} \sin \left (d x + c\right )^{2} + 5880 \, a^{6} b^{2} \sin \left (d x + c\right ) + 14700 \, a^{4} b^{4} \sin \left (d x + c\right ) + 5880 \, a^{2} b^{6} \sin \left (d x + c\right ) + 210 \, b^{8} \sin \left (d x + c\right ) - 105 \,{\left (a^{8} - 8 \, a^{7} b + 28 \, a^{6} b^{2} - 56 \, a^{5} b^{3} + 70 \, a^{4} b^{4} - 56 \, a^{3} b^{5} + 28 \, a^{2} b^{6} - 8 \, a b^{7} + b^{8}\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) + 105 \,{\left (a^{8} + 8 \, a^{7} b + 28 \, a^{6} b^{2} + 56 \, a^{5} b^{3} + 70 \, a^{4} b^{4} + 56 \, a^{3} b^{5} + 28 \, a^{2} b^{6} + 8 \, a b^{7} + b^{8}\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{210 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)*(a+b*sin(d*x+c))^8,x, algorithm="giac")

[Out]

-1/210*(30*b^8*sin(d*x + c)^7 + 280*a*b^7*sin(d*x + c)^6 + 1176*a^2*b^6*sin(d*x + c)^5 + 42*b^8*sin(d*x + c)^5
+ 2940*a^3*b^5*sin(d*x + c)^4 + 420*a*b^7*sin(d*x + c)^4 + 4900*a^4*b^4*sin(d*x + c)^3 + 1960*a^2*b^6*sin(d*x
+ c)^3 + 70*b^8*sin(d*x + c)^3 + 5880*a^5*b^3*sin(d*x + c)^2 + 5880*a^3*b^5*sin(d*x + c)^2 + 840*a*b^7*sin(d*
x + c)^2 + 5880*a^6*b^2*sin(d*x + c) + 14700*a^4*b^4*sin(d*x + c) + 5880*a^2*b^6*sin(d*x + c) + 210*b^8*sin(d*
x + c) - 105*(a^8 - 8*a^7*b + 28*a^6*b^2 - 56*a^5*b^3 + 70*a^4*b^4 - 56*a^3*b^5 + 28*a^2*b^6 - 8*a*b^7 + b^8)*
log(abs(sin(d*x + c) + 1)) + 105*(a^8 + 8*a^7*b + 28*a^6*b^2 + 56*a^5*b^3 + 70*a^4*b^4 + 56*a^3*b^5 + 28*a^2*b
^6 + 8*a*b^7 + b^8)*log(abs(sin(d*x + c) - 1)))/d