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

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

[Out]

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

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Rubi [A]  time = 0.18, antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, 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 31

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

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 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 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]

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*}

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Mathematica [A]  time = 0.21, size = 227, normalized size = 0.93 \[ \frac {b \left (-\frac {1}{5} b^5 \left (28 a^2+b^2\right ) \sin ^5(c+d x)-2 a b^4 \left (7 a^2+b^2\right ) \sin ^4(c+d x)-4 a b^2 \left (7 a^4+7 a^2 b^2+b^4\right ) \sin ^2(c+d x)-\frac {1}{3} b^3 \left (70 a^4+28 a^2 b^2+b^4\right ) \sin ^3(c+d x)-b \left (28 a^6+70 a^4 b^2+28 a^2 b^4+b^6\right ) \sin (c+d x)-\frac {4}{3} a b^6 \sin ^6(c+d x)+\frac {(a-b)^8 \log (\sin (c+d x)+1)}{2 b}-\frac {(a+b)^8 \log (1-\sin (c+d x))}{2 b}-\frac {1}{7} b^7 \sin ^7(c+d x)\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(-1/2*((a + b)^8*Log[1 - Sin[c + d*x]])/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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fricas [A]  time = 0.52, size = 327, normalized size = 1.33 \[ \frac {280 \, a b^{7} \cos \left (d x + c\right )^{6} - 420 \, {\left (7 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{4} + 840 \, {\left (7 \, a^{5} b^{3} + 14 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{2} + 105 \, {\left (a^{8} - 8 \, a^{7} b + 28 \, a^{6} b^{2} - 56 \, a^{5} b^{3} + 70 \, a^{4} b^{4} - 56 \, a^{3} b^{5} + 28 \, a^{2} b^{6} - 8 \, a b^{7} + b^{8}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left (a^{8} + 8 \, a^{7} b + 28 \, a^{6} b^{2} + 56 \, a^{5} b^{3} + 70 \, a^{4} b^{4} + 56 \, a^{3} b^{5} + 28 \, a^{2} b^{6} + 8 \, a b^{7} + b^{8}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (15 \, b^{8} \cos \left (d x + c\right )^{6} - 2940 \, a^{6} b^{2} - 9800 \, a^{4} b^{4} - 4508 \, a^{2} b^{6} - 176 \, b^{8} - 6 \, {\left (98 \, a^{2} b^{6} + 11 \, b^{8}\right )} \cos \left (d x + c\right )^{4} + 2 \, {\left (1225 \, a^{4} b^{4} + 1078 \, a^{2} b^{6} + 61 \, b^{8}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{210 \, d} \]

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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giac [A]  time = 1.99, size = 378, normalized size = 1.54 \[ -\frac {30 \, b^{8} \sin \left (d x + c\right )^{7} + 280 \, a b^{7} \sin \left (d x + c\right )^{6} + 1176 \, a^{2} b^{6} \sin \left (d x + c\right )^{5} + 42 \, b^{8} \sin \left (d x + c\right )^{5} + 2940 \, a^{3} b^{5} \sin \left (d x + c\right )^{4} + 420 \, a b^{7} \sin \left (d x + c\right )^{4} + 4900 \, a^{4} b^{4} \sin \left (d x + c\right )^{3} + 1960 \, a^{2} b^{6} \sin \left (d x + c\right )^{3} + 70 \, b^{8} \sin \left (d x + c\right )^{3} + 5880 \, a^{5} b^{3} \sin \left (d x + c\right )^{2} + 5880 \, a^{3} b^{5} \sin \left (d x + c\right )^{2} + 840 \, a b^{7} \sin \left (d x + c\right )^{2} + 5880 \, a^{6} b^{2} \sin \left (d x + c\right ) + 14700 \, a^{4} b^{4} \sin \left (d x + c\right ) + 5880 \, a^{2} b^{6} \sin \left (d x + c\right ) + 210 \, b^{8} \sin \left (d x + c\right ) - 105 \, {\left (a^{8} - 8 \, a^{7} b + 28 \, a^{6} b^{2} - 56 \, a^{5} b^{3} + 70 \, a^{4} b^{4} - 56 \, a^{3} b^{5} + 28 \, a^{2} b^{6} - 8 \, a b^{7} + b^{8}\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) + 105 \, {\left (a^{8} + 8 \, a^{7} b + 28 \, a^{6} b^{2} + 56 \, a^{5} b^{3} + 70 \, a^{4} b^{4} + 56 \, a^{3} b^{5} + 28 \, a^{2} b^{6} + 8 \, a b^{7} + b^{8}\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{210 \, d} \]

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

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maple [A]  time = 0.23, size = 465, normalized size = 1.90 \[ -\frac {b^{8} \left (\sin ^{7}\left (d x +c \right )\right )}{7 d}-\frac {28 a^{2} b^{6} \left (\sin ^{5}\left (d x +c \right )\right )}{5 d}-\frac {28 a^{2} b^{6} \left (\sin ^{3}\left (d x +c \right )\right )}{3 d}-\frac {2 a \,b^{7} \left (\sin ^{4}\left (d x +c \right )\right )}{d}-\frac {4 a \,b^{7} \left (\sin ^{2}\left (d x +c \right )\right )}{d}-\frac {14 a^{3} b^{5} \left (\sin ^{4}\left (d x +c \right )\right )}{d}-\frac {28 a^{3} b^{5} \left (\sin ^{2}\left (d x +c \right )\right )}{d}-\frac {70 a^{4} b^{4} \left (\sin ^{3}\left (d x +c \right )\right )}{3 d}-\frac {28 a^{5} b^{3} \left (\sin ^{2}\left (d x +c \right )\right )}{d}-\frac {\sin \left (d x +c \right ) b^{8}}{d}+\frac {b^{8} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {a^{8} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}-\frac {28 a^{2} b^{6} \sin \left (d x +c \right )}{d}+\frac {28 a^{2} b^{6} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}-\frac {8 a \,b^{7} \ln \left (\cos \left (d x +c \right )\right )}{d}-\frac {56 a^{3} b^{5} \ln \left (\cos \left (d x +c \right )\right )}{d}-\frac {70 a^{4} b^{4} \sin \left (d x +c \right )}{d}+\frac {70 a^{4} b^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}-\frac {8 a^{7} b \ln \left (\cos \left (d x +c \right )\right )}{d}-\frac {28 a^{6} b^{2} \sin \left (d x +c \right )}{d}+\frac {28 a^{6} b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}-\frac {56 a^{5} b^{3} \ln \left (\cos \left (d x +c \right )\right )}{d}-\frac {4 a \,b^{7} \left (\sin ^{6}\left (d x +c \right )\right )}{3 d}-\frac {b^{8} \left (\sin ^{3}\left (d x +c \right )\right )}{3 d}-\frac {b^{8} \left (\sin ^{5}\left (d x +c \right )\right )}{5 d} \]

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]

-4/3*a*b^7*sin(d*x+c)^6/d-1/7*b^8*sin(d*x+c)^7/d-1/d*sin(d*x+c)*b^8-1/5/d*b^8*sin(d*x+c)^5-1/3/d*b^8*sin(d*x+c
)^3+1/d*b^8*ln(sec(d*x+c)+tan(d*x+c))+1/d*a^8*ln(sec(d*x+c)+tan(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))-14/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))-70/3/d*a^4*b^4*sin(d*x+c)^3-70/d*a^4*b^4*sin(d*x+c)+70/d*a^4*b^4*ln(sec(d*x+c)+tan(d*x+c))-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))

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maxima [A]  time = 0.33, size = 317, normalized size = 1.29 \[ -\frac {30 \, b^{8} \sin \left (d x + c\right )^{7} + 280 \, a b^{7} \sin \left (d x + c\right )^{6} + 42 \, {\left (28 \, a^{2} b^{6} + b^{8}\right )} \sin \left (d x + c\right )^{5} + 420 \, {\left (7 \, a^{3} b^{5} + a b^{7}\right )} \sin \left (d x + c\right )^{4} + 70 \, {\left (70 \, a^{4} b^{4} + 28 \, a^{2} b^{6} + b^{8}\right )} \sin \left (d x + c\right )^{3} + 840 \, {\left (7 \, a^{5} b^{3} + 7 \, a^{3} b^{5} + a b^{7}\right )} \sin \left (d x + c\right )^{2} - 105 \, {\left (a^{8} - 8 \, a^{7} b + 28 \, a^{6} b^{2} - 56 \, a^{5} b^{3} + 70 \, a^{4} b^{4} - 56 \, a^{3} b^{5} + 28 \, a^{2} b^{6} - 8 \, a b^{7} + b^{8}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 105 \, {\left (a^{8} + 8 \, a^{7} b + 28 \, a^{6} b^{2} + 56 \, a^{5} b^{3} + 70 \, a^{4} b^{4} + 56 \, a^{3} b^{5} + 28 \, a^{2} b^{6} + 8 \, a b^{7} + b^{8}\right )} \log \left (\sin \left (d x + c\right ) - 1\right ) + 210 \, {\left (28 \, a^{6} b^{2} + 70 \, a^{4} b^{4} + 28 \, a^{2} b^{6} + b^{8}\right )} \sin \left (d x + c\right )}{210 \, d} \]

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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mupad [B]  time = 5.36, size = 212, normalized size = 0.87 \[ -\frac {\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )\,{\left (a+b\right )}^8}{2}+{\sin \left (c+d\,x\right )}^3\,\left (\frac {70\,a^4\,b^4}{3}+\frac {28\,a^2\,b^6}{3}+\frac {b^8}{3}\right )-\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )\,{\left (a-b\right )}^8}{2}+{\sin \left (c+d\,x\right )}^5\,\left (\frac {28\,a^2\,b^6}{5}+\frac {b^8}{5}\right )+\sin \left (c+d\,x\right )\,\left (28\,a^6\,b^2+70\,a^4\,b^4+28\,a^2\,b^6+b^8\right )+{\sin \left (c+d\,x\right )}^2\,\left (28\,a^5\,b^3+28\,a^3\,b^5+4\,a\,b^7\right )+\frac {b^8\,{\sin \left (c+d\,x\right )}^7}{7}+{\sin \left (c+d\,x\right )}^4\,\left (14\,a^3\,b^5+2\,a\,b^7\right )+\frac {4\,a\,b^7\,{\sin \left (c+d\,x\right )}^6}{3}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((log(sin(c + d*x) - 1)*(a + b)^8)/2 + sin(c + d*x)^3*(b^8/3 + (28*a^2*b^6)/3 + (70*a^4*b^4)/3) - (log(sin(c
+ d*x) + 1)*(a - b)^8)/2 + sin(c + d*x)^5*(b^8/5 + (28*a^2*b^6)/5) + sin(c + d*x)*(b^8 + 28*a^2*b^6 + 70*a^4*b
^4 + 28*a^6*b^2) + sin(c + d*x)^2*(4*a*b^7 + 28*a^3*b^5 + 28*a^5*b^3) + (b^8*sin(c + d*x)^7)/7 + sin(c + d*x)^
4*(2*a*b^7 + 14*a^3*b^5) + (4*a*b^7*sin(c + d*x)^6)/3)/d

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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