3.421 \(\int \sec ^4(c+d x) (a+b \sin (c+d x))^8 \, dx\)

Optimal. Leaf size=369 \[ \frac {b \left (2 a^2-7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{3 d}+\frac {a b \left (2 a^2-13 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{3 d}-\frac {\sec (c+d x) \left (5 a b-\left (2 a^2-7 b^2\right ) \sin (c+d x)\right ) (a+b \sin (c+d x))^6}{3 d}+\frac {b \left (8 a^4-72 a^2 b^2-35 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{12 d}+\frac {a b \left (8 a^4-88 a^2 b^2-151 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{12 d}+\frac {35}{8} b^4 x \left (16 a^4+16 a^2 b^2+b^4\right )+\frac {a b \left (8 a^6-104 a^4 b^2-803 a^2 b^4-256 b^6\right ) \cos (c+d x)}{6 d}+\frac {b^2 \left (16 a^6-200 a^4 b^2-866 a^2 b^4-105 b^6\right ) \sin (c+d x) \cos (c+d x)}{24 d}+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{3 d} \]

[Out]

35/8*b^4*(16*a^4+16*a^2*b^2+b^4)*x+1/6*a*b*(8*a^6-104*a^4*b^2-803*a^2*b^4-256*b^6)*cos(d*x+c)/d+1/24*b^2*(16*a
^6-200*a^4*b^2-866*a^2*b^4-105*b^6)*cos(d*x+c)*sin(d*x+c)/d+1/12*a*b*(8*a^4-88*a^2*b^2-151*b^4)*cos(d*x+c)*(a+
b*sin(d*x+c))^2/d+1/12*b*(8*a^4-72*a^2*b^2-35*b^4)*cos(d*x+c)*(a+b*sin(d*x+c))^3/d+1/3*a*b*(2*a^2-13*b^2)*cos(
d*x+c)*(a+b*sin(d*x+c))^4/d+1/3*b*(2*a^2-7*b^2)*cos(d*x+c)*(a+b*sin(d*x+c))^5/d+1/3*sec(d*x+c)^3*(b+a*sin(d*x+
c))*(a+b*sin(d*x+c))^7/d-1/3*sec(d*x+c)*(a+b*sin(d*x+c))^6*(5*a*b-(2*a^2-7*b^2)*sin(d*x+c))/d

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Rubi [A]  time = 0.65, antiderivative size = 369, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2691, 2861, 2753, 2734} \[ \frac {a b \left (-104 a^4 b^2-803 a^2 b^4+8 a^6-256 b^6\right ) \cos (c+d x)}{6 d}+\frac {b \left (2 a^2-7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{3 d}+\frac {a b \left (2 a^2-13 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{3 d}+\frac {b \left (-72 a^2 b^2+8 a^4-35 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{12 d}+\frac {a b \left (-88 a^2 b^2+8 a^4-151 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{12 d}+\frac {b^2 \left (-200 a^4 b^2-866 a^2 b^4+16 a^6-105 b^6\right ) \sin (c+d x) \cos (c+d x)}{24 d}-\frac {\sec (c+d x) \left (5 a b-\left (2 a^2-7 b^2\right ) \sin (c+d x)\right ) (a+b \sin (c+d x))^6}{3 d}+\frac {35}{8} b^4 x \left (16 a^2 b^2+16 a^4+b^4\right )+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^4*(a + b*Sin[c + d*x])^8,x]

[Out]

(35*b^4*(16*a^4 + 16*a^2*b^2 + b^4)*x)/8 + (a*b*(8*a^6 - 104*a^4*b^2 - 803*a^2*b^4 - 256*b^6)*Cos[c + d*x])/(6
*d) + (b^2*(16*a^6 - 200*a^4*b^2 - 866*a^2*b^4 - 105*b^6)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + (a*b*(8*a^4 - 88
*a^2*b^2 - 151*b^4)*Cos[c + d*x]*(a + b*Sin[c + d*x])^2)/(12*d) + (b*(8*a^4 - 72*a^2*b^2 - 35*b^4)*Cos[c + d*x
]*(a + b*Sin[c + d*x])^3)/(12*d) + (a*b*(2*a^2 - 13*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^4)/(3*d) + (b*(2*a^
2 - 7*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^5)/(3*d) + (Sec[c + d*x]^3*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*
x])^7)/(3*d) - (Sec[c + d*x]*(a + b*Sin[c + d*x])^6*(5*a*b - (2*a^2 - 7*b^2)*Sin[c + d*x]))/(3*d)

Rule 2691

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Simp[((g*C
os[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1)*(b + a*Sin[e + f*x]))/(f*g*(p + 1)), x] + Dist[1/(g^2*(p + 1
)), Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 2)*(b^2*(m - 1) + a^2*(p + 2) + a*b*(m + p + 1)*Sin
[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && LtQ[p, -1] && (IntegersQ[
2*m, 2*p] || IntegerQ[m])

Rule 2734

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[((2*a*c
+ b*d)*x)/2, x] + (-Simp[((b*c + a*d)*Cos[e + f*x])/f, x] - Simp[(b*d*Cos[e + f*x]*Sin[e + f*x])/(2*f), x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]

Rule 2753

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(d
*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(f*(m + 1)), x] + Dist[1/(m + 1), Int[(a + b*Sin[e + f*x])^(m - 1)*Simp[
b*d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*
c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && IntegerQ[2*m]

Rule 2861

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> -Simp[((g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m*(d + c*Sin[e + f*x]))/(f*
g*(p + 1)), x] + Dist[1/(g^2*(p + 1)), Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 1)*Simp[a*c*(p +
2) + b*d*m + b*c*(m + p + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[a^2 - b^2, 0]
&& GtQ[m, 0] && LtQ[p, -1] && IntegerQ[2*m] &&  !(EqQ[m, 1] && NeQ[c^2 - d^2, 0] && SimplerQ[c + d*x, a + b*x
])

Rubi steps

\begin {align*} \int \sec ^4(c+d x) (a+b \sin (c+d x))^8 \, dx &=\frac {\sec ^3(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{3 d}-\frac {1}{3} \int \sec ^2(c+d x) (a+b \sin (c+d x))^6 \left (-2 a^2+7 b^2+5 a b \sin (c+d x)\right ) \, dx\\ &=\frac {\sec ^3(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{3 d}-\frac {\sec (c+d x) (a+b \sin (c+d x))^6 \left (5 a b-\left (2 a^2-7 b^2\right ) \sin (c+d x)\right )}{3 d}+\frac {1}{3} \int (a+b \sin (c+d x))^5 \left (30 a b^2-6 b \left (2 a^2-7 b^2\right ) \sin (c+d x)\right ) \, dx\\ &=\frac {b \left (2 a^2-7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{3 d}+\frac {\sec ^3(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{3 d}-\frac {\sec (c+d x) (a+b \sin (c+d x))^6 \left (5 a b-\left (2 a^2-7 b^2\right ) \sin (c+d x)\right )}{3 d}+\frac {1}{18} \int (a+b \sin (c+d x))^4 \left (30 b^2 \left (4 a^2+7 b^2\right )-30 a b \left (2 a^2-13 b^2\right ) \sin (c+d x)\right ) \, dx\\ &=\frac {a b \left (2 a^2-13 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{3 d}+\frac {b \left (2 a^2-7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{3 d}+\frac {\sec ^3(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{3 d}-\frac {\sec (c+d x) (a+b \sin (c+d x))^6 \left (5 a b-\left (2 a^2-7 b^2\right ) \sin (c+d x)\right )}{3 d}+\frac {1}{90} \int (a+b \sin (c+d x))^3 \left (90 a b^2 \left (4 a^2+29 b^2\right )-30 b \left (8 a^4-72 a^2 b^2-35 b^4\right ) \sin (c+d x)\right ) \, dx\\ &=\frac {b \left (8 a^4-72 a^2 b^2-35 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{12 d}+\frac {a b \left (2 a^2-13 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{3 d}+\frac {b \left (2 a^2-7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{3 d}+\frac {\sec ^3(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{3 d}-\frac {\sec (c+d x) (a+b \sin (c+d x))^6 \left (5 a b-\left (2 a^2-7 b^2\right ) \sin (c+d x)\right )}{3 d}+\frac {1}{360} \int (a+b \sin (c+d x))^2 \left (90 b^2 \left (8 a^4+188 a^2 b^2+35 b^4\right )-90 a b \left (8 a^4-88 a^2 b^2-151 b^4\right ) \sin (c+d x)\right ) \, dx\\ &=\frac {a b \left (8 a^4-88 a^2 b^2-151 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{12 d}+\frac {b \left (8 a^4-72 a^2 b^2-35 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{12 d}+\frac {a b \left (2 a^2-13 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{3 d}+\frac {b \left (2 a^2-7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{3 d}+\frac {\sec ^3(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{3 d}-\frac {\sec (c+d x) (a+b \sin (c+d x))^6 \left (5 a b-\left (2 a^2-7 b^2\right ) \sin (c+d x)\right )}{3 d}+\frac {\int (a+b \sin (c+d x)) \left (90 a b^2 \left (8 a^4+740 a^2 b^2+407 b^4\right )-90 b \left (16 a^6-200 a^4 b^2-866 a^2 b^4-105 b^6\right ) \sin (c+d x)\right ) \, dx}{1080}\\ &=\frac {35}{8} b^4 \left (16 a^4+16 a^2 b^2+b^4\right ) x+\frac {a b \left (8 a^6-104 a^4 b^2-803 a^2 b^4-256 b^6\right ) \cos (c+d x)}{6 d}+\frac {b^2 \left (16 a^6-200 a^4 b^2-866 a^2 b^4-105 b^6\right ) \cos (c+d x) \sin (c+d x)}{24 d}+\frac {a b \left (8 a^4-88 a^2 b^2-151 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{12 d}+\frac {b \left (8 a^4-72 a^2 b^2-35 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{12 d}+\frac {a b \left (2 a^2-13 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{3 d}+\frac {b \left (2 a^2-7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{3 d}+\frac {\sec ^3(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{3 d}-\frac {\sec (c+d x) (a+b \sin (c+d x))^6 \left (5 a b-\left (2 a^2-7 b^2\right ) \sin (c+d x)\right )}{3 d}\\ \end {align*}

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Mathematica [A]  time = 1.11, size = 414, normalized size = 1.12 \[ \frac {\sec ^3(c+d x) \left (384 a^8 \sin (c+d x)+128 a^8 \sin (3 (c+d x))+2048 a^7 b+5376 a^6 b^2 \sin (c+d x)-1792 a^6 b^2 \sin (3 (c+d x))-21504 a^5 b^3 \cos (2 (c+d x))-7168 a^5 b^3-17920 a^4 b^4 \sin (3 (c+d x))+40320 a^4 b^4 (c+d x) \cos (c+d x)+13440 a^4 b^4 (c+d x) \cos (3 (c+d x))-64512 a^3 b^5 \cos (2 (c+d x))-5376 a^3 b^5 \cos (4 (c+d x))-44800 a^3 b^5-6720 a^2 b^6 \sin (c+d x)-14560 a^2 b^6 \sin (3 (c+d x))-672 a^2 b^6 \sin (5 (c+d x))+40320 a^2 b^6 (c+d x) \cos (c+d x)+13440 a^2 b^6 (c+d x) \cos (3 (c+d x))-17472 a b^7 \cos (2 (c+d x))-1920 a b^7 \cos (4 (c+d x))+64 a b^7 \cos (6 (c+d x))-13440 a b^7-525 b^8 \sin (c+d x)-847 b^8 \sin (3 (c+d x))-63 b^8 \sin (5 (c+d x))+3 b^8 \sin (7 (c+d x))+2520 b^8 (c+d x) \cos (c+d x)+840 b^8 (c+d x) \cos (3 (c+d x))\right )}{768 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^4*(a + b*Sin[c + d*x])^8,x]

[Out]

(Sec[c + d*x]^3*(2048*a^7*b - 7168*a^5*b^3 - 44800*a^3*b^5 - 13440*a*b^7 + 40320*a^4*b^4*(c + d*x)*Cos[c + d*x
] + 40320*a^2*b^6*(c + d*x)*Cos[c + d*x] + 2520*b^8*(c + d*x)*Cos[c + d*x] - 21504*a^5*b^3*Cos[2*(c + d*x)] -
64512*a^3*b^5*Cos[2*(c + d*x)] - 17472*a*b^7*Cos[2*(c + d*x)] + 13440*a^4*b^4*(c + d*x)*Cos[3*(c + d*x)] + 134
40*a^2*b^6*(c + d*x)*Cos[3*(c + d*x)] + 840*b^8*(c + d*x)*Cos[3*(c + d*x)] - 5376*a^3*b^5*Cos[4*(c + d*x)] - 1
920*a*b^7*Cos[4*(c + d*x)] + 64*a*b^7*Cos[6*(c + d*x)] + 384*a^8*Sin[c + d*x] + 5376*a^6*b^2*Sin[c + d*x] - 67
20*a^2*b^6*Sin[c + d*x] - 525*b^8*Sin[c + d*x] + 128*a^8*Sin[3*(c + d*x)] - 1792*a^6*b^2*Sin[3*(c + d*x)] - 17
920*a^4*b^4*Sin[3*(c + d*x)] - 14560*a^2*b^6*Sin[3*(c + d*x)] - 847*b^8*Sin[3*(c + d*x)] - 672*a^2*b^6*Sin[5*(
c + d*x)] - 63*b^8*Sin[5*(c + d*x)] + 3*b^8*Sin[7*(c + d*x)]))/(768*d)

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fricas [A]  time = 0.50, size = 268, normalized size = 0.73 \[ \frac {64 \, a b^{7} \cos \left (d x + c\right )^{6} + 64 \, a^{7} b + 448 \, a^{5} b^{3} + 448 \, a^{3} b^{5} + 64 \, a b^{7} + 105 \, {\left (16 \, a^{4} b^{4} + 16 \, a^{2} b^{6} + b^{8}\right )} d x \cos \left (d x + c\right )^{3} - 192 \, {\left (7 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{4} - 192 \, {\left (7 \, a^{5} b^{3} + 14 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{2} + {\left (6 \, b^{8} \cos \left (d x + c\right )^{6} + 8 \, a^{8} + 224 \, a^{6} b^{2} + 560 \, a^{4} b^{4} + 224 \, a^{2} b^{6} + 8 \, b^{8} - 3 \, {\left (112 \, a^{2} b^{6} + 13 \, b^{8}\right )} \cos \left (d x + c\right )^{4} + 16 \, {\left (a^{8} - 14 \, a^{6} b^{2} - 140 \, a^{4} b^{4} - 98 \, a^{2} b^{6} - 5 \, b^{8}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{24 \, d \cos \left (d x + c\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^4*(a+b*sin(d*x+c))^8,x, algorithm="fricas")

[Out]

1/24*(64*a*b^7*cos(d*x + c)^6 + 64*a^7*b + 448*a^5*b^3 + 448*a^3*b^5 + 64*a*b^7 + 105*(16*a^4*b^4 + 16*a^2*b^6
+ b^8)*d*x*cos(d*x + c)^3 - 192*(7*a^3*b^5 + 3*a*b^7)*cos(d*x + c)^4 - 192*(7*a^5*b^3 + 14*a^3*b^5 + 3*a*b^7)
*cos(d*x + c)^2 + (6*b^8*cos(d*x + c)^6 + 8*a^8 + 224*a^6*b^2 + 560*a^4*b^4 + 224*a^2*b^6 + 8*b^8 - 3*(112*a^2
*b^6 + 13*b^8)*cos(d*x + c)^4 + 16*(a^8 - 14*a^6*b^2 - 140*a^4*b^4 - 98*a^2*b^6 - 5*b^8)*cos(d*x + c)^2)*sin(d
*x + c))/(d*cos(d*x + c)^3)

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giac [A]  time = 0.76, size = 684, normalized size = 1.85 \[ \frac {105 \, {\left (16 \, a^{4} b^{4} + 16 \, a^{2} b^{6} + b^{8}\right )} {\left (d x + c\right )} - \frac {16 \, {\left (3 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 210 \, a^{4} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 168 \, a^{2} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 9 \, b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 24 \, a^{7} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 168 \, a^{3} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 48 \, a b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 2 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 112 \, a^{6} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 700 \, a^{4} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 448 \, a^{2} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 22 \, b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 336 \, a^{5} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 672 \, a^{3} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 144 \, a b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 210 \, a^{4} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 168 \, a^{2} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 9 \, b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 8 \, a^{7} b - 112 \, a^{5} b^{3} - 280 \, a^{3} b^{5} - 64 \, a b^{7}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}} + \frac {2 \, {\left (336 \, a^{2} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 33 \, b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1344 \, a^{3} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 384 \, a b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 336 \, a^{2} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 57 \, b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 4032 \, a^{3} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1536 \, a b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 336 \, a^{2} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 57 \, b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4032 \, a^{3} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1664 \, a b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 336 \, a^{2} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 33 \, b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1344 \, a^{3} b^{5} - 512 \, a b^{7}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4}}}{24 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^4*(a+b*sin(d*x+c))^8,x, algorithm="giac")

[Out]

1/24*(105*(16*a^4*b^4 + 16*a^2*b^6 + b^8)*(d*x + c) - 16*(3*a^8*tan(1/2*d*x + 1/2*c)^5 - 210*a^4*b^4*tan(1/2*d
*x + 1/2*c)^5 - 168*a^2*b^6*tan(1/2*d*x + 1/2*c)^5 - 9*b^8*tan(1/2*d*x + 1/2*c)^5 + 24*a^7*b*tan(1/2*d*x + 1/2
*c)^4 - 168*a^3*b^5*tan(1/2*d*x + 1/2*c)^4 - 48*a*b^7*tan(1/2*d*x + 1/2*c)^4 - 2*a^8*tan(1/2*d*x + 1/2*c)^3 +
112*a^6*b^2*tan(1/2*d*x + 1/2*c)^3 + 700*a^4*b^4*tan(1/2*d*x + 1/2*c)^3 + 448*a^2*b^6*tan(1/2*d*x + 1/2*c)^3 +
22*b^8*tan(1/2*d*x + 1/2*c)^3 + 336*a^5*b^3*tan(1/2*d*x + 1/2*c)^2 + 672*a^3*b^5*tan(1/2*d*x + 1/2*c)^2 + 144
*a*b^7*tan(1/2*d*x + 1/2*c)^2 + 3*a^8*tan(1/2*d*x + 1/2*c) - 210*a^4*b^4*tan(1/2*d*x + 1/2*c) - 168*a^2*b^6*ta
n(1/2*d*x + 1/2*c) - 9*b^8*tan(1/2*d*x + 1/2*c) + 8*a^7*b - 112*a^5*b^3 - 280*a^3*b^5 - 64*a*b^7)/(tan(1/2*d*x
+ 1/2*c)^2 - 1)^3 + 2*(336*a^2*b^6*tan(1/2*d*x + 1/2*c)^7 + 33*b^8*tan(1/2*d*x + 1/2*c)^7 - 1344*a^3*b^5*tan(
1/2*d*x + 1/2*c)^6 - 384*a*b^7*tan(1/2*d*x + 1/2*c)^6 + 336*a^2*b^6*tan(1/2*d*x + 1/2*c)^5 + 57*b^8*tan(1/2*d*
x + 1/2*c)^5 - 4032*a^3*b^5*tan(1/2*d*x + 1/2*c)^4 - 1536*a*b^7*tan(1/2*d*x + 1/2*c)^4 - 336*a^2*b^6*tan(1/2*d
*x + 1/2*c)^3 - 57*b^8*tan(1/2*d*x + 1/2*c)^3 - 4032*a^3*b^5*tan(1/2*d*x + 1/2*c)^2 - 1664*a*b^7*tan(1/2*d*x +
1/2*c)^2 - 336*a^2*b^6*tan(1/2*d*x + 1/2*c) - 33*b^8*tan(1/2*d*x + 1/2*c) - 1344*a^3*b^5 - 512*a*b^7)/(tan(1/
2*d*x + 1/2*c)^2 + 1)^4)/d

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maple [A]  time = 0.46, size = 495, normalized size = 1.34 \[ \frac {-a^{8} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+\frac {8 a^{7} b}{3 \cos \left (d x +c \right )^{3}}+\frac {28 a^{6} b^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )^{3}}+56 a^{5} b^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}\right )+70 a^{4} b^{4} \left (\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+d x +c \right )+56 a^{3} b^{5} \left (\frac {\sin ^{6}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )+28 a^{2} b^{6} \left (\frac {\sin ^{7}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {4 \left (\sin ^{7}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )}-\frac {4 \left (\sin ^{5}\left (d x +c \right )+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )+8 a \,b^{7} \left (\frac {\sin ^{8}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {5 \left (\sin ^{8}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )}-\frac {5 \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 )}{3}\right )+b^{8} \left (\frac {\sin ^{9}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {2 \left (\sin ^{9}\left (d x +c \right )\right )}{\cos \left (d x +c \right )}-2 \left (\sin ^{7}\left (d x +c \right )+\frac {7 \left (\sin ^{5}\left (d x +c \right )\right )}{6}+\frac {35 \left (\sin ^{3}\left (d x +c \right )\right )}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )+\frac {35 d x}{8}+\frac {35 c}{8}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^4*(a+b*sin(d*x+c))^8,x)

[Out]

1/d*(-a^8*(-2/3-1/3*sec(d*x+c)^2)*tan(d*x+c)+8/3*a^7*b/cos(d*x+c)^3+28/3*a^6*b^2*sin(d*x+c)^3/cos(d*x+c)^3+56*
a^5*b^3*(1/3*sin(d*x+c)^4/cos(d*x+c)^3-1/3*sin(d*x+c)^4/cos(d*x+c)-1/3*(2+sin(d*x+c)^2)*cos(d*x+c))+70*a^4*b^4
*(1/3*tan(d*x+c)^3-tan(d*x+c)+d*x+c)+56*a^3*b^5*(1/3*sin(d*x+c)^6/cos(d*x+c)^3-sin(d*x+c)^6/cos(d*x+c)-(8/3+si
n(d*x+c)^4+4/3*sin(d*x+c)^2)*cos(d*x+c))+28*a^2*b^6*(1/3*sin(d*x+c)^7/cos(d*x+c)^3-4/3*sin(d*x+c)^7/cos(d*x+c)
-4/3*(sin(d*x+c)^5+5/4*sin(d*x+c)^3+15/8*sin(d*x+c))*cos(d*x+c)+5/2*d*x+5/2*c)+8*a*b^7*(1/3*sin(d*x+c)^8/cos(d
*x+c)^3-5/3*sin(d*x+c)^8/cos(d*x+c)-5/3*(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))+b^8*
(1/3*sin(d*x+c)^9/cos(d*x+c)^3-2*sin(d*x+c)^9/cos(d*x+c)-2*(sin(d*x+c)^7+7/6*sin(d*x+c)^5+35/24*sin(d*x+c)^3+3
5/16*sin(d*x+c))*cos(d*x+c)+35/8*d*x+35/8*c))

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maxima [A]  time = 0.43, size = 328, normalized size = 0.89 \[ \frac {224 \, a^{6} b^{2} \tan \left (d x + c\right )^{3} + 8 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a^{8} + 560 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, d x + 3 \, c - 3 \, \tan \left (d x + c\right )\right )} a^{4} b^{4} + 112 \, {\left (2 \, \tan \left (d x + c\right )^{3} + 15 \, d x + 15 \, c - \frac {3 \, \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1} - 12 \, \tan \left (d x + c\right )\right )} a^{2} b^{6} + 64 \, {\left (\cos \left (d x + c\right )^{3} - \frac {9 \, \cos \left (d x + c\right )^{2} - 1}{\cos \left (d x + c\right )^{3}} - 9 \, \cos \left (d x + c\right )\right )} a b^{7} + {\left (8 \, \tan \left (d x + c\right )^{3} + 105 \, d x + 105 \, c - \frac {3 \, {\left (13 \, \tan \left (d x + c\right )^{3} + 11 \, \tan \left (d x + c\right )\right )}}{\tan \left (d x + c\right )^{4} + 2 \, \tan \left (d x + c\right )^{2} + 1} - 72 \, \tan \left (d x + c\right )\right )} b^{8} - 448 \, a^{3} b^{5} {\left (\frac {6 \, \cos \left (d x + c\right )^{2} - 1}{\cos \left (d x + c\right )^{3}} + 3 \, \cos \left (d x + c\right )\right )} - \frac {448 \, {\left (3 \, \cos \left (d x + c\right )^{2} - 1\right )} a^{5} b^{3}}{\cos \left (d x + c\right )^{3}} + \frac {64 \, a^{7} b}{\cos \left (d x + c\right )^{3}}}{24 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^4*(a+b*sin(d*x+c))^8,x, algorithm="maxima")

[Out]

1/24*(224*a^6*b^2*tan(d*x + c)^3 + 8*(tan(d*x + c)^3 + 3*tan(d*x + c))*a^8 + 560*(tan(d*x + c)^3 + 3*d*x + 3*c
- 3*tan(d*x + c))*a^4*b^4 + 112*(2*tan(d*x + c)^3 + 15*d*x + 15*c - 3*tan(d*x + c)/(tan(d*x + c)^2 + 1) - 12*
tan(d*x + c))*a^2*b^6 + 64*(cos(d*x + c)^3 - (9*cos(d*x + c)^2 - 1)/cos(d*x + c)^3 - 9*cos(d*x + c))*a*b^7 + (
8*tan(d*x + c)^3 + 105*d*x + 105*c - 3*(13*tan(d*x + c)^3 + 11*tan(d*x + c))/(tan(d*x + c)^4 + 2*tan(d*x + c)^
2 + 1) - 72*tan(d*x + c))*b^8 - 448*a^3*b^5*((6*cos(d*x + c)^2 - 1)/cos(d*x + c)^3 + 3*cos(d*x + c)) - 448*(3*
cos(d*x + c)^2 - 1)*a^5*b^3/cos(d*x + c)^3 + 64*a^7*b/cos(d*x + c)^3)/d

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mupad [B]  time = 7.82, size = 726, normalized size = 1.97 \[ \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {304\,a^7\,b}{3}+\frac {2464\,a^5\,b^3}{3}+\frac {1792\,a^3\,b^5}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (64\,a^7\,b+224\,a^5\,b^3\right )-\frac {256\,a\,b^7}{3}+\frac {16\,a^7\,b}{3}-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (-2\,a^8+140\,a^4\,b^4+140\,a^2\,b^6+\frac {35\,b^8}{4}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (-\frac {64\,a^7\,b}{3}+\frac {224\,a^5\,b^3}{3}+\frac {896\,a^3\,b^5}{3}+\frac {256\,a\,b^7}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (48\,a^7\,b+448\,a^5\,b^3+896\,a^3\,b^5+256\,a\,b^7\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {256\,a^7\,b}{3}+\frac {3136\,a^5\,b^3}{3}+\frac {4480\,a^3\,b^5}{3}+256\,a\,b^7\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (-\frac {20\,a^8}{3}-\frac {224\,a^6\,b^2}{3}+\frac {280\,a^4\,b^4}{3}+\frac {280\,a^2\,b^6}{3}+\frac {35\,b^8}{6}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (-\frac {20\,a^8}{3}-\frac {224\,a^6\,b^2}{3}+\frac {280\,a^4\,b^4}{3}+\frac {280\,a^2\,b^6}{3}+\frac {35\,b^8}{6}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (8\,a^8+448\,a^6\,b^2+1680\,a^4\,b^4+784\,a^2\,b^6+17\,b^8\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {26\,a^8}{3}+\frac {896\,a^6\,b^2}{3}+\frac {2660\,a^4\,b^4}{3}+\frac {1316\,a^2\,b^6}{3}+\frac {329\,b^8}{12}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (\frac {26\,a^8}{3}+\frac {896\,a^6\,b^2}{3}+\frac {2660\,a^4\,b^4}{3}+\frac {1316\,a^2\,b^6}{3}+\frac {329\,b^8}{12}\right )-\frac {896\,a^3\,b^5}{3}-\frac {224\,a^5\,b^3}{3}-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\left (-2\,a^8+140\,a^4\,b^4+140\,a^2\,b^6+\frac {35\,b^8}{4}\right )+16\,a^7\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{d\,\left (-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {35\,b^4\,\mathrm {atan}\left (\frac {35\,b^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (16\,a^4+16\,a^2\,b^2+b^4\right )}{560\,a^4\,b^4+560\,a^2\,b^6+35\,b^8}\right )\,\left (16\,a^4+16\,a^2\,b^2+b^4\right )}{4\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*sin(c + d*x))^8/cos(c + d*x)^4,x)

[Out]

(tan(c/2 + (d*x)/2)^8*((304*a^7*b)/3 + (1792*a^3*b^5)/3 + (2464*a^5*b^3)/3) + tan(c/2 + (d*x)/2)^10*(64*a^7*b
+ 224*a^5*b^3) - (256*a*b^7)/3 + (16*a^7*b)/3 - tan(c/2 + (d*x)/2)*((35*b^8)/4 - 2*a^8 + 140*a^2*b^6 + 140*a^4
*b^4) - tan(c/2 + (d*x)/2)^2*((256*a*b^7)/3 - (64*a^7*b)/3 + (896*a^3*b^5)/3 + (224*a^5*b^3)/3) + tan(c/2 + (d
*x)/2)^4*(256*a*b^7 + 48*a^7*b + 896*a^3*b^5 + 448*a^5*b^3) + tan(c/2 + (d*x)/2)^6*(256*a*b^7 + (256*a^7*b)/3
+ (4480*a^3*b^5)/3 + (3136*a^5*b^3)/3) - tan(c/2 + (d*x)/2)^3*((35*b^8)/6 - (20*a^8)/3 + (280*a^2*b^6)/3 + (28
0*a^4*b^4)/3 - (224*a^6*b^2)/3) - tan(c/2 + (d*x)/2)^11*((35*b^8)/6 - (20*a^8)/3 + (280*a^2*b^6)/3 + (280*a^4*
b^4)/3 - (224*a^6*b^2)/3) + tan(c/2 + (d*x)/2)^7*(8*a^8 + 17*b^8 + 784*a^2*b^6 + 1680*a^4*b^4 + 448*a^6*b^2) +
tan(c/2 + (d*x)/2)^5*((26*a^8)/3 + (329*b^8)/12 + (1316*a^2*b^6)/3 + (2660*a^4*b^4)/3 + (896*a^6*b^2)/3) + ta
n(c/2 + (d*x)/2)^9*((26*a^8)/3 + (329*b^8)/12 + (1316*a^2*b^6)/3 + (2660*a^4*b^4)/3 + (896*a^6*b^2)/3) - (896*
a^3*b^5)/3 - (224*a^5*b^3)/3 - tan(c/2 + (d*x)/2)^13*((35*b^8)/4 - 2*a^8 + 140*a^2*b^6 + 140*a^4*b^4) + 16*a^7
*b*tan(c/2 + (d*x)/2)^12)/(d*(tan(c/2 + (d*x)/2)^2 - 3*tan(c/2 + (d*x)/2)^4 - 3*tan(c/2 + (d*x)/2)^6 + 3*tan(c
/2 + (d*x)/2)^8 + 3*tan(c/2 + (d*x)/2)^10 - tan(c/2 + (d*x)/2)^12 - tan(c/2 + (d*x)/2)^14 + 1)) + (35*b^4*atan
((35*b^4*tan(c/2 + (d*x)/2)*(16*a^4 + b^4 + 16*a^2*b^2))/(35*b^8 + 560*a^2*b^6 + 560*a^4*b^4))*(16*a^4 + b^4 +
16*a^2*b^2))/(4*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)**4*(a+b*sin(d*x+c))**8,x)

[Out]

Timed out

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