Optimal. Leaf size=284 \[ \frac {7 b^6 \left (5 a^2+b^2\right ) \sin ^5(c+d x)}{10 d}+\frac {3 a b^5 \left (7 a^2+4 b^2\right ) \sin ^4(c+d x)}{2 d}+\frac {7 b^4 \left (15 a^4+20 a^2 b^2+b^4\right ) \sin ^3(c+d x)}{6 d}+\frac {a b^3 \left (35 a^4+112 a^2 b^2+24 b^4\right ) \sin ^2(c+d x)}{2 d}+\frac {7 b^2 \left (3 a^6+30 a^4 b^2+20 a^2 b^4+b^6\right ) \sin (c+d x)}{2 d}+\frac {a b^7 \sin ^6(c+d x)}{2 d}+\frac {(a+7 b) (a-b)^7 \log (\sin (c+d x)+1)}{4 d}-\frac {(a-7 b) (a+b)^7 \log (1-\sin (c+d x))}{4 d}+\frac {\sec ^2(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{2 d} \]
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Rubi [A] time = 0.24, antiderivative size = 284, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2668, 739, 801, 633, 31} \[ \frac {7 b^6 \left (5 a^2+b^2\right ) \sin ^5(c+d x)}{10 d}+\frac {3 a b^5 \left (7 a^2+4 b^2\right ) \sin ^4(c+d x)}{2 d}+\frac {7 b^4 \left (20 a^2 b^2+15 a^4+b^4\right ) \sin ^3(c+d x)}{6 d}+\frac {a b^3 \left (112 a^2 b^2+35 a^4+24 b^4\right ) \sin ^2(c+d x)}{2 d}+\frac {7 b^2 \left (30 a^4 b^2+20 a^2 b^4+3 a^6+b^6\right ) \sin (c+d x)}{2 d}+\frac {a b^7 \sin ^6(c+d x)}{2 d}+\frac {(a+7 b) (a-b)^7 \log (\sin (c+d x)+1)}{4 d}-\frac {(a-7 b) (a+b)^7 \log (1-\sin (c+d x))}{4 d}+\frac {\sec ^2(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{2 d} \]
Antiderivative was successfully verified.
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Rule 31
Rule 633
Rule 739
Rule 801
Rule 2668
Rubi steps
\begin {align*} \int \sec ^3(c+d x) (a+b \sin (c+d x))^8 \, dx &=\frac {b^3 \operatorname {Subst}\left (\int \frac {(a+x)^8}{\left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {\sec ^2(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{2 d}-\frac {b \operatorname {Subst}\left (\int \frac {(a+x)^6 \left (-a^2+7 b^2+6 a x\right )}{b^2-x^2} \, dx,x,b \sin (c+d x)\right )}{2 d}\\ &=\frac {\sec ^2(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{2 d}-\frac {b \operatorname {Subst}\left (\int \left (-7 \left (3 a^6+30 a^4 b^2+20 a^2 b^4+b^6\right )-2 a \left (35 a^4+112 a^2 b^2+24 b^4\right ) x-7 \left (15 a^4+20 a^2 b^2+b^4\right ) x^2-12 a \left (7 a^2+4 b^2\right ) x^3-7 \left (5 a^2+b^2\right ) x^4-6 a x^5-\frac {a^8-28 a^6 b^2-210 a^4 b^4-140 a^2 b^6-7 b^8-16 a b^2 \left (7 a^4+14 a^2 b^2+3 b^4\right ) x}{b^2-x^2}\right ) \, dx,x,b \sin (c+d x)\right )}{2 d}\\ &=\frac {7 b^2 \left (3 a^6+30 a^4 b^2+20 a^2 b^4+b^6\right ) \sin (c+d x)}{2 d}+\frac {a b^3 \left (35 a^4+112 a^2 b^2+24 b^4\right ) \sin ^2(c+d x)}{2 d}+\frac {7 b^4 \left (15 a^4+20 a^2 b^2+b^4\right ) \sin ^3(c+d x)}{6 d}+\frac {3 a b^5 \left (7 a^2+4 b^2\right ) \sin ^4(c+d x)}{2 d}+\frac {7 b^6 \left (5 a^2+b^2\right ) \sin ^5(c+d x)}{10 d}+\frac {a b^7 \sin ^6(c+d x)}{2 d}+\frac {\sec ^2(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{2 d}+\frac {b \operatorname {Subst}\left (\int \frac {a^8-28 a^6 b^2-210 a^4 b^4-140 a^2 b^6-7 b^8-16 a b^2 \left (7 a^4+14 a^2 b^2+3 b^4\right ) x}{b^2-x^2} \, dx,x,b \sin (c+d x)\right )}{2 d}\\ &=\frac {7 b^2 \left (3 a^6+30 a^4 b^2+20 a^2 b^4+b^6\right ) \sin (c+d x)}{2 d}+\frac {a b^3 \left (35 a^4+112 a^2 b^2+24 b^4\right ) \sin ^2(c+d x)}{2 d}+\frac {7 b^4 \left (15 a^4+20 a^2 b^2+b^4\right ) \sin ^3(c+d x)}{6 d}+\frac {3 a b^5 \left (7 a^2+4 b^2\right ) \sin ^4(c+d x)}{2 d}+\frac {7 b^6 \left (5 a^2+b^2\right ) \sin ^5(c+d x)}{10 d}+\frac {a b^7 \sin ^6(c+d x)}{2 d}+\frac {\sec ^2(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{2 d}+\frac {\left ((a-7 b) (a+b)^7\right ) \operatorname {Subst}\left (\int \frac {1}{b-x} \, dx,x,b \sin (c+d x)\right )}{4 d}-\frac {\left ((a-b)^7 (a+7 b)\right ) \operatorname {Subst}\left (\int \frac {1}{-b-x} \, dx,x,b \sin (c+d x)\right )}{4 d}\\ &=-\frac {(a-7 b) (a+b)^7 \log (1-\sin (c+d x))}{4 d}+\frac {(a-b)^7 (a+7 b) \log (1+\sin (c+d x))}{4 d}+\frac {7 b^2 \left (3 a^6+30 a^4 b^2+20 a^2 b^4+b^6\right ) \sin (c+d x)}{2 d}+\frac {a b^3 \left (35 a^4+112 a^2 b^2+24 b^4\right ) \sin ^2(c+d x)}{2 d}+\frac {7 b^4 \left (15 a^4+20 a^2 b^2+b^4\right ) \sin ^3(c+d x)}{6 d}+\frac {3 a b^5 \left (7 a^2+4 b^2\right ) \sin ^4(c+d x)}{2 d}+\frac {7 b^6 \left (5 a^2+b^2\right ) \sin ^5(c+d x)}{10 d}+\frac {a b^7 \sin ^6(c+d x)}{2 d}+\frac {\sec ^2(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{2 d}\\ \end {align*}
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Mathematica [A] time = 2.35, size = 366, normalized size = 1.29 \[ \frac {\frac {1}{2} b \left (a^2-b^2\right ) \left ((a-7 b) (a+b)^7 \log (1-\sin (c+d x))-(a-b)^7 (a+7 b) \log (\sin (c+d x)+1)\right )+b^9 \left (b^2-9 a^2\right ) \sin ^7(c+d x)-4 a b^8 \left (9 a^2-2 b^2\right ) \sin ^6(c+d x)+\frac {7}{5} b^7 \left (-60 a^4+19 a^2 b^2+b^4\right ) \sin ^5(c+d x)-2 a b^6 \left (63 a^4-22 a^2 b^2-6 b^4\right ) \sin ^4(c+d x)-4 a b^4 \left (21 a^6+14 a^4 b^2-22 a^2 b^4-6 b^6\right ) \sin ^2(c+d x)+\frac {7}{3} b^5 \left (-54 a^6+10 a^4 b^2+19 a^2 b^4+b^6\right ) \sin ^3(c+d x)+b^3 \left (-36 a^8-182 a^6 b^2+70 a^4 b^4+133 a^2 b^6+7 b^8\right ) \sin (c+d x)-a b^{10} \sin ^8(c+d x)+b \sec ^2(c+d x) (b-a \sin (c+d x)) (a+b \sin (c+d x))^9}{2 b d \left (b^2-a^2\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 368, normalized size = 1.30 \[ \frac {120 \, a b^{7} \cos \left (d x + c\right )^{6} + 240 \, a^{7} b + 1680 \, a^{5} b^{3} + 1680 \, a^{3} b^{5} + 240 \, a b^{7} - 240 \, {\left (7 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{4} + 15 \, {\left (a^{8} - 28 \, a^{6} b^{2} + 112 \, a^{5} b^{3} - 210 \, a^{4} b^{4} + 224 \, a^{3} b^{5} - 140 \, a^{2} b^{6} + 48 \, a b^{7} - 7 \, b^{8}\right )} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (a^{8} - 28 \, a^{6} b^{2} - 112 \, a^{5} b^{3} - 210 \, a^{4} b^{4} - 224 \, a^{3} b^{5} - 140 \, a^{2} b^{6} - 48 \, a b^{7} - 7 \, b^{8}\right )} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 105 \, {\left (8 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (6 \, b^{8} \cos \left (d x + c\right )^{6} + 15 \, a^{8} + 420 \, a^{6} b^{2} + 1050 \, a^{4} b^{4} + 420 \, a^{2} b^{6} + 15 \, b^{8} - 8 \, {\left (35 \, a^{2} b^{6} + 4 \, b^{8}\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (525 \, a^{4} b^{4} + 490 \, a^{2} b^{6} + 29 \, b^{8}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{60 \, d \cos \left (d x + c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.60, size = 408, normalized size = 1.44 \[ \frac {12 \, b^{8} \sin \left (d x + c\right )^{5} + 120 \, a b^{7} \sin \left (d x + c\right )^{4} + 560 \, a^{2} b^{6} \sin \left (d x + c\right )^{3} + 40 \, b^{8} \sin \left (d x + c\right )^{3} + 1680 \, a^{3} b^{5} \sin \left (d x + c\right )^{2} + 480 \, a b^{7} \sin \left (d x + c\right )^{2} + 4200 \, a^{4} b^{4} \sin \left (d x + c\right ) + 3360 \, a^{2} b^{6} \sin \left (d x + c\right ) + 180 \, b^{8} \sin \left (d x + c\right ) + 15 \, {\left (a^{8} - 28 \, a^{6} b^{2} + 112 \, a^{5} b^{3} - 210 \, a^{4} b^{4} + 224 \, a^{3} b^{5} - 140 \, a^{2} b^{6} + 48 \, a b^{7} - 7 \, b^{8}\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - 15 \, {\left (a^{8} - 28 \, a^{6} b^{2} - 112 \, a^{5} b^{3} - 210 \, a^{4} b^{4} - 224 \, a^{3} b^{5} - 140 \, a^{2} b^{6} - 48 \, a b^{7} - 7 \, b^{8}\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - \frac {30 \, {\left (56 \, a^{5} b^{3} \sin \left (d x + c\right )^{2} + 112 \, a^{3} b^{5} \sin \left (d x + c\right )^{2} + 24 \, a b^{7} \sin \left (d x + c\right )^{2} + a^{8} \sin \left (d x + c\right ) + 28 \, a^{6} b^{2} \sin \left (d x + c\right ) + 70 \, a^{4} b^{4} \sin \left (d x + c\right ) + 28 \, a^{2} b^{6} \sin \left (d x + c\right ) + b^{8} \sin \left (d x + c\right ) + 8 \, a^{7} b - 56 \, a^{3} b^{5} - 16 \, a b^{7}\right )}}{\sin \left (d x + c\right )^{2} - 1}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.33, size = 645, normalized size = 2.27 \[ \frac {b^{8} \left (\sin ^{7}\left (d x +c \right )\right )}{2 d}+\frac {14 a^{2} b^{6} \left (\sin ^{5}\left (d x +c \right )\right )}{d}+\frac {70 a^{2} b^{6} \left (\sin ^{3}\left (d x +c \right )\right )}{3 d}+\frac {6 a \,b^{7} \left (\sin ^{4}\left (d x +c \right )\right )}{d}+\frac {12 a \,b^{7} \left (\sin ^{2}\left (d x +c \right )\right )}{d}+\frac {28 a^{3} b^{5} \left (\sin ^{4}\left (d x +c \right )\right )}{d}+\frac {56 a^{3} b^{5} \left (\sin ^{2}\left (d x +c \right )\right )}{d}+\frac {35 a^{4} b^{4} \left (\sin ^{3}\left (d x +c \right )\right )}{d}+\frac {7 \sin \left (d x +c \right ) b^{8}}{2 d}-\frac {7 b^{8} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {a^{8} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {14 a^{6} b^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )^{2}}+\frac {35 a^{4} b^{4} \left (\sin ^{5}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )^{2}}+\frac {28 a^{3} b^{5} \left (\sin ^{6}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )^{2}}+\frac {14 a^{2} b^{6} \left (\sin ^{7}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )^{2}}+\frac {70 a^{2} b^{6} \sin \left (d x +c \right )}{d}-\frac {70 a^{2} b^{6} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {24 a \,b^{7} \ln \left (\cos \left (d x +c \right )\right )}{d}+\frac {112 a^{3} b^{5} \ln \left (\cos \left (d x +c \right )\right )}{d}+\frac {105 a^{4} b^{4} \sin \left (d x +c \right )}{d}-\frac {105 a^{4} b^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {14 a^{6} b^{2} \sin \left (d x +c \right )}{d}-\frac {14 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 )}{d}+\frac {7 b^{8} \left (\sin ^{3}\left (d x +c \right )\right )}{6 d}+\frac {7 b^{8} \left (\sin ^{5}\left (d x +c \right )\right )}{10 d}+\frac {4 a \,b^{7} \left (\sin ^{8}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )^{2}}+\frac {b^{8} \left (\sin ^{9}\left (d x +c \right )\right )}{2 d \cos \left (d x +c \right )^{2}}+\frac {28 a^{5} b^{3} \left (\tan ^{2}\left (d x +c \right )\right )}{d}+\frac {a^{8} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d}+\frac {4 a^{7} b}{d \cos \left (d x +c \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 323, normalized size = 1.14 \[ \frac {12 \, b^{8} \sin \left (d x + c\right )^{5} + 120 \, a b^{7} \sin \left (d x + c\right )^{4} + 40 \, {\left (14 \, a^{2} b^{6} + b^{8}\right )} \sin \left (d x + c\right )^{3} + 240 \, {\left (7 \, a^{3} b^{5} + 2 \, a b^{7}\right )} \sin \left (d x + c\right )^{2} + 15 \, {\left (a^{8} - 28 \, a^{6} b^{2} + 112 \, a^{5} b^{3} - 210 \, a^{4} b^{4} + 224 \, a^{3} b^{5} - 140 \, a^{2} b^{6} + 48 \, a b^{7} - 7 \, b^{8}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (a^{8} - 28 \, a^{6} b^{2} - 112 \, a^{5} b^{3} - 210 \, a^{4} b^{4} - 224 \, a^{3} b^{5} - 140 \, a^{2} b^{6} - 48 \, a b^{7} - 7 \, b^{8}\right )} \log \left (\sin \left (d x + c\right ) - 1\right ) + 60 \, {\left (70 \, a^{4} b^{4} + 56 \, a^{2} b^{6} + 3 \, b^{8}\right )} \sin \left (d x + c\right ) - \frac {30 \, {\left (8 \, a^{7} b + 56 \, a^{5} b^{3} + 56 \, a^{3} b^{5} + 8 \, a b^{7} + {\left (a^{8} + 28 \, a^{6} b^{2} + 70 \, a^{4} b^{4} + 28 \, a^{2} b^{6} + b^{8}\right )} \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{2} - 1}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.39, size = 257, normalized size = 0.90 \[ \frac {{\sin \left (c+d\,x\right )}^3\,\left (\frac {28\,a^2\,b^6}{3}+\frac {2\,b^8}{3}\right )}{d}+\frac {b^8\,{\sin \left (c+d\,x\right )}^5}{5\,d}+\frac {{\sin \left (c+d\,x\right )}^2\,\left (28\,a^3\,b^5+8\,a\,b^7\right )}{d}+\frac {\sin \left (c+d\,x\right )\,\left (70\,a^4\,b^4+56\,a^2\,b^6+3\,b^8\right )}{d}-\frac {\sin \left (c+d\,x\right )\,\left (\frac {a^8}{2}+14\,a^6\,b^2+35\,a^4\,b^4+14\,a^2\,b^6+\frac {b^8}{2}\right )+4\,a\,b^7+4\,a^7\,b+28\,a^3\,b^5+28\,a^5\,b^3}{d\,\left ({\sin \left (c+d\,x\right )}^2-1\right )}+\frac {2\,a\,b^7\,{\sin \left (c+d\,x\right )}^4}{d}-\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )\,{\left (a+b\right )}^7\,\left (a-7\,b\right )}{4\,d}+\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )\,{\left (a-b\right )}^7\,\left (a+7\,b\right )}{4\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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