Optimal. Leaf size=165 \[ \frac {4 a \left (2 a^2-b^2\right ) \tan ^3(c+d x)}{35 d}+\frac {12 a \left (2 a^2-b^2\right ) \tan (c+d x)}{35 d}+\frac {2 b \left (3 a^2-b^2\right ) \sec ^3(c+d x)}{35 d}+\frac {2 \sec ^5(c+d x) (a+b \sin (c+d x)) \left (\left (3 a^2-b^2\right ) \sin (c+d x)+2 a b\right )}{35 d}+\frac {\sec ^7(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{7 d} \]
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Rubi [A] time = 0.21, antiderivative size = 165, 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 = {2691, 2861, 2669, 3767} \[ \frac {4 a \left (2 a^2-b^2\right ) \tan ^3(c+d x)}{35 d}+\frac {12 a \left (2 a^2-b^2\right ) \tan (c+d x)}{35 d}+\frac {2 b \left (3 a^2-b^2\right ) \sec ^3(c+d x)}{35 d}+\frac {2 \sec ^5(c+d x) (a+b \sin (c+d x)) \left (\left (3 a^2-b^2\right ) \sin (c+d x)+2 a b\right )}{35 d}+\frac {\sec ^7(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{7 d} \]
Antiderivative was successfully verified.
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Rule 2669
Rule 2691
Rule 2861
Rule 3767
Rubi steps
\begin {align*} \int \sec ^8(c+d x) (a+b \sin (c+d x))^3 \, dx &=\frac {\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{7 d}-\frac {1}{7} \int \sec ^6(c+d x) (a+b \sin (c+d x)) \left (-6 a^2+2 b^2-4 a b \sin (c+d x)\right ) \, dx\\ &=\frac {\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{7 d}+\frac {2 \sec ^5(c+d x) (a+b \sin (c+d x)) \left (2 a b+\left (3 a^2-b^2\right ) \sin (c+d x)\right )}{35 d}+\frac {1}{35} \int \sec ^4(c+d x) \left (12 a \left (2 a^2-b^2\right )+6 b \left (3 a^2-b^2\right ) \sin (c+d x)\right ) \, dx\\ &=\frac {2 b \left (3 a^2-b^2\right ) \sec ^3(c+d x)}{35 d}+\frac {\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{7 d}+\frac {2 \sec ^5(c+d x) (a+b \sin (c+d x)) \left (2 a b+\left (3 a^2-b^2\right ) \sin (c+d x)\right )}{35 d}+\frac {1}{35} \left (12 a \left (2 a^2-b^2\right )\right ) \int \sec ^4(c+d x) \, dx\\ &=\frac {2 b \left (3 a^2-b^2\right ) \sec ^3(c+d x)}{35 d}+\frac {\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{7 d}+\frac {2 \sec ^5(c+d x) (a+b \sin (c+d x)) \left (2 a b+\left (3 a^2-b^2\right ) \sin (c+d x)\right )}{35 d}-\frac {\left (12 a \left (2 a^2-b^2\right )\right ) \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{35 d}\\ &=\frac {2 b \left (3 a^2-b^2\right ) \sec ^3(c+d x)}{35 d}+\frac {\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{7 d}+\frac {2 \sec ^5(c+d x) (a+b \sin (c+d x)) \left (2 a b+\left (3 a^2-b^2\right ) \sin (c+d x)\right )}{35 d}+\frac {12 a \left (2 a^2-b^2\right ) \tan (c+d x)}{35 d}+\frac {4 a \left (2 a^2-b^2\right ) \tan ^3(c+d x)}{35 d}\\ \end {align*}
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Mathematica [A] time = 0.90, size = 245, normalized size = 1.48 \[ \frac {\sec ^7(c+d x) \left (8960 a^3 \sin (c+d x)+5376 a^3 \sin (3 (c+d x))+1792 a^3 \sin (5 (c+d x))+256 a^3 \sin (7 (c+d x))+35 b \left (17 b^2-75 a^2\right ) \cos (c+d x)-1575 a^2 b \cos (3 (c+d x))-525 a^2 b \cos (5 (c+d x))-75 a^2 b \cos (7 (c+d x))+15360 a^2 b+13440 a b^2 \sin (c+d x)-2688 a b^2 \sin (3 (c+d x))-896 a b^2 \sin (5 (c+d x))-128 a b^2 \sin (7 (c+d x))-3584 b^3 \cos (2 (c+d x))+357 b^3 \cos (3 (c+d x))+119 b^3 \cos (5 (c+d x))+17 b^3 \cos (7 (c+d x))+1536 b^3\right )}{35840 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 124, normalized size = 0.75 \[ -\frac {7 \, b^{3} \cos \left (d x + c\right )^{2} - 15 \, a^{2} b - 5 \, b^{3} - {\left (8 \, {\left (2 \, a^{3} - a b^{2}\right )} \cos \left (d x + c\right )^{6} + 4 \, {\left (2 \, a^{3} - a b^{2}\right )} \cos \left (d x + c\right )^{4} + 5 \, a^{3} + 15 \, a b^{2} + 3 \, {\left (2 \, a^{3} - a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{35 \, d \cos \left (d x + c\right )^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.88, size = 358, normalized size = 2.17 \[ -\frac {2 \, {\left (35 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 105 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} - 70 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 140 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 70 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 301 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 112 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 525 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 70 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 212 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 456 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 140 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 301 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 112 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 315 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 28 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 70 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 140 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 14 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 35 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 15 \, a^{2} b - 2 \, b^{3}\right )}}{35 \, {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{7} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.37, size = 219, normalized size = 1.33 \[ \frac {-a^{3} \left (-\frac {16}{35}-\frac {\left (\sec ^{6}\left (d x +c \right )\right )}{7}-\frac {6 \left (\sec ^{4}\left (d x +c \right )\right )}{35}-\frac {8 \left (\sec ^{2}\left (d x +c \right )\right )}{35}\right ) \tan \left (d x +c \right )+\frac {3 a^{2} b}{7 \cos \left (d x +c \right )^{7}}+3 a \,b^{2} \left (\frac {\sin ^{3}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {4 \left (\sin ^{3}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{3}}\right )+b^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{4}\left (d x +c \right )}{35 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{35 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{35}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 124, normalized size = 0.75 \[ \frac {{\left (5 \, \tan \left (d x + c\right )^{7} + 21 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3} + 35 \, \tan \left (d x + c\right )\right )} a^{3} + {\left (15 \, \tan \left (d x + c\right )^{7} + 42 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3}\right )} a b^{2} - \frac {{\left (7 \, \cos \left (d x + c\right )^{2} - 5\right )} b^{3}}{\cos \left (d x + c\right )^{7}} + \frac {15 \, a^{2} b}{\cos \left (d x + c\right )^{7}}}{35 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.60, size = 152, normalized size = 0.92 \[ \frac {{\cos \left (c+d\,x\right )}^4\,\left (\frac {8\,a^3\,\sin \left (c+d\,x\right )}{35}-\frac {4\,a\,b^2\,\sin \left (c+d\,x\right )}{35}\right )+{\cos \left (c+d\,x\right )}^6\,\left (\frac {16\,a^3\,\sin \left (c+d\,x\right )}{35}-\frac {8\,a\,b^2\,\sin \left (c+d\,x\right )}{35}\right )-{\cos \left (c+d\,x\right )}^2\,\left (-\frac {6\,\sin \left (c+d\,x\right )\,a^3}{35}+\frac {3\,\sin \left (c+d\,x\right )\,a\,b^2}{35}+\frac {b^3}{5}\right )+\frac {3\,a^2\,b}{7}+\frac {a^3\,\sin \left (c+d\,x\right )}{7}+\frac {b^3}{7}+\frac {3\,a\,b^2\,\sin \left (c+d\,x\right )}{7}}{d\,{\cos \left (c+d\,x\right )}^7} \]
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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