Optimal. Leaf size=94 \[ \frac {\cos (a-c) \sec ^5(c+b x)}{5 b}+\frac {3 \tanh ^{-1}(\sin (c+b x)) \sin (a-c)}{8 b}+\frac {3 \sec (c+b x) \sin (a-c) \tan (c+b x)}{8 b}+\frac {\sec ^3(c+b x) \sin (a-c) \tan (c+b x)}{4 b} \]
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Rubi [A]
time = 0.04, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4676, 2686, 30,
3853, 3855} \begin {gather*} \frac {3 \sin (a-c) \tanh ^{-1}(\sin (b x+c))}{8 b}+\frac {\cos (a-c) \sec ^5(b x+c)}{5 b}+\frac {\sin (a-c) \tan (b x+c) \sec ^3(b x+c)}{4 b}+\frac {3 \sin (a-c) \tan (b x+c) \sec (b x+c)}{8 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2686
Rule 3853
Rule 3855
Rule 4676
Rubi steps
\begin {align*} \int \sec ^6(c+b x) \sin (a+b x) \, dx &=\cos (a-c) \int \sec ^5(c+b x) \tan (c+b x) \, dx+\sin (a-c) \int \sec ^5(c+b x) \, dx\\ &=\frac {\sec ^3(c+b x) \sin (a-c) \tan (c+b x)}{4 b}+\frac {\cos (a-c) \text {Subst}\left (\int x^4 \, dx,x,\sec (c+b x)\right )}{b}+\frac {1}{4} (3 \sin (a-c)) \int \sec ^3(c+b x) \, dx\\ &=\frac {\cos (a-c) \sec ^5(c+b x)}{5 b}+\frac {3 \sec (c+b x) \sin (a-c) \tan (c+b x)}{8 b}+\frac {\sec ^3(c+b x) \sin (a-c) \tan (c+b x)}{4 b}+\frac {1}{8} (3 \sin (a-c)) \int \sec (c+b x) \, dx\\ &=\frac {\cos (a-c) \sec ^5(c+b x)}{5 b}+\frac {3 \tanh ^{-1}(\sin (c+b x)) \sin (a-c)}{8 b}+\frac {3 \sec (c+b x) \sin (a-c) \tan (c+b x)}{8 b}+\frac {\sec ^3(c+b x) \sin (a-c) \tan (c+b x)}{4 b}\\ \end {align*}
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Mathematica [A]
time = 1.05, size = 78, normalized size = 0.83 \begin {gather*} \frac {480 \tanh ^{-1}\left (\sin (c)+\cos (c) \tan \left (\frac {b x}{2}\right )\right ) \sin (a-c)+2 \sec ^5(c+b x) (64 \cos (a-c)+5 \sin (a-c) (14 \sin (2 (c+b x))+3 \sin (4 (c+b x))))}{640 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(6741\) vs.
\(2(86)=172\).
time = 6.90, size = 6742, normalized size = 71.72
method | result | size |
risch | \(\frac {-15 \,{\mathrm e}^{i \left (9 b x +11 a +8 c \right )}+15 \,{\mathrm e}^{i \left (9 b x +9 a +10 c \right )}-70 \,{\mathrm e}^{i \left (7 b x +11 a +6 c \right )}+70 \,{\mathrm e}^{i \left (7 b x +9 a +8 c \right )}+128 \,{\mathrm e}^{i \left (5 b x +11 a +4 c \right )}+128 \,{\mathrm e}^{i \left (5 b x +9 a +6 c \right )}+70 \,{\mathrm e}^{i \left (3 b x +11 a +2 c \right )}-70 \,{\mathrm e}^{i \left (3 b x +9 a +4 c \right )}+15 \,{\mathrm e}^{i \left (b x +11 a \right )}-15 \,{\mathrm e}^{i \left (b x +9 a +2 c \right )}}{40 b \left ({\mathrm e}^{2 i \left (b x +a +c \right )}+{\mathrm e}^{2 i a}\right )^{5}}-\frac {3 \ln \left ({\mathrm e}^{i \left (b x +a \right )}-i {\mathrm e}^{i \left (a -c \right )}\right ) \sin \left (a -c \right )}{8 b}+\frac {3 \ln \left ({\mathrm e}^{i \left (b x +a \right )}+i {\mathrm e}^{i \left (a -c \right )}\right ) \sin \left (a -c \right )}{8 b}\) | \(259\) |
default | \(\text {Expression too large to display}\) | \(6742\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 3096 vs.
\(2 (86) = 172\).
time = 0.69, size = 3096, normalized size = 32.94 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.95, size = 107, normalized size = 1.14 \begin {gather*} -\frac {15 \, \cos \left (b x + c\right )^{5} \log \left (\sin \left (b x + c\right ) + 1\right ) \sin \left (-a + c\right ) - 15 \, \cos \left (b x + c\right )^{5} \log \left (-\sin \left (b x + c\right ) + 1\right ) \sin \left (-a + c\right ) + 10 \, {\left (3 \, \cos \left (b x + c\right )^{3} + 2 \, \cos \left (b x + c\right )\right )} \sin \left (b x + c\right ) \sin \left (-a + c\right ) - 16 \, \cos \left (-a + c\right )}{80 \, b \cos \left (b x + c\right )^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 756 vs.
\(2 (86) = 172\).
time = 0.44, size = 756, normalized size = 8.04 \begin {gather*} \frac {\frac {15 \, {\left (\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right ) - \tan \left (\frac {1}{2} \, c\right )\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right )^{2} + \tan \left (\frac {1}{2} \, c\right )^{2} + 1} - \frac {15 \, {\left (\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right ) - \tan \left (\frac {1}{2} \, c\right )\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right )^{2} + \tan \left (\frac {1}{2} \, c\right )^{2} + 1} + \frac {2 \, {\left (25 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{9} \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - 25 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{9} \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} - 20 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{8} \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + 25 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{9} \tan \left (\frac {1}{2} \, a\right ) + 20 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{8} \tan \left (\frac {1}{2} \, a\right )^{2} - 25 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{9} \tan \left (\frac {1}{2} \, c\right ) - 80 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{8} \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) - 10 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{7} \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) + 20 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{8} \tan \left (\frac {1}{2} \, c\right )^{2} + 10 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{7} \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} - 20 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{8} - 10 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{7} \tan \left (\frac {1}{2} \, a\right ) + 10 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{7} \tan \left (\frac {1}{2} \, c\right ) - 40 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{4} \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + 40 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{4} \tan \left (\frac {1}{2} \, a\right )^{2} - 160 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{4} \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) + 10 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{3} \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) + 40 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{4} \tan \left (\frac {1}{2} \, c\right )^{2} - 10 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{3} \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} - 40 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{4} + 10 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{3} \tan \left (\frac {1}{2} \, a\right ) - 10 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{3} \tan \left (\frac {1}{2} \, c\right ) - 25 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right ) \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) + 25 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right ) \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} - 4 \, \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - 25 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right ) \tan \left (\frac {1}{2} \, a\right ) + 4 \, \tan \left (\frac {1}{2} \, a\right )^{2} + 25 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right ) \tan \left (\frac {1}{2} \, c\right ) - 16 \, \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) + 4 \, \tan \left (\frac {1}{2} \, c\right )^{2} - 4\right )}}{{\left (\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right )^{2} + \tan \left (\frac {1}{2} \, c\right )^{2} + 1\right )} {\left (\tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{5}}}{20 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \text {Hanged} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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