Optimal. Leaf size=99 \[ \frac {a \tan ^4(c+d x)}{4 d}+\frac {a \tan ^2(c+d x)}{d}+\frac {a \log (\tan (c+d x))}{d}+\frac {3 b \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {b \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac {3 b \tan (c+d x) \sec (c+d x)}{8 d} \]
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Rubi [A] time = 0.11, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2834, 2620, 266, 43, 3768, 3770} \[ \frac {a \tan ^4(c+d x)}{4 d}+\frac {a \tan ^2(c+d x)}{d}+\frac {a \log (\tan (c+d x))}{d}+\frac {3 b \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {b \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac {3 b \tan (c+d x) \sec (c+d x)}{8 d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 2620
Rule 2834
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \csc (c+d x) \sec ^5(c+d x) (a+b \sin (c+d x)) \, dx &=a \int \csc (c+d x) \sec ^5(c+d x) \, dx+b \int \sec ^5(c+d x) \, dx\\ &=\frac {b \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{4} (3 b) \int \sec ^3(c+d x) \, dx+\frac {a \operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^2}{x} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {3 b \sec (c+d x) \tan (c+d x)}{8 d}+\frac {b \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{8} (3 b) \int \sec (c+d x) \, dx+\frac {a \operatorname {Subst}\left (\int \frac {(1+x)^2}{x} \, dx,x,\tan ^2(c+d x)\right )}{2 d}\\ &=\frac {3 b \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {3 b \sec (c+d x) \tan (c+d x)}{8 d}+\frac {b \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {a \operatorname {Subst}\left (\int \left (2+\frac {1}{x}+x\right ) \, dx,x,\tan ^2(c+d x)\right )}{2 d}\\ &=\frac {3 b \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {a \log (\tan (c+d x))}{d}+\frac {3 b \sec (c+d x) \tan (c+d x)}{8 d}+\frac {b \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {a \tan ^2(c+d x)}{d}+\frac {a \tan ^4(c+d x)}{4 d}\\ \end {align*}
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Mathematica [A] time = 1.26, size = 99, normalized size = 1.00 \[ -\frac {a \left (-\sec ^4(c+d x)-2 \sec ^2(c+d x)-4 \log (\sin (c+d x))+4 \log (\cos (c+d x))\right )}{4 d}+\frac {b \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac {3 b \left (\tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x)\right )}{8 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 125, normalized size = 1.26 \[ \frac {16 \, a \cos \left (d x + c\right )^{4} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - {\left (8 \, a - 3 \, b\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (8 \, a + 3 \, b\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 8 \, a \cos \left (d x + c\right )^{2} + 2 \, {\left (3 \, b \cos \left (d x + c\right )^{2} + 2 \, b\right )} \sin \left (d x + c\right ) + 4 \, a}{16 \, d \cos \left (d x + c\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 113, normalized size = 1.14 \[ -\frac {{\left (8 \, a - 3 \, b\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) + {\left (8 \, a + 3 \, b\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - 16 \, a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - \frac {2 \, {\left (6 \, a \sin \left (d x + c\right )^{4} - 3 \, b \sin \left (d x + c\right )^{3} - 16 \, a \sin \left (d x + c\right )^{2} + 5 \, b \sin \left (d x + c\right ) + 12 \, a\right )}}{{\left (\sin \left (d x + c\right )^{2} - 1\right )}^{2}}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.43, size = 100, normalized size = 1.01 \[ \frac {a}{4 d \cos \left (d x +c \right )^{4}}+\frac {a}{2 d \cos \left (d x +c \right )^{2}}+\frac {a \ln \left (\tan \left (d x +c \right )\right )}{d}+\frac {b \left (\sec ^{3}\left (d x +c \right )\right ) \tan \left (d x +c \right )}{4 d}+\frac {3 b \sec \left (d x +c \right ) \tan \left (d x +c \right )}{8 d}+\frac {3 b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 109, normalized size = 1.10 \[ -\frac {{\left (8 \, a - 3 \, b\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + {\left (8 \, a + 3 \, b\right )} \log \left (\sin \left (d x + c\right ) - 1\right ) - 16 \, a \log \left (\sin \left (d x + c\right )\right ) + \frac {2 \, {\left (3 \, b \sin \left (d x + c\right )^{3} + 4 \, a \sin \left (d x + c\right )^{2} - 5 \, b \sin \left (d x + c\right ) - 6 \, a\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.87, size = 116, normalized size = 1.17 \[ \frac {-\frac {3\,b\,{\sin \left (c+d\,x\right )}^3}{8}-\frac {a\,{\sin \left (c+d\,x\right )}^2}{2}+\frac {5\,b\,\sin \left (c+d\,x\right )}{8}+\frac {3\,a}{4}}{d\,\left ({\sin \left (c+d\,x\right )}^4-2\,{\sin \left (c+d\,x\right )}^2+1\right )}-\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )\,\left (\frac {a}{2}-\frac {3\,b}{16}\right )}{d}-\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )\,\left (\frac {a}{2}+\frac {3\,b}{16}\right )}{d}+\frac {a\,\ln \left (\sin \left (c+d\,x\right )\right )}{d} \]
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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