Optimal. Leaf size=48 \[ \frac {a \tan (c+d x)}{d}-\frac {a \cot (c+d x)}{d}+\frac {b \sec (c+d x)}{d}-\frac {b \tanh ^{-1}(\cos (c+d x))}{d} \]
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Rubi [A] time = 0.11, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2838, 2620, 14, 2622, 321, 207} \[ \frac {a \tan (c+d x)}{d}-\frac {a \cot (c+d x)}{d}+\frac {b \sec (c+d x)}{d}-\frac {b \tanh ^{-1}(\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 207
Rule 321
Rule 2620
Rule 2622
Rule 2838
Rubi steps
\begin {align*} \int \csc ^2(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x)) \, dx &=a \int \csc ^2(c+d x) \sec ^2(c+d x) \, dx+b \int \csc (c+d x) \sec ^2(c+d x) \, dx\\ &=\frac {a \operatorname {Subst}\left (\int \frac {1+x^2}{x^2} \, dx,x,\tan (c+d x)\right )}{d}+\frac {b \operatorname {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{d}\\ &=\frac {b \sec (c+d x)}{d}+\frac {a \operatorname {Subst}\left (\int \left (1+\frac {1}{x^2}\right ) \, dx,x,\tan (c+d x)\right )}{d}+\frac {b \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{d}\\ &=-\frac {b \tanh ^{-1}(\cos (c+d x))}{d}-\frac {a \cot (c+d x)}{d}+\frac {b \sec (c+d x)}{d}+\frac {a \tan (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 68, normalized size = 1.42 \[ \frac {a \tan (c+d x)}{d}-\frac {a \cot (c+d x)}{d}+\frac {b \sec (c+d x)}{d}+\frac {b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}-\frac {b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 96, normalized size = 2.00 \[ -\frac {b \cos \left (d x + c\right ) \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - b \cos \left (d x + c\right ) \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 4 \, a \cos \left (d x + c\right )^{2} - 2 \, b \sin \left (d x + c\right ) - 2 \, a}{2 \, d \cos \left (d x + c\right ) \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 103, normalized size = 2.15 \[ \frac {6 \, b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 10 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.40, size = 69, normalized size = 1.44 \[ \frac {a}{d \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {2 a \cot \left (d x +c \right )}{d}+\frac {b}{d \cos \left (d x +c \right )}+\frac {b \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 59, normalized size = 1.23 \[ \frac {b {\left (\frac {2}{\cos \left (d x + c\right )} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 2 \, a {\left (\frac {1}{\tan \left (d x + c\right )} - \tan \left (d x + c\right )\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.91, size = 92, normalized size = 1.92 \[ \frac {5\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+4\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-a}{d\,\left (2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )}+\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d}+\frac {b\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d} \]
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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