Optimal. Leaf size=59 \[ \frac {\tanh ^{-1}(\sin (c+d x))}{b d}-\frac {\sqrt {a-b} \tanh ^{-1}\left (\frac {\sqrt {a-b} \sin (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b d} \]
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Rubi [A] time = 0.08, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3676, 391, 206, 208} \[ \frac {\tanh ^{-1}(\sin (c+d x))}{b d}-\frac {\sqrt {a-b} \tanh ^{-1}\left (\frac {\sqrt {a-b} \sin (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 208
Rule 391
Rule 3676
Rubi steps
\begin {align*} \int \frac {\sec ^3(c+d x)}{a+b \tan ^2(c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \left (a-(a-b) x^2\right )} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sin (c+d x)\right )}{b d}-\frac {(a-b) \operatorname {Subst}\left (\int \frac {1}{a+(-a+b) x^2} \, dx,x,\sin (c+d x)\right )}{b d}\\ &=\frac {\tanh ^{-1}(\sin (c+d x))}{b d}-\frac {\sqrt {a-b} \tanh ^{-1}\left (\frac {\sqrt {a-b} \sin (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b d}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 53, normalized size = 0.90 \[ \frac {\tanh ^{-1}(\sin (c+d x))-\frac {\sqrt {a-b} \tanh ^{-1}\left (\frac {\sqrt {a-b} \sin (c+d x)}{\sqrt {a}}\right )}{\sqrt {a}}}{b d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 169, normalized size = 2.86 \[ \left [\frac {\sqrt {\frac {a - b}{a}} \log \left (-\frac {{\left (a - b\right )} \cos \left (d x + c\right )^{2} + 2 \, a \sqrt {\frac {a - b}{a}} \sin \left (d x + c\right ) - 2 \, a + b}{{\left (a - b\right )} \cos \left (d x + c\right )^{2} + b}\right ) + \log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (-\sin \left (d x + c\right ) + 1\right )}{2 \, b d}, \frac {2 \, \sqrt {-\frac {a - b}{a}} \arctan \left (\sqrt {-\frac {a - b}{a}} \sin \left (d x + c\right )\right ) + \log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (-\sin \left (d x + c\right ) + 1\right )}{2 \, b d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.84, size = 88, normalized size = 1.49 \[ -\frac {\frac {2 \, {\left (a - b\right )} \arctan \left (-\frac {a \sin \left (d x + c\right ) - b \sin \left (d x + c\right )}{\sqrt {-a^{2} + a b}}\right )}{\sqrt {-a^{2} + a b} b} - \frac {\log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{b} + \frac {\log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{b}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.61, size = 111, normalized size = 1.88 \[ -\frac {\arctanh \left (\frac {\left (a -b \right ) \sin \left (d x +c \right )}{\sqrt {a \left (a -b \right )}}\right ) a}{d b \sqrt {a \left (a -b \right )}}+\frac {\arctanh \left (\frac {\left (a -b \right ) \sin \left (d x +c \right )}{\sqrt {a \left (a -b \right )}}\right )}{d \sqrt {a \left (a -b \right )}}-\frac {\ln \left (-1+\sin \left (d x +c \right )\right )}{2 d b}+\frac {\ln \left (\sin \left (d x +c \right )+1\right )}{2 d b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.67, size = 67, normalized size = 1.14 \[ \frac {2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{b\,d}-\frac {\mathrm {atanh}\left (\frac {\sin \left (c+d\,x\right )\,\sqrt {a-b}}{\sqrt {a}}\right )\,\sqrt {a-b}}{\sqrt {a}\,b\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^{3}{\left (c + d x \right )}}{a + b \tan ^{2}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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