Optimal. Leaf size=94 \[ \frac {(2 a-b) \tanh ^{-1}\left (\frac {\sqrt {a-b} \sin (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} d (a-b)^{3/2}}-\frac {b \sin (c+d x)}{2 a d (a-b) \left (a-(a-b) \sin ^2(c+d x)\right )} \]
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Rubi [A] time = 0.08, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3676, 385, 208} \[ \frac {(2 a-b) \tanh ^{-1}\left (\frac {\sqrt {a-b} \sin (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} d (a-b)^{3/2}}-\frac {b \sin (c+d x)}{2 a d (a-b) \left (a-(a-b) \sin ^2(c+d x)\right )} \]
Antiderivative was successfully verified.
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Rule 208
Rule 385
Rule 3676
Rubi steps
\begin {align*} \int \frac {\sec (c+d x)}{\left (a+b \tan ^2(c+d x)\right )^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1-x^2}{\left (a-(a-b) x^2\right )^2} \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac {b \sin (c+d x)}{2 a (a-b) d \left (a-(a-b) \sin ^2(c+d x)\right )}+\frac {(2 a-b) \operatorname {Subst}\left (\int \frac {1}{a+(-a+b) x^2} \, dx,x,\sin (c+d x)\right )}{2 a (a-b) d}\\ &=\frac {(2 a-b) \tanh ^{-1}\left (\frac {\sqrt {a-b} \sin (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} (a-b)^{3/2} d}-\frac {b \sin (c+d x)}{2 a (a-b) d \left (a-(a-b) \sin ^2(c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 92, normalized size = 0.98 \[ \frac {\frac {\sqrt {a} b \sin (c+d x)}{(a-b) \left ((a-b) \sin ^2(c+d x)-a\right )}+\frac {(2 a-b) \tanh ^{-1}\left (\frac {\sqrt {a-b} \sin (c+d x)}{\sqrt {a}}\right )}{(a-b)^{3/2}}}{2 a^{3/2} d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 337, normalized size = 3.59 \[ \left [\frac {{\left ({\left (2 \, a^{2} - 3 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, a b - b^{2}\right )} \sqrt {a^{2} - a b} \log \left (-\frac {{\left (a - b\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {a^{2} - a b} \sin \left (d x + c\right ) - 2 \, a + b}{{\left (a - b\right )} \cos \left (d x + c\right )^{2} + b}\right ) - 2 \, {\left (a^{2} b - a b^{2}\right )} \sin \left (d x + c\right )}{4 \, {\left ({\left (a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right )} d \cos \left (d x + c\right )^{2} + {\left (a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} d\right )}}, -\frac {{\left ({\left (2 \, a^{2} - 3 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, a b - b^{2}\right )} \sqrt {-a^{2} + a b} \arctan \left (\frac {\sqrt {-a^{2} + a b} \sin \left (d x + c\right )}{a}\right ) + {\left (a^{2} b - a b^{2}\right )} \sin \left (d x + c\right )}{2 \, {\left ({\left (a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right )} d \cos \left (d x + c\right )^{2} + {\left (a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} d\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.85, size = 112, normalized size = 1.19 \[ -\frac {\frac {{\left (2 \, a - b\right )} \arctan \left (\frac {a \sin \left (d x + c\right ) - b \sin \left (d x + c\right )}{\sqrt {-a^{2} + a b}}\right )}{{\left (a^{2} - a b\right )} \sqrt {-a^{2} + a b}} - \frac {b \sin \left (d x + c\right )}{{\left (a \sin \left (d x + c\right )^{2} - b \sin \left (d x + c\right )^{2} - a\right )} {\left (a^{2} - a b\right )}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.59, size = 102, normalized size = 1.09 \[ \frac {\frac {b \sin \left (d x +c \right )}{2 a \left (a -b \right ) \left (a \left (\sin ^{2}\left (d x +c \right )\right )-b \left (\sin ^{2}\left (d x +c \right )\right )-a \right )}+\frac {\left (2 a -b \right ) \arctanh \left (\frac {\left (a -b \right ) \sin \left (d x +c \right )}{\sqrt {a \left (a -b \right )}}\right )}{2 a \left (a -b \right ) \sqrt {a \left (a -b \right )}}}{d} \]
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 = 12.78, size = 239, normalized size = 2.54 \[ -\frac {\left (a^2\,\mathrm {atanh}\left (\frac {\sin \left (c+d\,x\right )\,\sqrt {a-b}}{\sqrt {a}}\right )\,1{}\mathrm {i}-\frac {b^2\,\mathrm {atanh}\left (\frac {\sin \left (c+d\,x\right )\,\sqrt {a-b}}{\sqrt {a}}\right )\,1{}\mathrm {i}}{2}+a^2\,\cos \left (2\,c+2\,d\,x\right )\,\mathrm {atanh}\left (\frac {\sin \left (c+d\,x\right )\,\sqrt {a-b}}{\sqrt {a}}\right )\,1{}\mathrm {i}+\frac {b^2\,\cos \left (2\,c+2\,d\,x\right )\,\mathrm {atanh}\left (\frac {\sin \left (c+d\,x\right )\,\sqrt {a-b}}{\sqrt {a}}\right )\,1{}\mathrm {i}}{2}+\frac {a\,b\,\mathrm {atanh}\left (\frac {\sin \left (c+d\,x\right )\,\sqrt {a-b}}{\sqrt {a}}\right )\,1{}\mathrm {i}}{2}-\frac {a\,b\,\cos \left (2\,c+2\,d\,x\right )\,\mathrm {atanh}\left (\frac {\sin \left (c+d\,x\right )\,\sqrt {a-b}}{\sqrt {a}}\right )\,3{}\mathrm {i}}{2}-\sqrt {a}\,b\,\sin \left (c+d\,x\right )\,\sqrt {a-b}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,a^{3/2}\,d\,{\left (a-b\right )}^{3/2}\,\left (\frac {a}{2}+\frac {b}{2}+\frac {a\,\cos \left (2\,c+2\,d\,x\right )}{2}-\frac {b\,\cos \left (2\,c+2\,d\,x\right )}{2}\right )} \]
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 \[ \int \frac {\sec {\left (c + d x \right )}}{\left (a + b \tan ^{2}{\left (c + d x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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