Optimal. Leaf size=109 \[ -\frac {\sqrt {a-b} (2 a+b) \tanh ^{-1}\left (\frac {\sqrt {a-b} \sin (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} b^2 d}-\frac {(a-b) \sin (c+d x)}{2 a b d \left (a-(a-b) \sin ^2(c+d x)\right )}+\frac {\tanh ^{-1}(\sin (c+d x))}{b^2 d} \]
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Rubi [A] time = 0.14, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3676, 414, 522, 206, 208} \[ -\frac {\sqrt {a-b} (2 a+b) \tanh ^{-1}\left (\frac {\sqrt {a-b} \sin (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} b^2 d}-\frac {(a-b) \sin (c+d x)}{2 a b d \left (a-(a-b) \sin ^2(c+d x)\right )}+\frac {\tanh ^{-1}(\sin (c+d x))}{b^2 d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 208
Rule 414
Rule 522
Rule 3676
Rubi steps
\begin {align*} \int \frac {\sec ^5(c+d x)}{\left (a+b \tan ^2(c+d x)\right )^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \left (a-(a-b) x^2\right )^2} \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac {(a-b) \sin (c+d x)}{2 a b d \left (a-(a-b) \sin ^2(c+d x)\right )}-\frac {\operatorname {Subst}\left (\int \frac {-a-b+(-a+b) x^2}{\left (1-x^2\right ) \left (a+(-a+b) x^2\right )} \, dx,x,\sin (c+d x)\right )}{2 a b d}\\ &=-\frac {(a-b) \sin (c+d x)}{2 a b d \left (a-(a-b) \sin ^2(c+d x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sin (c+d x)\right )}{b^2 d}-\frac {((a-b) (2 a+b)) \operatorname {Subst}\left (\int \frac {1}{a+(-a+b) x^2} \, dx,x,\sin (c+d x)\right )}{2 a b^2 d}\\ &=\frac {\tanh ^{-1}(\sin (c+d x))}{b^2 d}-\frac {\sqrt {a-b} (2 a+b) \tanh ^{-1}\left (\frac {\sqrt {a-b} \sin (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} b^2 d}-\frac {(a-b) \sin (c+d x)}{2 a b d \left (a-(a-b) \sin ^2(c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.86, size = 191, normalized size = 1.75 \[ \frac {\frac {\sqrt {a-b} (2 a+b) \log \left (\sqrt {a}-\sqrt {a-b} \sin (c+d x)\right )}{a^{3/2}}+\frac {\left (-2 a^2+a b+b^2\right ) \log \left (\sqrt {a-b} \sin (c+d x)+\sqrt {a}\right )}{a^{3/2} \sqrt {a-b}}+\frac {4 b (b-a) \sin (c+d x)}{a ((a-b) \cos (2 (c+d x))+a+b)}-4 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+4 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{4 b^2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 407, normalized size = 3.73 \[ \left [\frac {{\left ({\left (2 \, a^{2} - a b - b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, a b + b^{2}\right )} \sqrt {\frac {a - b}{a}} \log \left (-\frac {{\left (a - b\right )} \cos \left (d x + c\right )^{2} + 2 \, a \sqrt {\frac {a - b}{a}} \sin \left (d x + c\right ) - 2 \, a + b}{{\left (a - b\right )} \cos \left (d x + c\right )^{2} + b}\right ) + 2 \, {\left ({\left (a^{2} - a b\right )} \cos \left (d x + c\right )^{2} + a b\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left ({\left (a^{2} - a b\right )} \cos \left (d x + c\right )^{2} + a b\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (a b - b^{2}\right )} \sin \left (d x + c\right )}{4 \, {\left (a b^{3} d + {\left (a^{2} b^{2} - a b^{3}\right )} d \cos \left (d x + c\right )^{2}\right )}}, \frac {{\left ({\left (2 \, a^{2} - a b - b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, a b + b^{2}\right )} \sqrt {-\frac {a - b}{a}} \arctan \left (\sqrt {-\frac {a - b}{a}} \sin \left (d x + c\right )\right ) + {\left ({\left (a^{2} - a b\right )} \cos \left (d x + c\right )^{2} + a b\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left ({\left (a^{2} - a b\right )} \cos \left (d x + c\right )^{2} + a b\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - {\left (a b - b^{2}\right )} \sin \left (d x + c\right )}{2 \, {\left (a b^{3} d + {\left (a^{2} b^{2} - a b^{3}\right )} d \cos \left (d x + c\right )^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.11, size = 153, normalized size = 1.40 \[ \frac {\frac {\log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{b^{2}} - \frac {\log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{b^{2}} - \frac {{\left (2 \, a^{2} - a b - b^{2}\right )} \arctan \left (-\frac {a \sin \left (d x + c\right ) - b \sin \left (d x + c\right )}{\sqrt {-a^{2} + a b}}\right )}{\sqrt {-a^{2} + a b} a b^{2}} + \frac {a \sin \left (d x + c\right ) - b \sin \left (d x + c\right )}{{\left (a \sin \left (d x + c\right )^{2} - b \sin \left (d x + c\right )^{2} - a\right )} a b}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.77, size = 236, normalized size = 2.17 \[ \frac {\sin \left (d x +c \right )}{2 d b \left (a \left (\sin ^{2}\left (d x +c \right )\right )-b \left (\sin ^{2}\left (d x +c \right )\right )-a \right )}-\frac {\arctanh \left (\frac {\left (a -b \right ) \sin \left (d x +c \right )}{\sqrt {a \left (a -b \right )}}\right ) a}{d \,b^{2} \sqrt {a \left (a -b \right )}}+\frac {\arctanh \left (\frac {\left (a -b \right ) \sin \left (d x +c \right )}{\sqrt {a \left (a -b \right )}}\right )}{2 d b \sqrt {a \left (a -b \right )}}-\frac {\sin \left (d x +c \right )}{2 d a \left (a \left (\sin ^{2}\left (d x +c \right )\right )-b \left (\sin ^{2}\left (d x +c \right )\right )-a \right )}+\frac {\arctanh \left (\frac {\left (a -b \right ) \sin \left (d x +c \right )}{\sqrt {a \left (a -b \right )}}\right )}{2 d a \sqrt {a \left (a -b \right )}}-\frac {\ln \left (-1+\sin \left (d x +c \right )\right )}{2 d \,b^{2}}+\frac {\ln \left (\sin \left (d x +c \right )+1\right )}{2 d \,b^{2}} \]
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 = 14.37, size = 946, normalized size = 8.68 \[ \text {result too large to display} \]
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 \[ \int \frac {\sec ^{5}{\left (c + d x \right )}}{\left (a + b \tan ^{2}{\left (c + d x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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