Optimal. Leaf size=167 \[ \frac{(4 a+b) (a-b)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right )}{2 a^{3/2} b^3 d}+\frac{(2 a-b) (a-b) \sin (c+d x)}{2 a b^2 d \left (a-(a-b) \sin ^2(c+d x)\right )}-\frac{(4 a-5 b) \tanh ^{-1}(\sin (c+d x))}{2 b^3 d}+\frac{\tan (c+d x) \sec (c+d x)}{2 b d \left (a-(a-b) \sin ^2(c+d x)\right )} \]
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Rubi [A] time = 0.267281, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {3676, 414, 527, 522, 206, 208} \[ \frac{(4 a+b) (a-b)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right )}{2 a^{3/2} b^3 d}+\frac{(2 a-b) (a-b) \sin (c+d x)}{2 a b^2 d \left (a-(a-b) \sin ^2(c+d x)\right )}-\frac{(4 a-5 b) \tanh ^{-1}(\sin (c+d x))}{2 b^3 d}+\frac{\tan (c+d x) \sec (c+d x)}{2 b d \left (a-(a-b) \sin ^2(c+d x)\right )} \]
Antiderivative was successfully verified.
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Rule 3676
Rule 414
Rule 527
Rule 522
Rule 206
Rule 208
Rubi steps
\begin{align*} \int \frac{\sec ^7(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 )^2 \left (a-(a-b) x^2\right )^2} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac{\sec (c+d x) \tan (c+d x)}{2 b d \left (a-(a-b) \sin ^2(c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{-a+2 b-3 (a-b) x^2}{\left (1-x^2\right ) \left (a+(-a+b) x^2\right )^2} \, dx,x,\sin (c+d x)\right )}{2 b d}\\ &=\frac{(a-b) (2 a-b) \sin (c+d x)}{2 a b^2 d \left (a-(a-b) \sin ^2(c+d x)\right )}+\frac{\sec (c+d x) \tan (c+d x)}{2 b d \left (a-(a-b) \sin ^2(c+d x)\right )}-\frac{\operatorname{Subst}\left (\int \frac{2 \left (2 a^2-2 a b-b^2\right )+2 (a-b) (2 a-b) x^2}{\left (1-x^2\right ) \left (a+(-a+b) x^2\right )} \, dx,x,\sin (c+d x)\right )}{4 a b^2 d}\\ &=\frac{(a-b) (2 a-b) \sin (c+d x)}{2 a b^2 d \left (a-(a-b) \sin ^2(c+d x)\right )}+\frac{\sec (c+d x) \tan (c+d x)}{2 b d \left (a-(a-b) \sin ^2(c+d x)\right )}-\frac{(4 a-5 b) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sin (c+d x)\right )}{2 b^3 d}+\frac{\left ((a-b)^2 (4 a+b)\right ) \operatorname{Subst}\left (\int \frac{1}{a+(-a+b) x^2} \, dx,x,\sin (c+d x)\right )}{2 a b^3 d}\\ &=-\frac{(4 a-5 b) \tanh ^{-1}(\sin (c+d x))}{2 b^3 d}+\frac{(a-b)^{3/2} (4 a+b) \tanh ^{-1}\left (\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right )}{2 a^{3/2} b^3 d}+\frac{(a-b) (2 a-b) \sin (c+d x)}{2 a b^2 d \left (a-(a-b) \sin ^2(c+d x)\right )}+\frac{\sec (c+d x) \tan (c+d x)}{2 b d \left (a-(a-b) \sin ^2(c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 4.10419, size = 254, normalized size = 1.52 \[ \frac{-\frac{(4 a+b) (a-b)^{3/2} \log \left (\sqrt{a}-\sqrt{a-b} \sin (c+d x)\right )}{a^{3/2}}+\frac{(4 a+b) (a-b)^{3/2} \log \left (\sqrt{a-b} \sin (c+d x)+\sqrt{a}\right )}{a^{3/2}}+\frac{4 b (a-b)^2 \sin (c+d x)}{a ((a-b) \cos (2 (c+d x))+a+b)}+2 (4 a-5 b) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+2 (5 b-4 a) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+\frac{b}{\left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^2}-\frac{b}{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2}}{4 b^3 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.109, size = 389, normalized size = 2.3 \begin{align*} -{\frac{\sin \left ( dx+c \right ) a}{2\,d{b}^{2} \left ( a \left ( \sin \left ( dx+c \right ) \right ) ^{2}-b \left ( \sin \left ( dx+c \right ) \right ) ^{2}-a \right ) }}+{\frac{\sin \left ( dx+c \right ) }{db \left ( a \left ( \sin \left ( dx+c \right ) \right ) ^{2}-b \left ( \sin \left ( dx+c \right ) \right ) ^{2}-a \right ) }}-{\frac{\sin \left ( dx+c \right ) }{2\,da \left ( a \left ( \sin \left ( dx+c \right ) \right ) ^{2}-b \left ( \sin \left ( dx+c \right ) \right ) ^{2}-a \right ) }}+2\,{\frac{{a}^{2}}{d{b}^{3}\sqrt{a \left ( a-b \right ) }}{\it Artanh} \left ({\frac{ \left ( a-b \right ) \sin \left ( dx+c \right ) }{\sqrt{a \left ( a-b \right ) }}} \right ) }-{\frac{7\,a}{2\,d{b}^{2}}{\it Artanh} \left ({ \left ( a-b \right ) \sin \left ( dx+c \right ){\frac{1}{\sqrt{a \left ( a-b \right ) }}}} \right ){\frac{1}{\sqrt{a \left ( a-b \right ) }}}}+{\frac{1}{db}{\it Artanh} \left ({ \left ( a-b \right ) \sin \left ( dx+c \right ){\frac{1}{\sqrt{a \left ( a-b \right ) }}}} \right ){\frac{1}{\sqrt{a \left ( a-b \right ) }}}}+{\frac{1}{2\,da}{\it Artanh} \left ({ \left ( a-b \right ) \sin \left ( dx+c \right ){\frac{1}{\sqrt{a \left ( a-b \right ) }}}} \right ){\frac{1}{\sqrt{a \left ( a-b \right ) }}}}-{\frac{1}{4\,d{b}^{2} \left ( \sin \left ( dx+c \right ) +1 \right ) }}-{\frac{\ln \left ( \sin \left ( dx+c \right ) +1 \right ) a}{d{b}^{3}}}+{\frac{5\,\ln \left ( \sin \left ( dx+c \right ) +1 \right ) }{4\,d{b}^{2}}}-{\frac{1}{4\,d{b}^{2} \left ( \sin \left ( dx+c \right ) -1 \right ) }}+{\frac{\ln \left ( \sin \left ( dx+c \right ) -1 \right ) a}{d{b}^{3}}}-{\frac{5\,\ln \left ( \sin \left ( dx+c \right ) -1 \right ) }{4\,d{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.42098, size = 1447, normalized size = 8.66 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.8084, size = 331, normalized size = 1.98 \begin{align*} -\frac{\frac{{\left (4 \, a - 5 \, b\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{b^{3}} - \frac{{\left (4 \, a - 5 \, b\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{b^{3}} - \frac{2 \,{\left (4 \, a^{3} - 7 \, a^{2} b + 2 \, a b^{2} + b^{3}\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^{3}} + \frac{2 \,{\left (2 \, a^{2} \sin \left (d x + c\right )^{3} - 3 \, a b \sin \left (d x + c\right )^{3} + b^{2} \sin \left (d x + c\right )^{3} - 2 \, a^{2} \sin \left (d x + c\right ) + 2 \, a b \sin \left (d x + c\right ) - b^{2} \sin \left (d x + c\right )\right )}}{{\left (a \sin \left (d x + c\right )^{4} - b \sin \left (d x + c\right )^{4} - 2 \, a \sin \left (d x + c\right )^{2} + b \sin \left (d x + c\right )^{2} + a\right )} a b^{2}}}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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