Optimal. Leaf size=239 \[ \frac{5 \sec (c+d x) \left (4 a^2-2 a b \tan (c+d x)+b^2\right )}{2 b^5 d}-\frac{5 \sqrt{a^2+b^2} \left (4 a^2+b^2\right ) \sec (c+d x) \tanh ^{-1}\left (\frac{b-a \tan (c+d x)}{\sqrt{a^2+b^2} \sqrt{\sec ^2(c+d x)}}\right )}{2 b^6 d \sqrt{\sec ^2(c+d x)}}-\frac{5 a \left (4 a^2+3 b^2\right ) \sec (c+d x) \sinh ^{-1}(\tan (c+d x))}{2 b^6 d \sqrt{\sec ^2(c+d x)}}+\frac{5 \sec ^3(c+d x) (4 a+b \tan (c+d x))}{6 b^3 d (a+b \tan (c+d x))}-\frac{\sec ^5(c+d x)}{2 b d (a+b \tan (c+d x))^2} \]
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Rubi [A] time = 0.243291, antiderivative size = 239, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {3512, 733, 813, 815, 844, 215, 725, 206} \[ \frac{5 \sec (c+d x) \left (4 a^2-2 a b \tan (c+d x)+b^2\right )}{2 b^5 d}-\frac{5 \sqrt{a^2+b^2} \left (4 a^2+b^2\right ) \sec (c+d x) \tanh ^{-1}\left (\frac{b-a \tan (c+d x)}{\sqrt{a^2+b^2} \sqrt{\sec ^2(c+d x)}}\right )}{2 b^6 d \sqrt{\sec ^2(c+d x)}}-\frac{5 a \left (4 a^2+3 b^2\right ) \sec (c+d x) \sinh ^{-1}(\tan (c+d x))}{2 b^6 d \sqrt{\sec ^2(c+d x)}}+\frac{5 \sec ^3(c+d x) (4 a+b \tan (c+d x))}{6 b^3 d (a+b \tan (c+d x))}-\frac{\sec ^5(c+d x)}{2 b d (a+b \tan (c+d x))^2} \]
Antiderivative was successfully verified.
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Rule 3512
Rule 733
Rule 813
Rule 815
Rule 844
Rule 215
Rule 725
Rule 206
Rubi steps
\begin{align*} \int \frac{\sec ^7(c+d x)}{(a+b \tan (c+d x))^3} \, dx &=\frac{\sec (c+d x) \operatorname{Subst}\left (\int \frac{\left (1+\frac{x^2}{b^2}\right )^{5/2}}{(a+x)^3} \, dx,x,b \tan (c+d x)\right )}{b d \sqrt{\sec ^2(c+d x)}}\\ &=-\frac{\sec ^5(c+d x)}{2 b d (a+b \tan (c+d x))^2}+\frac{(5 \sec (c+d x)) \operatorname{Subst}\left (\int \frac{x \left (1+\frac{x^2}{b^2}\right )^{3/2}}{(a+x)^2} \, dx,x,b \tan (c+d x)\right )}{2 b^3 d \sqrt{\sec ^2(c+d x)}}\\ &=-\frac{\sec ^5(c+d x)}{2 b d (a+b \tan (c+d x))^2}+\frac{5 \sec ^3(c+d x) (4 a+b \tan (c+d x))}{6 b^3 d (a+b \tan (c+d x))}-\frac{(5 \sec (c+d x)) \operatorname{Subst}\left (\int \frac{\left (-2+\frac{8 a x}{b^2}\right ) \sqrt{1+\frac{x^2}{b^2}}}{a+x} \, dx,x,b \tan (c+d x)\right )}{4 b^3 d \sqrt{\sec ^2(c+d x)}}\\ &=-\frac{\sec ^5(c+d x)}{2 b d (a+b \tan (c+d x))^2}+\frac{5 \sec ^3(c+d x) (4 a+b \tan (c+d x))}{6 b^3 d (a+b \tan (c+d x))}+\frac{5 \sec (c+d x) \left (4 a^2+b^2-2 a b \tan (c+d x)\right )}{2 b^5 d}-\frac{(5 \sec (c+d x)) \operatorname{Subst}\left (\int \frac{-\frac{4 \left (2 a^2+b^2\right )}{b^4}+\frac{4 a \left (4 a^2+3 b^2\right ) x}{b^6}}{(a+x) \sqrt{1+\frac{x^2}{b^2}}} \, dx,x,b \tan (c+d x)\right )}{8 b d \sqrt{\sec ^2(c+d x)}}\\ &=-\frac{\sec ^5(c+d x)}{2 b d (a+b \tan (c+d x))^2}+\frac{5 \sec ^3(c+d x) (4 a+b \tan (c+d x))}{6 b^3 d (a+b \tan (c+d x))}+\frac{5 \sec (c+d x) \left (4 a^2+b^2-2 a b \tan (c+d x)\right )}{2 b^5 d}+\frac{\left (5 \left (a^2+b^2\right ) \left (4 a^2+b^2\right ) \sec (c+d x)\right ) \operatorname{Subst}\left (\int \frac{1}{(a+x) \sqrt{1+\frac{x^2}{b^2}}} \, dx,x,b \tan (c+d x)\right )}{2 b^7 d \sqrt{\sec ^2(c+d x)}}-\frac{\left (5 a \left (4 a^2+3 b^2\right ) \sec (c+d x)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{b^2}}} \, dx,x,b \tan (c+d x)\right )}{2 b^7 d \sqrt{\sec ^2(c+d x)}}\\ &=-\frac{5 a \left (4 a^2+3 b^2\right ) \sinh ^{-1}(\tan (c+d x)) \sec (c+d x)}{2 b^6 d \sqrt{\sec ^2(c+d x)}}-\frac{\sec ^5(c+d x)}{2 b d (a+b \tan (c+d x))^2}+\frac{5 \sec ^3(c+d x) (4 a+b \tan (c+d x))}{6 b^3 d (a+b \tan (c+d x))}+\frac{5 \sec (c+d x) \left (4 a^2+b^2-2 a b \tan (c+d x)\right )}{2 b^5 d}-\frac{\left (5 \left (a^2+b^2\right ) \left (4 a^2+b^2\right ) \sec (c+d x)\right ) \operatorname{Subst}\left (\int \frac{1}{1+\frac{a^2}{b^2}-x^2} \, dx,x,\frac{1-\frac{a \tan (c+d x)}{b}}{\sqrt{\sec ^2(c+d x)}}\right )}{2 b^7 d \sqrt{\sec ^2(c+d x)}}\\ &=-\frac{5 a \left (4 a^2+3 b^2\right ) \sinh ^{-1}(\tan (c+d x)) \sec (c+d x)}{2 b^6 d \sqrt{\sec ^2(c+d x)}}-\frac{5 \sqrt{a^2+b^2} \left (4 a^2+b^2\right ) \tanh ^{-1}\left (\frac{b \left (1-\frac{a \tan (c+d x)}{b}\right )}{\sqrt{a^2+b^2} \sqrt{\sec ^2(c+d x)}}\right ) \sec (c+d x)}{2 b^6 d \sqrt{\sec ^2(c+d x)}}-\frac{\sec ^5(c+d x)}{2 b d (a+b \tan (c+d x))^2}+\frac{5 \sec ^3(c+d x) (4 a+b \tan (c+d x))}{6 b^3 d (a+b \tan (c+d x))}+\frac{5 \sec (c+d x) \left (4 a^2+b^2-2 a b \tan (c+d x)\right )}{2 b^5 d}\\ \end{align*}
Mathematica [C] time = 2.21019, size = 688, normalized size = 2.88 \[ \frac{\sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \left (\frac{6 b^2 \left (a^2+b^2\right )^2 \sin (c+d x)}{a}+2 b \left (36 a^2+13 b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2+\frac{2 b \left (36 a^2+13 b^2\right ) \sin \left (\frac{1}{2} (c+d x)\right ) (a \cos (c+d x)+b \sin (c+d x))^2}{\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )}-\frac{2 b \left (36 a^2+13 b^2\right ) \sin \left (\frac{1}{2} (c+d x)\right ) (a \cos (c+d x)+b \sin (c+d x))^2}{\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )}+\frac{6 b (a-i b) (a+i b) \left (8 a^2-b^2\right ) (a \cos (c+d x)+b \sin (c+d x))}{a}+30 a \left (4 a^2+3 b^2\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b \sin (c+d x))^2-30 a \left (4 a^2+3 b^2\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b \sin (c+d x))^2+60 \sqrt{a^2+b^2} \left (4 a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2 \tanh ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )-b}{\sqrt{a^2+b^2}}\right )+\frac{2 b^3 \sin \left (\frac{1}{2} (c+d x)\right ) (a \cos (c+d x)+b \sin (c+d x))^2}{\left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^3}-\frac{2 b^3 \sin \left (\frac{1}{2} (c+d x)\right ) (a \cos (c+d x)+b \sin (c+d x))^2}{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^3}+\frac{b^2 (b-9 a) (a \cos (c+d x)+b \sin (c+d x))^2}{\left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^2}+\frac{b^2 (9 a+b) (a \cos (c+d x)+b \sin (c+d x))^2}{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2}\right )}{12 b^6 d (a+b \tan (c+d x))^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.148, size = 1125, normalized size = 4.7 \begin{align*} \text{result too large to display} \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 [B] time = 3.12065, size = 1299, normalized size = 5.44 \begin{align*} \frac{4 \, b^{5} + 30 \,{\left (4 \, a^{4} b + a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{4} + 20 \,{\left (2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{2} + 15 \,{\left ({\left (4 \, a^{4} - 3 \, a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{5} + 2 \,{\left (4 \, a^{3} b + a b^{3}\right )} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) +{\left (4 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{3}\right )} \sqrt{a^{2} + b^{2}} \log \left (-\frac{2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) +{\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a^{2} - b^{2} + 2 \, \sqrt{a^{2} + b^{2}}{\left (b \cos \left (d x + c\right ) - a \sin \left (d x + c\right )\right )}}{2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) +{\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}}\right ) - 15 \,{\left ({\left (4 \, a^{5} - a^{3} b^{2} - 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} + 2 \,{\left (4 \, a^{4} b + 3 \, a^{2} b^{3}\right )} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) +{\left (4 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \,{\left ({\left (4 \, a^{5} - a^{3} b^{2} - 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} + 2 \,{\left (4 \, a^{4} b + 3 \, a^{2} b^{3}\right )} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) +{\left (4 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 10 \,{\left (a b^{4} \cos \left (d x + c\right ) - 6 \,{\left (3 \, a^{3} b^{2} + 2 \, a b^{4}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{12 \,{\left (2 \, a b^{7} d \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) + b^{8} d \cos \left (d x + c\right )^{3} +{\left (a^{2} b^{6} - b^{8}\right )} d \cos \left (d x + c\right )^{5}\right )}} \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 [B] time = 1.63557, size = 689, normalized size = 2.88 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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