Optimal. Leaf size=520 \[ -\frac{7 a b^{5/2} \sec ^2(e+f x)^{3/4} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{2 f \left (a^2+b^2\right )^{11/4} (d \sec (e+f x))^{3/2}}+\frac{b \left (2 a^2-5 b^2\right ) \sec ^2(e+f x)}{3 f \left (a^2+b^2\right )^2 (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}-\frac{7 a b^{5/2} \sec ^2(e+f x)^{3/4} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{2 f \left (a^2+b^2\right )^{11/4} (d \sec (e+f x))^{3/2}}+\frac{2 (a \tan (e+f x)+b)}{3 f \left (a^2+b^2\right ) (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}+\frac{\left (2 a^2-5 b^2\right ) \sec ^2(e+f x)^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right )}{3 f \left (a^2+b^2\right )^2 (d \sec (e+f x))^{3/2}}+\frac{7 a^2 b^2 \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sec ^2(e+f x)^{3/4} \Pi \left (-\frac{b}{\sqrt{a^2+b^2}};\left .\sin ^{-1}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right )}{2 f \left (a^2+b^2\right )^3 (d \sec (e+f x))^{3/2}}+\frac{7 a^2 b^2 \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sec ^2(e+f x)^{3/4} \Pi \left (\frac{b}{\sqrt{a^2+b^2}};\left .\sin ^{-1}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right )}{2 f \left (a^2+b^2\right )^3 (d \sec (e+f x))^{3/2}} \]
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Rubi [A] time = 0.566093, antiderivative size = 520, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 16, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.64, Rules used = {3512, 741, 835, 844, 231, 747, 401, 108, 409, 1213, 537, 444, 63, 212, 208, 205} \[ -\frac{7 a b^{5/2} \sec ^2(e+f x)^{3/4} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{2 f \left (a^2+b^2\right )^{11/4} (d \sec (e+f x))^{3/2}}+\frac{b \left (2 a^2-5 b^2\right ) \sec ^2(e+f x)}{3 f \left (a^2+b^2\right )^2 (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}-\frac{7 a b^{5/2} \sec ^2(e+f x)^{3/4} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{2 f \left (a^2+b^2\right )^{11/4} (d \sec (e+f x))^{3/2}}+\frac{2 (a \tan (e+f x)+b)}{3 f \left (a^2+b^2\right ) (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}+\frac{\left (2 a^2-5 b^2\right ) \sec ^2(e+f x)^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right )}{3 f \left (a^2+b^2\right )^2 (d \sec (e+f x))^{3/2}}+\frac{7 a^2 b^2 \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sec ^2(e+f x)^{3/4} \Pi \left (-\frac{b}{\sqrt{a^2+b^2}};\left .\sin ^{-1}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right )}{2 f \left (a^2+b^2\right )^3 (d \sec (e+f x))^{3/2}}+\frac{7 a^2 b^2 \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sec ^2(e+f x)^{3/4} \Pi \left (\frac{b}{\sqrt{a^2+b^2}};\left .\sin ^{-1}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right )}{2 f \left (a^2+b^2\right )^3 (d \sec (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3512
Rule 741
Rule 835
Rule 844
Rule 231
Rule 747
Rule 401
Rule 108
Rule 409
Rule 1213
Rule 537
Rule 444
Rule 63
Rule 212
Rule 208
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{(d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^2} \, dx &=\frac{\sec ^2(e+f x)^{3/4} \operatorname{Subst}\left (\int \frac{1}{(a+x)^2 \left (1+\frac{x^2}{b^2}\right )^{7/4}} \, dx,x,b \tan (e+f x)\right )}{b f (d \sec (e+f x))^{3/2}}\\ &=\frac{2 (b+a \tan (e+f x))}{3 \left (a^2+b^2\right ) f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}-\frac{\left (2 b \sec ^2(e+f x)^{3/4}\right ) \operatorname{Subst}\left (\int \frac{\frac{1}{2} \left (-5-\frac{a^2}{b^2}\right )-\frac{3 a x}{2 b^2}}{(a+x)^2 \left (1+\frac{x^2}{b^2}\right )^{3/4}} \, dx,x,b \tan (e+f x)\right )}{3 \left (a^2+b^2\right ) f (d \sec (e+f x))^{3/2}}\\ &=\frac{b \left (2 a^2-5 b^2\right ) \sec ^2(e+f x)}{3 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}+\frac{2 (b+a \tan (e+f x))}{3 \left (a^2+b^2\right ) f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}+\frac{\left (2 b^3 \sec ^2(e+f x)^{3/4}\right ) \operatorname{Subst}\left (\int \frac{\frac{a \left (a^2+8 b^2\right )}{2 b^4}+\frac{\left (2 a^2-5 b^2\right ) x}{4 b^4}}{(a+x) \left (1+\frac{x^2}{b^2}\right )^{3/4}} \, dx,x,b \tan (e+f x)\right )}{3 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2}}\\ &=\frac{b \left (2 a^2-5 b^2\right ) \sec ^2(e+f x)}{3 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}+\frac{2 (b+a \tan (e+f x))}{3 \left (a^2+b^2\right ) f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}+\frac{\left (7 a b \sec ^2(e+f x)^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{(a+x) \left (1+\frac{x^2}{b^2}\right )^{3/4}} \, dx,x,b \tan (e+f x)\right )}{2 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2}}+\frac{\left (\left (2 a^2-5 b^2\right ) \sec ^2(e+f x)^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{x^2}{b^2}\right )^{3/4}} \, dx,x,b \tan (e+f x)\right )}{6 b \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2}}\\ &=\frac{\left (2 a^2-5 b^2\right ) F\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) \sec ^2(e+f x)^{3/4}}{3 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2}}+\frac{b \left (2 a^2-5 b^2\right ) \sec ^2(e+f x)}{3 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}+\frac{2 (b+a \tan (e+f x))}{3 \left (a^2+b^2\right ) f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}-\frac{\left (7 a b \sec ^2(e+f x)^{3/4}\right ) \operatorname{Subst}\left (\int \frac{x}{\left (a^2-x^2\right ) \left (1+\frac{x^2}{b^2}\right )^{3/4}} \, dx,x,b \tan (e+f x)\right )}{2 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2}}+\frac{\left (7 a^2 b \sec ^2(e+f x)^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (a^2-x^2\right ) \left (1+\frac{x^2}{b^2}\right )^{3/4}} \, dx,x,b \tan (e+f x)\right )}{2 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2}}\\ &=\frac{\left (2 a^2-5 b^2\right ) F\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) \sec ^2(e+f x)^{3/4}}{3 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2}}+\frac{b \left (2 a^2-5 b^2\right ) \sec ^2(e+f x)}{3 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}+\frac{2 (b+a \tan (e+f x))}{3 \left (a^2+b^2\right ) f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}-\frac{\left (7 a b \sec ^2(e+f x)^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (a^2-x\right ) \left (1+\frac{x}{b^2}\right )^{3/4}} \, dx,x,b^2 \tan ^2(e+f x)\right )}{4 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2}}+\frac{\left (7 a^2 \cot (e+f x) \sec ^2(e+f x)^{3/4} \sqrt{-\tan ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (a^2-x\right ) \sqrt{-\frac{x}{b^2}} \left (1+\frac{x}{b^2}\right )^{3/4}} \, dx,x,b^2 \tan ^2(e+f x)\right )}{4 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2}}\\ &=\frac{\left (2 a^2-5 b^2\right ) F\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) \sec ^2(e+f x)^{3/4}}{3 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2}}+\frac{b \left (2 a^2-5 b^2\right ) \sec ^2(e+f x)}{3 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}+\frac{2 (b+a \tan (e+f x))}{3 \left (a^2+b^2\right ) f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}-\frac{\left (7 a b^3 \sec ^2(e+f x)^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{a^2+b^2-b^2 x^4} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{\left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2}}-\frac{\left (7 a^2 \cot (e+f x) \sec ^2(e+f x)^{3/4} \sqrt{-\tan ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^4} \left (-1-\frac{a^2}{b^2}+x^4\right )} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{\left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2}}\\ &=\frac{\left (2 a^2-5 b^2\right ) F\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) \sec ^2(e+f x)^{3/4}}{3 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2}}+\frac{b \left (2 a^2-5 b^2\right ) \sec ^2(e+f x)}{3 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}+\frac{2 (b+a \tan (e+f x))}{3 \left (a^2+b^2\right ) f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}-\frac{\left (7 a b^3 \sec ^2(e+f x)^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a^2+b^2}-b x^2} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{2 \left (a^2+b^2\right )^{5/2} f (d \sec (e+f x))^{3/2}}-\frac{\left (7 a b^3 \sec ^2(e+f x)^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a^2+b^2}+b x^2} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{2 \left (a^2+b^2\right )^{5/2} f (d \sec (e+f x))^{3/2}}+\frac{\left (7 a^2 b^2 \cot (e+f x) \sec ^2(e+f x)^{3/4} \sqrt{-\tan ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{b x^2}{\sqrt{a^2+b^2}}\right ) \sqrt{1-x^4}} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{2 \left (a^2+b^2\right )^3 f (d \sec (e+f x))^{3/2}}+\frac{\left (7 a^2 b^2 \cot (e+f x) \sec ^2(e+f x)^{3/4} \sqrt{-\tan ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{b x^2}{\sqrt{a^2+b^2}}\right ) \sqrt{1-x^4}} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{2 \left (a^2+b^2\right )^3 f (d \sec (e+f x))^{3/2}}\\ &=-\frac{7 a b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sec ^2(e+f x)^{3/4}}{2 \left (a^2+b^2\right )^{11/4} f (d \sec (e+f x))^{3/2}}-\frac{7 a b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sec ^2(e+f x)^{3/4}}{2 \left (a^2+b^2\right )^{11/4} f (d \sec (e+f x))^{3/2}}+\frac{\left (2 a^2-5 b^2\right ) F\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) \sec ^2(e+f x)^{3/4}}{3 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2}}+\frac{b \left (2 a^2-5 b^2\right ) \sec ^2(e+f x)}{3 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}+\frac{2 (b+a \tan (e+f x))}{3 \left (a^2+b^2\right ) f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}+\frac{\left (7 a^2 b^2 \cot (e+f x) \sec ^2(e+f x)^{3/4} \sqrt{-\tan ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+x^2} \left (1-\frac{b x^2}{\sqrt{a^2+b^2}}\right )} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{2 \left (a^2+b^2\right )^3 f (d \sec (e+f x))^{3/2}}+\frac{\left (7 a^2 b^2 \cot (e+f x) \sec ^2(e+f x)^{3/4} \sqrt{-\tan ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+x^2} \left (1+\frac{b x^2}{\sqrt{a^2+b^2}}\right )} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{2 \left (a^2+b^2\right )^3 f (d \sec (e+f x))^{3/2}}\\ &=-\frac{7 a b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sec ^2(e+f x)^{3/4}}{2 \left (a^2+b^2\right )^{11/4} f (d \sec (e+f x))^{3/2}}-\frac{7 a b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sec ^2(e+f x)^{3/4}}{2 \left (a^2+b^2\right )^{11/4} f (d \sec (e+f x))^{3/2}}+\frac{\left (2 a^2-5 b^2\right ) F\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) \sec ^2(e+f x)^{3/4}}{3 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2}}+\frac{7 a^2 b^2 \cot (e+f x) \Pi \left (-\frac{b}{\sqrt{a^2+b^2}};\left .\sin ^{-1}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right ) \sec ^2(e+f x)^{3/4} \sqrt{-\tan ^2(e+f x)}}{2 \left (a^2+b^2\right )^3 f (d \sec (e+f x))^{3/2}}+\frac{7 a^2 b^2 \cot (e+f x) \Pi \left (\frac{b}{\sqrt{a^2+b^2}};\left .\sin ^{-1}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right ) \sec ^2(e+f x)^{3/4} \sqrt{-\tan ^2(e+f x)}}{2 \left (a^2+b^2\right )^3 f (d \sec (e+f x))^{3/2}}+\frac{b \left (2 a^2-5 b^2\right ) \sec ^2(e+f x)}{3 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}+\frac{2 (b+a \tan (e+f x))}{3 \left (a^2+b^2\right ) f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}\\ \end{align*}
Mathematica [C] time = 27.6345, size = 11962, normalized size = 23. \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.838, size = 15455, normalized size = 29.7 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (d \sec \left (f x + e\right )\right )^{\frac{3}{2}}{\left (b \tan \left (f x + e\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (d \sec{\left (e + f x \right )}\right )^{\frac{3}{2}} \left (a + b \tan{\left (e + f x \right )}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (d \sec \left (f x + e\right )\right )^{\frac{3}{2}}{\left (b \tan \left (f x + e\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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