Optimal. Leaf size=882 \[ -\frac{e^2 \Pi \left (\frac{2 a}{a-\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{\sin (c+d x)} b^4}{2 a^4 \left (a^2-\sqrt{a^2-b^2} a-b^2\right ) d \sqrt{e \sin (c+d x)}}-\frac{e^2 \Pi \left (\frac{2 a}{a+\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{\sin (c+d x)} b^4}{2 a^4 \left (a^2+\sqrt{a^2-b^2} a-b^2\right ) d \sqrt{e \sin (c+d x)}}+\frac{e^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right ) b^3}{2 a^{7/2} \left (a^2-b^2\right )^{3/4} d}+\frac{e^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right ) b^3}{2 a^{7/2} \left (a^2-b^2\right )^{3/4} d}+\frac{e \sqrt{e \sin (c+d x)} b^2}{a^3 d (b+a \cos (c+d x))}-\frac{5 e^2 \text{EllipticF}\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right ),2\right ) \sqrt{\sin (c+d x)} b^2}{a^4 d \sqrt{e \sin (c+d x)}}+\frac{2 \left (a^2-b^2\right ) e^2 \Pi \left (\frac{2 a}{a-\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{\sin (c+d x)} b^2}{a^4 \left (a^2-\sqrt{a^2-b^2} a-b^2\right ) d \sqrt{e \sin (c+d x)}}+\frac{2 \left (a^2-b^2\right ) e^2 \Pi \left (\frac{2 a}{a+\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{\sin (c+d x)} b^2}{a^4 \left (a^2+\sqrt{a^2-b^2} a-b^2\right ) d \sqrt{e \sin (c+d x)}}-\frac{2 \sqrt [4]{a^2-b^2} e^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right ) b}{a^{7/2} d}-\frac{2 \sqrt [4]{a^2-b^2} e^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right ) b}{a^{7/2} d}+\frac{4 e \sqrt{e \sin (c+d x)} b}{a^3 d}-\frac{2 e \cos (c+d x) \sqrt{e \sin (c+d x)}}{3 a^2 d}+\frac{2 e^2 \text{EllipticF}\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right ),2\right ) \sqrt{\sin (c+d x)}}{3 a^2 d \sqrt{e \sin (c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 2.16782, antiderivative size = 882, normalized size of antiderivative = 1., number of steps used = 32, number of rules used = 15, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {3872, 2912, 2635, 2642, 2641, 2693, 2867, 2702, 2807, 2805, 329, 212, 208, 205, 2695} \[ -\frac{e^2 \Pi \left (\frac{2 a}{a-\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{\sin (c+d x)} b^4}{2 a^4 \left (a^2-\sqrt{a^2-b^2} a-b^2\right ) d \sqrt{e \sin (c+d x)}}-\frac{e^2 \Pi \left (\frac{2 a}{a+\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{\sin (c+d x)} b^4}{2 a^4 \left (a^2+\sqrt{a^2-b^2} a-b^2\right ) d \sqrt{e \sin (c+d x)}}+\frac{e^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right ) b^3}{2 a^{7/2} \left (a^2-b^2\right )^{3/4} d}+\frac{e^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right ) b^3}{2 a^{7/2} \left (a^2-b^2\right )^{3/4} d}+\frac{e \sqrt{e \sin (c+d x)} b^2}{a^3 d (b+a \cos (c+d x))}-\frac{5 e^2 F\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{\sin (c+d x)} b^2}{a^4 d \sqrt{e \sin (c+d x)}}+\frac{2 \left (a^2-b^2\right ) e^2 \Pi \left (\frac{2 a}{a-\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{\sin (c+d x)} b^2}{a^4 \left (a^2-\sqrt{a^2-b^2} a-b^2\right ) d \sqrt{e \sin (c+d x)}}+\frac{2 \left (a^2-b^2\right ) e^2 \Pi \left (\frac{2 a}{a+\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{\sin (c+d x)} b^2}{a^4 \left (a^2+\sqrt{a^2-b^2} a-b^2\right ) d \sqrt{e \sin (c+d x)}}-\frac{2 \sqrt [4]{a^2-b^2} e^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right ) b}{a^{7/2} d}-\frac{2 \sqrt [4]{a^2-b^2} e^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right ) b}{a^{7/2} d}+\frac{4 e \sqrt{e \sin (c+d x)} b}{a^3 d}-\frac{2 e \cos (c+d x) \sqrt{e \sin (c+d x)}}{3 a^2 d}+\frac{2 e^2 F\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{\sin (c+d x)}}{3 a^2 d \sqrt{e \sin (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3872
Rule 2912
Rule 2635
Rule 2642
Rule 2641
Rule 2693
Rule 2867
Rule 2702
Rule 2807
Rule 2805
Rule 329
Rule 212
Rule 208
Rule 205
Rule 2695
Rubi steps
\begin{align*} \int \frac{(e \sin (c+d x))^{3/2}}{(a+b \sec (c+d x))^2} \, dx &=\int \frac{\cos ^2(c+d x) (e \sin (c+d x))^{3/2}}{(-b-a \cos (c+d x))^2} \, dx\\ &=\int \left (\frac{(e \sin (c+d x))^{3/2}}{a^2}+\frac{b^2 (e \sin (c+d x))^{3/2}}{a^2 (b+a \cos (c+d x))^2}-\frac{2 b (e \sin (c+d x))^{3/2}}{a^2 (b+a \cos (c+d x))}\right ) \, dx\\ &=\frac{\int (e \sin (c+d x))^{3/2} \, dx}{a^2}-\frac{(2 b) \int \frac{(e \sin (c+d x))^{3/2}}{b+a \cos (c+d x)} \, dx}{a^2}+\frac{b^2 \int \frac{(e \sin (c+d x))^{3/2}}{(b+a \cos (c+d x))^2} \, dx}{a^2}\\ &=\frac{4 b e \sqrt{e \sin (c+d x)}}{a^3 d}-\frac{2 e \cos (c+d x) \sqrt{e \sin (c+d x)}}{3 a^2 d}+\frac{b^2 e \sqrt{e \sin (c+d x)}}{a^3 d (b+a \cos (c+d x))}+\frac{e^2 \int \frac{1}{\sqrt{e \sin (c+d x)}} \, dx}{3 a^2}+\frac{\left (2 b e^2\right ) \int \frac{-a-b \cos (c+d x)}{(b+a \cos (c+d x)) \sqrt{e \sin (c+d x)}} \, dx}{a^3}-\frac{\left (b^2 e^2\right ) \int \frac{\cos (c+d x)}{(b+a \cos (c+d x)) \sqrt{e \sin (c+d x)}} \, dx}{2 a^3}\\ &=\frac{4 b e \sqrt{e \sin (c+d x)}}{a^3 d}-\frac{2 e \cos (c+d x) \sqrt{e \sin (c+d x)}}{3 a^2 d}+\frac{b^2 e \sqrt{e \sin (c+d x)}}{a^3 d (b+a \cos (c+d x))}-\frac{\left (b^2 e^2\right ) \int \frac{1}{\sqrt{e \sin (c+d x)}} \, dx}{2 a^4}-\frac{\left (2 b^2 e^2\right ) \int \frac{1}{\sqrt{e \sin (c+d x)}} \, dx}{a^4}+\frac{\left (b^3 e^2\right ) \int \frac{1}{(b+a \cos (c+d x)) \sqrt{e \sin (c+d x)}} \, dx}{2 a^4}-\frac{\left (2 b \left (a^2-b^2\right ) e^2\right ) \int \frac{1}{(b+a \cos (c+d x)) \sqrt{e \sin (c+d x)}} \, dx}{a^4}+\frac{\left (e^2 \sqrt{\sin (c+d x)}\right ) \int \frac{1}{\sqrt{\sin (c+d x)}} \, dx}{3 a^2 \sqrt{e \sin (c+d x)}}\\ &=\frac{2 e^2 F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{3 a^2 d \sqrt{e \sin (c+d x)}}+\frac{4 b e \sqrt{e \sin (c+d x)}}{a^3 d}-\frac{2 e \cos (c+d x) \sqrt{e \sin (c+d x)}}{3 a^2 d}+\frac{b^2 e \sqrt{e \sin (c+d x)}}{a^3 d (b+a \cos (c+d x))}-\frac{\left (b^4 e^2\right ) \int \frac{1}{\sqrt{e \sin (c+d x)} \left (\sqrt{a^2-b^2}-a \sin (c+d x)\right )} \, dx}{4 a^4 \sqrt{a^2-b^2}}-\frac{\left (b^4 e^2\right ) \int \frac{1}{\sqrt{e \sin (c+d x)} \left (\sqrt{a^2-b^2}+a \sin (c+d x)\right )} \, dx}{4 a^4 \sqrt{a^2-b^2}}+\frac{\left (b^2 \sqrt{a^2-b^2} e^2\right ) \int \frac{1}{\sqrt{e \sin (c+d x)} \left (\sqrt{a^2-b^2}-a \sin (c+d x)\right )} \, dx}{a^4}+\frac{\left (b^2 \sqrt{a^2-b^2} e^2\right ) \int \frac{1}{\sqrt{e \sin (c+d x)} \left (\sqrt{a^2-b^2}+a \sin (c+d x)\right )} \, dx}{a^4}-\frac{\left (b^3 e^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (\left (-a^2+b^2\right ) e^2+a^2 x^2\right )} \, dx,x,e \sin (c+d x)\right )}{2 a^3 d}+\frac{\left (2 b \left (a^2-b^2\right ) e^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (\left (-a^2+b^2\right ) e^2+a^2 x^2\right )} \, dx,x,e \sin (c+d x)\right )}{a^3 d}-\frac{\left (b^2 e^2 \sqrt{\sin (c+d x)}\right ) \int \frac{1}{\sqrt{\sin (c+d x)}} \, dx}{2 a^4 \sqrt{e \sin (c+d x)}}-\frac{\left (2 b^2 e^2 \sqrt{\sin (c+d x)}\right ) \int \frac{1}{\sqrt{\sin (c+d x)}} \, dx}{a^4 \sqrt{e \sin (c+d x)}}\\ &=\frac{2 e^2 F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{3 a^2 d \sqrt{e \sin (c+d x)}}-\frac{5 b^2 e^2 F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{a^4 d \sqrt{e \sin (c+d x)}}+\frac{4 b e \sqrt{e \sin (c+d x)}}{a^3 d}-\frac{2 e \cos (c+d x) \sqrt{e \sin (c+d x)}}{3 a^2 d}+\frac{b^2 e \sqrt{e \sin (c+d x)}}{a^3 d (b+a \cos (c+d x))}-\frac{\left (b^3 e^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (-a^2+b^2\right ) e^2+a^2 x^4} \, dx,x,\sqrt{e \sin (c+d x)}\right )}{a^3 d}+\frac{\left (4 b \left (a^2-b^2\right ) e^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (-a^2+b^2\right ) e^2+a^2 x^4} \, dx,x,\sqrt{e \sin (c+d x)}\right )}{a^3 d}-\frac{\left (b^4 e^2 \sqrt{\sin (c+d x)}\right ) \int \frac{1}{\sqrt{\sin (c+d x)} \left (\sqrt{a^2-b^2}-a \sin (c+d x)\right )} \, dx}{4 a^4 \sqrt{a^2-b^2} \sqrt{e \sin (c+d x)}}-\frac{\left (b^4 e^2 \sqrt{\sin (c+d x)}\right ) \int \frac{1}{\sqrt{\sin (c+d x)} \left (\sqrt{a^2-b^2}+a \sin (c+d x)\right )} \, dx}{4 a^4 \sqrt{a^2-b^2} \sqrt{e \sin (c+d x)}}+\frac{\left (b^2 \sqrt{a^2-b^2} e^2 \sqrt{\sin (c+d x)}\right ) \int \frac{1}{\sqrt{\sin (c+d x)} \left (\sqrt{a^2-b^2}-a \sin (c+d x)\right )} \, dx}{a^4 \sqrt{e \sin (c+d x)}}+\frac{\left (b^2 \sqrt{a^2-b^2} e^2 \sqrt{\sin (c+d x)}\right ) \int \frac{1}{\sqrt{\sin (c+d x)} \left (\sqrt{a^2-b^2}+a \sin (c+d x)\right )} \, dx}{a^4 \sqrt{e \sin (c+d x)}}\\ &=\frac{2 e^2 F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{3 a^2 d \sqrt{e \sin (c+d x)}}-\frac{5 b^2 e^2 F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{a^4 d \sqrt{e \sin (c+d x)}}-\frac{2 b^2 \sqrt{a^2-b^2} e^2 \Pi \left (\frac{2 a}{a-\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{a^4 \left (a-\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}-\frac{b^4 e^2 \Pi \left (\frac{2 a}{a-\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{2 a^4 \left (a^2-b^2-a \sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}-\frac{b^4 e^2 \Pi \left (\frac{2 a}{a+\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{2 a^4 \sqrt{a^2-b^2} \left (a+\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}+\frac{2 b^2 \sqrt{a^2-b^2} e^2 \Pi \left (\frac{2 a}{a+\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{a^4 \left (a+\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}+\frac{4 b e \sqrt{e \sin (c+d x)}}{a^3 d}-\frac{2 e \cos (c+d x) \sqrt{e \sin (c+d x)}}{3 a^2 d}+\frac{b^2 e \sqrt{e \sin (c+d x)}}{a^3 d (b+a \cos (c+d x))}+\frac{\left (b^3 e^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a^2-b^2} e-a x^2} \, dx,x,\sqrt{e \sin (c+d x)}\right )}{2 a^3 \sqrt{a^2-b^2} d}+\frac{\left (b^3 e^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a^2-b^2} e+a x^2} \, dx,x,\sqrt{e \sin (c+d x)}\right )}{2 a^3 \sqrt{a^2-b^2} d}-\frac{\left (2 b \sqrt{a^2-b^2} e^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a^2-b^2} e-a x^2} \, dx,x,\sqrt{e \sin (c+d x)}\right )}{a^3 d}-\frac{\left (2 b \sqrt{a^2-b^2} e^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a^2-b^2} e+a x^2} \, dx,x,\sqrt{e \sin (c+d x)}\right )}{a^3 d}\\ &=\frac{b^3 e^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right )}{2 a^{7/2} \left (a^2-b^2\right )^{3/4} d}-\frac{2 b \sqrt [4]{a^2-b^2} e^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right )}{a^{7/2} d}+\frac{b^3 e^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right )}{2 a^{7/2} \left (a^2-b^2\right )^{3/4} d}-\frac{2 b \sqrt [4]{a^2-b^2} e^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right )}{a^{7/2} d}+\frac{2 e^2 F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{3 a^2 d \sqrt{e \sin (c+d x)}}-\frac{5 b^2 e^2 F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{a^4 d \sqrt{e \sin (c+d x)}}-\frac{2 b^2 \sqrt{a^2-b^2} e^2 \Pi \left (\frac{2 a}{a-\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{a^4 \left (a-\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}-\frac{b^4 e^2 \Pi \left (\frac{2 a}{a-\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{2 a^4 \left (a^2-b^2-a \sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}-\frac{b^4 e^2 \Pi \left (\frac{2 a}{a+\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{2 a^4 \sqrt{a^2-b^2} \left (a+\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}+\frac{2 b^2 \sqrt{a^2-b^2} e^2 \Pi \left (\frac{2 a}{a+\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{a^4 \left (a+\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}+\frac{4 b e \sqrt{e \sin (c+d x)}}{a^3 d}-\frac{2 e \cos (c+d x) \sqrt{e \sin (c+d x)}}{3 a^2 d}+\frac{b^2 e \sqrt{e \sin (c+d x)}}{a^3 d (b+a \cos (c+d x))}\\ \end{align*}
Mathematica [C] time = 16.0484, size = 2012, normalized size = 2.28 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 7.666, size = 2282, normalized size = 2.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e \sin \left (d x + c\right )\right )^{\frac{3}{2}}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]