Optimal. Leaf size=292 \[ -\frac{4 a \tan (e+f x)}{3 f (a+b)^2 \sqrt{a+b \sin ^2(e+f x)}}+\frac{b (7 a-b) \sin (e+f x) \cos (e+f x)}{3 f (a+b)^3 \sqrt{a+b \sin ^2(e+f x)}}+\frac{\tan (e+f x) \sec ^2(e+f x)}{3 f (a+b) \sqrt{a+b \sin ^2(e+f x)}}-\frac{4 a \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left (\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right )}{3 f (a+b)^2 \sqrt{a+b \sin ^2(e+f x)}}+\frac{(7 a-b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left (\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right )}{3 f (a+b)^3 \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.314868, antiderivative size = 292, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {3196, 470, 527, 524, 426, 424, 421, 419} \[ -\frac{4 a \tan (e+f x)}{3 f (a+b)^2 \sqrt{a+b \sin ^2(e+f x)}}+\frac{b (7 a-b) \sin (e+f x) \cos (e+f x)}{3 f (a+b)^3 \sqrt{a+b \sin ^2(e+f x)}}+\frac{\tan (e+f x) \sec ^2(e+f x)}{3 f (a+b) \sqrt{a+b \sin ^2(e+f x)}}-\frac{4 a \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left (\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right )}{3 f (a+b)^2 \sqrt{a+b \sin ^2(e+f x)}}+\frac{(7 a-b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left (\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right )}{3 f (a+b)^3 \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3196
Rule 470
Rule 527
Rule 524
Rule 426
Rule 424
Rule 421
Rule 419
Rubi steps
\begin{align*} \int \frac{\tan ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx &=\frac{\left (\sqrt{\cos ^2(e+f x)} \sec (e+f x)\right ) \operatorname{Subst}\left (\int \frac{x^4}{\left (1-x^2\right )^{5/2} \left (a+b x^2\right )^{3/2}} \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac{\sec ^2(e+f x) \tan (e+f x)}{3 (a+b) f \sqrt{a+b \sin ^2(e+f x)}}-\frac{\left (\sqrt{\cos ^2(e+f x)} \sec (e+f x)\right ) \operatorname{Subst}\left (\int \frac{a+3 a x^2}{\left (1-x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}} \, dx,x,\sin (e+f x)\right )}{3 (a+b) f}\\ &=-\frac{4 a \tan (e+f x)}{3 (a+b)^2 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{\sec ^2(e+f x) \tan (e+f x)}{3 (a+b) f \sqrt{a+b \sin ^2(e+f x)}}-\frac{\left (\sqrt{\cos ^2(e+f x)} \sec (e+f x)\right ) \operatorname{Subst}\left (\int \frac{-a (3 a-b)+4 a b x^2}{\sqrt{1-x^2} \left (a+b x^2\right )^{3/2}} \, dx,x,\sin (e+f x)\right )}{3 (a+b)^2 f}\\ &=\frac{(7 a-b) b \cos (e+f x) \sin (e+f x)}{3 (a+b)^3 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{4 a \tan (e+f x)}{3 (a+b)^2 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{\sec ^2(e+f x) \tan (e+f x)}{3 (a+b) f \sqrt{a+b \sin ^2(e+f x)}}+\frac{\left (\sqrt{\cos ^2(e+f x)} \sec (e+f x)\right ) \operatorname{Subst}\left (\int \frac{a^2 (3 a-5 b)+a (7 a-b) b x^2}{\sqrt{1-x^2} \sqrt{a+b x^2}} \, dx,x,\sin (e+f x)\right )}{3 a (a+b)^3 f}\\ &=\frac{(7 a-b) b \cos (e+f x) \sin (e+f x)}{3 (a+b)^3 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{4 a \tan (e+f x)}{3 (a+b)^2 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{\sec ^2(e+f x) \tan (e+f x)}{3 (a+b) f \sqrt{a+b \sin ^2(e+f x)}}+\frac{\left ((7 a-b) \sqrt{\cos ^2(e+f x)} \sec (e+f x)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x^2}}{\sqrt{1-x^2}} \, dx,x,\sin (e+f x)\right )}{3 (a+b)^3 f}-\frac{\left (4 a \sqrt{\cos ^2(e+f x)} \sec (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{a+b x^2}} \, dx,x,\sin (e+f x)\right )}{3 (a+b)^2 f}\\ &=\frac{(7 a-b) b \cos (e+f x) \sin (e+f x)}{3 (a+b)^3 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{4 a \tan (e+f x)}{3 (a+b)^2 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{\sec ^2(e+f x) \tan (e+f x)}{3 (a+b) f \sqrt{a+b \sin ^2(e+f x)}}+\frac{\left ((7 a-b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{b x^2}{a}}}{\sqrt{1-x^2}} \, dx,x,\sin (e+f x)\right )}{3 (a+b)^3 f \sqrt{1+\frac{b \sin ^2(e+f x)}{a}}}-\frac{\left (4 a \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{1+\frac{b \sin ^2(e+f x)}{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+\frac{b x^2}{a}}} \, dx,x,\sin (e+f x)\right )}{3 (a+b)^2 f \sqrt{a+b \sin ^2(e+f x)}}\\ &=\frac{(7 a-b) b \cos (e+f x) \sin (e+f x)}{3 (a+b)^3 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{(7 a-b) \sqrt{\cos ^2(e+f x)} E\left (\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right ) \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 (a+b)^3 f \sqrt{1+\frac{b \sin ^2(e+f x)}{a}}}-\frac{4 a \sqrt{\cos ^2(e+f x)} F\left (\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right ) \sec (e+f x) \sqrt{1+\frac{b \sin ^2(e+f x)}{a}}}{3 (a+b)^2 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{4 a \tan (e+f x)}{3 (a+b)^2 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{\sec ^2(e+f x) \tan (e+f x)}{3 (a+b) f \sqrt{a+b \sin ^2(e+f x)}}\\ \end{align*}
Mathematica [A] time = 2.3262, size = 197, normalized size = 0.67 \[ \frac{-\frac{\tan (e+f x) \sec ^2(e+f x) \left (4 \left (4 a^2-3 a b+b^2\right ) \cos (2 (e+f x))+8 a^2+b (b-7 a) \cos (4 (e+f x))-21 a b-5 b^2\right )}{2 \sqrt{2}}-8 a (a+b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left (e+f x\left |-\frac{b}{a}\right .\right )+2 a (7 a-b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left (e+f x\left |-\frac{b}{a}\right .\right )}{6 f (a+b)^3 \sqrt{2 a-b \cos (2 (e+f x))+b}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 2.556, size = 368, normalized size = 1.3 \begin{align*} -{\frac{1}{ \left ( -3+3\,\sin \left ( fx+e \right ) \right ) \left ( 1+\sin \left ( fx+e \right ) \right ) \left ( a+b \right ) ^{3}\cos \left ( fx+e \right ) f} \left ( \sqrt{-b \left ( \cos \left ( fx+e \right ) \right ) ^{4}+ \left ( a+b \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}}b \left ( 7\,a-b \right ) \sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{4}-4\,\sqrt{-b \left ( \cos \left ( fx+e \right ) \right ) ^{4}+ \left ( a+b \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}}a \left ( a+b \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) +\sqrt{-b \left ( \cos \left ( fx+e \right ) \right ) ^{4}+ \left ( a+b \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}} \left ({a}^{2}+2\,ab+{b}^{2} \right ) \sin \left ( fx+e \right ) -\sqrt{-{\frac{b \left ( \cos \left ( fx+e \right ) \right ) ^{2}}{a}}+{\frac{a+b}{a}}}\sqrt{-b \left ( \cos \left ( fx+e \right ) \right ) ^{4}+ \left ( a+b \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}}\sqrt{ \left ( \cos \left ( fx+e \right ) \right ) ^{2}}a \left ( 4\,{\it EllipticF} \left ( \sin \left ( fx+e \right ) ,\sqrt{-{\frac{b}{a}}} \right ) a+4\,{\it EllipticF} \left ( \sin \left ( fx+e \right ) ,\sqrt{-{\frac{b}{a}}} \right ) b-7\,{\it EllipticE} \left ( \sin \left ( fx+e \right ) ,\sqrt{-{\frac{b}{a}}} \right ) a+{\it EllipticE} \left ( \sin \left ( fx+e \right ) ,\sqrt{-{\frac{b}{a}}} \right ) b \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2} \right ){\frac{1}{\sqrt{- \left ( a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) \left ( -1+\sin \left ( fx+e \right ) \right ) \left ( 1+\sin \left ( fx+e \right ) \right ) }}}{\frac{1}{\sqrt{a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2}}}}} \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] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-b \cos \left (f x + e\right )^{2} + a + b} \tan \left (f x + e\right )^{4}}{b^{2} \cos \left (f x + e\right )^{4} - 2 \,{\left (a b + b^{2}\right )} \cos \left (f x + e\right )^{2} + a^{2} + 2 \, a b + b^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan ^{4}{\left (e + f x \right )}}{\left (a + b \sin ^{2}{\left (e + f x \right )}\right )^{\frac{3}{2}}}\, dx \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{\tan \left (f x + e\right )^{4}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]