Optimal. Leaf size=91 \[ -\frac{1}{f (a+b)^2 \sqrt{a+b \sin ^2(e+f x)}}-\frac{1}{3 f (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a+b}}\right )}{f (a+b)^{5/2}} \]
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Rubi [A] time = 0.0845923, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {3194, 51, 63, 208} \[ -\frac{1}{f (a+b)^2 \sqrt{a+b \sin ^2(e+f x)}}-\frac{1}{3 f (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a+b}}\right )}{f (a+b)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 3194
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\tan (e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{(1-x) (a+b x)^{5/2}} \, dx,x,\sin ^2(e+f x)\right )}{2 f}\\ &=-\frac{1}{3 (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac{\operatorname{Subst}\left (\int \frac{1}{(1-x) (a+b x)^{3/2}} \, dx,x,\sin ^2(e+f x)\right )}{2 (a+b) f}\\ &=-\frac{1}{3 (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac{1}{(a+b)^2 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{\operatorname{Subst}\left (\int \frac{1}{(1-x) \sqrt{a+b x}} \, dx,x,\sin ^2(e+f x)\right )}{2 (a+b)^2 f}\\ &=-\frac{1}{3 (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac{1}{(a+b)^2 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{\operatorname{Subst}\left (\int \frac{1}{1+\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sin ^2(e+f x)}\right )}{b (a+b)^2 f}\\ &=\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a+b}}\right )}{(a+b)^{5/2} f}-\frac{1}{3 (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac{1}{(a+b)^2 f \sqrt{a+b \sin ^2(e+f x)}}\\ \end{align*}
Mathematica [C] time = 0.0821613, size = 56, normalized size = 0.62 \[ -\frac{\, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};1-\frac{b \cos ^2(e+f x)}{a+b}\right )}{3 f (a+b) \left (a-b \cos ^2(e+f x)+b\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 5.415, size = 895, normalized size = 9.8 \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 = 2.62347, size = 1226, normalized size = 13.47 \begin{align*} \left [\frac{3 \,{\left (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}\right )} \sqrt{a + b} \log \left (\frac{b \cos \left (f x + e\right )^{2} - 2 \, \sqrt{-b \cos \left (f x + e\right )^{2} + a + b} \sqrt{a + b} - 2 \, a - 2 \, b}{\cos \left (f x + e\right )^{2}}\right ) + 2 \,{\left (3 \,{\left (a b + b^{2}\right )} \cos \left (f x + e\right )^{2} - 4 \, a^{2} - 8 \, a b - 4 \, b^{2}\right )} \sqrt{-b \cos \left (f x + e\right )^{2} + a + b}}{6 \,{\left ({\left (a^{3} b^{2} + 3 \, a^{2} b^{3} + 3 \, a b^{4} + b^{5}\right )} f \cos \left (f x + e\right )^{4} - 2 \,{\left (a^{4} b + 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} + 4 \, a b^{4} + b^{5}\right )} f \cos \left (f x + e\right )^{2} +{\left (a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 10 \, a^{2} b^{3} + 5 \, a b^{4} + b^{5}\right )} f\right )}}, -\frac{3 \,{\left (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}\right )} \sqrt{-a - b} \arctan \left (\frac{\sqrt{-b \cos \left (f x + e\right )^{2} + a + b} \sqrt{-a - b}}{a + b}\right ) -{\left (3 \,{\left (a b + b^{2}\right )} \cos \left (f x + e\right )^{2} - 4 \, a^{2} - 8 \, a b - 4 \, b^{2}\right )} \sqrt{-b \cos \left (f x + e\right )^{2} + a + b}}{3 \,{\left ({\left (a^{3} b^{2} + 3 \, a^{2} b^{3} + 3 \, a b^{4} + b^{5}\right )} f \cos \left (f x + e\right )^{4} - 2 \,{\left (a^{4} b + 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} + 4 \, a b^{4} + b^{5}\right )} f \cos \left (f x + e\right )^{2} +{\left (a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 10 \, a^{2} b^{3} + 5 \, a b^{4} + b^{5}\right )} f\right )}}\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan \left (f x + e\right )}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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