Optimal. Leaf size=180 \[ \frac{4 b \sqrt{a+b \tan (c+d x)}}{3 a^2 d \sqrt{\tan (c+d x)}}-\frac{\tan ^{-1}\left (\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d \sqrt{-b+i a}}-\frac{2 \sqrt{a+b \tan (c+d x)}}{3 a d \tan ^{\frac{3}{2}}(c+d x)}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d \sqrt{b+i a}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.348155, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {3569, 3649, 12, 3575, 912, 93, 205, 208} \[ \frac{4 b \sqrt{a+b \tan (c+d x)}}{3 a^2 d \sqrt{\tan (c+d x)}}-\frac{\tan ^{-1}\left (\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d \sqrt{-b+i a}}-\frac{2 \sqrt{a+b \tan (c+d x)}}{3 a d \tan ^{\frac{3}{2}}(c+d x)}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d \sqrt{b+i a}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3569
Rule 3649
Rule 12
Rule 3575
Rule 912
Rule 93
Rule 205
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{\tan ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}} \, dx &=-\frac{2 \sqrt{a+b \tan (c+d x)}}{3 a d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2 \int \frac{b+\frac{3}{2} a \tan (c+d x)+b \tan ^2(c+d x)}{\tan ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}} \, dx}{3 a}\\ &=-\frac{2 \sqrt{a+b \tan (c+d x)}}{3 a d \tan ^{\frac{3}{2}}(c+d x)}+\frac{4 b \sqrt{a+b \tan (c+d x)}}{3 a^2 d \sqrt{\tan (c+d x)}}+\frac{4 \int -\frac{3 a^2}{4 \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}} \, dx}{3 a^2}\\ &=-\frac{2 \sqrt{a+b \tan (c+d x)}}{3 a d \tan ^{\frac{3}{2}}(c+d x)}+\frac{4 b \sqrt{a+b \tan (c+d x)}}{3 a^2 d \sqrt{\tan (c+d x)}}-\int \frac{1}{\sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}} \, dx\\ &=-\frac{2 \sqrt{a+b \tan (c+d x)}}{3 a d \tan ^{\frac{3}{2}}(c+d x)}+\frac{4 b \sqrt{a+b \tan (c+d x)}}{3 a^2 d \sqrt{\tan (c+d x)}}-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \sqrt{a+b x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac{2 \sqrt{a+b \tan (c+d x)}}{3 a d \tan ^{\frac{3}{2}}(c+d x)}+\frac{4 b \sqrt{a+b \tan (c+d x)}}{3 a^2 d \sqrt{\tan (c+d x)}}-\frac{\operatorname{Subst}\left (\int \left (\frac{i}{2 (i-x) \sqrt{x} \sqrt{a+b x}}+\frac{i}{2 \sqrt{x} (i+x) \sqrt{a+b x}}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac{2 \sqrt{a+b \tan (c+d x)}}{3 a d \tan ^{\frac{3}{2}}(c+d x)}+\frac{4 b \sqrt{a+b \tan (c+d x)}}{3 a^2 d \sqrt{\tan (c+d x)}}-\frac{i \operatorname{Subst}\left (\int \frac{1}{(i-x) \sqrt{x} \sqrt{a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d}-\frac{i \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} (i+x) \sqrt{a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=-\frac{2 \sqrt{a+b \tan (c+d x)}}{3 a d \tan ^{\frac{3}{2}}(c+d x)}+\frac{4 b \sqrt{a+b \tan (c+d x)}}{3 a^2 d \sqrt{\tan (c+d x)}}-\frac{i \operatorname{Subst}\left (\int \frac{1}{i-(-a+i b) x^2} \, dx,x,\frac{\sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}-\frac{i \operatorname{Subst}\left (\int \frac{1}{i-(a+i b) x^2} \, dx,x,\frac{\sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt{i a-b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{\sqrt{i a-b} d}-\frac{\tanh ^{-1}\left (\frac{\sqrt{i a+b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{\sqrt{i a+b} d}-\frac{2 \sqrt{a+b \tan (c+d x)}}{3 a d \tan ^{\frac{3}{2}}(c+d x)}+\frac{4 b \sqrt{a+b \tan (c+d x)}}{3 a^2 d \sqrt{\tan (c+d x)}}\\ \end{align*}
Mathematica [A] time = 1.53462, size = 172, normalized size = 0.96 \[ \frac{-\frac{2 (a-2 b \tan (c+d x)) \sqrt{a+b \tan (c+d x)}}{a^2 \tan ^{\frac{3}{2}}(c+d x)}+\frac{3 (-1)^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{-1} \sqrt{-a-i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{\sqrt{-a-i b}}+\frac{3 (-1)^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{-1} \sqrt{a-i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{\sqrt{a-i b}}}{3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.286, size = 944677, normalized size = 5248.2 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b \tan \left (d x + c\right ) + a} \tan \left (d x + c\right )^{\frac{5}{2}}}\,{d x} \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a + b \tan{\left (c + d x \right )}} \tan ^{\frac{5}{2}}{\left (c + d x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]