Optimal. Leaf size=174 \[ \frac{2 \sqrt{2 \pi } b^{3/2} e^{\frac{2 b c}{d}-2 a} \text{Erf}\left (\frac{\sqrt{2} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{3 d^{5/2}}+\frac{2 \sqrt{2 \pi } b^{3/2} e^{2 a-\frac{2 b c}{d}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{3 d^{5/2}}-\frac{8 b \sinh (a+b x) \cosh (a+b x)}{3 d^2 \sqrt{c+d x}}-\frac{2 \sinh ^2(a+b x)}{3 d (c+d x)^{3/2}} \]
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Rubi [A] time = 0.321269, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389, Rules used = {3314, 32, 3312, 3307, 2180, 2204, 2205} \[ \frac{2 \sqrt{2 \pi } b^{3/2} e^{\frac{2 b c}{d}-2 a} \text{Erf}\left (\frac{\sqrt{2} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{3 d^{5/2}}+\frac{2 \sqrt{2 \pi } b^{3/2} e^{2 a-\frac{2 b c}{d}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{3 d^{5/2}}-\frac{8 b \sinh (a+b x) \cosh (a+b x)}{3 d^2 \sqrt{c+d x}}-\frac{2 \sinh ^2(a+b x)}{3 d (c+d x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3314
Rule 32
Rule 3312
Rule 3307
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int \frac{\sinh ^2(a+b x)}{(c+d x)^{5/2}} \, dx &=-\frac{8 b \cosh (a+b x) \sinh (a+b x)}{3 d^2 \sqrt{c+d x}}-\frac{2 \sinh ^2(a+b x)}{3 d (c+d x)^{3/2}}+\frac{\left (8 b^2\right ) \int \frac{1}{\sqrt{c+d x}} \, dx}{3 d^2}+\frac{\left (16 b^2\right ) \int \frac{\sinh ^2(a+b x)}{\sqrt{c+d x}} \, dx}{3 d^2}\\ &=\frac{16 b^2 \sqrt{c+d x}}{3 d^3}-\frac{8 b \cosh (a+b x) \sinh (a+b x)}{3 d^2 \sqrt{c+d x}}-\frac{2 \sinh ^2(a+b x)}{3 d (c+d x)^{3/2}}-\frac{\left (16 b^2\right ) \int \left (\frac{1}{2 \sqrt{c+d x}}-\frac{\cosh (2 a+2 b x)}{2 \sqrt{c+d x}}\right ) \, dx}{3 d^2}\\ &=-\frac{8 b \cosh (a+b x) \sinh (a+b x)}{3 d^2 \sqrt{c+d x}}-\frac{2 \sinh ^2(a+b x)}{3 d (c+d x)^{3/2}}+\frac{\left (8 b^2\right ) \int \frac{\cosh (2 a+2 b x)}{\sqrt{c+d x}} \, dx}{3 d^2}\\ &=-\frac{8 b \cosh (a+b x) \sinh (a+b x)}{3 d^2 \sqrt{c+d x}}-\frac{2 \sinh ^2(a+b x)}{3 d (c+d x)^{3/2}}+\frac{\left (4 b^2\right ) \int \frac{e^{-i (2 i a+2 i b x)}}{\sqrt{c+d x}} \, dx}{3 d^2}+\frac{\left (4 b^2\right ) \int \frac{e^{i (2 i a+2 i b x)}}{\sqrt{c+d x}} \, dx}{3 d^2}\\ &=-\frac{8 b \cosh (a+b x) \sinh (a+b x)}{3 d^2 \sqrt{c+d x}}-\frac{2 \sinh ^2(a+b x)}{3 d (c+d x)^{3/2}}+\frac{\left (8 b^2\right ) \operatorname{Subst}\left (\int e^{i \left (2 i a-\frac{2 i b c}{d}\right )-\frac{2 b x^2}{d}} \, dx,x,\sqrt{c+d x}\right )}{3 d^3}+\frac{\left (8 b^2\right ) \operatorname{Subst}\left (\int e^{-i \left (2 i a-\frac{2 i b c}{d}\right )+\frac{2 b x^2}{d}} \, dx,x,\sqrt{c+d x}\right )}{3 d^3}\\ &=\frac{2 b^{3/2} e^{-2 a+\frac{2 b c}{d}} \sqrt{2 \pi } \text{erf}\left (\frac{\sqrt{2} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{3 d^{5/2}}+\frac{2 b^{3/2} e^{2 a-\frac{2 b c}{d}} \sqrt{2 \pi } \text{erfi}\left (\frac{\sqrt{2} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{3 d^{5/2}}-\frac{8 b \cosh (a+b x) \sinh (a+b x)}{3 d^2 \sqrt{c+d x}}-\frac{2 \sinh ^2(a+b x)}{3 d (c+d x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 1.21472, size = 156, normalized size = 0.9 \[ -\frac{2 e^{-2 \left (a+\frac{b c}{d}\right )} \left (\sqrt{2} e^{4 a} d \left (-\frac{b (c+d x)}{d}\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-\frac{2 b (c+d x)}{d}\right )+\sqrt{2} d e^{\frac{4 b c}{d}} \left (\frac{b (c+d x)}{d}\right )^{3/2} \text{Gamma}\left (\frac{1}{2},\frac{2 b (c+d x)}{d}\right )+e^{2 \left (a+\frac{b c}{d}\right )} \left (2 b (c+d x) \sinh (2 (a+b x))+d \sinh ^2(a+b x)\right )\right )}{3 d^2 (c+d x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.066, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \sinh \left ( bx+a \right ) \right ) ^{2} \left ( dx+c \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.30908, size = 159, normalized size = 0.91 \begin{align*} -\frac{\frac{3 \, \sqrt{2} \left (\frac{{\left (d x + c\right )} b}{d}\right )^{\frac{3}{2}} e^{\left (\frac{2 \,{\left (b c - a d\right )}}{d}\right )} \Gamma \left (-\frac{3}{2}, \frac{2 \,{\left (d x + c\right )} b}{d}\right )}{{\left (d x + c\right )}^{\frac{3}{2}}} + \frac{3 \, \sqrt{2} \left (-\frac{{\left (d x + c\right )} b}{d}\right )^{\frac{3}{2}} e^{\left (-\frac{2 \,{\left (b c - a d\right )}}{d}\right )} \Gamma \left (-\frac{3}{2}, -\frac{2 \,{\left (d x + c\right )} b}{d}\right )}{{\left (d x + c\right )}^{\frac{3}{2}}} - \frac{2}{{\left (d x + c\right )}^{\frac{3}{2}}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 3.01618, size = 2053, normalized size = 11.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh ^{2}{\left (a + b x \right )}}{\left (c + d x\right )^{\frac{5}{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{\sinh \left (b x + a\right )^{2}}{{\left (d x + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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